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The Project Gutenberg eBook of Encyclopaedia Britannica,
11th Edition, "Matter" to "Mecklenburg"
This eBook is for the use of anyone anywhere in the United States and
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Title: Encyclopaedia Britannica, 11th Edition, "Matter" to
"Mecklenburg"
Author: Various
Release date: April 7, 2013 [eBook #42473]
Most recently updated: October 23, 2024
Language: English
Other information and formats: www.gutenberg.org/ebooks/42473
Credits: Produced by Marius Masi, Don Kretz and the Online
Distributed Proofreading Team at http://www.pgdp.net
*** START OF THE PROJECT GUTENBERG EBOOK
ENCYCLOPAEDIA BRITANNICA, 11TH EDITION, "MATTER" TO
"MECKLENBURG" ***
11th Edition, "Matter" to "Mecklenburg"
This eBook is for the use of anyone anywhere in the United States and
most other parts of the world at no cost and with almost no restrictions
whatsoever. You may copy it, give it away or re-use it under the terms
of the Project Gutenberg License included with this eBook or online at
www.gutenberg.org. If you are not located in the United States, you
will have to check the laws of the country where you are located
before using this eBook.
Title: Encyclopaedia Britannica, 11th Edition, "Matter" to
"Mecklenburg"
Author: Various
Release date: April 7, 2013 [eBook #42473]
Most recently updated: October 23, 2024
Language: English
Other information and formats: www.gutenberg.org/ebooks/42473
Credits: Produced by Marius Masi, Don Kretz and the Online
Distributed Proofreading Team at http://www.pgdp.net
*** START OF THE PROJECT GUTENBERG EBOOK
ENCYCLOPAEDIA BRITANNICA, 11TH EDITION, "MATTER" TO
"MECKLENBURG" ***
Page 4
Transcriber’s note: A few typographical errors have been corrected.
They appear in the text like this, and the
explanation will appear when the mouse pointer is
moved over the marked passage. Sections in Greek
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They appear in the text like this, and the
explanation will appear when the mouse pointer is
moved over the marked passage. Sections in Greek
will yield a transliteration when the pointer is
moved over them, and words using diacritic
characters in the Latin Extended Additional block,
which may not display in some fonts or browsers,
will display an unaccented version.
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Page 5
THE ENCYCLOPÆDIA BRITANNICA
Page 6
A DICTIONARY OF ARTS, SCIENCES,
LITERATURE AND GENERAL
INFORMATION
ELEVENTH EDITION
VOLUME XVII SLICE VIII
Matter to Mecklenburg
Articles in This Slice
MATTER MAX MÜLLER, FRIEDRICH
MATTERHORN MAXWELL
MATTEUCCI, CARLO MAXWELL, JAMES CLERK
MATTHEW, ST MAXWELLTOWN
MATTHEW, TOBIAS MAY, PHIL
MATTHEW, GOSPEL OF ST MAY, THOMAS
MATTHEW CANTACUZENUS MAY, WILLIAM
MATTHEW OF PARIS MAY (month)
LITERATURE AND GENERAL
INFORMATION
ELEVENTH EDITION
VOLUME XVII SLICE VIII
Matter to Mecklenburg
Articles in This Slice
MATTER MAX MÜLLER, FRIEDRICH
MATTERHORN MAXWELL
MATTEUCCI, CARLO MAXWELL, JAMES CLERK
MATTHEW, ST MAXWELLTOWN
MATTHEW, TOBIAS MAY, PHIL
MATTHEW, GOSPEL OF ST MAY, THOMAS
MATTHEW CANTACUZENUS MAY, WILLIAM
MATTHEW OF PARIS MAY (month)
Page 7
MATTHEW OF WESTMINSTER MAY, ISLE OF
MATTHEWS, STANLEY MAYA
MATTHIAE, AUGUST HEINRICH MAYAGUEZ
MATTHIAS (disciple) MAYAVARAM
MATTHIAS (Roman emperor) MAYBOLE
MATTHIAS I., HUNYADI MAYEN
MATTHISSON, FRIEDRICH VON MAYENNE, CHARLES OF
LORRAINE
MATTING MAYENNE (department of France)
MATTOCK MAYENNE (town of France)
MATTO GROSSO MAYER, JOHANN TOBIAS
MATTOON MAYER, JULIUS ROBERT
MATTRESS MAYFLOWER
MATURIN, CHARLES ROBERT MAY-FLY
MATVYEEV, ARTAMON MAYHEM
SERGYEEVICH
MAUBEUGE MAYHEW, HENRY
MAUCH CHUNK MAYHEW, JONATHAN
MAUCHLINE MAYHEW, THOMAS
MAUDE, CYRIL MAYMYO
MAULE MAYNARD, FRANÇOIS DE
MAULÉON, SAVARI DE MAYNE, JASPER
MAULSTICK MAYNOOTH
MAUNDY THURSDAY MAYO, RICHARD SOUTHWELL
BOURKE
MAUPASSANT, HENRI RENÉ ALBERT MAYO
GUY DE
MAUPEOU, RENÉ NICOLAS MAYOR, JOHN EYTON
CHARLES AUGUSTIN BICKERSTETH
MAUPERTUIS, PIERRE LOUIS MAYOR
MOREAU DE
MAU RANIPUR MAYOR OF THE PALACE
MAUREL, ABDIAS MAYORUNA
MAUREL, VICTOR MAYO-SMITH, RICHMOND
MATTHEWS, STANLEY MAYA
MATTHIAE, AUGUST HEINRICH MAYAGUEZ
MATTHIAS (disciple) MAYAVARAM
MATTHIAS (Roman emperor) MAYBOLE
MATTHIAS I., HUNYADI MAYEN
MATTHISSON, FRIEDRICH VON MAYENNE, CHARLES OF
LORRAINE
MATTING MAYENNE (department of France)
MATTOCK MAYENNE (town of France)
MATTO GROSSO MAYER, JOHANN TOBIAS
MATTOON MAYER, JULIUS ROBERT
MATTRESS MAYFLOWER
MATURIN, CHARLES ROBERT MAY-FLY
MATVYEEV, ARTAMON MAYHEM
SERGYEEVICH
MAUBEUGE MAYHEW, HENRY
MAUCH CHUNK MAYHEW, JONATHAN
MAUCHLINE MAYHEW, THOMAS
MAUDE, CYRIL MAYMYO
MAULE MAYNARD, FRANÇOIS DE
MAULÉON, SAVARI DE MAYNE, JASPER
MAULSTICK MAYNOOTH
MAUNDY THURSDAY MAYO, RICHARD SOUTHWELL
BOURKE
MAUPASSANT, HENRI RENÉ ALBERT MAYO
GUY DE
MAUPEOU, RENÉ NICOLAS MAYOR, JOHN EYTON
CHARLES AUGUSTIN BICKERSTETH
MAUPERTUIS, PIERRE LOUIS MAYOR
MOREAU DE
MAU RANIPUR MAYOR OF THE PALACE
MAUREL, ABDIAS MAYORUNA
MAUREL, VICTOR MAYO-SMITH, RICHMOND
Page 8
MAURENBRECHER, KARL PETER MAYOTTE
WILHELM
MAUREPAS, JEAN FRÉDÉRIC MAYOW, JOHN
PHÉLYPEAUX
MAURER, GEORG LUDWIG VON MAYSVILLE
MAURETANIA MAZAGAN
MAURIAC MAZAMET
MAURICE, ST MAZANDARAN
MAURICE (Roman emperor) MAZARIN, JULES
MAURICE (elector of Saxony) MAZAR-I-SHARIF
MAURICE, JOHN FREDERICK MAZARRÓN
DENISON
MAURICE OF NASSAU MAZATLÁN
MAURISTS MAZE
MAURITIUS MAZEPA-KOLEDINSKY, IVAN
STEPANOVICH
MAURY, JEAN SIFFREIN MAZER
MAURY, LOUIS FERDINAND ALFRED MAZURKA
MAURY, MATTHEW FONTAINE MAZZARA DEL VALLO
MAUSOLEUM MAZZINI, GIUSEPPE
MAUSOLUS MAZZONI, GIACOMO
MAUVE, ANTON MAZZONI, GUIDO
MAVROCORDATO MEAD, LARKIN GOLDSMITH
MAWKMAI MEAD, RICHARD
MAXENTIUS, MARCUS AURELIUS MEAD
VALERIUS
MAXIM, SIR HIRAM STEVENS MEADE, GEORGE GORDON
MAXIMA AND MINIMA MEADE, WILLIAM
MAXIMIANUS MEADVILLE
MAXIMIANUS, MARCUS AURELIUS MEAGHER, THOMAS FRANCIS
VALERIUS
MAXIMILIAN I. (elector of Bavaria) MEAL
MAXIMILIAN I. (king of Bavaria) MEALIE
MAXIMILIAN II. (king of Bavaria) MEAN
WILHELM
MAUREPAS, JEAN FRÉDÉRIC MAYOW, JOHN
PHÉLYPEAUX
MAURER, GEORG LUDWIG VON MAYSVILLE
MAURETANIA MAZAGAN
MAURIAC MAZAMET
MAURICE, ST MAZANDARAN
MAURICE (Roman emperor) MAZARIN, JULES
MAURICE (elector of Saxony) MAZAR-I-SHARIF
MAURICE, JOHN FREDERICK MAZARRÓN
DENISON
MAURICE OF NASSAU MAZATLÁN
MAURISTS MAZE
MAURITIUS MAZEPA-KOLEDINSKY, IVAN
STEPANOVICH
MAURY, JEAN SIFFREIN MAZER
MAURY, LOUIS FERDINAND ALFRED MAZURKA
MAURY, MATTHEW FONTAINE MAZZARA DEL VALLO
MAUSOLEUM MAZZINI, GIUSEPPE
MAUSOLUS MAZZONI, GIACOMO
MAUVE, ANTON MAZZONI, GUIDO
MAVROCORDATO MEAD, LARKIN GOLDSMITH
MAWKMAI MEAD, RICHARD
MAXENTIUS, MARCUS AURELIUS MEAD
VALERIUS
MAXIM, SIR HIRAM STEVENS MEADE, GEORGE GORDON
MAXIMA AND MINIMA MEADE, WILLIAM
MAXIMIANUS MEADVILLE
MAXIMIANUS, MARCUS AURELIUS MEAGHER, THOMAS FRANCIS
VALERIUS
MAXIMILIAN I. (elector of Bavaria) MEAL
MAXIMILIAN I. (king of Bavaria) MEALIE
MAXIMILIAN II. (king of Bavaria) MEAN
Page 9
MAXIMILIAN I. (Roman emperor) MEASLES
MAXIMILIAN II. (Roman emperor) MEAT
MAXIMILIAN (emperor of Mexico) MEATH
MAXIMINUS, GAIUS JULIUS VERUS MEAUX
MAXIMINUS, GALERIUS VALERIUS MECCA
MAXIMS, LEGAL MECHANICS
MAXIMUS MECHANICVILLE
MAXIMUS, ST MECHITHARISTS
MAXIMUS OF SMYRNA MECKLENBURG
MAXIMUS OF TYRE
MATTER. Our conceptions of the nature and structure of matter
have been profoundly influenced in recent years by investigations on the
Conduction of Electricity through Gases (see Conduction, Electric) and on
Radio-activity (q.v.). These researches and the ideas which they have
suggested have already thrown much light on some of the most fundamental
questions connected with matter; they have, too, furnished us with far more
powerful methods for investigating many problems connected with the
structure of matter than those hitherto available. There is thus every reason
to believe that our knowledge of the structure of matter will soon become
far more precise and complete than it is at present, for now we have the
means of settling by testing directly many points which are still doubtful,
but which formerly seemed far beyond the reach of experiment.
MAXIMILIAN II. (Roman emperor) MEAT
MAXIMILIAN (emperor of Mexico) MEATH
MAXIMINUS, GAIUS JULIUS VERUS MEAUX
MAXIMINUS, GALERIUS VALERIUS MECCA
MAXIMS, LEGAL MECHANICS
MAXIMUS MECHANICVILLE
MAXIMUS, ST MECHITHARISTS
MAXIMUS OF SMYRNA MECKLENBURG
MAXIMUS OF TYRE
MATTER. Our conceptions of the nature and structure of matter
have been profoundly influenced in recent years by investigations on the
Conduction of Electricity through Gases (see Conduction, Electric) and on
Radio-activity (q.v.). These researches and the ideas which they have
suggested have already thrown much light on some of the most fundamental
questions connected with matter; they have, too, furnished us with far more
powerful methods for investigating many problems connected with the
structure of matter than those hitherto available. There is thus every reason
to believe that our knowledge of the structure of matter will soon become
far more precise and complete than it is at present, for now we have the
means of settling by testing directly many points which are still doubtful,
but which formerly seemed far beyond the reach of experiment.
Page 10
The Molecular Theory of Matter—the only theory ever seriously
advocated—supposes that all visible forms of matter are collocations of
simpler and smaller portions. There has been a continuous tendency as
science has advanced to reduce further and further the number of the
different kinds of things of which all matter is supposed to be built up. First
came the molecular theory teaching us to regard matter as made up of an
enormous number of small particles, each kind of matter having its
characteristic particle, thus the particles of water were supposed to be
different from those of air and indeed from those of any other substance.
Then came Dalton’s Atomic Theory which taught that these molecules, in
spite of their almost infinite variety, were all built up of still smaller bodies,
the atoms of the chemical elements, and that the number of different types
of these smaller bodies was limited to the sixty or seventy types which
represent the atoms of the substance regarded by chemists as elements.
In 1815 Prout suggested that the atoms of the heavier chemical elements
were themselves composite and that they were all built up of atoms of the
lightest element, hydrogen, so that all the different forms of matter are
edifices built of the same material—the atom of hydrogen. If the atoms of
hydrogen do not alter in weight when they combine to form atoms of other
elements the atomic weights of all elements would be multiples of that of
hydrogen; though the number of elements whose atomic weights are
multiples or very nearly so of hydrogen is very striking, there are several
which are universally admitted to have atomic weights differing largely
from whole numbers. We do not know enough about gravity to say whether
this is due to the change of weight of the hydrogen atoms when they
combine to form other atoms, or whether the primordial form from which
all matter is built up is something other than the hydrogen atom. Whatever
may be the nature of this primordial form, the tendency of all recent
discoveries has been to emphasize the truth of the conception of a common
basis of matter of all kinds. That the atoms of the different elements have a
advocated—supposes that all visible forms of matter are collocations of
simpler and smaller portions. There has been a continuous tendency as
science has advanced to reduce further and further the number of the
different kinds of things of which all matter is supposed to be built up. First
came the molecular theory teaching us to regard matter as made up of an
enormous number of small particles, each kind of matter having its
characteristic particle, thus the particles of water were supposed to be
different from those of air and indeed from those of any other substance.
Then came Dalton’s Atomic Theory which taught that these molecules, in
spite of their almost infinite variety, were all built up of still smaller bodies,
the atoms of the chemical elements, and that the number of different types
of these smaller bodies was limited to the sixty or seventy types which
represent the atoms of the substance regarded by chemists as elements.
In 1815 Prout suggested that the atoms of the heavier chemical elements
were themselves composite and that they were all built up of atoms of the
lightest element, hydrogen, so that all the different forms of matter are
edifices built of the same material—the atom of hydrogen. If the atoms of
hydrogen do not alter in weight when they combine to form atoms of other
elements the atomic weights of all elements would be multiples of that of
hydrogen; though the number of elements whose atomic weights are
multiples or very nearly so of hydrogen is very striking, there are several
which are universally admitted to have atomic weights differing largely
from whole numbers. We do not know enough about gravity to say whether
this is due to the change of weight of the hydrogen atoms when they
combine to form other atoms, or whether the primordial form from which
all matter is built up is something other than the hydrogen atom. Whatever
may be the nature of this primordial form, the tendency of all recent
discoveries has been to emphasize the truth of the conception of a common
basis of matter of all kinds. That the atoms of the different elements have a
Page 11
common basis, that they behave as if they consisted of different numbers of
small particles of the same kind, is proved to most minds by the Periodic
Law of Mendeléeff and Newlands (see Element). This law shows that the
physical and chemical properties of the different elements are determined
by their atomic weights, or to use the language of mathematics, the
properties of an element are functions of its atomic weight. Now if we
constructed models of the atoms out of different materials, the atomic
weight would be but one factor out of many which would influence the
physical and chemical properties of the model, we should require to know
more than the atomic weight to fix its behaviour. If we were to plot a curve
representing the variation of some property of the substance with the atomic
weight we should not expect the curve to be a smooth one, for instance two
atoms might have the same atomic weight and yet if they were made of
different materials have no other property in common. The influence of the
atomic weight on the properties of the elements is nowhere more strikingly
shown than in the recent developments of physics connected with the
discharge of electricity through gases and with radio-activity. The
transparency of bodies to Röntgen rays, to cathode rays, to the rays emitted
by radio-active substances, the quality of the secondary radiation emitted by
the different elements are all determined by the atomic weight of the
element. So much is this the case that the behaviour of the element with
respect to these rays has been used to determine its atomic weight, when as
in the case of Indium, uncertainty as to the valency of the element makes
the result of ordinary chemical methods ambiguous.
The radio-active elements indeed furnish us with direct evidence of this
unity of composition of matter, for not only does one element uranium,
produce another, radium, but all the radio-active substances give rise to
helium, so that the substance of the atoms of this gas must be contained in
the atoms of the radio-active elements.
small particles of the same kind, is proved to most minds by the Periodic
Law of Mendeléeff and Newlands (see Element). This law shows that the
physical and chemical properties of the different elements are determined
by their atomic weights, or to use the language of mathematics, the
properties of an element are functions of its atomic weight. Now if we
constructed models of the atoms out of different materials, the atomic
weight would be but one factor out of many which would influence the
physical and chemical properties of the model, we should require to know
more than the atomic weight to fix its behaviour. If we were to plot a curve
representing the variation of some property of the substance with the atomic
weight we should not expect the curve to be a smooth one, for instance two
atoms might have the same atomic weight and yet if they were made of
different materials have no other property in common. The influence of the
atomic weight on the properties of the elements is nowhere more strikingly
shown than in the recent developments of physics connected with the
discharge of electricity through gases and with radio-activity. The
transparency of bodies to Röntgen rays, to cathode rays, to the rays emitted
by radio-active substances, the quality of the secondary radiation emitted by
the different elements are all determined by the atomic weight of the
element. So much is this the case that the behaviour of the element with
respect to these rays has been used to determine its atomic weight, when as
in the case of Indium, uncertainty as to the valency of the element makes
the result of ordinary chemical methods ambiguous.
The radio-active elements indeed furnish us with direct evidence of this
unity of composition of matter, for not only does one element uranium,
produce another, radium, but all the radio-active substances give rise to
helium, so that the substance of the atoms of this gas must be contained in
the atoms of the radio-active elements.
Page 12
It is not radio-active atoms alone that contain a common constituent, for
it has been found that all bodies can by suitable treatment, such as raising
them to incandescence or exposing them to ultra-violet light, be made to
emit negatively electrified particles, and that these particles are the same
from whatever source they may be derived. These particles all carry the
same charge of negative electricity and all have the same mass, this mass is
exceedingly small even when compared with the mass of an atom of
hydrogen, which until the discovery of these particles was the smallest mass
known to science. These particles are called corpuscles or electrons; their
mass according to the most recent determinations is only about 1⁄1700 of that
of an atom of hydrogen, and their radius is only about one hundred-
thousandth part of the radius of the hydrogen atom. As corpuscles of this
kind can be obtained from all substances, we infer that they form a
constituent of the atoms of all bodies. The atoms of the different elements
do not all contain the same number of corpuscles—there are more
corpuscles in the atoms of the heavier elements than in the atoms of the
lighter ones; in fact, many different considerations point to the conclusion
that the number of corpuscles in the atom of any element is proportional to
the atomic weight of the element. Different methods of estimating the exact
number of corpuscles in the atom have all led to the conclusion that this
number is of the same order as the atomic weight; that, for instance, the
number of corpuscles in the atom of oxygen is not a large multiple of 16.
Some methods indicate that the number of corpuscles in the atom is equal to
the atomic weight, while the maximum value obtained by any method is
only about four times the atomic weight. This is one of the points on which
further experiments will enable us to speak with greater precision. Thus one
of the constituents of all atoms is the negatively charged corpuscle; since
the atoms are electrically neutral, this negative charge must be accompanied
by an equal positive one, so that on this view the atoms must contain a
charge of positive electricity proportional to the atomic weight; the way in
it has been found that all bodies can by suitable treatment, such as raising
them to incandescence or exposing them to ultra-violet light, be made to
emit negatively electrified particles, and that these particles are the same
from whatever source they may be derived. These particles all carry the
same charge of negative electricity and all have the same mass, this mass is
exceedingly small even when compared with the mass of an atom of
hydrogen, which until the discovery of these particles was the smallest mass
known to science. These particles are called corpuscles or electrons; their
mass according to the most recent determinations is only about 1⁄1700 of that
of an atom of hydrogen, and their radius is only about one hundred-
thousandth part of the radius of the hydrogen atom. As corpuscles of this
kind can be obtained from all substances, we infer that they form a
constituent of the atoms of all bodies. The atoms of the different elements
do not all contain the same number of corpuscles—there are more
corpuscles in the atoms of the heavier elements than in the atoms of the
lighter ones; in fact, many different considerations point to the conclusion
that the number of corpuscles in the atom of any element is proportional to
the atomic weight of the element. Different methods of estimating the exact
number of corpuscles in the atom have all led to the conclusion that this
number is of the same order as the atomic weight; that, for instance, the
number of corpuscles in the atom of oxygen is not a large multiple of 16.
Some methods indicate that the number of corpuscles in the atom is equal to
the atomic weight, while the maximum value obtained by any method is
only about four times the atomic weight. This is one of the points on which
further experiments will enable us to speak with greater precision. Thus one
of the constituents of all atoms is the negatively charged corpuscle; since
the atoms are electrically neutral, this negative charge must be accompanied
by an equal positive one, so that on this view the atoms must contain a
charge of positive electricity proportional to the atomic weight; the way in
Page 13
which this positive electricity is arranged is a matter of great importance in
the consideration of the constitution of matter. The question naturally arises,
is the positive electricity done up into definite units like the negative, or
does it merely indicate a property acquired by an atom when one or more
corpuscles leave it? It is very remarkable that we have up to the present
(1910), in spite of many investigations on this point, no direct evidence of
the existence of positively charged particles with a mass comparable with
that of a corpuscle; the smallest positive particle of which we have any
direct indication has a mass equal to the mass of an atom of hydrogen, and
it is a most remarkable fact that we get positively charged particles having
this mass when we send the electric discharge through gases at low
pressures, whatever be the kind of gas. It is no doubt exceedingly difficult
to get rid of traces of hydrogen in vessels containing gases at low pressures
through which an electric discharge is passing, but the circumstances under
which the positively electrified particles just alluded to appear, and the way
in which they remain unaltered in spite of all efforts to clear out any traces
of hydrogen, all seem to indicate that these positively electrified particles,
whose mass is equal to that of an atom of hydrogen, do not come from
minute traces of hydrogen present as an impurity but from the oxygen,
nitrogen, or helium, or whatever may be the gas through which the
discharge passes. If this is so, then the most natural conclusion we can come
to is that these positively electrified particles with the mass of the atom of
hydrogen are the natural units of positive electricity, just as the corpuscles
are those of negative, and that these positive particles form a part of all
atoms.
Thus in this way we are led to an electrical view of the constitution of the
atom. We regard the atom as built up of units of negative electricity and of
an equal number of units of positive electricity; these two units are of very
different mass, the mass of the negative unit being only 1⁄1700 of that of the
positive. The number of units of either kind is proportional to the atomic
the consideration of the constitution of matter. The question naturally arises,
is the positive electricity done up into definite units like the negative, or
does it merely indicate a property acquired by an atom when one or more
corpuscles leave it? It is very remarkable that we have up to the present
(1910), in spite of many investigations on this point, no direct evidence of
the existence of positively charged particles with a mass comparable with
that of a corpuscle; the smallest positive particle of which we have any
direct indication has a mass equal to the mass of an atom of hydrogen, and
it is a most remarkable fact that we get positively charged particles having
this mass when we send the electric discharge through gases at low
pressures, whatever be the kind of gas. It is no doubt exceedingly difficult
to get rid of traces of hydrogen in vessels containing gases at low pressures
through which an electric discharge is passing, but the circumstances under
which the positively electrified particles just alluded to appear, and the way
in which they remain unaltered in spite of all efforts to clear out any traces
of hydrogen, all seem to indicate that these positively electrified particles,
whose mass is equal to that of an atom of hydrogen, do not come from
minute traces of hydrogen present as an impurity but from the oxygen,
nitrogen, or helium, or whatever may be the gas through which the
discharge passes. If this is so, then the most natural conclusion we can come
to is that these positively electrified particles with the mass of the atom of
hydrogen are the natural units of positive electricity, just as the corpuscles
are those of negative, and that these positive particles form a part of all
atoms.
Thus in this way we are led to an electrical view of the constitution of the
atom. We regard the atom as built up of units of negative electricity and of
an equal number of units of positive electricity; these two units are of very
different mass, the mass of the negative unit being only 1⁄1700 of that of the
positive. The number of units of either kind is proportional to the atomic
Page 14
weight of the element and of the same order as this quantity. Whether this is
anything besides the positive and negative electricity in the atom we do not
know. In the present state of our knowledge of the properties of matter it is
unnecessary to postulate the existence of anything besides these positive
and negative units.
The atom of a chemical element on this view of the constitution of matter
is a system formed by n corpuscles and n units of positive electricity which
is in equilibrium or in a state of steady motion under the electrical forces
which the charged 2n constituents exert upon each other. Sir J. J. Thomson
(Phil. Mag., March 1904, “Corpuscular Theory of Matter”) has investigated
the systems in steady motion which can be formed by various numbers of
negatively electrified particles immersed in a sphere of uniform positive
electrification, a case, which in consequence of the enormous volume of the
units of positive electricity in comparison with that of the negative has
much in common with the problem under consideration, and has shown that
some of the properties of n systems of corpuscles vary in a periodic way
suggestive of the Periodic Law in Chemistry as n is continually increased.
Mass on the Electrical Theory of Matter.—One of the most characteristic
things about matter is the possession of mass. When we take the electrical
theory of matter the idea of mass takes new and interesting forms. This
point may be illustrated by the case of a single electrified particle; when this
moves it produces in the region around it a magnetic field, the magnetic
force being proportional to the velocity of the electrified particle.1 In a
magnetic field, however, there is energy, and the amount of energy per unit
volume at any place is proportional to the square of the magnetic force at
that place. Thus there will be energy distributed through the space around
the moving particle, and when the velocity of the particle is small compared
with that of light we can easily show that the energy in the region around
the charged particle is μe2/3a, when v is the velocity of the particle, e its
anything besides the positive and negative electricity in the atom we do not
know. In the present state of our knowledge of the properties of matter it is
unnecessary to postulate the existence of anything besides these positive
and negative units.
The atom of a chemical element on this view of the constitution of matter
is a system formed by n corpuscles and n units of positive electricity which
is in equilibrium or in a state of steady motion under the electrical forces
which the charged 2n constituents exert upon each other. Sir J. J. Thomson
(Phil. Mag., March 1904, “Corpuscular Theory of Matter”) has investigated
the systems in steady motion which can be formed by various numbers of
negatively electrified particles immersed in a sphere of uniform positive
electrification, a case, which in consequence of the enormous volume of the
units of positive electricity in comparison with that of the negative has
much in common with the problem under consideration, and has shown that
some of the properties of n systems of corpuscles vary in a periodic way
suggestive of the Periodic Law in Chemistry as n is continually increased.
Mass on the Electrical Theory of Matter.—One of the most characteristic
things about matter is the possession of mass. When we take the electrical
theory of matter the idea of mass takes new and interesting forms. This
point may be illustrated by the case of a single electrified particle; when this
moves it produces in the region around it a magnetic field, the magnetic
force being proportional to the velocity of the electrified particle.1 In a
magnetic field, however, there is energy, and the amount of energy per unit
volume at any place is proportional to the square of the magnetic force at
that place. Thus there will be energy distributed through the space around
the moving particle, and when the velocity of the particle is small compared
with that of light we can easily show that the energy in the region around
the charged particle is μe2/3a, when v is the velocity of the particle, e its
Page 15
charge, a its radius, and μ the magnetic permeability of the region round the
particle. If m is the ordinary mass of the particle, the part of the kinetic
energy due to the motion of this mass is 1⁄2 mv2, thus the total kinetic energy
is 1⁄2 (m + 2⁄3μe2/a). Thus the electric charge on the particle makes it behave
as if its mass were increased by 2⁄3μe2/a. Since this increase in mass is due
to the energy in the region outside the charged particle, it is natural to look
to that region for this additional mass. This region is traversed by the tubes
of force which start from the electrified body and move with it, and a very
simple calculation shows that we should get the increase in the mass which
is due to the electrification if we suppose that these tubes of force as they
move carry with them a certain amount of the ether, and that this ether had
mass. The mass of ether thus carried along must be such that the amount of
it in unit volume at any part of the field is such that if this were to move
with the velocity of light its kinetic energy would be equal to the potential
energy of the electric field in the unit volume under consideration. When a
tube moves this mass of ether only participates in the motion at right angles
to the tube, it is not set in motion by a movement of the tube along its
length. We may compare the mass which a charged body acquires in virtue
of its charge with the additional mass which a ball apparently acquires when
it is placed in water; a ball placed in water behaves as if its mass were
greater than its mass when moving in vacuo; we can easily understand why
this should be the case, because when the ball in the water moves the water
around it must move as well; so that when a force acting on the ball sets it
in motion it has to move some of the water as well as the ball, and thus the
ball behaves as if its mass were increased. Similarly in the case of the
electrified particle, which when it moves carries with it its lines of force,
which grip the ether and carry some of it along with them. When the
electrified particle is moved a mass of ether has to be moved too, and thus
the apparent mass of the particle is increased. The mass of the electrified
particle is thus resident in every part of space reached by its lines of force;
particle. If m is the ordinary mass of the particle, the part of the kinetic
energy due to the motion of this mass is 1⁄2 mv2, thus the total kinetic energy
is 1⁄2 (m + 2⁄3μe2/a). Thus the electric charge on the particle makes it behave
as if its mass were increased by 2⁄3μe2/a. Since this increase in mass is due
to the energy in the region outside the charged particle, it is natural to look
to that region for this additional mass. This region is traversed by the tubes
of force which start from the electrified body and move with it, and a very
simple calculation shows that we should get the increase in the mass which
is due to the electrification if we suppose that these tubes of force as they
move carry with them a certain amount of the ether, and that this ether had
mass. The mass of ether thus carried along must be such that the amount of
it in unit volume at any part of the field is such that if this were to move
with the velocity of light its kinetic energy would be equal to the potential
energy of the electric field in the unit volume under consideration. When a
tube moves this mass of ether only participates in the motion at right angles
to the tube, it is not set in motion by a movement of the tube along its
length. We may compare the mass which a charged body acquires in virtue
of its charge with the additional mass which a ball apparently acquires when
it is placed in water; a ball placed in water behaves as if its mass were
greater than its mass when moving in vacuo; we can easily understand why
this should be the case, because when the ball in the water moves the water
around it must move as well; so that when a force acting on the ball sets it
in motion it has to move some of the water as well as the ball, and thus the
ball behaves as if its mass were increased. Similarly in the case of the
electrified particle, which when it moves carries with it its lines of force,
which grip the ether and carry some of it along with them. When the
electrified particle is moved a mass of ether has to be moved too, and thus
the apparent mass of the particle is increased. The mass of the electrified
particle is thus resident in every part of space reached by its lines of force;
Page 16
in this sense an electrified body may be said to extend to an infinite
distance; the amount of the mass of the ether attached to the particle
diminishes so rapidly as we recede from it that the contributions of regions
remote from the particle are quite insignificant, and in the case of a particle
as small as a corpuscle not one millionth part of its mass will be farther
away from it than the radius of an atom.
The increase in the mass of a particle due to given charges varies as we
have seen inversely as the radius of the particle; thus the smaller the particle
the greater the increase in the mass. For bodies of appreciable size or even
for those as small as ordinary atoms the effect of any realizable electric
charge is quite insignificant, on the other hand for the smallest bodies
known, the corpuscle, there is evidence that the whole of the mass is due to
the electric charge. This result has been deduced by the help of an
extremely interesting property of the mass due to a charge of electricity,
which is that this mass is not constant but varies with the velocity. This
comes about in the following way. When the charged particle, which for
simplicity we shall suppose to be spherical, is at rest or moving very slowly
the lines of electric force are distributed uniformly around it in all
directions; when the sphere moves, however, magnetic forces are produced
in the region around it, while these, in consequence of electro-magnetic
induction in a moving magnetic field, give rise to electric forces which
displace the tubes of electric force in such a way as to make them set
themselves so as to be more at right angles to the direction in which they
are moving than they were before. Thus if the charged sphere were moving
along the line AB, the tubes of force would, when the sphere was in motion,
tend to leave the region near AB and crowd towards a plane through the
centre of the sphere and at right angles to AB, where they would be moving
more nearly at right angles to themselves. This crowding of the lines of
force increases, however, the potential energy of the electric field, and since
the mass of the ether carried along by the lines of force is proportional to
distance; the amount of the mass of the ether attached to the particle
diminishes so rapidly as we recede from it that the contributions of regions
remote from the particle are quite insignificant, and in the case of a particle
as small as a corpuscle not one millionth part of its mass will be farther
away from it than the radius of an atom.
The increase in the mass of a particle due to given charges varies as we
have seen inversely as the radius of the particle; thus the smaller the particle
the greater the increase in the mass. For bodies of appreciable size or even
for those as small as ordinary atoms the effect of any realizable electric
charge is quite insignificant, on the other hand for the smallest bodies
known, the corpuscle, there is evidence that the whole of the mass is due to
the electric charge. This result has been deduced by the help of an
extremely interesting property of the mass due to a charge of electricity,
which is that this mass is not constant but varies with the velocity. This
comes about in the following way. When the charged particle, which for
simplicity we shall suppose to be spherical, is at rest or moving very slowly
the lines of electric force are distributed uniformly around it in all
directions; when the sphere moves, however, magnetic forces are produced
in the region around it, while these, in consequence of electro-magnetic
induction in a moving magnetic field, give rise to electric forces which
displace the tubes of electric force in such a way as to make them set
themselves so as to be more at right angles to the direction in which they
are moving than they were before. Thus if the charged sphere were moving
along the line AB, the tubes of force would, when the sphere was in motion,
tend to leave the region near AB and crowd towards a plane through the
centre of the sphere and at right angles to AB, where they would be moving
more nearly at right angles to themselves. This crowding of the lines of
force increases, however, the potential energy of the electric field, and since
the mass of the ether carried along by the lines of force is proportional to
Page 17
the potential energy, the mass of the charged particle will also be increased.
The amount of variation of the mass with the velocity depends to some
extent on the assumptions we make as to the shape of the corpuscle and the
way in which it is electrified. The simplest expression connecting the mass
with the velocity is that when the velocity is v the mass is equal to 2⁄3μe2/a
[1/(1 − v2/c2)1/2] where c is the velocity of light. We see from this that the
variation of mass with velocity is very small unless the velocity of the body
approaches that of light, but when, as in the case of the β particles emitted
by radium, the velocity is only a few per cent less than that of light, the
effect of velocity on the mass becomes very considerable; the formula
indicates that if the particles were moving with a velocity equal to that of
light they would behave as if their mass were infinite. By observing the
variation in the mass of a corpuscle as its velocity changes we can
determine how much of the mass depends upon the electric charge and how
much is independent of it. For since the latter part of the mass is
independent of the velocity, if it predominates the variation with velocity of
the mass of a corpuscle will be small; if on the other hand it is negligible
the variation in mass with velocity will be that indicated by theory given
above. The experiment of Kaufmann (Göttingen Nach., Nov. 8, 1901),
Bucherer (Ann. der Physik., xxviii. 513, 1909) on the masses of the β
particles shot out by radium, as well as those by Hupka (Berichte der
deutsch. physik. Gesell., 1909, p. 249) on the masses of the corpuscle in
cathode rays are in agreement with the view that the whole of the mass of
these particles is due to their electric charge.
The alteration in the mass of a moving charge with its velocity is
primarily due to the increase in the potential energy which accompanies the
increase in velocity. The connexion between potential energy and mass is
general and holds for any arrangement of electrified particles; thus if we
assume the electrical constitution of matter, there will be a part of the mass
of any system dependent upon the potential energy and in fact proportional
The amount of variation of the mass with the velocity depends to some
extent on the assumptions we make as to the shape of the corpuscle and the
way in which it is electrified. The simplest expression connecting the mass
with the velocity is that when the velocity is v the mass is equal to 2⁄3μe2/a
[1/(1 − v2/c2)1/2] where c is the velocity of light. We see from this that the
variation of mass with velocity is very small unless the velocity of the body
approaches that of light, but when, as in the case of the β particles emitted
by radium, the velocity is only a few per cent less than that of light, the
effect of velocity on the mass becomes very considerable; the formula
indicates that if the particles were moving with a velocity equal to that of
light they would behave as if their mass were infinite. By observing the
variation in the mass of a corpuscle as its velocity changes we can
determine how much of the mass depends upon the electric charge and how
much is independent of it. For since the latter part of the mass is
independent of the velocity, if it predominates the variation with velocity of
the mass of a corpuscle will be small; if on the other hand it is negligible
the variation in mass with velocity will be that indicated by theory given
above. The experiment of Kaufmann (Göttingen Nach., Nov. 8, 1901),
Bucherer (Ann. der Physik., xxviii. 513, 1909) on the masses of the β
particles shot out by radium, as well as those by Hupka (Berichte der
deutsch. physik. Gesell., 1909, p. 249) on the masses of the corpuscle in
cathode rays are in agreement with the view that the whole of the mass of
these particles is due to their electric charge.
The alteration in the mass of a moving charge with its velocity is
primarily due to the increase in the potential energy which accompanies the
increase in velocity. The connexion between potential energy and mass is
general and holds for any arrangement of electrified particles; thus if we
assume the electrical constitution of matter, there will be a part of the mass
of any system dependent upon the potential energy and in fact proportional
Page 18
to it. Thus every change in potential energy, such for example as occurs
when two elements combine with evolution or absorption of heat, must be
attended by a change in mass. The amount of this change can be calculated
by the rule that if a mass equal to the change in mass were to move with the
velocity of light its kinetic energy would equal the change in the potential
energy. If we apply this result to the case of the combination of hydrogen
and oxygen, where the evolution of heat, about 1.6 × 1011 ergs per gramme
of water, is greater than in any other known case of chemical combination,
we see that the change in mass would only amount to one part in 3000
million, which is far beyond the reach of experiment. The evolution of
energy by radio-active substances is enormously larger than in ordinary
chemical transformations; thus one gramme of radium emits per day about
as much energy as is evolved in the formation of one gramme of water, and
goes on doing this for thousands of years. We see, however, that even in this
case it would require hundreds of years before the changes in mass became
appreciable.
The evolution of energy from the gaseous emanation given off by radium
is more rapid than that from radium itself, since according to the
experiments of Rutherford (Rutherford, Radio-activity, p. 432) a gramme of
the emanation would evolve about 2.1 × 1016 ergs in four days; this by the
rule given above would diminish the mass by about one part in 20,000; but
since only very small quantities of the emanation could be used the
detection of the change of mass does not seem feasible even in this case.
On the view we have been discussing the existence of potential energy
due to an electric field is always associated with mass; wherever there is
potential energy there is mass. On the electro-magnetic theory of light,
however, a wave of light is accompanied by electric forces, and therefore by
potential energy; thus waves of light must behave as if they possessed mass.
It may be shown that it follows from the same principles that they must also
when two elements combine with evolution or absorption of heat, must be
attended by a change in mass. The amount of this change can be calculated
by the rule that if a mass equal to the change in mass were to move with the
velocity of light its kinetic energy would equal the change in the potential
energy. If we apply this result to the case of the combination of hydrogen
and oxygen, where the evolution of heat, about 1.6 × 1011 ergs per gramme
of water, is greater than in any other known case of chemical combination,
we see that the change in mass would only amount to one part in 3000
million, which is far beyond the reach of experiment. The evolution of
energy by radio-active substances is enormously larger than in ordinary
chemical transformations; thus one gramme of radium emits per day about
as much energy as is evolved in the formation of one gramme of water, and
goes on doing this for thousands of years. We see, however, that even in this
case it would require hundreds of years before the changes in mass became
appreciable.
The evolution of energy from the gaseous emanation given off by radium
is more rapid than that from radium itself, since according to the
experiments of Rutherford (Rutherford, Radio-activity, p. 432) a gramme of
the emanation would evolve about 2.1 × 1016 ergs in four days; this by the
rule given above would diminish the mass by about one part in 20,000; but
since only very small quantities of the emanation could be used the
detection of the change of mass does not seem feasible even in this case.
On the view we have been discussing the existence of potential energy
due to an electric field is always associated with mass; wherever there is
potential energy there is mass. On the electro-magnetic theory of light,
however, a wave of light is accompanied by electric forces, and therefore by
potential energy; thus waves of light must behave as if they possessed mass.
It may be shown that it follows from the same principles that they must also
Page 19
possess momentum, the direction of the momentum being the direction
along which the light is travelling; when the light is absorbed by an opaque
substance the momentum in the light is communicated to the substance,
which therefore behaves as if the light pressed upon it. The pressure exerted
by light was shown by Maxwell (Electricity and Magnetism, 3rd ed., p.
440) to be a consequence of his electro-magnetic theory, its existence has
been established by the experiment of Lebedew, of Nichols and Hull, and of
Poynting.
We have hitherto been considering mass from the point of view that the
constitution of matter is electrical; we shall proceed to consider the question
of weight from the same point of view. The relation between mass and
weight is, while the simplest in expression, perhaps the
Weight. most fundamental and mysterious property possessed by
matter. The weight of a body is proportional to its mass,
that is if the weights of a number of substances are equal the masses will be
equal, whatever the substances may be. This result was verified to a
considerable degree of approximation by Newton by means of experiments
with pendulums; later, in 1830 Bessel by a very extensive and accurate
series of experiments, also made on pendulums, showed that the ratio of
mass to weight was certainly to one part in 60,000 the same for all the
substances examined by him, these included brass, silver, iron, lead, copper,
ivory, water.
The constancy of this ratio acquires new interest when looked at from the
point of view of the electrical constitution of matter. We have seen that the
atoms of all bodies contain corpuscles, that the mass of a corpuscle is only
1⁄1700 of the mass of an atom of hydrogen, that it carries a constant charge of
negative electricity, and that its mass is entirely due to this charge, and can
be regarded as arising from ether gripped by the lines of force starting from
the electrical charge. The question at once suggests itself, Is this kind of
along which the light is travelling; when the light is absorbed by an opaque
substance the momentum in the light is communicated to the substance,
which therefore behaves as if the light pressed upon it. The pressure exerted
by light was shown by Maxwell (Electricity and Magnetism, 3rd ed., p.
440) to be a consequence of his electro-magnetic theory, its existence has
been established by the experiment of Lebedew, of Nichols and Hull, and of
Poynting.
We have hitherto been considering mass from the point of view that the
constitution of matter is electrical; we shall proceed to consider the question
of weight from the same point of view. The relation between mass and
weight is, while the simplest in expression, perhaps the
Weight. most fundamental and mysterious property possessed by
matter. The weight of a body is proportional to its mass,
that is if the weights of a number of substances are equal the masses will be
equal, whatever the substances may be. This result was verified to a
considerable degree of approximation by Newton by means of experiments
with pendulums; later, in 1830 Bessel by a very extensive and accurate
series of experiments, also made on pendulums, showed that the ratio of
mass to weight was certainly to one part in 60,000 the same for all the
substances examined by him, these included brass, silver, iron, lead, copper,
ivory, water.
The constancy of this ratio acquires new interest when looked at from the
point of view of the electrical constitution of matter. We have seen that the
atoms of all bodies contain corpuscles, that the mass of a corpuscle is only
1⁄1700 of the mass of an atom of hydrogen, that it carries a constant charge of
negative electricity, and that its mass is entirely due to this charge, and can
be regarded as arising from ether gripped by the lines of force starting from
the electrical charge. The question at once suggests itself, Is this kind of
Page 20
mass ponderable? does it add to the weight of the body? and, if so, is the
proportion between mass and weight the same as for ordinary bodies? Let
us suppose for a moment that this mass is not ponderable, so that the
corpuscles increase the mass but not the weight of an atom. Then, since the
mass of a corpuscle is 1⁄1700 that of an atom of hydrogen, the addition or
removal of one corpuscle would in the case of an atom of atomic weight x
alter the mass by one part in 1700 x, without altering the weight, this would
produce an effect of the same magnitude on the ratio of mass to weight and
would in the case of the atoms of the lighter elements be easily measurable
in experiments of the same order of accuracy as those made by Bessel. If
the number of corpuscles in the atom were proportional to the atomic
weight, then the ratio of mass to weight would be constant whether the
corpuscles were ponderable or not. If the number were not proportional
there would be greater discrepancies in the ratio of mass to weight than is
consistent with Bessel’s experiments if the corpuscles had no weight. We
have seen there are other grounds for concluding that the number of
corpuscles in an atom is proportional to the atom weight, so that the
constancy of the ratio of mass to weight for a large number of substances
does not enable us to determine whether or not mass due to charges of
electricity is ponderable or not.
There seems some hope that the determination of this ratio for radio-
active substances may throw some light on this point. The enormous
amount of heat evolved by these bodies may indicate that they possess
much greater stores of potential energy than other substances. If we suppose
that the heat developed by one gramme of a radio-active substance in the
transformations which it undergoes before it reaches the non-radio-active
stage is a measure of the excess of the potential energy in a gramme of this
substance above that in a gramme of non-radio-active substance, it would
follow that a larger part of the mass was due to electric charges in radio-
active than in non-radio-active substances; in the case of uranium this
proportion between mass and weight the same as for ordinary bodies? Let
us suppose for a moment that this mass is not ponderable, so that the
corpuscles increase the mass but not the weight of an atom. Then, since the
mass of a corpuscle is 1⁄1700 that of an atom of hydrogen, the addition or
removal of one corpuscle would in the case of an atom of atomic weight x
alter the mass by one part in 1700 x, without altering the weight, this would
produce an effect of the same magnitude on the ratio of mass to weight and
would in the case of the atoms of the lighter elements be easily measurable
in experiments of the same order of accuracy as those made by Bessel. If
the number of corpuscles in the atom were proportional to the atomic
weight, then the ratio of mass to weight would be constant whether the
corpuscles were ponderable or not. If the number were not proportional
there would be greater discrepancies in the ratio of mass to weight than is
consistent with Bessel’s experiments if the corpuscles had no weight. We
have seen there are other grounds for concluding that the number of
corpuscles in an atom is proportional to the atom weight, so that the
constancy of the ratio of mass to weight for a large number of substances
does not enable us to determine whether or not mass due to charges of
electricity is ponderable or not.
There seems some hope that the determination of this ratio for radio-
active substances may throw some light on this point. The enormous
amount of heat evolved by these bodies may indicate that they possess
much greater stores of potential energy than other substances. If we suppose
that the heat developed by one gramme of a radio-active substance in the
transformations which it undergoes before it reaches the non-radio-active
stage is a measure of the excess of the potential energy in a gramme of this
substance above that in a gramme of non-radio-active substance, it would
follow that a larger part of the mass was due to electric charges in radio-
active than in non-radio-active substances; in the case of uranium this
Page 21
difference would amount to at least one part in 20,000 of the total mass. If
this extra mass had no weight the ratio of mass to weight for uranium would
differ from the normal amount by more than one part in 20,000, a quantity
quite within the range of pendulum experiments. It thus appears very
desirable to make experiments on the ratio of mass to weight for radio-
active substances. Sir J. J. Thomson, by swinging a small pendulum whose
bob was made of radium bromide, has shown that this ratio for radium does
not differ from the normal by one part in 2000. The small quantity of
radium available prevented the attainment of greater accuracy. Experiments
just completed (1910) by Southerns at the Cavendish Laboratory on this
ratio for uranium show that it is normal to an accuracy of one part in
200,000; indicating that in non-radio-active, as in radio-active, substances
the electrical mass is proportional to the atomic weight.
Though but few experiments have been made in recent years on the value
of the ratio of mass to weight, many important investigations have been
made on the effect of alterations in the chemical and physical conditions on
the weight of bodies. These have all led to the conclusion that no change
which can be detected by our present means of investigation occurs in the
weight of a body in consequence of any physical or chemical changes yet
investigated. Thus Landolt, who devoted a great number of years to the
question whether any change in weight occurs during chemical
combination, came finally to the conclusion that in no case out of the many
he investigated did any measurable change of weight occur during chemical
combination. Poynting and Phillips (Proc. Roy. Soc., 76, p. 445), as well as
Southerns (78, p. 392), have shown that change in temperature produces no
change in the weight of a body; and Poynting has also shown that neither
the weight of a crystal nor the attraction between two crystals depends at all
upon the direction in which the axis of the crystal points. The result of these
laborious and very carefully made experiments has been to strengthen the
conviction that the weight of a given portion of matter is absolutely
this extra mass had no weight the ratio of mass to weight for uranium would
differ from the normal amount by more than one part in 20,000, a quantity
quite within the range of pendulum experiments. It thus appears very
desirable to make experiments on the ratio of mass to weight for radio-
active substances. Sir J. J. Thomson, by swinging a small pendulum whose
bob was made of radium bromide, has shown that this ratio for radium does
not differ from the normal by one part in 2000. The small quantity of
radium available prevented the attainment of greater accuracy. Experiments
just completed (1910) by Southerns at the Cavendish Laboratory on this
ratio for uranium show that it is normal to an accuracy of one part in
200,000; indicating that in non-radio-active, as in radio-active, substances
the electrical mass is proportional to the atomic weight.
Though but few experiments have been made in recent years on the value
of the ratio of mass to weight, many important investigations have been
made on the effect of alterations in the chemical and physical conditions on
the weight of bodies. These have all led to the conclusion that no change
which can be detected by our present means of investigation occurs in the
weight of a body in consequence of any physical or chemical changes yet
investigated. Thus Landolt, who devoted a great number of years to the
question whether any change in weight occurs during chemical
combination, came finally to the conclusion that in no case out of the many
he investigated did any measurable change of weight occur during chemical
combination. Poynting and Phillips (Proc. Roy. Soc., 76, p. 445), as well as
Southerns (78, p. 392), have shown that change in temperature produces no
change in the weight of a body; and Poynting has also shown that neither
the weight of a crystal nor the attraction between two crystals depends at all
upon the direction in which the axis of the crystal points. The result of these
laborious and very carefully made experiments has been to strengthen the
conviction that the weight of a given portion of matter is absolutely
Page 22
independent of its physical condition or state of chemical combinations. It
should, however, be noticed that we have as yet no accurate investigation as
to whether or not any changes of weight occur during radio-active
transformations, such for example as the emanation from radium undergoes
when the atoms themselves of the substance are disrupted.
It is a matter of some interest in connexion with a discussion of any
views of the constitution of matter to consider the theories of gravitation
which have been put forward to explain that apparently invariable property
of matter—its weight. It would be impossible to consider in detail the
numerous theories which have been put forward to account for gravitation;
a concise summary of many of these has been given by Drude (Wied. Ann.
62, p. 1);2 there is no dearth of theories as to the cause of gravitation, what
is lacking is the means of putting any of them to a decisive test.
There are, however, two theories of gravitation, both old, which seem to
be especially closely connected with the idea of the electrical constitution of
matter. The first of these is the theory, associated with the two fluid theory
of electricity, that gravity is a kind of residual electrical effect, due to the
attraction between the units of positive and negative electricity being a little
greater than the repulsion between the units of electricity of the same kind.
Thus on this view two charges of equal magnitude, but of opposite sign,
would exert an attraction varying inversely as the square of the distance on
a charge of electricity of either sign, and therefore an attraction on a system
consisting of two charges equal in magnitude but opposite in sign forming
an electrically neutral system. Thus if we had two neutral systems, A and B,
A consisting of m positive units of electricity and an equal number of
negative, while B has n units of each kind, then the gravitational attraction
between A and B would be inversely proportional to the square of the
distance and proportional to n m. The connexion between this view of
gravity and that of the electrical constitution of matter is evidently very
should, however, be noticed that we have as yet no accurate investigation as
to whether or not any changes of weight occur during radio-active
transformations, such for example as the emanation from radium undergoes
when the atoms themselves of the substance are disrupted.
It is a matter of some interest in connexion with a discussion of any
views of the constitution of matter to consider the theories of gravitation
which have been put forward to explain that apparently invariable property
of matter—its weight. It would be impossible to consider in detail the
numerous theories which have been put forward to account for gravitation;
a concise summary of many of these has been given by Drude (Wied. Ann.
62, p. 1);2 there is no dearth of theories as to the cause of gravitation, what
is lacking is the means of putting any of them to a decisive test.
There are, however, two theories of gravitation, both old, which seem to
be especially closely connected with the idea of the electrical constitution of
matter. The first of these is the theory, associated with the two fluid theory
of electricity, that gravity is a kind of residual electrical effect, due to the
attraction between the units of positive and negative electricity being a little
greater than the repulsion between the units of electricity of the same kind.
Thus on this view two charges of equal magnitude, but of opposite sign,
would exert an attraction varying inversely as the square of the distance on
a charge of electricity of either sign, and therefore an attraction on a system
consisting of two charges equal in magnitude but opposite in sign forming
an electrically neutral system. Thus if we had two neutral systems, A and B,
A consisting of m positive units of electricity and an equal number of
negative, while B has n units of each kind, then the gravitational attraction
between A and B would be inversely proportional to the square of the
distance and proportional to n m. The connexion between this view of
gravity and that of the electrical constitution of matter is evidently very
Page 23
close, for if gravity arose in this way the weight of a body would only
depend upon the number of units of electricity in the body. On the view that
the constitution of matter is electrical, the fundamental units which build up
matter are the units of electric charge, and as the magnitude of these charges
does not change, whatever chemical or physical vicissitudes matter, the
weight of matter ought not to be affected by such changes. There is one
result of this theory which might possibly afford a means of testing it: since
the charge on a corpuscle is equal to that on a positive unit, the weights of
the two are equal; but the mass of the corpuscle is only 1⁄1700 of that of the
positive unit, so that the acceleration of the corpuscle under gravity will be
1700 times that of the positive unit, which we should expect to be the same
as that for ponderable matter or 981.
The acceleration of the corpuscle under gravity on this view would be 1.6
× 106. It does not seem altogether impossible that with methods slightly
more powerful than those we now possess we might measure the effect of
gravity on a corpuscle if the acceleration were as large as this.
The other theory of gravitation to which we call attention is that due to
Le Sage of Geneva and published in 1818. Le Sage supposed that the
universe was thronged with exceedingly small particles moving with very
great velocities. These particles he called ultra-mundane corpuscles,
because they came to us from regions far beyond the solar system. He
assumed that these were so penetrating that they could pass through masses
as large as the sun or the earth without being absorbed to more than a very
small extent. There is, however, some absorption, and if bodies are made up
of the same kind of atoms, whose dimensions are small compared with the
distances between them, the absorption will be proportional to the mass of
the body. So that as the ultra-mundane corpuscles stream through the body a
small fraction, proportional to the mass of the body, of their momentum is
communicated to it. If the direction of the ultra-mundane corpuscles passing
depend upon the number of units of electricity in the body. On the view that
the constitution of matter is electrical, the fundamental units which build up
matter are the units of electric charge, and as the magnitude of these charges
does not change, whatever chemical or physical vicissitudes matter, the
weight of matter ought not to be affected by such changes. There is one
result of this theory which might possibly afford a means of testing it: since
the charge on a corpuscle is equal to that on a positive unit, the weights of
the two are equal; but the mass of the corpuscle is only 1⁄1700 of that of the
positive unit, so that the acceleration of the corpuscle under gravity will be
1700 times that of the positive unit, which we should expect to be the same
as that for ponderable matter or 981.
The acceleration of the corpuscle under gravity on this view would be 1.6
× 106. It does not seem altogether impossible that with methods slightly
more powerful than those we now possess we might measure the effect of
gravity on a corpuscle if the acceleration were as large as this.
The other theory of gravitation to which we call attention is that due to
Le Sage of Geneva and published in 1818. Le Sage supposed that the
universe was thronged with exceedingly small particles moving with very
great velocities. These particles he called ultra-mundane corpuscles,
because they came to us from regions far beyond the solar system. He
assumed that these were so penetrating that they could pass through masses
as large as the sun or the earth without being absorbed to more than a very
small extent. There is, however, some absorption, and if bodies are made up
of the same kind of atoms, whose dimensions are small compared with the
distances between them, the absorption will be proportional to the mass of
the body. So that as the ultra-mundane corpuscles stream through the body a
small fraction, proportional to the mass of the body, of their momentum is
communicated to it. If the direction of the ultra-mundane corpuscles passing
Page 24
through the body were uniformly distributed, the momentum communicated
by them to the body would not tend to move it in one direction rather than
in another, so that a body, A, alone in the universe and exposed to
bombardment by the ultra-mundane corpuscles would remain at rest. If,
however, there were a second body, B, in the neighbourhood of A, B will
shield A from some of the corpuscles moving in the direction BA; thus A
will not receive as much momentum in this direction as when it was alone;
but in this case it only received just enough to keep it in equilibrium, so that
when B is present the momentum in the opposite direction will get the
upper hand and A will move in the direction AB, and will thus be attracted
by B. Similarly, we see that B will be attracted by A. Le Sage proved that
the rate at which momentum was being communicated to A or B by the
passage through them of his corpuscles was proportional to the product of
the masses of A and B, and if the distance between A and B was large
compared with their dimensions, inversely proportional to the square of the
distance between them; in fact, that the forces acting on them would obey
the same laws as the gravitational attraction between them. Clerk Maxwell
(article “Atom,” Ency. Brit., 9th ed.) pointed out that this transference of
momentum from the ultra-mundane corpuscles to the body through which
they passed involved the loss of kinetic energy by the corpuscles, and if the
loss of momentum were large enough to account for the gravitational
attraction, the loss of kinetic energy would be so large that if converted into
heat it would be sufficient to keep the body white hot. We need not,
however, suppose that this energy is converted into heat; it might, as in the
case where Röntgen rays are produced by the passage of electrified
corpuscles through matter, be transformed into the energy of a still more
penetrating form of radiation, which might escape from the gravitating body
without heating it. It is a very interesting result of recent discoveries that the
machinery which Le Sage introduced for the purpose of his theory has a
very close analogy with things for which we have now direct experimental
by them to the body would not tend to move it in one direction rather than
in another, so that a body, A, alone in the universe and exposed to
bombardment by the ultra-mundane corpuscles would remain at rest. If,
however, there were a second body, B, in the neighbourhood of A, B will
shield A from some of the corpuscles moving in the direction BA; thus A
will not receive as much momentum in this direction as when it was alone;
but in this case it only received just enough to keep it in equilibrium, so that
when B is present the momentum in the opposite direction will get the
upper hand and A will move in the direction AB, and will thus be attracted
by B. Similarly, we see that B will be attracted by A. Le Sage proved that
the rate at which momentum was being communicated to A or B by the
passage through them of his corpuscles was proportional to the product of
the masses of A and B, and if the distance between A and B was large
compared with their dimensions, inversely proportional to the square of the
distance between them; in fact, that the forces acting on them would obey
the same laws as the gravitational attraction between them. Clerk Maxwell
(article “Atom,” Ency. Brit., 9th ed.) pointed out that this transference of
momentum from the ultra-mundane corpuscles to the body through which
they passed involved the loss of kinetic energy by the corpuscles, and if the
loss of momentum were large enough to account for the gravitational
attraction, the loss of kinetic energy would be so large that if converted into
heat it would be sufficient to keep the body white hot. We need not,
however, suppose that this energy is converted into heat; it might, as in the
case where Röntgen rays are produced by the passage of electrified
corpuscles through matter, be transformed into the energy of a still more
penetrating form of radiation, which might escape from the gravitating body
without heating it. It is a very interesting result of recent discoveries that the
machinery which Le Sage introduced for the purpose of his theory has a
very close analogy with things for which we have now direct experimental
Page 25
evidence. We know that small particles moving with very high speeds do
exist, that they possess considerable powers of penetrating solids, though
not, as far as we know at present, to an extent comparable with that
postulated by Le Sage; and we know that the energy lost by them as they
pass through a solid is to a large extent converted into a still more
penetrating form of radiation, Röntgen rays. In Le Sage’s theory the only
function of the corpuscles is to act as carriers of momentum, any systems
which possessed momentum, moved with a high velocity and had the power
of penetrating solids, might be substituted for them; now waves of electric
and magnetic force, such as light waves or Röntgen rays, possess
momentum, move with a high velocity, and the latter at any rate possess
considerable powers of penetration; so that we might formulate a theory in
which penetrating Röntgen rays replaced Le Sage’s corpuscles. Röntgen
rays, however, when absorbed do not, as far as we know, give rise to more
penetrating Röntgen rays as they should to explain attraction, but either to
less penetrating rays or to rays of the same kind.
We have confined our attention in this article to the view that the
constitution of matter is electrical; we have done so because this view is
more closely in touch with experiment than any other yet advanced. The
units of which matter is built up on this theory have been isolated and
detected in the laboratory, and we may hope to discover more and more of
their properties. By seeing whether the properties of matter are or are not
such as would arise from a collection of units having these properties, we
can apply to this theory tests of a much more definite and rigorous character
than we can apply to any other theory of matter.
(J. J. T.)
1 We may measure this velocity with reference to any axes, provided we refer the
motion of all the bodies which come into consideration to the same axes.
2 A theory published after Drude’s paper in that of Professor Osborne Reynolds, given
in his Rede lecture “On an Inversion of Ideas as to the Structure of the Universe.”
exist, that they possess considerable powers of penetrating solids, though
not, as far as we know at present, to an extent comparable with that
postulated by Le Sage; and we know that the energy lost by them as they
pass through a solid is to a large extent converted into a still more
penetrating form of radiation, Röntgen rays. In Le Sage’s theory the only
function of the corpuscles is to act as carriers of momentum, any systems
which possessed momentum, moved with a high velocity and had the power
of penetrating solids, might be substituted for them; now waves of electric
and magnetic force, such as light waves or Röntgen rays, possess
momentum, move with a high velocity, and the latter at any rate possess
considerable powers of penetration; so that we might formulate a theory in
which penetrating Röntgen rays replaced Le Sage’s corpuscles. Röntgen
rays, however, when absorbed do not, as far as we know, give rise to more
penetrating Röntgen rays as they should to explain attraction, but either to
less penetrating rays or to rays of the same kind.
We have confined our attention in this article to the view that the
constitution of matter is electrical; we have done so because this view is
more closely in touch with experiment than any other yet advanced. The
units of which matter is built up on this theory have been isolated and
detected in the laboratory, and we may hope to discover more and more of
their properties. By seeing whether the properties of matter are or are not
such as would arise from a collection of units having these properties, we
can apply to this theory tests of a much more definite and rigorous character
than we can apply to any other theory of matter.
(J. J. T.)
1 We may measure this velocity with reference to any axes, provided we refer the
motion of all the bodies which come into consideration to the same axes.
2 A theory published after Drude’s paper in that of Professor Osborne Reynolds, given
in his Rede lecture “On an Inversion of Ideas as to the Structure of the Universe.”
Page 26
MATTERHORN, one of the best known mountains (14,782 ft.) in
the Alps. It rises S.W. of the village of Zermatt, and on the frontier between
Switzerland (canton of the Valais) and Italy. Though on the Swiss side it
appears to be an isolated obelisk, it is really but the butt end of a ridge,
while the Swiss slope is not nearly as steep or difficult as the grand terraced
walls of the Italian slope. It was first conquered, after a number of attempts
chiefly on the Italian side, on the 14th of July 1865, by Mr E. Whymper’s
party, three members of which (Lord Francis Douglas, the Rev. C. Hudson
and Mr Hadow) with the guide, Michel Croz, perished by a slip on the
descent. Three days later it was scaled from the Italian side by a party of
men from Val Tournanche. Nowadays it is frequently ascended in summer,
especially from Zermatt.
MATTEUCCI, CARLO (1811-1868), Italian physicist, was
born at Forlì on the 20th of June 1811. After attending the École
Polytechnique at Paris, he became professor of physics successively at
Bologna (1832), Ravenna (1837) and Pisa (1840). From 1847 he took an
the Alps. It rises S.W. of the village of Zermatt, and on the frontier between
Switzerland (canton of the Valais) and Italy. Though on the Swiss side it
appears to be an isolated obelisk, it is really but the butt end of a ridge,
while the Swiss slope is not nearly as steep or difficult as the grand terraced
walls of the Italian slope. It was first conquered, after a number of attempts
chiefly on the Italian side, on the 14th of July 1865, by Mr E. Whymper’s
party, three members of which (Lord Francis Douglas, the Rev. C. Hudson
and Mr Hadow) with the guide, Michel Croz, perished by a slip on the
descent. Three days later it was scaled from the Italian side by a party of
men from Val Tournanche. Nowadays it is frequently ascended in summer,
especially from Zermatt.
MATTEUCCI, CARLO (1811-1868), Italian physicist, was
born at Forlì on the 20th of June 1811. After attending the École
Polytechnique at Paris, he became professor of physics successively at
Bologna (1832), Ravenna (1837) and Pisa (1840). From 1847 he took an
Page 27
active part in politics, and in 1860 was chosen an Italian senator, at the
same time becoming inspector-general of the Italian telegraph lines. Two
years later he was minister of education. He died near Leghorn on the 25th
of June 1868.
He was the author of four scientific treatises: Lezioni di fisica (2
vols., Pisa, 1841), Lezioni sui fenomeni fisicochimici dei corpi viventi
(Pisa, 1844), Manuale di telegrafia elettrica (Pisa, 1850) and Cours
spécial sur l’induction, le magnetisme de rotation, &c. (Paris, 1854).
His numerous papers were published in the Annales de chimie et de
physique (1829-1858); and most of them also appeared at the time in
the Italian scientific journals. They relate almost entirely to electrical
phenomena, such as the magnetic rotation of light, the action of gas
batteries, the effects of torsion on magnetism, the polarization of
electrodes, &c., sufficiently complete accounts of which are given in
Wiedemann’s Galvanismus. Nine memoirs, entitled “Electro-
Physiological Researches,” were published in the Philosophical
Transactions, 1845-1860. See Bianchi’s Carlo Matteucci e l’Italia del
suo tempo (Rome, 1874).
MATTHEW, ST (Μαθθαῖος or Ματθαῖος, probably a shortened
form of the Hebrew equivalent to Theodorus), one of the twelve apostles,
and the traditional author of the First Gospel, where he is described as
having been a tax-gatherer or customs-officer (τελώνης, x. 3), in the
same time becoming inspector-general of the Italian telegraph lines. Two
years later he was minister of education. He died near Leghorn on the 25th
of June 1868.
He was the author of four scientific treatises: Lezioni di fisica (2
vols., Pisa, 1841), Lezioni sui fenomeni fisicochimici dei corpi viventi
(Pisa, 1844), Manuale di telegrafia elettrica (Pisa, 1850) and Cours
spécial sur l’induction, le magnetisme de rotation, &c. (Paris, 1854).
His numerous papers were published in the Annales de chimie et de
physique (1829-1858); and most of them also appeared at the time in
the Italian scientific journals. They relate almost entirely to electrical
phenomena, such as the magnetic rotation of light, the action of gas
batteries, the effects of torsion on magnetism, the polarization of
electrodes, &c., sufficiently complete accounts of which are given in
Wiedemann’s Galvanismus. Nine memoirs, entitled “Electro-
Physiological Researches,” were published in the Philosophical
Transactions, 1845-1860. See Bianchi’s Carlo Matteucci e l’Italia del
suo tempo (Rome, 1874).
MATTHEW, ST (Μαθθαῖος or Ματθαῖος, probably a shortened
form of the Hebrew equivalent to Theodorus), one of the twelve apostles,
and the traditional author of the First Gospel, where he is described as
having been a tax-gatherer or customs-officer (τελώνης, x. 3), in the
Page 28
service of the tetrarch Herod. The circumstances of his call to become a
follower of Jesus, received as he sat in the “customs house” in one of the
towns by the Sea of Galilee—apparently Capernaum (Mark ii. 1, 13), are
briefly related in ix. 9. We should gather from the parallel narrative in Mark
ii. 14, Luke v. 27, that he was at the time known as “Levi the son of
Alphaeus” (compare Simon Cephas, Joseph Barnabas): if so, “James the
son of Alphaeus” may have been his brother. Possibly “Matthew”
(Yahweh’s gift) was his Christian surname, since two native names, neither
being a patronymic, is contrary to Jewish usage. It must be noted, however,
that Matthew and Levi were sometimes distinguished in early times, as by
Heracleon (c. 170 a.d.), and more dubiously by Origen (c. Celsum, i. 62),
also apparently in the Syriac Didascalia (sec. iii.), V. xiv. 14. It has
generally been supposed, on the strength of Luke’s account (v. 29), that
Matthew gave a feast in Jesus’ honour (like Zacchaeus, Luke xix. 6 seq.).
But Mark (ii. 15), followed by Matthew (ix. 10), may mean that the meal in
question was one in Jesus’ own home at Capernaum (cf. v. 1). In the lists of
the Apostles given in the Synoptic Gospels and in Acts, Matthew ranks third
or fourth in the second group of four—a fair index of his relative
importance in the apostolic age. The only other facts related of Matthew on
good authority concern him as Evangelist. Eusebius (H.E. iii. 24) says that
he, like John, wrote only at the spur of necessity. “For Matthew, after
preaching to Hebrews, when about to go also to others, committed to
writing in his native tongue the Gospel that bears his name; and so by his
writing supplied, for those whom he was leaving, the loss of his presence.”
The value of this tradition, which may be based on Papias, who certainly
reported that “Matthew compiled the Oracles (of the Lord) in Hebrew,” can
be estimated only in connexion with the study of the Gospel itself (see
below). No historical use can be made of the artificial story, in Sanhedrin
43a, that Matthew was condemned to death by a Jewish court (see Laihle,
Christ in the Talmud, 71 seq.). According to the Gnostic Heracleon, quoted
follower of Jesus, received as he sat in the “customs house” in one of the
towns by the Sea of Galilee—apparently Capernaum (Mark ii. 1, 13), are
briefly related in ix. 9. We should gather from the parallel narrative in Mark
ii. 14, Luke v. 27, that he was at the time known as “Levi the son of
Alphaeus” (compare Simon Cephas, Joseph Barnabas): if so, “James the
son of Alphaeus” may have been his brother. Possibly “Matthew”
(Yahweh’s gift) was his Christian surname, since two native names, neither
being a patronymic, is contrary to Jewish usage. It must be noted, however,
that Matthew and Levi were sometimes distinguished in early times, as by
Heracleon (c. 170 a.d.), and more dubiously by Origen (c. Celsum, i. 62),
also apparently in the Syriac Didascalia (sec. iii.), V. xiv. 14. It has
generally been supposed, on the strength of Luke’s account (v. 29), that
Matthew gave a feast in Jesus’ honour (like Zacchaeus, Luke xix. 6 seq.).
But Mark (ii. 15), followed by Matthew (ix. 10), may mean that the meal in
question was one in Jesus’ own home at Capernaum (cf. v. 1). In the lists of
the Apostles given in the Synoptic Gospels and in Acts, Matthew ranks third
or fourth in the second group of four—a fair index of his relative
importance in the apostolic age. The only other facts related of Matthew on
good authority concern him as Evangelist. Eusebius (H.E. iii. 24) says that
he, like John, wrote only at the spur of necessity. “For Matthew, after
preaching to Hebrews, when about to go also to others, committed to
writing in his native tongue the Gospel that bears his name; and so by his
writing supplied, for those whom he was leaving, the loss of his presence.”
The value of this tradition, which may be based on Papias, who certainly
reported that “Matthew compiled the Oracles (of the Lord) in Hebrew,” can
be estimated only in connexion with the study of the Gospel itself (see
below). No historical use can be made of the artificial story, in Sanhedrin
43a, that Matthew was condemned to death by a Jewish court (see Laihle,
Christ in the Talmud, 71 seq.). According to the Gnostic Heracleon, quoted
Page 29
by Clement of Alexandria (Strom. iv. 9), Matthew died a natural death. The
tradition as to his ascetic diet (in Clem. Alex. Paedag. ii. 16) maybe due to
confusion with Matthias (cf. Mart. Matthaei, i.). The earliest legend as to
his later labours, one of Syrian origin, places them in the Parthian kingdom,
where it represents him as dying a natural death at Hierapolis (= Mabog on
the Euphrates). This agrees with his legend as known to Ambrose and
Paulinus of Nola, and is the most probable in itself. The legends which
make him work with Andrew among the Anthropophagi near the Black Sea,
or again in Ethiopia (Rufinus, and Socrates, H.E. i. 19), are due to
confusion with Matthias, who from the first was associated in his Acts with
Andrew (see M. Bonnet, Acta Apost. apocr., 1808, II. i. 65). Another
legend, his Martyrium, makes him labour and suffer in Mysore. He is
commemorated as a martyr by the Greek Church on the 16th of November,
and by the Roman on the 21st of September, the scene of his martyrdom
being placed in Ethiopia. The Latin Breviary also affirms that his body was
afterwards translated to Salerno, where it is said to lie in the church built by
Robert Guiscard. In Christian art (following Jerome) the Evangelist
Matthew is generally symbolized by the “man” in the imagery of Ezek. i.
10, Rev. iv. 7.
For the historical Matthew, see Ency. Bibl. and Zahn, Introd. to New
Test., ii. 506 seq., 522 seq. For his legends, as under Mark.
(J. V. B.)
tradition as to his ascetic diet (in Clem. Alex. Paedag. ii. 16) maybe due to
confusion with Matthias (cf. Mart. Matthaei, i.). The earliest legend as to
his later labours, one of Syrian origin, places them in the Parthian kingdom,
where it represents him as dying a natural death at Hierapolis (= Mabog on
the Euphrates). This agrees with his legend as known to Ambrose and
Paulinus of Nola, and is the most probable in itself. The legends which
make him work with Andrew among the Anthropophagi near the Black Sea,
or again in Ethiopia (Rufinus, and Socrates, H.E. i. 19), are due to
confusion with Matthias, who from the first was associated in his Acts with
Andrew (see M. Bonnet, Acta Apost. apocr., 1808, II. i. 65). Another
legend, his Martyrium, makes him labour and suffer in Mysore. He is
commemorated as a martyr by the Greek Church on the 16th of November,
and by the Roman on the 21st of September, the scene of his martyrdom
being placed in Ethiopia. The Latin Breviary also affirms that his body was
afterwards translated to Salerno, where it is said to lie in the church built by
Robert Guiscard. In Christian art (following Jerome) the Evangelist
Matthew is generally symbolized by the “man” in the imagery of Ezek. i.
10, Rev. iv. 7.
For the historical Matthew, see Ency. Bibl. and Zahn, Introd. to New
Test., ii. 506 seq., 522 seq. For his legends, as under Mark.
(J. V. B.)
Page 30
MATTHEW, TOBIAS, or Tobie (1546-1628), archbishop of
York, was the son of Sir John Matthew of Ross in Herefordshire, and of his
wife Eleanor Crofton of Ludlow. He was born at Bristol in 1546. He was
educated at Wells, and then in succession at University College and Christ
Church, Oxford. He proceeded B.A. in 1564, and M.A. in 1566. He
attracted the favourable notice of Queen Elizabeth, and his rise was steady
though not very rapid. He was public orator in 1569, president of St John’s
College, Oxford, in 1572, dean of Christ Church in 1576, vice-chancellor of
the university in 1579, dean of Durham in 1583, bishop of Durham in 1595,
and archbishop of York in 1606. In 1581 he had a controversy with the
Jesuit Edmund Campion, and published at Oxford his arguments in 1638
under the title, Piissimi et eminentissimi viri Tobiae Matthew, archiepiscopi
olim Eboracencis concio apologetica adversus Campianam. While in the
north he was active in forcing the recusants to conform to the Church of
England, preaching hundreds of sermons and carrying out thorough
visitations. During his later years he was to some extent in opposition to the
administration of James I. He was exempted from attendance in the
parliament of 1625 on the ground of age and infirmities, and died on the
29th of March 1628. His wife, Frances, was the daughter of William
Barlow, bishop of Chichester.
His son, Sir Tobias, or Tobie, Matthew (1577-1655), is remembered as
the correspondent and friend of Francis Bacon. He was educated at Christ
Church, and was early attached to the court, serving in the embassy at Paris.
His debts and dissipations were a great source of sorrow to his father, from
whom he is known to have received at different times £14,000, the modern
equivalent of which is much larger. He was chosen member for Newport in
Cornwall in the parliament of 1601, and member for St Albans in 1604.
Before this time he had become the intimate friend of Bacon, whom he
replaced as member for St Albans. When peace was made with Spain, on
the accession of James I., he wished to travel abroad. His family, who
York, was the son of Sir John Matthew of Ross in Herefordshire, and of his
wife Eleanor Crofton of Ludlow. He was born at Bristol in 1546. He was
educated at Wells, and then in succession at University College and Christ
Church, Oxford. He proceeded B.A. in 1564, and M.A. in 1566. He
attracted the favourable notice of Queen Elizabeth, and his rise was steady
though not very rapid. He was public orator in 1569, president of St John’s
College, Oxford, in 1572, dean of Christ Church in 1576, vice-chancellor of
the university in 1579, dean of Durham in 1583, bishop of Durham in 1595,
and archbishop of York in 1606. In 1581 he had a controversy with the
Jesuit Edmund Campion, and published at Oxford his arguments in 1638
under the title, Piissimi et eminentissimi viri Tobiae Matthew, archiepiscopi
olim Eboracencis concio apologetica adversus Campianam. While in the
north he was active in forcing the recusants to conform to the Church of
England, preaching hundreds of sermons and carrying out thorough
visitations. During his later years he was to some extent in opposition to the
administration of James I. He was exempted from attendance in the
parliament of 1625 on the ground of age and infirmities, and died on the
29th of March 1628. His wife, Frances, was the daughter of William
Barlow, bishop of Chichester.
His son, Sir Tobias, or Tobie, Matthew (1577-1655), is remembered as
the correspondent and friend of Francis Bacon. He was educated at Christ
Church, and was early attached to the court, serving in the embassy at Paris.
His debts and dissipations were a great source of sorrow to his father, from
whom he is known to have received at different times £14,000, the modern
equivalent of which is much larger. He was chosen member for Newport in
Cornwall in the parliament of 1601, and member for St Albans in 1604.
Before this time he had become the intimate friend of Bacon, whom he
replaced as member for St Albans. When peace was made with Spain, on
the accession of James I., he wished to travel abroad. His family, who
Page 31
feared his conversion to Roman Catholicism, opposed his wish, but he
promised not to go beyond France. When once safe out of England he broke
his word and went to Italy. The persuasion of some of his countrymen in
Florence, one of whom is said to have been the Jesuit Robert Parsons, and a
story he heard of the miraculous liquefaction of the blood of San Januarius
at Naples, led to his conversion in 1606. When he returned to England he
was imprisoned, and many efforts were made to obtain his reconversion
without success. He would not take the oath of allegiance to the king. In
1608 he was exiled, and remained out of England for ten years, mostly in
Flanders and Spain. He returned in 1617, but went abroad again in 1619.
His friends obtained his leave to return in 1621. At home he was known as
the intimate friend of Gondomar, the Spanish ambassador. In 1623 he was
sent to join Prince Charles, afterwards Charles I., at Madrid, and was
knighted on the 23rd of October of that year. He remained in England till
1640, when he was finally driven abroad by the parliament, which looked
upon him as an agent of the pope. He died in the English college in Ghent
on the 13th of October 1655. In 1618 he published an Italian translation of
Bacon’s essays. The “Essay on Friendship” was written for him. He was
also the author of a translation of The Confessions of the Incomparable
Doctor St Augustine, which led him into controversy. His correspondence
was published in London in 1660.
For the father, see John Le Neve’s Fasti ecclesiae anglicanae
(London, 1716), and Anthony Wood’s Athenae oxonienses. For the son,
the notice in Athenae oxonienses, an abridgment of his
autobiographical Historical Relation of his own life, published by
Alban Butler in 1795, and A. H. Matthew and A. Calthrop, Life of Sir
Tobie Matthew (London, 1907).
promised not to go beyond France. When once safe out of England he broke
his word and went to Italy. The persuasion of some of his countrymen in
Florence, one of whom is said to have been the Jesuit Robert Parsons, and a
story he heard of the miraculous liquefaction of the blood of San Januarius
at Naples, led to his conversion in 1606. When he returned to England he
was imprisoned, and many efforts were made to obtain his reconversion
without success. He would not take the oath of allegiance to the king. In
1608 he was exiled, and remained out of England for ten years, mostly in
Flanders and Spain. He returned in 1617, but went abroad again in 1619.
His friends obtained his leave to return in 1621. At home he was known as
the intimate friend of Gondomar, the Spanish ambassador. In 1623 he was
sent to join Prince Charles, afterwards Charles I., at Madrid, and was
knighted on the 23rd of October of that year. He remained in England till
1640, when he was finally driven abroad by the parliament, which looked
upon him as an agent of the pope. He died in the English college in Ghent
on the 13th of October 1655. In 1618 he published an Italian translation of
Bacon’s essays. The “Essay on Friendship” was written for him. He was
also the author of a translation of The Confessions of the Incomparable
Doctor St Augustine, which led him into controversy. His correspondence
was published in London in 1660.
For the father, see John Le Neve’s Fasti ecclesiae anglicanae
(London, 1716), and Anthony Wood’s Athenae oxonienses. For the son,
the notice in Athenae oxonienses, an abridgment of his
autobiographical Historical Relation of his own life, published by
Alban Butler in 1795, and A. H. Matthew and A. Calthrop, Life of Sir
Tobie Matthew (London, 1907).
Page 32
MATTHEW, GOSPEL OF ST, the first of the four canonical
Gospels of the Christian Church. The indications of the use of this Gospel
in the two or three generations following the Apostolic Age (see Gospel) are
more plentiful than of any of the others. Throughout the history of the
Church, also, it has held a place second to none of the Gospels alike in
public instruction and in the private reading of Christians. The reasons for
its having impressed itself in this way and become thus familiar are in large
part to be found in the characteristics noticed below. But in addition there
has been from an early time the belief that it was the work of one of those
publicans whose heart Jesus touched and of whose call to follow Him the
three Synoptics contain an interesting account, but who is identified as
Matthew (q.v.) only in this one (Matt. ix. 9-13 = Mark ii. 13-17 = Luke v.
27-32).
1. The Connexion of our Greek Gospel of Matthew with the Apostle
whose name it bears.—The earliest reference to a writing by Matthew
occurs in a fragment taken by Eusebius from the same work of Papias from
which he has given an account of the composition of a record by Mark
(Euseb. Hist. Eccl. iii. 39; see Mark, Gospel of St). The statement about
Matthew is much briefer and is harder to interpret. In spite of much
controversy, the same measure of agreement as to its meaning cannot be
said to have been attained. This is the fragment: “Matthew, however, put
together and wrote down the Oracles (τὰ λόγια συνέγραψεν) in the
Hebrew language, and each man interpreted them as he was able.” Whether
Gospels of the Christian Church. The indications of the use of this Gospel
in the two or three generations following the Apostolic Age (see Gospel) are
more plentiful than of any of the others. Throughout the history of the
Church, also, it has held a place second to none of the Gospels alike in
public instruction and in the private reading of Christians. The reasons for
its having impressed itself in this way and become thus familiar are in large
part to be found in the characteristics noticed below. But in addition there
has been from an early time the belief that it was the work of one of those
publicans whose heart Jesus touched and of whose call to follow Him the
three Synoptics contain an interesting account, but who is identified as
Matthew (q.v.) only in this one (Matt. ix. 9-13 = Mark ii. 13-17 = Luke v.
27-32).
1. The Connexion of our Greek Gospel of Matthew with the Apostle
whose name it bears.—The earliest reference to a writing by Matthew
occurs in a fragment taken by Eusebius from the same work of Papias from
which he has given an account of the composition of a record by Mark
(Euseb. Hist. Eccl. iii. 39; see Mark, Gospel of St). The statement about
Matthew is much briefer and is harder to interpret. In spite of much
controversy, the same measure of agreement as to its meaning cannot be
said to have been attained. This is the fragment: “Matthew, however, put
together and wrote down the Oracles (τὰ λόγια συνέγραψεν) in the
Hebrew language, and each man interpreted them as he was able.” Whether
Page 33
“the elder” referred to in the passage on Mark, or some other like authority,
was the source of this statement also does not appear; but it is probable that
this was the case from the context in which Eusebius gives it. Conservative
writers on the Gospels have frequently maintained that the writing here
referred to was virtually the Hebrew original of our Greek Gospel which
bears his name. And it is indeed likely that Papias himself closely
associated the latter with the Hebrew (or Aramaic) work by Matthew, of
which he had been told, since the traditional connexion of this Greek
Gospel with Matthew can hardly have begun later than this time. It is
reasonable also to suppose that there was some ground for it. The
description, however, of what Matthew did suits better the making of a
collection of Christ’s discourses and sayings than the composition of a work
corresponding in form and character to our Gospel of Matthew.
The next reference in Christian literature to a Gospel-record by Matthew
is that of Irenaeus in his famous passage on the four Gospels (Adv. haer. iii.
i. r). He says that it was written in Hebrew; but in all probability he
regarded the Greek Gospel, which stood first in his, as it does in our,
enumeration, as in the strict sense a translation of the Apostle’s work; and
this was the view of it universally taken till the 16th century, when some of
the scholars of the Reformation maintained that the Greek Gospel itself was
by Matthew.
The actual phenomena, however, of this Gospel, and of its relation to
sources that have been used in it, cannot be explained consistently with
either of the two views just mentioned. It is a composite work in which two
chief sources, known in Greek to the author of our present Gospel, have,
together with some other matter, been combined. It is inconceivable that
one of the Twelve should have proceeded in this way in giving an account
of Christ’s ministry. One of the chief documents, however, here referred to
seems to correspond in character with the description given in Papias’
was the source of this statement also does not appear; but it is probable that
this was the case from the context in which Eusebius gives it. Conservative
writers on the Gospels have frequently maintained that the writing here
referred to was virtually the Hebrew original of our Greek Gospel which
bears his name. And it is indeed likely that Papias himself closely
associated the latter with the Hebrew (or Aramaic) work by Matthew, of
which he had been told, since the traditional connexion of this Greek
Gospel with Matthew can hardly have begun later than this time. It is
reasonable also to suppose that there was some ground for it. The
description, however, of what Matthew did suits better the making of a
collection of Christ’s discourses and sayings than the composition of a work
corresponding in form and character to our Gospel of Matthew.
The next reference in Christian literature to a Gospel-record by Matthew
is that of Irenaeus in his famous passage on the four Gospels (Adv. haer. iii.
i. r). He says that it was written in Hebrew; but in all probability he
regarded the Greek Gospel, which stood first in his, as it does in our,
enumeration, as in the strict sense a translation of the Apostle’s work; and
this was the view of it universally taken till the 16th century, when some of
the scholars of the Reformation maintained that the Greek Gospel itself was
by Matthew.
The actual phenomena, however, of this Gospel, and of its relation to
sources that have been used in it, cannot be explained consistently with
either of the two views just mentioned. It is a composite work in which two
chief sources, known in Greek to the author of our present Gospel, have,
together with some other matter, been combined. It is inconceivable that
one of the Twelve should have proceeded in this way in giving an account
of Christ’s ministry. One of the chief documents, however, here referred to
seems to correspond in character with the description given in Papias’
Page 34
fragment of a record of the compilation of “the divine utterances” made by
Matthew; and the use made of it in our first Gospel may explain the
connexion of this Apostle’s name with it. In the Gospel of Luke also, it is
true, this same source has been used for the teaching of Jesus. But the
original Aramaic Logian document may have been more largely reproduced
in our Greek Matthew. Indeed, in the case of one important passage (v. 17-
48) this is suggested by a comparison with Luke itself, and there are one or
two others where from the character of the matter it seems not improbable,
especially vi. 1-18 and xxiii. 1-5, 7b-10, 15-22. On the whole, as will be
seen below, what appears to be a Palestinian form of the Gospel-tradition is
most fully represented in this Gospel; but in many instances at least this
may well be due to some other cause than the use of the original Logian
document.
2. The Plan on which the Contents is arranged.—In two respects the
arrangement of the book itself is significant.
(a) As to the general outline in the first half of the account of the
Galilean ministry (iv. 23-xi. 30). Immediately after relating the call of
the first four disciples (iv. 18-22) the evangelist gives in iv. 23 a
comprehensive summary of Christ’s work in Galilee under its two chief
aspects, teaching and healing. In the sequel both these are illustrated.
First, he gives in the Sermon on the Mount (v.-vii.) a considerable body
of teaching, of the kind required by the disciples of Jesus generally, and
a large portion of which probably also stood not far from the beginning
of the Logian document. After this he turns to the other aspect. Up to
this point he has mentioned no miracle. He now describes a number in
succession, introducing all but the first of those told between Mark i.
23 and ii. 12, and also four specially remarkable ones, which occurred
a good deal later according to Mark’s order (Matt. viii. 23-34 = Mark
iv. 35-v. 20; Matt. ix. 18-26 = Mark v. 21-43); and he also adds some
Matthew; and the use made of it in our first Gospel may explain the
connexion of this Apostle’s name with it. In the Gospel of Luke also, it is
true, this same source has been used for the teaching of Jesus. But the
original Aramaic Logian document may have been more largely reproduced
in our Greek Matthew. Indeed, in the case of one important passage (v. 17-
48) this is suggested by a comparison with Luke itself, and there are one or
two others where from the character of the matter it seems not improbable,
especially vi. 1-18 and xxiii. 1-5, 7b-10, 15-22. On the whole, as will be
seen below, what appears to be a Palestinian form of the Gospel-tradition is
most fully represented in this Gospel; but in many instances at least this
may well be due to some other cause than the use of the original Logian
document.
2. The Plan on which the Contents is arranged.—In two respects the
arrangement of the book itself is significant.
(a) As to the general outline in the first half of the account of the
Galilean ministry (iv. 23-xi. 30). Immediately after relating the call of
the first four disciples (iv. 18-22) the evangelist gives in iv. 23 a
comprehensive summary of Christ’s work in Galilee under its two chief
aspects, teaching and healing. In the sequel both these are illustrated.
First, he gives in the Sermon on the Mount (v.-vii.) a considerable body
of teaching, of the kind required by the disciples of Jesus generally, and
a large portion of which probably also stood not far from the beginning
of the Logian document. After this he turns to the other aspect. Up to
this point he has mentioned no miracle. He now describes a number in
succession, introducing all but the first of those told between Mark i.
23 and ii. 12, and also four specially remarkable ones, which occurred
a good deal later according to Mark’s order (Matt. viii. 23-34 = Mark
iv. 35-v. 20; Matt. ix. 18-26 = Mark v. 21-43); and he also adds some
Page 35
derived from another source, or other sources (viii. 5-13; ix. 27-34).
Then, after another general description at ix. 35, similar to that at iv.
23, he brings strikingly before us the needs of the masses of the people
and Christ’s compassion for them, and so introduces the mission of the
Twelve (which again occurs later according to Mark’s order, viz. at vi.
7 seq.), whereby the ministry both of teaching and of healing was
further extended (ix. 36-x. 42). Finally, the message of John the
Baptist, and the reply of Jesus, and the reflections that follow (xi.),
bring out the significance of the preceding narrative. It should be
observed that examples have been given of every kind of mighty work
referred to in the reply of Jesus to the messengers of the Baptist; and
that in the discourse which follows their departure the perversity and
unbelief of the people generally are condemned, and the faith of the
humble-minded is contrasted therewith. The greater part of the matter
from ix. 37 to end of xi. is taken from the Logian document. After this
point, i.e. from xii. 1 onwards, the first evangelist follows Mark almost
step by step down to the point (Mark xvi. 8), after which Mark’s
Gospel breaks off, and another ending has been supplied; and gives in
substance almost the whole of Mark’s contents, with the exception that
he passes over the few narratives that he has (as we have seen) placed
earlier. At the same time he brings in additional matter in connexion
with most of the Marcan sections.
(b) With the accounts of the words of Jesus spoken on certain
occasions, which our first evangelist found given in one or another of
his sources, he has combined other pieces, taken from other parts of the
same source or from different sources, which seemed to him connected
in subject, e.g. into the discourse spoken on a mountain, when crowds
from all parts were present, given in the Logian document, he has
introduced some pieces which, as we infer from Luke, stood separately
in that document (cf. Matt. vi. 19-21 with Luke xii. 33, 34; Matt. vi. 22,
Then, after another general description at ix. 35, similar to that at iv.
23, he brings strikingly before us the needs of the masses of the people
and Christ’s compassion for them, and so introduces the mission of the
Twelve (which again occurs later according to Mark’s order, viz. at vi.
7 seq.), whereby the ministry both of teaching and of healing was
further extended (ix. 36-x. 42). Finally, the message of John the
Baptist, and the reply of Jesus, and the reflections that follow (xi.),
bring out the significance of the preceding narrative. It should be
observed that examples have been given of every kind of mighty work
referred to in the reply of Jesus to the messengers of the Baptist; and
that in the discourse which follows their departure the perversity and
unbelief of the people generally are condemned, and the faith of the
humble-minded is contrasted therewith. The greater part of the matter
from ix. 37 to end of xi. is taken from the Logian document. After this
point, i.e. from xii. 1 onwards, the first evangelist follows Mark almost
step by step down to the point (Mark xvi. 8), after which Mark’s
Gospel breaks off, and another ending has been supplied; and gives in
substance almost the whole of Mark’s contents, with the exception that
he passes over the few narratives that he has (as we have seen) placed
earlier. At the same time he brings in additional matter in connexion
with most of the Marcan sections.
(b) With the accounts of the words of Jesus spoken on certain
occasions, which our first evangelist found given in one or another of
his sources, he has combined other pieces, taken from other parts of the
same source or from different sources, which seemed to him connected
in subject, e.g. into the discourse spoken on a mountain, when crowds
from all parts were present, given in the Logian document, he has
introduced some pieces which, as we infer from Luke, stood separately
in that document (cf. Matt. vi. 19-21 with Luke xii. 33, 34; Matt. vi. 22,
Page 36
23 with Luke xi. 34-36; Matt. vi. 24 with Luke xvi. 13; Matt. vi. 25-34
with Luke xii. 22-32; Matt. vii. 7-11 with Luke xi. 9-13). Again, the
address to the Twelve in Mark vi. 7-11, which in Matthew is combined
with an address to disciples, from the Logian document, is connected
by Luke with the sending out of seventy disciples (Luke x. 1-16). Our
first evangelist has also added here various other sayings (Matt. x. 17-
39, 42). Again, with the Marcan account of the charge of collusion with
Satan and Christ’s reply (Mark iii. 22-30), the first evangelist (xii. 24-
45) combines the parallel account in the Logian document and adds
Christ’s reply to another attack (Luke xi. 14-16, 17-26, 29-32). These
are some examples. He has in all in this manner constructed eight
discourses or collections of sayings, into which the greater part of
Christ’s teaching is gathered: (1) On the character of the heirs of the
kingdom (v.-vii.); (2) The Mission address (x.); (3) Teaching suggested
by the message of John the Baptist (xi.); (4) The reply to an accusation
and a challenge (xii. 22-45); (5) The teaching by parables (xiii.); (6) On
offences (xviii.); (7) Concerning the Scribes and Pharisees (xxiii.); (8)
On the Last Things (xxiv., xxv.). In this arrangement of his material the
writer has in many instances disregarded chronological considerations.
But his documents also gave only very imperfect indications of the
occasions of many of the utterances; and the result of his method of
procedure has been to give us an exceedingly effective representation
of the teaching of Jesus.
In the concluding verses of the Gospel, where the original Marcan
parallel is wanting, the evangelist may still have followed in part that
document while making additions as before. The account of the
silencing of the Roman guard by the chief priests is the sequel to the
setting of this guard and their presence at the Resurrection, which at an
earlier point arc peculiar to Matthew (xxvii. 62-66, xxviii. 4). And,
further, this matter seems to belong to the same cycle of tradition as the
with Luke xii. 22-32; Matt. vii. 7-11 with Luke xi. 9-13). Again, the
address to the Twelve in Mark vi. 7-11, which in Matthew is combined
with an address to disciples, from the Logian document, is connected
by Luke with the sending out of seventy disciples (Luke x. 1-16). Our
first evangelist has also added here various other sayings (Matt. x. 17-
39, 42). Again, with the Marcan account of the charge of collusion with
Satan and Christ’s reply (Mark iii. 22-30), the first evangelist (xii. 24-
45) combines the parallel account in the Logian document and adds
Christ’s reply to another attack (Luke xi. 14-16, 17-26, 29-32). These
are some examples. He has in all in this manner constructed eight
discourses or collections of sayings, into which the greater part of
Christ’s teaching is gathered: (1) On the character of the heirs of the
kingdom (v.-vii.); (2) The Mission address (x.); (3) Teaching suggested
by the message of John the Baptist (xi.); (4) The reply to an accusation
and a challenge (xii. 22-45); (5) The teaching by parables (xiii.); (6) On
offences (xviii.); (7) Concerning the Scribes and Pharisees (xxiii.); (8)
On the Last Things (xxiv., xxv.). In this arrangement of his material the
writer has in many instances disregarded chronological considerations.
But his documents also gave only very imperfect indications of the
occasions of many of the utterances; and the result of his method of
procedure has been to give us an exceedingly effective representation
of the teaching of Jesus.
In the concluding verses of the Gospel, where the original Marcan
parallel is wanting, the evangelist may still have followed in part that
document while making additions as before. The account of the
silencing of the Roman guard by the chief priests is the sequel to the
setting of this guard and their presence at the Resurrection, which at an
earlier point arc peculiar to Matthew (xxvii. 62-66, xxviii. 4). And,
further, this matter seems to belong to the same cycle of tradition as the
Page 37
story of Pilate’s wife and his throwing the guilt of the Crucifixion of
Jesus upon the Jews, and the testimony borne by the Roman guard (as
well as the centurion) who kept watch by the cross (xxvii. 15-26, 54),
all which also are peculiar to this Gospel. It cannot but seem probable
that these are legendary additions which had arisen through the desire
to commend the Gospel to the Romans.
On the other hand, the meeting of Jesus with the disciples in Galilee
(Matt. xxviii. 16 seq.) is the natural sequel to the message to them
related in Mark xvi. 7, as well as in Matt, xxviii. 7. Again, the
commission to them to preach throughout the world is supported by
Luke xxiv. 47, and by the present ending of Mark (xvi. 15), though
neither of these mention Galilee as the place where it was given. The
baptismal formula in Matt. xxviii. 19, is, however, peculiar, and in
view of its non-occurrence in the Acts and Epistles of the New
Testament must be regarded as probably an addition in accordance with
Church usage at the time the Gospel was written.
3. The Palestinian Element.—Teaching is preserved in this Gospel which
would have peculiar interest and be specially required in the home of
Judaism. The best examples of this are the passages already referred to near
end of § 1, as probably derived from the Logian document. There are,
besides, a good many turns of expression and sayings peculiar to this
Gospel which have a Semitic cast, or which suggest a point of view that
would be natural to Palestinian Christians, e.g. “kingdom of heaven”
frequently for “kingdom of God”; xiii. 52 (“every scribe”); xxiv. 20
(“neither on a Sabbath”). See also v. 35 and xix. 9; x. 5, 23. Again, several
of the quotations which are peculiar to this Gospel are not taken from the
LXX., as those in the other Gospels and in the corresponding contexts in
this Gospel commonly are, but are wholly or partly independent renderings
from the Hebrew (ii. 6, 15, 18; viii. 17, xii. 17-21, &c.). Once more, there is
Jesus upon the Jews, and the testimony borne by the Roman guard (as
well as the centurion) who kept watch by the cross (xxvii. 15-26, 54),
all which also are peculiar to this Gospel. It cannot but seem probable
that these are legendary additions which had arisen through the desire
to commend the Gospel to the Romans.
On the other hand, the meeting of Jesus with the disciples in Galilee
(Matt. xxviii. 16 seq.) is the natural sequel to the message to them
related in Mark xvi. 7, as well as in Matt, xxviii. 7. Again, the
commission to them to preach throughout the world is supported by
Luke xxiv. 47, and by the present ending of Mark (xvi. 15), though
neither of these mention Galilee as the place where it was given. The
baptismal formula in Matt. xxviii. 19, is, however, peculiar, and in
view of its non-occurrence in the Acts and Epistles of the New
Testament must be regarded as probably an addition in accordance with
Church usage at the time the Gospel was written.
3. The Palestinian Element.—Teaching is preserved in this Gospel which
would have peculiar interest and be specially required in the home of
Judaism. The best examples of this are the passages already referred to near
end of § 1, as probably derived from the Logian document. There are,
besides, a good many turns of expression and sayings peculiar to this
Gospel which have a Semitic cast, or which suggest a point of view that
would be natural to Palestinian Christians, e.g. “kingdom of heaven”
frequently for “kingdom of God”; xiii. 52 (“every scribe”); xxiv. 20
(“neither on a Sabbath”). See also v. 35 and xix. 9; x. 5, 23. Again, several
of the quotations which are peculiar to this Gospel are not taken from the
LXX., as those in the other Gospels and in the corresponding contexts in
this Gospel commonly are, but are wholly or partly independent renderings
from the Hebrew (ii. 6, 15, 18; viii. 17, xii. 17-21, &c.). Once more, there is
Page 38
somewhat more parallelism between the fragments of the Gospel according
to the Hebrews and this Gospel than is the case with Luke, not to say Mark.
4. Doctrinal Character.—In this Gospel, more decidedly than in either of
the other two Synoptics, there is a doctrinal point of view from which the
whole history is regarded. Certain aspects which are of profound
significance are dwelt upon, and this without there being any great
difference between this Gospel and the two other Synoptics in respect to the
facts recorded or the beliefs implied. The effect is produced partly by the
comments of the evangelist, which especially take the form of citations
from the Old Testament; partly by the frequency with which certain
expressions are used, and the prominence that is given in this and other
ways to particular traits and topics.
He sets forth the restriction of the mission of Jesus during His life on
earth to the people of Israel in a way which suggests at first sight a spirit of
Jewish exclusiveness. But there are various indications that this is not the
true explanation. In particular the evangelist brings out more strongly than
either Mark or Luke the national rejection of Jesus, while the Gospel ends
with the commission of Jesus to His disciples after His resurrection to
“make disciples of all the peoples.” One may divine in all this an intention
to “justify the ways of God” to the Jew, by proving that God in His
faithfulness to His ancient people had given them the first opportunity of
salvation through Christ, but that now their national privilege had been
rightly forfeited. He was also specially concerned to show that prophecy is
fulfilled in the life and work of Jesus, but the conception of this fulfilment
which is presented to us is a large one; it is to be seen not merely in
particular events or features of Christ’s ministry, but in the whole new
dispensation, new relations between God and men, and new rules of
conduct which Christ has introduced. The divine meaning of the work of
to the Hebrews and this Gospel than is the case with Luke, not to say Mark.
4. Doctrinal Character.—In this Gospel, more decidedly than in either of
the other two Synoptics, there is a doctrinal point of view from which the
whole history is regarded. Certain aspects which are of profound
significance are dwelt upon, and this without there being any great
difference between this Gospel and the two other Synoptics in respect to the
facts recorded or the beliefs implied. The effect is produced partly by the
comments of the evangelist, which especially take the form of citations
from the Old Testament; partly by the frequency with which certain
expressions are used, and the prominence that is given in this and other
ways to particular traits and topics.
He sets forth the restriction of the mission of Jesus during His life on
earth to the people of Israel in a way which suggests at first sight a spirit of
Jewish exclusiveness. But there are various indications that this is not the
true explanation. In particular the evangelist brings out more strongly than
either Mark or Luke the national rejection of Jesus, while the Gospel ends
with the commission of Jesus to His disciples after His resurrection to
“make disciples of all the peoples.” One may divine in all this an intention
to “justify the ways of God” to the Jew, by proving that God in His
faithfulness to His ancient people had given them the first opportunity of
salvation through Christ, but that now their national privilege had been
rightly forfeited. He was also specially concerned to show that prophecy is
fulfilled in the life and work of Jesus, but the conception of this fulfilment
which is presented to us is a large one; it is to be seen not merely in
particular events or features of Christ’s ministry, but in the whole new
dispensation, new relations between God and men, and new rules of
conduct which Christ has introduced. The divine meaning of the work of
Page 39
Jesus is thus made apparent, while of the majesty and glory of His person a
peculiarly strong impression is conveyed.
Some illustrations in detail of these points are subjoined. Where there are
parallels in the other Gospels they should be compared and the words in
Matthew noted which in many instances serve to emphasize the points in
question.
(a) The Ministry of Jesus among the Jewish People as their promised
Messiah, their rejection of Him, and the extension of the Gospel to the
Gentiles. The mission to Israel: Matt. i. 21; iv. 23 (note in these
passages the use of ὁ λαός, which here, as generally in Matthew,
denotes the chosen nation), ix. 33, 35, xv. 31. For the rule limiting the
work of Jesus while on earth see xv. 24 (and note ἰξελθοῦσα in verse
22, which implies that Jesus had not himself entered the heathen
borders), and for a similar rule prescribed to the disciples, x. 5, 6 and
23.
The rejection of Jesus by the people in Galilee, xi. 21; xiii. 13-15,
and by the heads of “the nation,” xxvi. 3, 47 and by “the whole nation,”
xxvii. 25; their condemnation xxiii. 38.
Mercy to the Gentiles and the punishment of “the sons of the
kingdom” is foretold viii. 11, 12. The commission to go and convert
Gentile peoples (ἔθνη) is given after Christ’s resurrection (xxviii. 19).
(b) The Fulfilment of Prophecy.—In the birth and childhood of Jesus,
i. 23; ii. 6, 15, 18, 23. By these citations attention is drawn to the
lowliness of the beginnings of the Saviour’s life, the unexpected and
secret manner of His appearing, the dangers to which from the first He
was exposed and from which He escaped.
peculiarly strong impression is conveyed.
Some illustrations in detail of these points are subjoined. Where there are
parallels in the other Gospels they should be compared and the words in
Matthew noted which in many instances serve to emphasize the points in
question.
(a) The Ministry of Jesus among the Jewish People as their promised
Messiah, their rejection of Him, and the extension of the Gospel to the
Gentiles. The mission to Israel: Matt. i. 21; iv. 23 (note in these
passages the use of ὁ λαός, which here, as generally in Matthew,
denotes the chosen nation), ix. 33, 35, xv. 31. For the rule limiting the
work of Jesus while on earth see xv. 24 (and note ἰξελθοῦσα in verse
22, which implies that Jesus had not himself entered the heathen
borders), and for a similar rule prescribed to the disciples, x. 5, 6 and
23.
The rejection of Jesus by the people in Galilee, xi. 21; xiii. 13-15,
and by the heads of “the nation,” xxvi. 3, 47 and by “the whole nation,”
xxvii. 25; their condemnation xxiii. 38.
Mercy to the Gentiles and the punishment of “the sons of the
kingdom” is foretold viii. 11, 12. The commission to go and convert
Gentile peoples (ἔθνη) is given after Christ’s resurrection (xxviii. 19).
(b) The Fulfilment of Prophecy.—In the birth and childhood of Jesus,
i. 23; ii. 6, 15, 18, 23. By these citations attention is drawn to the
lowliness of the beginnings of the Saviour’s life, the unexpected and
secret manner of His appearing, the dangers to which from the first He
was exposed and from which He escaped.
Page 40
The ministry of Christ’s forerunner, iii. 3. (The same prophecy, Isa.
xl. 3, is also quoted in the other Gospels.)
The ministry of Jesus. The quotations serve to bring out the
significance of important events, especially such as were turning-
points, and also to mark the broad features of Christ’s life and work, iv.
15, 16; viii. 17; xii. 18 seq.; xiii. 35; xxi. 5; xxvii. 9.
(c) The Teaching on the Kingdom of God.—Note the collection of
parables “of the Kingdom” in xiii.; also the use of ἡ βασιλεία (“the
Kingdom”) without further definition as a term the reference of which
could not be misunderstood, especially in the following phrases
peculiar to this Gospel: τὁ εὐαγγέλιον τῆς βασιλείας (“the Gospel of
the Kingdom”) iv. 23, ix. 35, xxiv. 14; and ὁ λόγος τῆς βασιλείας
(“the word of the kingdom”) xiii. 19. The following descriptions of the
kingdom, peculiar to this Gospel, are also interesting ἡ βασιλεία τοῦ
πατρὁς αὐτῶν (“the kingdom of their father”) xiii. 43 and τοῦ
πατρός μου(“of my father”) xxvi. 29.
(d) The Relation of the New Law to the Old.—Verses 17-48, cf. also,
addition at xxii. 40 and xix. 19b. Further, his use of δικαιοσύνη
(“righteousness”) and δίκαιος(“righteous”) (specially frequent in this
Gospel) is such as to connect the New with the Old; the standard in
mind is the law which “fulfilled” that previously given.
(e) The Christian Ecclesia.—Chap. xvi. 18, xviii. 17.
(f) The Messianic Dignity and Glory of Jesus.—The narrative in i.
and ii. show the royalty of the new-born child. The title “Son of David”
occurs with special frequency in this Gospel. The following instances
are without parallels in the other Gospels: ix. 27; xii. 23; xv. 22; xxi. 9;
xxi. 15. The title “Son of God” is also used with somewhat greater
xl. 3, is also quoted in the other Gospels.)
The ministry of Jesus. The quotations serve to bring out the
significance of important events, especially such as were turning-
points, and also to mark the broad features of Christ’s life and work, iv.
15, 16; viii. 17; xii. 18 seq.; xiii. 35; xxi. 5; xxvii. 9.
(c) The Teaching on the Kingdom of God.—Note the collection of
parables “of the Kingdom” in xiii.; also the use of ἡ βασιλεία (“the
Kingdom”) without further definition as a term the reference of which
could not be misunderstood, especially in the following phrases
peculiar to this Gospel: τὁ εὐαγγέλιον τῆς βασιλείας (“the Gospel of
the Kingdom”) iv. 23, ix. 35, xxiv. 14; and ὁ λόγος τῆς βασιλείας
(“the word of the kingdom”) xiii. 19. The following descriptions of the
kingdom, peculiar to this Gospel, are also interesting ἡ βασιλεία τοῦ
πατρὁς αὐτῶν (“the kingdom of their father”) xiii. 43 and τοῦ
πατρός μου(“of my father”) xxvi. 29.
(d) The Relation of the New Law to the Old.—Verses 17-48, cf. also,
addition at xxii. 40 and xix. 19b. Further, his use of δικαιοσύνη
(“righteousness”) and δίκαιος(“righteous”) (specially frequent in this
Gospel) is such as to connect the New with the Old; the standard in
mind is the law which “fulfilled” that previously given.
(e) The Christian Ecclesia.—Chap. xvi. 18, xviii. 17.
(f) The Messianic Dignity and Glory of Jesus.—The narrative in i.
and ii. show the royalty of the new-born child. The title “Son of David”
occurs with special frequency in this Gospel. The following instances
are without parallels in the other Gospels: ix. 27; xii. 23; xv. 22; xxi. 9;
xxi. 15. The title “Son of God” is also used with somewhat greater
Page 41
frequency than in Mark and Luke: ii. 15; xiv. 33; xvi. 16; xxii. 2 seq.
(where it is implied); xxvii. 40, 43.
The thought of the future coming of Christ, and in particular of the
judgment to be executed by Him then, is much more prominent in this
Gospel than in the others. Some of the following predictions are
peculiar to it, while in several others there are additional touches: vii.
22, 23; x. 23, 32, 33; xiii. 39-43; xvi. 27, 28; xix. 28; xxiv. 3, 27, 30,
31, 37, 39; xxv. 31-46; xxvi. 64.
The majesty of Christ is also impressed upon us by the signs at His
crucifixion, some of which are related only in this Gospel, xxvii. 51-53,
and by the sublime vision of the Risen Christ at the close, xxviii. 16-
20.
(5) Time of Composition and Readers addressed.—The signs of dogmatic
reflection in this Gospel point to its having been composed somewhat late
in the 1st century, probably after Luke’s Gospel, and this is in accord with
the conclusion that some insertions had been made in the Marcan document
used by this evangelist which were not in that used by Luke (see Luke,
Gospel of St). We may assign a.d. 80-100 as a probable time for the
composition.
The author was in all probability a Jew by race, and he would seem to
have addressed himself especially to Jewish readers; but they were Jews of
the Dispersion. For although he was in specially close touch with Palestine,
either personally or through the sources at his command, or both, his book
was composed in Greek by the aid of Greek documents.
See commentaries by Th. Zahn (1903) and W. C. Allen (in the series
of International Critical Commentaries, 1907); also books on the Four
Gospels or the Synoptic Gospels cited at the end of Gospel.
(V. H. S.)
(where it is implied); xxvii. 40, 43.
The thought of the future coming of Christ, and in particular of the
judgment to be executed by Him then, is much more prominent in this
Gospel than in the others. Some of the following predictions are
peculiar to it, while in several others there are additional touches: vii.
22, 23; x. 23, 32, 33; xiii. 39-43; xvi. 27, 28; xix. 28; xxiv. 3, 27, 30,
31, 37, 39; xxv. 31-46; xxvi. 64.
The majesty of Christ is also impressed upon us by the signs at His
crucifixion, some of which are related only in this Gospel, xxvii. 51-53,
and by the sublime vision of the Risen Christ at the close, xxviii. 16-
20.
(5) Time of Composition and Readers addressed.—The signs of dogmatic
reflection in this Gospel point to its having been composed somewhat late
in the 1st century, probably after Luke’s Gospel, and this is in accord with
the conclusion that some insertions had been made in the Marcan document
used by this evangelist which were not in that used by Luke (see Luke,
Gospel of St). We may assign a.d. 80-100 as a probable time for the
composition.
The author was in all probability a Jew by race, and he would seem to
have addressed himself especially to Jewish readers; but they were Jews of
the Dispersion. For although he was in specially close touch with Palestine,
either personally or through the sources at his command, or both, his book
was composed in Greek by the aid of Greek documents.
See commentaries by Th. Zahn (1903) and W. C. Allen (in the series
of International Critical Commentaries, 1907); also books on the Four
Gospels or the Synoptic Gospels cited at the end of Gospel.
(V. H. S.)
Page 42
MATTHEW CANTACUZENUS, Byzantine emperor, was
the son of John VI. Cantacuzenus (q.v.). In return for the support he gave to
his father during his struggle with John V. he was allowed to annex part of
Thrace under his own dominion and in 1353 was proclaimed joint emperor.
From his Thracian principality he levied several wars against the Servians.
An attack which he prepared in 1350 was frustrated by the defection of his
Turkish auxiliaries. In 1357 he was captured by his enemies, who delivered
him to the rival emperor, John V. Compelled to abdicate, he withdrew to a
monastery, where he busied himself with writing commentaries on the
Scriptures.
MATTHEW OF PARIS (d. 1259), English monk and chronicler
known to us only through his voluminous writings. In spite of his surname,
and of his knowledge of the French language, his attitude towards
foreigners attests that he was of English birth. He may have studied at Paris
in his youth, but the earliest fact which he records of himself is his
admission as a monk at St Albans in the year 1217. His life was mainly
the son of John VI. Cantacuzenus (q.v.). In return for the support he gave to
his father during his struggle with John V. he was allowed to annex part of
Thrace under his own dominion and in 1353 was proclaimed joint emperor.
From his Thracian principality he levied several wars against the Servians.
An attack which he prepared in 1350 was frustrated by the defection of his
Turkish auxiliaries. In 1357 he was captured by his enemies, who delivered
him to the rival emperor, John V. Compelled to abdicate, he withdrew to a
monastery, where he busied himself with writing commentaries on the
Scriptures.
MATTHEW OF PARIS (d. 1259), English monk and chronicler
known to us only through his voluminous writings. In spite of his surname,
and of his knowledge of the French language, his attitude towards
foreigners attests that he was of English birth. He may have studied at Paris
in his youth, but the earliest fact which he records of himself is his
admission as a monk at St Albans in the year 1217. His life was mainly
Page 43
spent in this religious house. In 1248, however, he was sent to Norway as
the bearer of a message from Louis IX. of France to Haakon VI.; he made
himself so agreeable to the Norwegian sovereign that he was invited, a little
later, to superintend the reformation of the Benedictine monastery of St
Benet Holme at Trondhjem. Apart from these missions, his activities were
devoted to the composition of history, a pursuit for which the monks of St
Albans had long been famous. Matthew edited anew the works of Abbot
John de Cella and Roger of Wendover, which in their altered form
constitute the first part of his most important work, the Chronica majora.
From 1235, the point at which Wendover dropped his pen, Matthew
continued the history on the plan which his predecessors had followed. He
derived much of his information from the letters of important personages,
which he sometimes inserts, but much more from conversation with the
eye-witnesses of events. Among his informants were Earl Richard of
Cornwall and Henry III. With the latter he appears to have been on terms of
intimacy. The king knew that Matthew was writing a history, and showed
some anxiety that it should be as exact as possible. In 1257, in the course of
a week’s visit to St Albans, Henry kept the chronicler beside him night and
day, “and guided my pen,” says Paris, “with much good will and diligence.”
It is therefore curious that the Chronica majora should give so unfavourable
an account of the king’s policy. Luard supposes that Matthew never
intended his work to see the light in its present form, and many passages of
the autograph have against them the note offendiculum, which shows that
the writer understood the danger which he ran. On the other hand,
unexpurgated copies were made in Matthew’s lifetime; though the
offending passages are duly omitted or softened in his abridgment of his
longer work, the Historia Anglorum (written about 1253), the real
sentiments of the author must have been an open secret. In any case there is
no ground for the old theory that he was an official historiographer.
the bearer of a message from Louis IX. of France to Haakon VI.; he made
himself so agreeable to the Norwegian sovereign that he was invited, a little
later, to superintend the reformation of the Benedictine monastery of St
Benet Holme at Trondhjem. Apart from these missions, his activities were
devoted to the composition of history, a pursuit for which the monks of St
Albans had long been famous. Matthew edited anew the works of Abbot
John de Cella and Roger of Wendover, which in their altered form
constitute the first part of his most important work, the Chronica majora.
From 1235, the point at which Wendover dropped his pen, Matthew
continued the history on the plan which his predecessors had followed. He
derived much of his information from the letters of important personages,
which he sometimes inserts, but much more from conversation with the
eye-witnesses of events. Among his informants were Earl Richard of
Cornwall and Henry III. With the latter he appears to have been on terms of
intimacy. The king knew that Matthew was writing a history, and showed
some anxiety that it should be as exact as possible. In 1257, in the course of
a week’s visit to St Albans, Henry kept the chronicler beside him night and
day, “and guided my pen,” says Paris, “with much good will and diligence.”
It is therefore curious that the Chronica majora should give so unfavourable
an account of the king’s policy. Luard supposes that Matthew never
intended his work to see the light in its present form, and many passages of
the autograph have against them the note offendiculum, which shows that
the writer understood the danger which he ran. On the other hand,
unexpurgated copies were made in Matthew’s lifetime; though the
offending passages are duly omitted or softened in his abridgment of his
longer work, the Historia Anglorum (written about 1253), the real
sentiments of the author must have been an open secret. In any case there is
no ground for the old theory that he was an official historiographer.
Page 44
Matthew Paris was unfortunate in living at a time when English
politics were peculiarly involved and tedious. His talent is for narrative
and description. Though he took a keen interest in the personal side of
politics he has no claim to be considered a judge of character. His
appreciations of his contemporaries throw more light on his own
prejudices than on their aims and ideas. His work is always vigorous,
but he imputes motives in the spirit of a partisan who never pauses to
weigh the evidence or to take a comprehensive view of the situation.
His redeeming feature is his generous admiration for strength of
character, even when it goes along with a policy of which he
disapproves. Thus he praises Grosseteste, while he denounces
Grosseteste’s scheme of monastic reform. Matthew is a vehement
supporter of the monastic orders against their rivals, the secular clergy
and the mendicant friars. He is violently opposed to the court and the
foreign favourites. He despises the king as a statesman, though for the
man he has some kindly feeling. The frankness with which he attacks
the court of Rome for its exactions is remarkable; so, too, is the intense
nationalism which he displays in dealing with this topic. His faults of
presentment are more often due to carelessness and narrow views than
to deliberate purpose. But he is sometimes guilty of inserting rhetorical
speeches which are not only fictitious, but also misleading as an
account of the speaker’s sentiments. In other cases he tampers with the
documents which he inserts (as, for instance, with the text of Magna
Carta). His chronology is, for a contemporary, inexact; and he
occasionally inserts duplicate versions of the same incident in different
places. Hence he must always be rigorously checked where other
authorities exist and used with caution where he is our sole informant.
None the less, he gives a more vivid impression of his age than any
other English chronicler; and it is a matter for regret that his great
politics were peculiarly involved and tedious. His talent is for narrative
and description. Though he took a keen interest in the personal side of
politics he has no claim to be considered a judge of character. His
appreciations of his contemporaries throw more light on his own
prejudices than on their aims and ideas. His work is always vigorous,
but he imputes motives in the spirit of a partisan who never pauses to
weigh the evidence or to take a comprehensive view of the situation.
His redeeming feature is his generous admiration for strength of
character, even when it goes along with a policy of which he
disapproves. Thus he praises Grosseteste, while he denounces
Grosseteste’s scheme of monastic reform. Matthew is a vehement
supporter of the monastic orders against their rivals, the secular clergy
and the mendicant friars. He is violently opposed to the court and the
foreign favourites. He despises the king as a statesman, though for the
man he has some kindly feeling. The frankness with which he attacks
the court of Rome for its exactions is remarkable; so, too, is the intense
nationalism which he displays in dealing with this topic. His faults of
presentment are more often due to carelessness and narrow views than
to deliberate purpose. But he is sometimes guilty of inserting rhetorical
speeches which are not only fictitious, but also misleading as an
account of the speaker’s sentiments. In other cases he tampers with the
documents which he inserts (as, for instance, with the text of Magna
Carta). His chronology is, for a contemporary, inexact; and he
occasionally inserts duplicate versions of the same incident in different
places. Hence he must always be rigorously checked where other
authorities exist and used with caution where he is our sole informant.
None the less, he gives a more vivid impression of his age than any
other English chronicler; and it is a matter for regret that his great
Page 45
history breaks off in 1259, on the eve of the crowning struggle between
Henry III and the baronage.
Authorities.—The relation of Matthew Paris’s work to those of John
de Cella and Roger of Wendover may best be studied in H. R. Luard’s
edition of the Chronica majora (7 vols., Rolls series, 1872-1883),
which contains valuable prefaces. The Historia Anglorum sive historia
minor (1067-1253) has been edited by F. Madden (3 vols., Rolls series,
1866-1869). Matthew Paris is often confused with “Matthew of
Westminster,” the reputed author of the Flores historiarum edited by H.
R. Luard (3 vols., Rolls series, 1890). This work, compiled by various
hands, is an edition of Matthew Paris, with continuations extending to
1326. Matthew Paris also wrote a life of Edmund Rich (q.v.), which is
probably the work printed in W. Wallace’s St Edmund of Canterbury
(London, 1893) pp. 543-588, though this is attributed by the editor to
the monk Eustace; Vitae abbatum S Albani (up to 1225) which have
been edited by W. Watts (1640, &c.); and (possibly) the Abbreviatio
chronicorum (1000-1255), edited by F. Madden, in the third volume of
the Historia Anglorum. On the value of Matthew as an historian see F.
Liebermann in G. H. Pertz’s Scriptores xxviii. pp. 74-106; A. Jessopp’s
Studies by a Recluse (London, 1893); H. Plehn’s Politische Character
Matheus Parisiensis (Leipzig, 1897). (H. W. C. D.)
MATTHEW OF WESTMINSTER, the name of an
imaginary person who was long regarded as the author of the Flores
Henry III and the baronage.
Authorities.—The relation of Matthew Paris’s work to those of John
de Cella and Roger of Wendover may best be studied in H. R. Luard’s
edition of the Chronica majora (7 vols., Rolls series, 1872-1883),
which contains valuable prefaces. The Historia Anglorum sive historia
minor (1067-1253) has been edited by F. Madden (3 vols., Rolls series,
1866-1869). Matthew Paris is often confused with “Matthew of
Westminster,” the reputed author of the Flores historiarum edited by H.
R. Luard (3 vols., Rolls series, 1890). This work, compiled by various
hands, is an edition of Matthew Paris, with continuations extending to
1326. Matthew Paris also wrote a life of Edmund Rich (q.v.), which is
probably the work printed in W. Wallace’s St Edmund of Canterbury
(London, 1893) pp. 543-588, though this is attributed by the editor to
the monk Eustace; Vitae abbatum S Albani (up to 1225) which have
been edited by W. Watts (1640, &c.); and (possibly) the Abbreviatio
chronicorum (1000-1255), edited by F. Madden, in the third volume of
the Historia Anglorum. On the value of Matthew as an historian see F.
Liebermann in G. H. Pertz’s Scriptores xxviii. pp. 74-106; A. Jessopp’s
Studies by a Recluse (London, 1893); H. Plehn’s Politische Character
Matheus Parisiensis (Leipzig, 1897). (H. W. C. D.)
MATTHEW OF WESTMINSTER, the name of an
imaginary person who was long regarded as the author of the Flores
Page 46
Historiarum. The error was first discovered in 1826 by Sir F. Palgrave, who
said that Matthew was “a phantom who never existed,” and later the truth of
this statement was completely proved by H. R. Luard. The name appears to
have been taken from that of Matthew of Paris, from whose Chronica
majora the earlier part of the work was mainly copied, and from
Westminster, the abbey in which the work was partially written.
The Flores historiarum is a Latin chronicle dealing with English
history from the creation to 1326, although some of the earlier
manuscripts end at 1306; it was compiled by various persons, and
written partly at St Albans and partly at Westminster. The part from
1306 to 1326 was written by Robert of Reading (d. 1325) and another
Westminster monk. Except for parts dealing with the reign of Edward I.
its value is not great. It was first printed by Matthew Parker, archbishop
of Canterbury, in 1567, and the best edition is the one edited with
introduction by H. R. Luard for the Rolls series (London, 1890). It has
been translated into English by C. D. Yonge (London, 1853). See
Luard’s introduction, and C. Bémont in the Revue critique d’histoire
(Paris, 1891).
MATTHEWS, STANLEY (1824-1889), American jurist, was
born in Cincinnati, Ohio, on the 21st of July 1824. He graduated from
Kenyon College in 1840, studied law, and in 1842 was admitted to the bar
of Maury county, Tennessee. In 1844 he became assistant prosecuting
said that Matthew was “a phantom who never existed,” and later the truth of
this statement was completely proved by H. R. Luard. The name appears to
have been taken from that of Matthew of Paris, from whose Chronica
majora the earlier part of the work was mainly copied, and from
Westminster, the abbey in which the work was partially written.
The Flores historiarum is a Latin chronicle dealing with English
history from the creation to 1326, although some of the earlier
manuscripts end at 1306; it was compiled by various persons, and
written partly at St Albans and partly at Westminster. The part from
1306 to 1326 was written by Robert of Reading (d. 1325) and another
Westminster monk. Except for parts dealing with the reign of Edward I.
its value is not great. It was first printed by Matthew Parker, archbishop
of Canterbury, in 1567, and the best edition is the one edited with
introduction by H. R. Luard for the Rolls series (London, 1890). It has
been translated into English by C. D. Yonge (London, 1853). See
Luard’s introduction, and C. Bémont in the Revue critique d’histoire
(Paris, 1891).
MATTHEWS, STANLEY (1824-1889), American jurist, was
born in Cincinnati, Ohio, on the 21st of July 1824. He graduated from
Kenyon College in 1840, studied law, and in 1842 was admitted to the bar
of Maury county, Tennessee. In 1844 he became assistant prosecuting
Page 47
attorney of Hamilton county, Ohio; and in 1846-1849 edited a short-lived
anti-slavery paper, the Cincinnati Herald. He was clerk of the Ohio House
of Representatives in 1848-1849, a judge of common pleas of Hamilton
county in 1850-1853, state senator in 1856-1858, and U.S. district-attorney
for the southern district of Ohio in 1858-1861. First a Whig and then a Free-
Soiler, he joined the Republican party in 1861. After the outbreak of the
Civil War he was commissioned a lieutenant of the 23rd Ohio, of which
Rutherford B. Hayes was major; but saw service only with the 57th Ohio, of
which he was colonel, and with a brigade which he commanded in the
Army of the Cumberland. He resigned from the army in 1863, and was
judge of the Cincinnati superior court in 1863-1864. He was a Republican
presidential elector in 1864 and 1868. In 1872 he joined the Liberal
Republican movement, and was temporary chairman of the Cincinnati
convention which nominated Horace Greeley for the presidency, but in the
campaign he supported Grant. In 1877, as counsel before the Electoral
Commission, he opened the argument for the Republican electors of Florida
and made the principal argument for the Republican electors of Oregon. In
March of the same year he succeeded John Sherman as senator from Ohio,
and served until March 1879. In 1881 President Hayes nominated him as
associate justice of the Supreme Court, to succeed Noah H. Swayne; there
was much opposition, especially in the press, to this appointment, because
Matthews had been a prominent railway and corporation lawyer and had
been one of the Republican “visiting statesmen” who witnessed the canvass
of the vote of Louisiana1 in 1876; and the nomination had not been
approved when the session of Congress expired. Matthews was
renominated by President Garfield on the 15th of March, and the
nomination was confirmed by the Senate (22 for, 21 against) on the 12th of
May. He was an honest, impartial and conscientious judge. He died in
Washington, on the 22nd of March 1889.
anti-slavery paper, the Cincinnati Herald. He was clerk of the Ohio House
of Representatives in 1848-1849, a judge of common pleas of Hamilton
county in 1850-1853, state senator in 1856-1858, and U.S. district-attorney
for the southern district of Ohio in 1858-1861. First a Whig and then a Free-
Soiler, he joined the Republican party in 1861. After the outbreak of the
Civil War he was commissioned a lieutenant of the 23rd Ohio, of which
Rutherford B. Hayes was major; but saw service only with the 57th Ohio, of
which he was colonel, and with a brigade which he commanded in the
Army of the Cumberland. He resigned from the army in 1863, and was
judge of the Cincinnati superior court in 1863-1864. He was a Republican
presidential elector in 1864 and 1868. In 1872 he joined the Liberal
Republican movement, and was temporary chairman of the Cincinnati
convention which nominated Horace Greeley for the presidency, but in the
campaign he supported Grant. In 1877, as counsel before the Electoral
Commission, he opened the argument for the Republican electors of Florida
and made the principal argument for the Republican electors of Oregon. In
March of the same year he succeeded John Sherman as senator from Ohio,
and served until March 1879. In 1881 President Hayes nominated him as
associate justice of the Supreme Court, to succeed Noah H. Swayne; there
was much opposition, especially in the press, to this appointment, because
Matthews had been a prominent railway and corporation lawyer and had
been one of the Republican “visiting statesmen” who witnessed the canvass
of the vote of Louisiana1 in 1876; and the nomination had not been
approved when the session of Congress expired. Matthews was
renominated by President Garfield on the 15th of March, and the
nomination was confirmed by the Senate (22 for, 21 against) on the 12th of
May. He was an honest, impartial and conscientious judge. He died in
Washington, on the 22nd of March 1889.
Page 48
1 It seems certain that Matthews and Charles Foster of Ohio gave their written promise
that Hayes, if elected, would recognize the Democratic governors in Louisiana and South
Carolina.
MATTHIAE, AUGUST HEINRICH (1769-1835),
German classical scholar, was born at Göttingen, on the 25th of December
1769, and educated at the university. He then spent some years as a tutor in
Amsterdam. In 1798 he returned to Germany, and in 1802 was appointed
director of the Friedrichsgymnasium at Altenburg, which post he held till
his death, on the 6th of January 1835. Of his numerous important works the
best-known are his Greek Grammar (3rd ed., 1835), translated into English
by E. V. Blomfield (5th ed., by J. Kenrick, 1832), his edition of Euripides (9
vols., 1813-1829), Grundriss der Geschichte der griechischen und
römischen Litteratur (3rd ed., 1834, Eng. trans., Oxford, 1841) Lehrbuch
für den ersten Unterricht in der Philosophie (3rd ed., 1833), Encyklopädie
und Methodologie der Philologie (1835). His Life was written by his son
Constantin (1845).
His brother, Friedrich Christian Matthiae (1763-1822), rector of the
Frankfort gymnasium, published valuable editions of Seneca’s Letters,
Aratus, and Dionysius Periegetes.
that Hayes, if elected, would recognize the Democratic governors in Louisiana and South
Carolina.
MATTHIAE, AUGUST HEINRICH (1769-1835),
German classical scholar, was born at Göttingen, on the 25th of December
1769, and educated at the university. He then spent some years as a tutor in
Amsterdam. In 1798 he returned to Germany, and in 1802 was appointed
director of the Friedrichsgymnasium at Altenburg, which post he held till
his death, on the 6th of January 1835. Of his numerous important works the
best-known are his Greek Grammar (3rd ed., 1835), translated into English
by E. V. Blomfield (5th ed., by J. Kenrick, 1832), his edition of Euripides (9
vols., 1813-1829), Grundriss der Geschichte der griechischen und
römischen Litteratur (3rd ed., 1834, Eng. trans., Oxford, 1841) Lehrbuch
für den ersten Unterricht in der Philosophie (3rd ed., 1833), Encyklopädie
und Methodologie der Philologie (1835). His Life was written by his son
Constantin (1845).
His brother, Friedrich Christian Matthiae (1763-1822), rector of the
Frankfort gymnasium, published valuable editions of Seneca’s Letters,
Aratus, and Dionysius Periegetes.
Page 49
MATTHIAS, the disciple elected by the primitive Christian
community to fill the place in the Twelve vacated by Judas Iscariot (Acts i.
21-26). Nothing further is recorded of him in the New Testament. Eusebius
(Hist. Eccl., I. xii.) says he was, like his competitor, Barsabas Justus, one of
the seventy, and the Syriac version of Eusebius calls him throughout not
Matthias but Tolmai, i.e. Bartholomew, without confusing him with the
Bartholomew who was originally one of the Twelve, and is often identified
with the Nathanael mentioned in the Fourth Gospel (Expository Times, ix.
566). Clement of Alexandria says some identified him with Zacchaeus, the
Clementine Recognitions identify him with Barnabas, Hilgenfeld thinks he
is the same as Nathanael.
Various works—a Gospel, Traditions and Apocryphal Words—were
ascribed to him; and there is also extant The Acts of Andrew and
Matthias, which places his activity in “the city of the cannibals” in
Ethiopia. Clement of Alexandria quotes two sayings from the
Traditions: (1) Wonder at the things before you (suggesting, like Plato,
that wonder is the first step to new knowledge); (2) If an elect man’s
neighbour sin, the elect man has sinned.
community to fill the place in the Twelve vacated by Judas Iscariot (Acts i.
21-26). Nothing further is recorded of him in the New Testament. Eusebius
(Hist. Eccl., I. xii.) says he was, like his competitor, Barsabas Justus, one of
the seventy, and the Syriac version of Eusebius calls him throughout not
Matthias but Tolmai, i.e. Bartholomew, without confusing him with the
Bartholomew who was originally one of the Twelve, and is often identified
with the Nathanael mentioned in the Fourth Gospel (Expository Times, ix.
566). Clement of Alexandria says some identified him with Zacchaeus, the
Clementine Recognitions identify him with Barnabas, Hilgenfeld thinks he
is the same as Nathanael.
Various works—a Gospel, Traditions and Apocryphal Words—were
ascribed to him; and there is also extant The Acts of Andrew and
Matthias, which places his activity in “the city of the cannibals” in
Ethiopia. Clement of Alexandria quotes two sayings from the
Traditions: (1) Wonder at the things before you (suggesting, like Plato,
that wonder is the first step to new knowledge); (2) If an elect man’s
neighbour sin, the elect man has sinned.
Page 50
MATTHIAS (1557-1619), Roman emperor, son of the emperor
Maximilian II. and Maria, daughter of the emperor Charles V., was born in
Vienna, on the 24th of February 1557. Educated by the diplomatist O. G. de
Busbecq, he began his public life in 1577, soon after his father’s death,
when he was invited to assume the governorship of the Netherlands, then in
the midst of the long struggle with Spain. He eagerly accepted this
invitation, although it involved a definite breach with his Spanish kinsman,
Philip II., and entering Brussels in January 1578 was named governor-
general; but he was merely a cipher, and only held the position for about
three years, returning to Germany in October 1581. Matthias was appointed
governor of Austria in 1593 by his brother, the emperor Rudolph II.; and
two years later, when another brother, the archduke Ernest, died, he became
a person of more importance as the eldest surviving brother of the
unmarried emperor. As governor of Austria Matthias continued the policy of
crushing the Protestants, although personally he appears to have been
inclined to religious tolerance; and he dealt with the rising of the peasants in
1595, in addition to representing Rudolph at the imperial diets, and gaining
some fame as a soldier during the Turkish War. A few years later the
discontent felt by the members of the Habsburg family at the incompetence
of the emperor became very acute, and the lead was taken by Matthias.
Obtaining in May 1605 a reluctant consent from his brother, he took over
the conduct of affairs in Hungary, where a revolt had broken out, and was
formally recognized by the Habsburgs as their head in April 1606, and was
Maximilian II. and Maria, daughter of the emperor Charles V., was born in
Vienna, on the 24th of February 1557. Educated by the diplomatist O. G. de
Busbecq, he began his public life in 1577, soon after his father’s death,
when he was invited to assume the governorship of the Netherlands, then in
the midst of the long struggle with Spain. He eagerly accepted this
invitation, although it involved a definite breach with his Spanish kinsman,
Philip II., and entering Brussels in January 1578 was named governor-
general; but he was merely a cipher, and only held the position for about
three years, returning to Germany in October 1581. Matthias was appointed
governor of Austria in 1593 by his brother, the emperor Rudolph II.; and
two years later, when another brother, the archduke Ernest, died, he became
a person of more importance as the eldest surviving brother of the
unmarried emperor. As governor of Austria Matthias continued the policy of
crushing the Protestants, although personally he appears to have been
inclined to religious tolerance; and he dealt with the rising of the peasants in
1595, in addition to representing Rudolph at the imperial diets, and gaining
some fame as a soldier during the Turkish War. A few years later the
discontent felt by the members of the Habsburg family at the incompetence
of the emperor became very acute, and the lead was taken by Matthias.
Obtaining in May 1605 a reluctant consent from his brother, he took over
the conduct of affairs in Hungary, where a revolt had broken out, and was
formally recognized by the Habsburgs as their head in April 1606, and was
Page 51
promised the succession to the Empire. In June 1606 he concluded the
peace of Vienna with the rebellious Hungarians, and was thus in a better
position to treat with the sultan, with whom peace was made in November.
This pacific policy was displeasing to Rudolph, who prepared to renew the
Turkish War; but having secured the support of the national party in
Hungary and gathered an army, Matthias forced his brother to cede to him
this kingdom, together with Austria and Moravia, both of which had thrown
in their lot with Hungary (1608). The king of Hungary, as Matthias now
became, was reluctantly compelled to grant religious liberty to the
inhabitants of Austria. The strained relations which had arisen between
Rudolph and Matthias as a result of these proceedings were temporarily
improved, and a formal reconciliation took place in 1610; but affairs in
Bohemia soon destroyed this fraternal peace. In spite of the letter of majesty
(Majestätsbrief) which the Bohemians had extorted from Rudolph, they
were very dissatisfied with their ruler, whose troops were ravaging their
land; and in 1611 they invited Matthias to come to their aid. Accepting this
invitation, he inflicted another humiliation upon his brother, and was
crowned king of Bohemia in May 1611. Rudolph, however, was successful
in preventing the election of Matthias as German king, or king of the
Romans, and when he died, in January 1612, no provision had been made
for a successor. Already king of Hungary and Bohemia, however, Matthias
obtained the remaining hereditary dominions of the Habsburgs, and in June
1612 was crowned emperor, although the ecclesiastical electors favoured
his younger brother, the archduke Albert (1559-1621).
The short reign of the new emperor was troubled by the religious
dissensions of Germany. His health became impaired and his indolence
increased, and he fell completely under the influence of Melchior Klesl
(q.v.), who practically conducted the imperial business. By Klesl’s advice he
took up an attitude of moderation and sought to reconcile the contending
religious parties; but the proceedings at the diet of Regensburg in 1613
peace of Vienna with the rebellious Hungarians, and was thus in a better
position to treat with the sultan, with whom peace was made in November.
This pacific policy was displeasing to Rudolph, who prepared to renew the
Turkish War; but having secured the support of the national party in
Hungary and gathered an army, Matthias forced his brother to cede to him
this kingdom, together with Austria and Moravia, both of which had thrown
in their lot with Hungary (1608). The king of Hungary, as Matthias now
became, was reluctantly compelled to grant religious liberty to the
inhabitants of Austria. The strained relations which had arisen between
Rudolph and Matthias as a result of these proceedings were temporarily
improved, and a formal reconciliation took place in 1610; but affairs in
Bohemia soon destroyed this fraternal peace. In spite of the letter of majesty
(Majestätsbrief) which the Bohemians had extorted from Rudolph, they
were very dissatisfied with their ruler, whose troops were ravaging their
land; and in 1611 they invited Matthias to come to their aid. Accepting this
invitation, he inflicted another humiliation upon his brother, and was
crowned king of Bohemia in May 1611. Rudolph, however, was successful
in preventing the election of Matthias as German king, or king of the
Romans, and when he died, in January 1612, no provision had been made
for a successor. Already king of Hungary and Bohemia, however, Matthias
obtained the remaining hereditary dominions of the Habsburgs, and in June
1612 was crowned emperor, although the ecclesiastical electors favoured
his younger brother, the archduke Albert (1559-1621).
The short reign of the new emperor was troubled by the religious
dissensions of Germany. His health became impaired and his indolence
increased, and he fell completely under the influence of Melchior Klesl
(q.v.), who practically conducted the imperial business. By Klesl’s advice he
took up an attitude of moderation and sought to reconcile the contending
religious parties; but the proceedings at the diet of Regensburg in 1613
Page 52
proved the hopelessness of these attempts, while their author was regarded
with general distrust. Meanwhile the younger Habsburgs, led by the
emperor’s brother, the archduke Maximilian, and his cousin, Ferdinand,
archduke of Styria, afterwards the emperor Ferdinand II., disliking the
peaceful policy of Klesl, had allied themselves with the unyielding Roman
Catholics, while the question of the imperial succession was forcing its way
to the front. In 1611 Matthias had married his cousin Anna (d. 1618),
daughter of the archduke Ferdinand (d. 1595), but he was old and childless
and the Habsburgs were anxious to retain his extensive possessions in the
family. Klesl, on the one hand, wished the settlement of the religious
difficulties to precede any arrangement about the imperial succession; the
Habsburgs, on the other, regarded the question of the succession as urgent
and vital. Meanwhile the disputed succession to the duchies of Cleves and
Jülich again threatened a European war; the imperial commands were
flouted in Cologne and Aix-la-Chapelle, and the Bohemians were again
becoming troublesome. Having decided that Ferdinand should succeed
Matthias as emperor, the Habsburgs had secured his election as king of
Bohemia in June 1617, but were unable to stem the rising tide of disorder in
that country. Matthias and Klesl were in favour of concessions, but
Ferdinand and Maximilian met this move by seizing and imprisoning Klesl.
Ferdinand had just secured his coronation as king of Hungary when there
broke out in Bohemia those struggles which heralded the Thirty Years’ War;
and on the 20th of March 1619 the emperor died at Vienna.
For the life and reign of Matthias the following works may be
consulted: J. Heling, Die Wahl des römischen Königs Matthias
(Belgrade, 1892); A. Gindely, Rudolf II. und seine Zeit (Prague, 1862-
1868); F. Stieve, Die Verhandlungen über die Nachfolge Kaisers
Rudolf II. (Munich, 1880); P. von Chlumecky, Karl von Zierotin und
seine Zeit (Brünn, 1862-1879); A. Kerschbaumer, Kardinal Klesel
(Vienna, 1865); M. Ritter, Quellenbeiträge zur Geschichte des Kaisers
with general distrust. Meanwhile the younger Habsburgs, led by the
emperor’s brother, the archduke Maximilian, and his cousin, Ferdinand,
archduke of Styria, afterwards the emperor Ferdinand II., disliking the
peaceful policy of Klesl, had allied themselves with the unyielding Roman
Catholics, while the question of the imperial succession was forcing its way
to the front. In 1611 Matthias had married his cousin Anna (d. 1618),
daughter of the archduke Ferdinand (d. 1595), but he was old and childless
and the Habsburgs were anxious to retain his extensive possessions in the
family. Klesl, on the one hand, wished the settlement of the religious
difficulties to precede any arrangement about the imperial succession; the
Habsburgs, on the other, regarded the question of the succession as urgent
and vital. Meanwhile the disputed succession to the duchies of Cleves and
Jülich again threatened a European war; the imperial commands were
flouted in Cologne and Aix-la-Chapelle, and the Bohemians were again
becoming troublesome. Having decided that Ferdinand should succeed
Matthias as emperor, the Habsburgs had secured his election as king of
Bohemia in June 1617, but were unable to stem the rising tide of disorder in
that country. Matthias and Klesl were in favour of concessions, but
Ferdinand and Maximilian met this move by seizing and imprisoning Klesl.
Ferdinand had just secured his coronation as king of Hungary when there
broke out in Bohemia those struggles which heralded the Thirty Years’ War;
and on the 20th of March 1619 the emperor died at Vienna.
For the life and reign of Matthias the following works may be
consulted: J. Heling, Die Wahl des römischen Königs Matthias
(Belgrade, 1892); A. Gindely, Rudolf II. und seine Zeit (Prague, 1862-
1868); F. Stieve, Die Verhandlungen über die Nachfolge Kaisers
Rudolf II. (Munich, 1880); P. von Chlumecky, Karl von Zierotin und
seine Zeit (Brünn, 1862-1879); A. Kerschbaumer, Kardinal Klesel
(Vienna, 1865); M. Ritter, Quellenbeiträge zur Geschichte des Kaisers
Page 53
Rudolf II. (Munich, 1872); Deutsche Geschichte im Zeitalter der
Gegenreformation und des dreissigjährigen Krieges (Stuttgart, 1887,
seq.); and the article on Matthias in the Allgemeine deutsche
Biographie, Bd. XX. (Leipzig, 1884); L. von Ranke, Zur deutschen
Geschichte vom Religionsfrieden bis zum 30-jährigen Kriege (Leipzig,
1888); and J. Janssen, Geschichte des deutschen Volks seit dem
Ausgang des Mittelalters (Freiburg, 1878 seq.), Eng. trans. by M. A.
Mitchell and A. M. Christie (London, 1896, seq.).
MATTHIAS I., HUNYADI (1440-1490), king of Hungary,
also known as Matthias Corvinus, a surname which he received from the
raven (corvus) on his escutcheon, second son of János Hunyadi and
Elizabeth Szilágyi, was born at Kolozsvár, probably on
the 23rd of February 1440. His tutors were the learned János Vitéz,
bishop of Nagyvárad, whom he subsequently raised to the primacy, and the
Polish humanist Gregory Sanocki. The precocious lad quickly mastered the
German, Latin and principal Slavonic languages, frequently acting as his
father’s interpreter at the reception of ambassadors. His military training
proceeded under the eye of his father, whom he began to follow on his
campaigns when only twelve years of age. In 1453 he was created count of
Bistercze, and was knighted at the siege of Belgrade in 1454. The same care
for his welfare led his father to choose him a bride in the powerful Cilli
family, but the young Elizabeth died before the marriage was consummated,
leaving Matthias a widower at the age of fifteen. On the death of his father
Gegenreformation und des dreissigjährigen Krieges (Stuttgart, 1887,
seq.); and the article on Matthias in the Allgemeine deutsche
Biographie, Bd. XX. (Leipzig, 1884); L. von Ranke, Zur deutschen
Geschichte vom Religionsfrieden bis zum 30-jährigen Kriege (Leipzig,
1888); and J. Janssen, Geschichte des deutschen Volks seit dem
Ausgang des Mittelalters (Freiburg, 1878 seq.), Eng. trans. by M. A.
Mitchell and A. M. Christie (London, 1896, seq.).
MATTHIAS I., HUNYADI (1440-1490), king of Hungary,
also known as Matthias Corvinus, a surname which he received from the
raven (corvus) on his escutcheon, second son of János Hunyadi and
Elizabeth Szilágyi, was born at Kolozsvár, probably on
the 23rd of February 1440. His tutors were the learned János Vitéz,
bishop of Nagyvárad, whom he subsequently raised to the primacy, and the
Polish humanist Gregory Sanocki. The precocious lad quickly mastered the
German, Latin and principal Slavonic languages, frequently acting as his
father’s interpreter at the reception of ambassadors. His military training
proceeded under the eye of his father, whom he began to follow on his
campaigns when only twelve years of age. In 1453 he was created count of
Bistercze, and was knighted at the siege of Belgrade in 1454. The same care
for his welfare led his father to choose him a bride in the powerful Cilli
family, but the young Elizabeth died before the marriage was consummated,
leaving Matthias a widower at the age of fifteen. On the death of his father
Page 54
he was inveigled to Buda by the enemies of his house, and, on the pretext of
being concerned in a purely imaginary conspiracy against Ladislaus V., was
condemned to decapitation, but was spared on account of his youth, and on
the king’s death fell into the hands of George Poděbrad, governor of
Bohemia, the friend of the Hunyadis, in whose interests it was that a
national king should sit on the Magyar throne. Poděbrad treated Matthias
hospitably and affianced him with his daughter Catherine, but still detained
him, for safety’s sake, in Prague, even after a Magyar deputation had
hastened thither to offer the youth the crown. Matthias was the elect of the
Hungarian people, gratefully mindful of his father’s services to the state and
inimical to all foreign candidates; and though an influential section of the
magnates, headed by the palatine László Garai and the voivode of
Transylvania, Miklós Ujlaki, who had been concerned in the judicial
murder of Matthias’s brother László, and hated the Hunyadis as semi-
foreign upstarts, were fiercely opposed to Matthias’s election, they were not
strong enough to resist the manifest wish of the nation, supported as it was
by Matthias’s uncle Mihály Szilágyi at the head of 15,000 veterans. On the
24th of January 1458, 40,000 Hungarian noblemen, assembled on the ice of
the frozen Danube, unanimously elected Matthias Hunyadi king of
Hungary, and on the 14th of February the new king made his state entry into
Buda.
The realm at this time was environed by perils. The Turks and the
Venetians threatened it from the south, the emperor Frederick III. from the
west, and Casimir IV. of Poland from the north, both Frederick and Casimir
claiming the throne. The Czech mercenaries under Giszkra held the
northern counties and from thence plundered those in the centre. Meanwhile
Matthias’s friends had only pacified the hostile dignitaries by engaging to
marry the daughter of the palatine Garai to their nominee, whereas Matthias
not unnaturally refused to marry into the family of one of his brother’s
murderers, and on the 9th of February confirmed his previous nuptial
being concerned in a purely imaginary conspiracy against Ladislaus V., was
condemned to decapitation, but was spared on account of his youth, and on
the king’s death fell into the hands of George Poděbrad, governor of
Bohemia, the friend of the Hunyadis, in whose interests it was that a
national king should sit on the Magyar throne. Poděbrad treated Matthias
hospitably and affianced him with his daughter Catherine, but still detained
him, for safety’s sake, in Prague, even after a Magyar deputation had
hastened thither to offer the youth the crown. Matthias was the elect of the
Hungarian people, gratefully mindful of his father’s services to the state and
inimical to all foreign candidates; and though an influential section of the
magnates, headed by the palatine László Garai and the voivode of
Transylvania, Miklós Ujlaki, who had been concerned in the judicial
murder of Matthias’s brother László, and hated the Hunyadis as semi-
foreign upstarts, were fiercely opposed to Matthias’s election, they were not
strong enough to resist the manifest wish of the nation, supported as it was
by Matthias’s uncle Mihály Szilágyi at the head of 15,000 veterans. On the
24th of January 1458, 40,000 Hungarian noblemen, assembled on the ice of
the frozen Danube, unanimously elected Matthias Hunyadi king of
Hungary, and on the 14th of February the new king made his state entry into
Buda.
The realm at this time was environed by perils. The Turks and the
Venetians threatened it from the south, the emperor Frederick III. from the
west, and Casimir IV. of Poland from the north, both Frederick and Casimir
claiming the throne. The Czech mercenaries under Giszkra held the
northern counties and from thence plundered those in the centre. Meanwhile
Matthias’s friends had only pacified the hostile dignitaries by engaging to
marry the daughter of the palatine Garai to their nominee, whereas Matthias
not unnaturally refused to marry into the family of one of his brother’s
murderers, and on the 9th of February confirmed his previous nuptial
Page 55
contract with the daughter of George Poděbrad, who shortly afterwards was
elected king of Bohemia (March 2, 1458). Throughout 1458 the struggle
between the young king and the magnates, reinforced by Matthias’s own
uncle and guardian Szilágyi, was acute. But Matthias, who began by
deposing Garai and dismissing Szilágyi, and then proceeded to levy a tax,
without the consent of the Diet, in order to hire mercenaries, easily
prevailed. Nor did these complications prevent him from recovering the
fortress of Galamboc from the Turks, successfully invading Servia, and
reasserting the suzerainty of the Hungarian crown over Bosnia. In the
following year there was a fresh rebellion, when the emperor Frederick was
actually crowned king by the malcontents at Vienna-Neustadt (March 4,
1459); but Matthias drove him out, and Pope Pius II. intervened so as to
leave Matthias free to engage in a projected crusade against the Turks,
which subsequent political complications, however, rendered impossible.
From 1461 to 1465 the career of Matthias was a perpetual struggle
punctuated by truces. Having come to an understanding with his father-in-
law Poděbrad, he was able to turn his arms against the emperor Frederick,
and in April 1462 Frederick restored the holy crown for 60,000 ducats and
was allowed to retain certain Hungarian counties with the title of king; in
return for which concessions, extorted from Matthias by the necessity of
coping with a simultaneous rebellion of the Magyar noble in league with
Poděbrad’s son Victorinus, the emperor recognized Matthias as the actual
sovereign of Hungary. Only now was Matthias able to turn against the
Turks, who were again threatening the southern provinces. He began by
defeating Ali Pasha, and then penetrated into Bosnia, and captured the
newly built fortress of Jajce after a long and obstinate defence (Dec. 1463).
On returning home he was crowned with the holy crown on the 29th of
March 1464, and, after driving the Czechs out of his northern counties,
turned southwards again, this time recovering all the parts of Bosnia which
still remained in Turkish hands.
elected king of Bohemia (March 2, 1458). Throughout 1458 the struggle
between the young king and the magnates, reinforced by Matthias’s own
uncle and guardian Szilágyi, was acute. But Matthias, who began by
deposing Garai and dismissing Szilágyi, and then proceeded to levy a tax,
without the consent of the Diet, in order to hire mercenaries, easily
prevailed. Nor did these complications prevent him from recovering the
fortress of Galamboc from the Turks, successfully invading Servia, and
reasserting the suzerainty of the Hungarian crown over Bosnia. In the
following year there was a fresh rebellion, when the emperor Frederick was
actually crowned king by the malcontents at Vienna-Neustadt (March 4,
1459); but Matthias drove him out, and Pope Pius II. intervened so as to
leave Matthias free to engage in a projected crusade against the Turks,
which subsequent political complications, however, rendered impossible.
From 1461 to 1465 the career of Matthias was a perpetual struggle
punctuated by truces. Having come to an understanding with his father-in-
law Poděbrad, he was able to turn his arms against the emperor Frederick,
and in April 1462 Frederick restored the holy crown for 60,000 ducats and
was allowed to retain certain Hungarian counties with the title of king; in
return for which concessions, extorted from Matthias by the necessity of
coping with a simultaneous rebellion of the Magyar noble in league with
Poděbrad’s son Victorinus, the emperor recognized Matthias as the actual
sovereign of Hungary. Only now was Matthias able to turn against the
Turks, who were again threatening the southern provinces. He began by
defeating Ali Pasha, and then penetrated into Bosnia, and captured the
newly built fortress of Jajce after a long and obstinate defence (Dec. 1463).
On returning home he was crowned with the holy crown on the 29th of
March 1464, and, after driving the Czechs out of his northern counties,
turned southwards again, this time recovering all the parts of Bosnia which
still remained in Turkish hands.
Page 56
A political event of the first importance now riveted his attention upon
the north. Poděbrad, who had gained the throne of Bohemia with the aid of
the Hussites and Utraquists, had long been in ill odour at Rome, and in
1465 Pope Paul II. determined to depose the semi-Catholic monarch. All the
neighbouring princes, the emperor, Casimir IV. of Poland and Matthias,
were commanded in turn to execute the papal decree of deposition, and
Matthias gladly placed his army at the disposal of the Holy See. The war
began on the 31st of May 1468, but, as early as the 27th of February 1469,
Matthias anticipated an alliance between George and Frederick by himself
concluding an armistice with the former. On the 3rd of May the Czech
Catholics elected Matthias king of Bohemia, but this was contrary to the
wishes of both pope and emperor, who preferred to partition Bohemia. But
now George discomfited all his enemies by suddenly excluding his own son
from the throne in favour of Ladislaus, the eldest son of Casimir IV., thus
skilfully enlisting Poland on his side. The sudden death of Poděbrad on the
22nd of March 1471 led to fresh complications. At the very moment when
Matthias was about to profit by the disappearance of his most capable rival,
another dangerous rebellion, headed by the primate and the chief dignitaries
of the state, with the object of placing Casimir, son of Casimir IV., on the
throne, paralysed Matthias’s foreign policy during the critical years 1470-
1471. He suppressed this domestic rebellion indeed, but in the meantime the
Poles had invaded the Bohemian domains with 60,000 men, and when in
1474 Matthias was at last able to take the field against them in order to raise
the siege of Breslau, he was obliged to fortify himself in an entrenched
camp, whence he so skilfully harried the enemy that the Poles, impatient to
return to their own country, made peace at Breslau (Feb. 1475) on an uti
possidetis basis, a peace subsequently confirmed by the congress of Olmütz
(July 1479). During the interval between these peaces, Matthias, in self-
defence, again made war on the emperor, reducing Frederick to such
extremities that he was glad to accept peace on any terms. By the final
the north. Poděbrad, who had gained the throne of Bohemia with the aid of
the Hussites and Utraquists, had long been in ill odour at Rome, and in
1465 Pope Paul II. determined to depose the semi-Catholic monarch. All the
neighbouring princes, the emperor, Casimir IV. of Poland and Matthias,
were commanded in turn to execute the papal decree of deposition, and
Matthias gladly placed his army at the disposal of the Holy See. The war
began on the 31st of May 1468, but, as early as the 27th of February 1469,
Matthias anticipated an alliance between George and Frederick by himself
concluding an armistice with the former. On the 3rd of May the Czech
Catholics elected Matthias king of Bohemia, but this was contrary to the
wishes of both pope and emperor, who preferred to partition Bohemia. But
now George discomfited all his enemies by suddenly excluding his own son
from the throne in favour of Ladislaus, the eldest son of Casimir IV., thus
skilfully enlisting Poland on his side. The sudden death of Poděbrad on the
22nd of March 1471 led to fresh complications. At the very moment when
Matthias was about to profit by the disappearance of his most capable rival,
another dangerous rebellion, headed by the primate and the chief dignitaries
of the state, with the object of placing Casimir, son of Casimir IV., on the
throne, paralysed Matthias’s foreign policy during the critical years 1470-
1471. He suppressed this domestic rebellion indeed, but in the meantime the
Poles had invaded the Bohemian domains with 60,000 men, and when in
1474 Matthias was at last able to take the field against them in order to raise
the siege of Breslau, he was obliged to fortify himself in an entrenched
camp, whence he so skilfully harried the enemy that the Poles, impatient to
return to their own country, made peace at Breslau (Feb. 1475) on an uti
possidetis basis, a peace subsequently confirmed by the congress of Olmütz
(July 1479). During the interval between these peaces, Matthias, in self-
defence, again made war on the emperor, reducing Frederick to such
extremities that he was glad to accept peace on any terms. By the final
Page 57
arrangement made between the contending princes, Matthias recognized
Ladislaus as king of Bohemia proper in return for the surrender of Moravia,
Silesia and Upper and Lower Lusatia, hitherto component parts of the
Czech monarchy, till he should have redeemed them for 400,000 florins.
The emperor promised to pay Matthias 100,000 florins as a war indemnity,
and recognized him as the legitimate king of Hungary on the understanding
that he should succeed him if he died without male issue, a contingency at
this time somewhat improbable, as Matthias, only three years previously
(Dec. 15, 1476), had married his third wife, Beatrice of Naples, daughter of
Ferdinand of Aragon.
The endless tergiversations and depredations of the emperor speedily
induced Matthias to declare war against him for the third time (1481), the
Magyar king conquering all the fortresses in Frederick’s hereditary
domains. Finally, on the 1st of June 1485, at the head of 8000 veterans, he
made his triumphal entry into Vienna, which he henceforth made his
capital. Styria, Carinthia and Carniola were next subdued, and Trieste was
only saved by the intervention of the Venetians. Matthias consolidated his
position by alliances with the dukes of Saxony and Bavaria, with the Swiss
Confederation, and the archbishop of Salzburg, and was henceforth the
greatest potentate in central Europe. His far-reaching hand even extended to
Italy. Thus, in 1480, when a Turkish fleet seized Otranto, Matthias, at the
earnest solicitation of the pope, sent Balasz Magyar to recover the fortress,
which surrendered to him on the 10th of May 1481. Again in 1488,
Matthias took Ancona under his protection for a time and occupied it with a
Hungarian garrison.
Though Matthias’s policy was so predominantly occidental that he soon
abandoned his youthful idea of driving the Turks out of Europe, he at least
succeeded in making them respect Hungarian territory. Thus in 1479 a huge
Turkish army, on its return home from ravaging Transylvania, was
Ladislaus as king of Bohemia proper in return for the surrender of Moravia,
Silesia and Upper and Lower Lusatia, hitherto component parts of the
Czech monarchy, till he should have redeemed them for 400,000 florins.
The emperor promised to pay Matthias 100,000 florins as a war indemnity,
and recognized him as the legitimate king of Hungary on the understanding
that he should succeed him if he died without male issue, a contingency at
this time somewhat improbable, as Matthias, only three years previously
(Dec. 15, 1476), had married his third wife, Beatrice of Naples, daughter of
Ferdinand of Aragon.
The endless tergiversations and depredations of the emperor speedily
induced Matthias to declare war against him for the third time (1481), the
Magyar king conquering all the fortresses in Frederick’s hereditary
domains. Finally, on the 1st of June 1485, at the head of 8000 veterans, he
made his triumphal entry into Vienna, which he henceforth made his
capital. Styria, Carinthia and Carniola were next subdued, and Trieste was
only saved by the intervention of the Venetians. Matthias consolidated his
position by alliances with the dukes of Saxony and Bavaria, with the Swiss
Confederation, and the archbishop of Salzburg, and was henceforth the
greatest potentate in central Europe. His far-reaching hand even extended to
Italy. Thus, in 1480, when a Turkish fleet seized Otranto, Matthias, at the
earnest solicitation of the pope, sent Balasz Magyar to recover the fortress,
which surrendered to him on the 10th of May 1481. Again in 1488,
Matthias took Ancona under his protection for a time and occupied it with a
Hungarian garrison.
Though Matthias’s policy was so predominantly occidental that he soon
abandoned his youthful idea of driving the Turks out of Europe, he at least
succeeded in making them respect Hungarian territory. Thus in 1479 a huge
Turkish army, on its return home from ravaging Transylvania, was
Page 58
annihilated at Szászváros (Oct. 13), and in 1480 Matthias recaptured Jajce,
drove the Turks from Servia and erected two new military banates, Jajce
and Srebernik, out of reconquered Bosnian territory. On the death of
Mahommed II. in 1481, a unique opportunity for the intervention of Europe
in Turkish affairs presented itself. A civil war ensued in Turkey between his
sons Bayezid and Jem, and the latter, being worsted, fled to the knights of
Rhodes, by whom he was kept in custody in France (see Bayezid II.).
Matthias, as the next-door neighbour of the Turks, claimed the custody of
so valuable a hostage, and would have used him as a means of extorting
concessions from Bayezid. But neither the pope nor the Venetians would
hear of such a transfer, and the negotiations on this subject greatly
embittered Matthias against the Curia. The last days of Matthias were
occupied in endeavouring to secure the succession to the throne for his
illegitimate son János (see Corvinus, János); but Queen Beatrice, though
childless, fiercely and openly opposed the idea and the matter was still
pending when Matthias, who had long been crippled by gout, expired very
suddenly on Palm Sunday, the 4th of April 1490.
Matthias Hunyadi was indisputably the greatest man of his day, and one
of the greatest monarchs who ever reigned. The precocity and universality
of his genius impress one the most. Like Napoleon, with whom he has often
been compared, he was equally illustrious as a soldier, a statesman, an
orator, a legislator and an administrator. But in all moral qualities the
brilliant adventurer of the 15th was infinitely superior to the brilliant
adventurer of the 19th century. Though naturally passionate, Matthias’s
self-control was almost superhuman, and throughout his stormy life, with
his innumerable experiences of ingratitude and treachery, he never was
guilty of a single cruel or vindictive action. His capacity for work was
inexhaustible. Frequently half his nights were spent in reading, after the
labour of his most strenuous days. There was no branch of knowledge in
which he did not take an absorbing interest, no polite art which he did not
drove the Turks from Servia and erected two new military banates, Jajce
and Srebernik, out of reconquered Bosnian territory. On the death of
Mahommed II. in 1481, a unique opportunity for the intervention of Europe
in Turkish affairs presented itself. A civil war ensued in Turkey between his
sons Bayezid and Jem, and the latter, being worsted, fled to the knights of
Rhodes, by whom he was kept in custody in France (see Bayezid II.).
Matthias, as the next-door neighbour of the Turks, claimed the custody of
so valuable a hostage, and would have used him as a means of extorting
concessions from Bayezid. But neither the pope nor the Venetians would
hear of such a transfer, and the negotiations on this subject greatly
embittered Matthias against the Curia. The last days of Matthias were
occupied in endeavouring to secure the succession to the throne for his
illegitimate son János (see Corvinus, János); but Queen Beatrice, though
childless, fiercely and openly opposed the idea and the matter was still
pending when Matthias, who had long been crippled by gout, expired very
suddenly on Palm Sunday, the 4th of April 1490.
Matthias Hunyadi was indisputably the greatest man of his day, and one
of the greatest monarchs who ever reigned. The precocity and universality
of his genius impress one the most. Like Napoleon, with whom he has often
been compared, he was equally illustrious as a soldier, a statesman, an
orator, a legislator and an administrator. But in all moral qualities the
brilliant adventurer of the 15th was infinitely superior to the brilliant
adventurer of the 19th century. Though naturally passionate, Matthias’s
self-control was almost superhuman, and throughout his stormy life, with
his innumerable experiences of ingratitude and treachery, he never was
guilty of a single cruel or vindictive action. His capacity for work was
inexhaustible. Frequently half his nights were spent in reading, after the
labour of his most strenuous days. There was no branch of knowledge in
which he did not take an absorbing interest, no polite art which he did not
Page 59
cultivate and encourage. His camp was a school of chivalry, his court a
nursery of poets and artists. Matthias was a middle-sized, broad-shouldered
man of martial bearing, with a large fleshy nose, hair reaching to his heels,
and the clean-shaven, heavy chinned face of an early Roman emperor.
See Vilmós Fraknói, King Matthias Hunyadi (Hung., Budapest,
1890, German ed., Freiburg, 1891); Ignácz Acsády, History of the
Hungarian Realm (Hung. vol. i., Budapest, 1904); József Teleki, The
Age of the Hunyadis in Hungary (Hung., vols. 3-5, Budapest, 1852-
1890); V. Fraknói, Life of János Vitéz (Hung. Budapest 1879); Karl
Schober, Die Eroberung Niederösterreichs durch Matthias Corvinus
(Vienna, 1879); János Huszár, Matthias’s Black Army (Hung. Budapest,
1890); Antonio Bonfini, Rerum hungaricarum decades (7th ed.,
Leipzig, 1771); Aeneas Sylvius, Opera (Frankfort, 1707); The
Correspondence of King Matthias (Hung. and Lat., Budapest, 1893); V.
Fraknói, The Embassies of Cardinal Carvajal to Hungary (Hung.,
Budapest, 1889); Marzio Galeotti, De egregie sapienter et jocose,
dictis ac factis Matthiae regis (Script. reg. hung. I.) (Vienna, 1746). Of
the above the first is the best general sketch and is rich in notes; the
second somewhat chauvinistic but excellently written; the third the best
work for scholars; the seventh, eighth and eleventh are valuable as
being by contemporaries.
(R. N. B.)
nursery of poets and artists. Matthias was a middle-sized, broad-shouldered
man of martial bearing, with a large fleshy nose, hair reaching to his heels,
and the clean-shaven, heavy chinned face of an early Roman emperor.
See Vilmós Fraknói, King Matthias Hunyadi (Hung., Budapest,
1890, German ed., Freiburg, 1891); Ignácz Acsády, History of the
Hungarian Realm (Hung. vol. i., Budapest, 1904); József Teleki, The
Age of the Hunyadis in Hungary (Hung., vols. 3-5, Budapest, 1852-
1890); V. Fraknói, Life of János Vitéz (Hung. Budapest 1879); Karl
Schober, Die Eroberung Niederösterreichs durch Matthias Corvinus
(Vienna, 1879); János Huszár, Matthias’s Black Army (Hung. Budapest,
1890); Antonio Bonfini, Rerum hungaricarum decades (7th ed.,
Leipzig, 1771); Aeneas Sylvius, Opera (Frankfort, 1707); The
Correspondence of King Matthias (Hung. and Lat., Budapest, 1893); V.
Fraknói, The Embassies of Cardinal Carvajal to Hungary (Hung.,
Budapest, 1889); Marzio Galeotti, De egregie sapienter et jocose,
dictis ac factis Matthiae regis (Script. reg. hung. I.) (Vienna, 1746). Of
the above the first is the best general sketch and is rich in notes; the
second somewhat chauvinistic but excellently written; the third the best
work for scholars; the seventh, eighth and eleventh are valuable as
being by contemporaries.
(R. N. B.)
Page 60
MATTHISSON, FRIEDRICH VON (1761-1831), German
poet, was born at Hohendodeleben near Magdeburg, the son of the village
pastor, on the 23rd of January 1761. After studying theology and philology
at the university of Halle, he was appointed in 1781 master at the classical
school Philanthropin in Dessau. This once famous seminary was, however,
then rapidly decaying in public favour, and in 1784 Matthisson was glad to
accept a travelling tutorship. He lived for two years with the Swiss author
Bonstetten at Nyon on the lake of Geneva. In 1794 he was appointed reader
and travelling companion to the princess Louisa of Anhalt-Dessau. In 1812
he entered the service of the king of Württemberg, was ennobled, created
counsellor of legation, appointed intendant of the court theatre and chief
librarian of the royal library at Stuttgart. In 1828 he retired and settled at
Wörlitz near Dessau, where he died on the 12th of March 1831. Matthisson
enjoyed for a time a great popularity on account of his poems, Gedichte
(1787; 15th ed., 1851; new ed., 1876), which Schiller extravagantly praised
for their melancholy sweetness and their fine descriptions of scenery. The
verse is melodious and the language musical, but the thought and
sentiments they express are too often artificial and insincere. His Adelaide
has been rendered famous owing to Beethoven’s setting of the song. Of his
elegies, Die Elegie in den Ruinen eines alten Bergschlosses is still a
favourite. His reminiscences, Erinnerungen (5 vols., 1810-1816), contain
interesting accounts of his travels.
Matthisson’s Schriften appeared in eight volumes (1825-1829), of
which the first contains his poems, the remainder his Erinnerungen; a
ninth volume was added in 1833 containing his biography by H.
Döring. His Literarischer Nachlass, with a selection from his
correspondence, was published in four volumes by F. R. Schoch in
1832.
poet, was born at Hohendodeleben near Magdeburg, the son of the village
pastor, on the 23rd of January 1761. After studying theology and philology
at the university of Halle, he was appointed in 1781 master at the classical
school Philanthropin in Dessau. This once famous seminary was, however,
then rapidly decaying in public favour, and in 1784 Matthisson was glad to
accept a travelling tutorship. He lived for two years with the Swiss author
Bonstetten at Nyon on the lake of Geneva. In 1794 he was appointed reader
and travelling companion to the princess Louisa of Anhalt-Dessau. In 1812
he entered the service of the king of Württemberg, was ennobled, created
counsellor of legation, appointed intendant of the court theatre and chief
librarian of the royal library at Stuttgart. In 1828 he retired and settled at
Wörlitz near Dessau, where he died on the 12th of March 1831. Matthisson
enjoyed for a time a great popularity on account of his poems, Gedichte
(1787; 15th ed., 1851; new ed., 1876), which Schiller extravagantly praised
for their melancholy sweetness and their fine descriptions of scenery. The
verse is melodious and the language musical, but the thought and
sentiments they express are too often artificial and insincere. His Adelaide
has been rendered famous owing to Beethoven’s setting of the song. Of his
elegies, Die Elegie in den Ruinen eines alten Bergschlosses is still a
favourite. His reminiscences, Erinnerungen (5 vols., 1810-1816), contain
interesting accounts of his travels.
Matthisson’s Schriften appeared in eight volumes (1825-1829), of
which the first contains his poems, the remainder his Erinnerungen; a
ninth volume was added in 1833 containing his biography by H.
Döring. His Literarischer Nachlass, with a selection from his
correspondence, was published in four volumes by F. R. Schoch in
1832.
Page 61
MATTING, a general term embracing many coarse woven or plaited
fibrous materials used for covering floors or furniture, for hanging as
screens, for wrapping up heavy merchandise and for other miscellaneous
purposes. In the United Kingdom, under the name of “coir” matting, a large
amount of a coarse kind of carpet is made from coco-nut fibre; and the
same material, as well as strips of cane, Manila hemp, various grasses and
rushes, is largely employed in various forms for making door mats. Large
quantities of the coco-nut fibre are woven in heavy looms, then cut up into
various sizes, and finally bound round the edges by a kind of rope made
from the same material. The mats may be of one colour only, or they may
be made of different colours and in different designs. Sometimes the names
of institutions are introduced into the mats. Another type of mat is made
exclusively from the above-mentioned rope by arranging alternate layers in
sinuous and straight paths, and then stitching the parts together. It is also
largely used for the outer covering of ships’ fenders. Perforated and
otherwise prepared rubber, as well as wire-woven material, are also largely
utilized for door and floor mats. Matting of various kinds is very
extensively employed throughout India for floor coverings, the bottoms of
bedsteads, fans and fly-flaps, &c.; and a considerable export trade in such
manufactures is carried on. The materials used are numerous; but the
principal substances are straw, the bulrushes Typha elephantina and T.
angustifolia, leaves of the date palm (Phoenix sylvestris), of the dwarf palm
(Chamaerops Ritchiana), of the Palmyra palm (Borassus flabelliformis), of
fibrous materials used for covering floors or furniture, for hanging as
screens, for wrapping up heavy merchandise and for other miscellaneous
purposes. In the United Kingdom, under the name of “coir” matting, a large
amount of a coarse kind of carpet is made from coco-nut fibre; and the
same material, as well as strips of cane, Manila hemp, various grasses and
rushes, is largely employed in various forms for making door mats. Large
quantities of the coco-nut fibre are woven in heavy looms, then cut up into
various sizes, and finally bound round the edges by a kind of rope made
from the same material. The mats may be of one colour only, or they may
be made of different colours and in different designs. Sometimes the names
of institutions are introduced into the mats. Another type of mat is made
exclusively from the above-mentioned rope by arranging alternate layers in
sinuous and straight paths, and then stitching the parts together. It is also
largely used for the outer covering of ships’ fenders. Perforated and
otherwise prepared rubber, as well as wire-woven material, are also largely
utilized for door and floor mats. Matting of various kinds is very
extensively employed throughout India for floor coverings, the bottoms of
bedsteads, fans and fly-flaps, &c.; and a considerable export trade in such
manufactures is carried on. The materials used are numerous; but the
principal substances are straw, the bulrushes Typha elephantina and T.
angustifolia, leaves of the date palm (Phoenix sylvestris), of the dwarf palm
(Chamaerops Ritchiana), of the Palmyra palm (Borassus flabelliformis), of
Page 62
the coco-nut palm (Cocos nucifera) and of the screw pine (Pandanus
odoratissimus), the munja or munj grass (Saccharum Munja) and allied
grasses, and the mat grasses Cyperus textilis and C. Pangorei, from the last
of which the well-known Palghat mats of the Madras Presidency are made.
Many of these Indian grass-mats are admirable examples of elegant design,
and the colours in which they are woven are rich, harmonious and effective
in the highest degree. Several useful household articles are made from the
different kinds of grasses. The grasses are dyed in all shades and plaited to
form attractive designs suitable for the purposes to which they are to be
applied. This class of work obtains in India, Japan and other Eastern
countries. Vast quantities of coarse matting used for packing furniture,
heavy and coarse goods, flax and other plants, &c., are made in Russia from
the bast or inner bark of the lime tree. This industry centres in the great
forest governments of Viatka, Nizhniy-Novgorod, Kostroma, Kazan, Perm
and Simbirsk.
MATTOCK (O.E. mattuc, of uncertain origin), a tool having a
double iron head, of which one end is shaped like an adze, and the other
like a pickaxe. The head has a socket in the centre in which the handle is
inserted transversely to the blades. It is used chiefly for grubbing and
rooting among tree stumps in plantations and copses, where the roots are
too close for the use of a spade, or for loosening hard soil.
odoratissimus), the munja or munj grass (Saccharum Munja) and allied
grasses, and the mat grasses Cyperus textilis and C. Pangorei, from the last
of which the well-known Palghat mats of the Madras Presidency are made.
Many of these Indian grass-mats are admirable examples of elegant design,
and the colours in which they are woven are rich, harmonious and effective
in the highest degree. Several useful household articles are made from the
different kinds of grasses. The grasses are dyed in all shades and plaited to
form attractive designs suitable for the purposes to which they are to be
applied. This class of work obtains in India, Japan and other Eastern
countries. Vast quantities of coarse matting used for packing furniture,
heavy and coarse goods, flax and other plants, &c., are made in Russia from
the bast or inner bark of the lime tree. This industry centres in the great
forest governments of Viatka, Nizhniy-Novgorod, Kostroma, Kazan, Perm
and Simbirsk.
MATTOCK (O.E. mattuc, of uncertain origin), a tool having a
double iron head, of which one end is shaped like an adze, and the other
like a pickaxe. The head has a socket in the centre in which the handle is
inserted transversely to the blades. It is used chiefly for grubbing and
rooting among tree stumps in plantations and copses, where the roots are
too close for the use of a spade, or for loosening hard soil.
Page 63
MATTO GROSSO, an inland state of Brazil, bounded N. by
Amazonas and Pará, E. by Goyaz, Minas Geraes, São Paulo and Paraná, S.
by Paraguay and S.W. and W. by Bolivia. It ranks next to Amazonas in size,
its area, which is largely unsettled and unexplored, being 532,370 sq. m.,
and its population only 92,827 in 1890 and 118,025 in 1900. No satisfactory
estimate of its Indian population can be made. The greater part of the state
belongs to the western extension of the Brazilian plateau, across which,
between the 14th and 16th parallels, runs the watershed which separates the
drainage basins of the Amazon and La Plata. This elevated region is known
as the plateau of Matto Grosso, and its elevations so far as known rarely
exceed 3000 ft. The northern slope of this great plateau is drained by the
Araguaya-Tocantins, Xingú, Tapajos and Guaporé-Mamoré-Madeira, which
flow northward, and, except the first, empty into the Amazon; the southern
slope drains southward through a multitude of streams flowing into the
Paraná and Paraguay. The general elevation in the south part of the state is
much lower, and large areas bordering the Paraguay are swampy, partially
submerged plains which the sluggish rivers are unable to drain. The lowland
elevations in this part of the state range from 300 to 400 ft. above sea-level,
the climate is hot, humid and unhealthy, and the conditions for permanent
settlement are apparently unfavourable. On the highlands, however, which
contain extensive open campos, the climate, though dry and hot, is
considered healthy. The basins of the Paraná and Paraguay are separated by
low mountain ranges extending north from the sierras of Paraguay. In the
Amazonas and Pará, E. by Goyaz, Minas Geraes, São Paulo and Paraná, S.
by Paraguay and S.W. and W. by Bolivia. It ranks next to Amazonas in size,
its area, which is largely unsettled and unexplored, being 532,370 sq. m.,
and its population only 92,827 in 1890 and 118,025 in 1900. No satisfactory
estimate of its Indian population can be made. The greater part of the state
belongs to the western extension of the Brazilian plateau, across which,
between the 14th and 16th parallels, runs the watershed which separates the
drainage basins of the Amazon and La Plata. This elevated region is known
as the plateau of Matto Grosso, and its elevations so far as known rarely
exceed 3000 ft. The northern slope of this great plateau is drained by the
Araguaya-Tocantins, Xingú, Tapajos and Guaporé-Mamoré-Madeira, which
flow northward, and, except the first, empty into the Amazon; the southern
slope drains southward through a multitude of streams flowing into the
Paraná and Paraguay. The general elevation in the south part of the state is
much lower, and large areas bordering the Paraguay are swampy, partially
submerged plains which the sluggish rivers are unable to drain. The lowland
elevations in this part of the state range from 300 to 400 ft. above sea-level,
the climate is hot, humid and unhealthy, and the conditions for permanent
settlement are apparently unfavourable. On the highlands, however, which
contain extensive open campos, the climate, though dry and hot, is
considered healthy. The basins of the Paraná and Paraguay are separated by
low mountain ranges extending north from the sierras of Paraguay. In the
Page 64
north, however, the ranges which separate the river valleys are apparently
the remains of the table-land through which deep valleys have been eroded.
The resources of Matto Grosso are practically undeveloped, owing to the
isolated situation of the state, the costs of transportation and the small
population.
The first industry was that of mining, gold having been discovered in the
river valleys on the southern slopes of the plateau, and diamonds on the
head-waters of the Paraguay, about Diamantino and in two or three other
districts. Gold is found chiefly in placers, and in colonial times the output
was large, but the deposits were long ago exhausted and the industry is now
comparatively unimportant. As to other minerals little is definitely known.
Agriculture exists only for the supply of local needs, though tobacco of a
superior quality is grown. Cattle-raising, however, has received some
attention and is the principal industry of the landowners. The forest
products of the state include fine woods, rubber, ipecacuanha, sarsaparilla,
jaborandi, vanilla and copaiba. There is little export, however, the only
means of communication being down the Paraguay and Paraná rivers by
means of subsidized steamers. The capital of the state is Cuyabá, and the
chief commercial town is Corumbá at the head of navigation for the larger
river boats, and 1986 m. from the mouth of the La Plata. Communication
between these two towns is maintained by a line of smaller boats, the
distance being 517 m.
The first permanent settlements in Matto Grosso seem to have been made
in 1718 and 1719, in the first year at Forquilha and in the second at or near
the site of Cuyabá, where rich placer mines had been found. At this time all
this inland region was considered a part of São Paulo, but in 1748 it was
made a separate capitania and was named Matto Grosso (“great woods”).
In 1752 its capital was situated on the right bank of the Guaporé river and
was named Villa Bella da Santissima Trindade de Matto Grosso, but in
the remains of the table-land through which deep valleys have been eroded.
The resources of Matto Grosso are practically undeveloped, owing to the
isolated situation of the state, the costs of transportation and the small
population.
The first industry was that of mining, gold having been discovered in the
river valleys on the southern slopes of the plateau, and diamonds on the
head-waters of the Paraguay, about Diamantino and in two or three other
districts. Gold is found chiefly in placers, and in colonial times the output
was large, but the deposits were long ago exhausted and the industry is now
comparatively unimportant. As to other minerals little is definitely known.
Agriculture exists only for the supply of local needs, though tobacco of a
superior quality is grown. Cattle-raising, however, has received some
attention and is the principal industry of the landowners. The forest
products of the state include fine woods, rubber, ipecacuanha, sarsaparilla,
jaborandi, vanilla and copaiba. There is little export, however, the only
means of communication being down the Paraguay and Paraná rivers by
means of subsidized steamers. The capital of the state is Cuyabá, and the
chief commercial town is Corumbá at the head of navigation for the larger
river boats, and 1986 m. from the mouth of the La Plata. Communication
between these two towns is maintained by a line of smaller boats, the
distance being 517 m.
The first permanent settlements in Matto Grosso seem to have been made
in 1718 and 1719, in the first year at Forquilha and in the second at or near
the site of Cuyabá, where rich placer mines had been found. At this time all
this inland region was considered a part of São Paulo, but in 1748 it was
made a separate capitania and was named Matto Grosso (“great woods”).
In 1752 its capital was situated on the right bank of the Guaporé river and
was named Villa Bella da Santissima Trindade de Matto Grosso, but in
Page 65
1820 the seat of government was removed to Cuyabá and Villa Bella has
fallen into decay. In 1822 Matto Grosso became a province of the empire
and in 1889 a republican state. It was invaded by the Paraguayans in the
war of 1860-65.
MATTOON, a city of Coles county, Illinois, U.S.A., in the east
central part of the state, about 12 m. south-east of Peoria. Pop. (1890),
6833; (1900), 9622, of whom 430 were foreign-born; (1910 census) 11,456.
It is served by the Illinois Central and Cleveland, Cincinnati, Chicago & St
Louis railways, which have repair shops here, and by inter-urban electric
lines. The city has a public library, a Methodist Episcopal Hospital, and an
Old Folks’ Home, the last supported by the Independent Order of Odd
Fellows. Mattoon is an important shipping point for Indian corn and broom
corn, extensively grown in the vicinity, and for fruit and livestock. Among
its manufactures are foundry and machine shop products, stoves and bricks;
in 1905 the factory product was valued at $1,308,781, an increase of 71.2%
over that in 1900. The municipality owns the waterworks and an electric
lighting plant. Mattoon was first settled about 1855, was named in honour
of William Mattoon, an early landowner, was first chartered as a city in
1857, and was reorganized under a general state law in 1879.
fallen into decay. In 1822 Matto Grosso became a province of the empire
and in 1889 a republican state. It was invaded by the Paraguayans in the
war of 1860-65.
MATTOON, a city of Coles county, Illinois, U.S.A., in the east
central part of the state, about 12 m. south-east of Peoria. Pop. (1890),
6833; (1900), 9622, of whom 430 were foreign-born; (1910 census) 11,456.
It is served by the Illinois Central and Cleveland, Cincinnati, Chicago & St
Louis railways, which have repair shops here, and by inter-urban electric
lines. The city has a public library, a Methodist Episcopal Hospital, and an
Old Folks’ Home, the last supported by the Independent Order of Odd
Fellows. Mattoon is an important shipping point for Indian corn and broom
corn, extensively grown in the vicinity, and for fruit and livestock. Among
its manufactures are foundry and machine shop products, stoves and bricks;
in 1905 the factory product was valued at $1,308,781, an increase of 71.2%
over that in 1900. The municipality owns the waterworks and an electric
lighting plant. Mattoon was first settled about 1855, was named in honour
of William Mattoon, an early landowner, was first chartered as a city in
1857, and was reorganized under a general state law in 1879.
Page 66
MATTRESS (O.Fr. materas, mod. matelas; the origin is the Arab.
al-materah, cushion, whence Span. and Port. almadraque, Ital. materasso),
the padded foundation of a bed, formed of canvas or other stout material
stuffed with wool, hair, flock or straw; in the last case it is properly known
as a “palliasse” (Fr. paille, straw; Lat. palea); but this term is often applied
to an under-mattress stuffed with substances other than straw. The padded
mattress on which lay the feather-bed has been replaced by the “wire-
mattress,” a network of wire stretched on a light wooden or iron frame,
which is either a separate structure or a component part of the bedstead
itself. The “wire-mattress” has taken the place of the “spring mattress,” in
which spiral springs support the stuffing. The term “mattress” is used in
engineering for a mat of brushwood, faggots, &c., corded together and used
as a foundation or as surface in the construction of dams, jetties, dikes, &c.
MATURIN, CHARLES ROBERT (1782-1824), Irish
novelist and dramatist, was born in Dublin in 1782. His grandfather, Gabriel
Jasper Maturin, had been Swift’s successor in the deanery of St Patrick.
al-materah, cushion, whence Span. and Port. almadraque, Ital. materasso),
the padded foundation of a bed, formed of canvas or other stout material
stuffed with wool, hair, flock or straw; in the last case it is properly known
as a “palliasse” (Fr. paille, straw; Lat. palea); but this term is often applied
to an under-mattress stuffed with substances other than straw. The padded
mattress on which lay the feather-bed has been replaced by the “wire-
mattress,” a network of wire stretched on a light wooden or iron frame,
which is either a separate structure or a component part of the bedstead
itself. The “wire-mattress” has taken the place of the “spring mattress,” in
which spiral springs support the stuffing. The term “mattress” is used in
engineering for a mat of brushwood, faggots, &c., corded together and used
as a foundation or as surface in the construction of dams, jetties, dikes, &c.
MATURIN, CHARLES ROBERT (1782-1824), Irish
novelist and dramatist, was born in Dublin in 1782. His grandfather, Gabriel
Jasper Maturin, had been Swift’s successor in the deanery of St Patrick.
Page 67
Charles Maturin was educated at Trinity College, Dublin, and became
curate of Loughrea and then of St Peter’s, Dublin. His first novels, The
Fatal Revenge; or, the Family of Montorio (1807), The Wild Irish Boy
(1808), The Milesian Chief (1812), were issued under the pseudonym of
“Dennis Jasper Murphy.” All these were mercilessly ridiculed, but the
irregular power displayed in them attracted the notice of Sir Walter Scott,
who recommended the author to Byron. Through their influence Maturin’s
tragedy of Bertram was produced at Drury Lane in 1816, with Kean and
Miss Kelly in the leading parts. A French version by Charles Nodier and
Baron Taylor was produced in Paris at the Théâtre Favart. Two more
tragedies, Manuel (1817) and Fredolfo (1819), were failures, and his poem
The Universe (1821) fell flat. He wrote three more novels, Women (1818),
Melmoth, the Wanderer (1820), and The Albigenses (1824). Melmoth, which
forms its author’s title to remembrance, is the best of them, and has for hero
a kind of “Wandering Jew.” Honoré de Balzac wrote a sequel to it under the
title of Melmoth réconcilié à l’église (1835). Maturin died in Dublin on the
30th of October 1824.
MATVYEEV, ARTAMON SERGYEEVICH
( -1682), Russian statesman and reformer, was one of the greatest of the
precursors of Peter the Great. His parentage and the date of his birth are
uncertain. Apparently his birth was humble, but when the obscure figure of
the young Artamon emerges into the light of history we find him equipped
at all points with the newest ideas, absolutely free from the worst prejudices
curate of Loughrea and then of St Peter’s, Dublin. His first novels, The
Fatal Revenge; or, the Family of Montorio (1807), The Wild Irish Boy
(1808), The Milesian Chief (1812), were issued under the pseudonym of
“Dennis Jasper Murphy.” All these were mercilessly ridiculed, but the
irregular power displayed in them attracted the notice of Sir Walter Scott,
who recommended the author to Byron. Through their influence Maturin’s
tragedy of Bertram was produced at Drury Lane in 1816, with Kean and
Miss Kelly in the leading parts. A French version by Charles Nodier and
Baron Taylor was produced in Paris at the Théâtre Favart. Two more
tragedies, Manuel (1817) and Fredolfo (1819), were failures, and his poem
The Universe (1821) fell flat. He wrote three more novels, Women (1818),
Melmoth, the Wanderer (1820), and The Albigenses (1824). Melmoth, which
forms its author’s title to remembrance, is the best of them, and has for hero
a kind of “Wandering Jew.” Honoré de Balzac wrote a sequel to it under the
title of Melmoth réconcilié à l’église (1835). Maturin died in Dublin on the
30th of October 1824.
MATVYEEV, ARTAMON SERGYEEVICH
( -1682), Russian statesman and reformer, was one of the greatest of the
precursors of Peter the Great. His parentage and the date of his birth are
uncertain. Apparently his birth was humble, but when the obscure figure of
the young Artamon emerges into the light of history we find him equipped
at all points with the newest ideas, absolutely free from the worst prejudices
Page 68
of his age, a ripe scholar, and even an author of some distinction. In 1671
the tsar Alexius and Artamon were already on intimate terms, and on the
retirement of Orduin-Nashchokin Matvyeev became the tsar’s chief
counsellor. It was at his house, full of all the wondrous, half-forbidden
novelties of the west, that Alexius, after the death of his first consort,
Martha, met Matvyeev’s favourite pupil, the beautiful Natalia Naruishkina,
whom he married on the 21st of January 1672. At the end of the year
Matvyeev was raised to the rank of okolnichy, and on the 1st of September
1674 attained the still higher dignity of boyar. Matvyeev remained
paramount to the end of the reign and introduced play-acting and all sorts of
refining western novelties into Muscovy. The deplorable physical condition
of Alexius’s immediate successor, Theodore III. suggested to Matvyeev the
desirability of elevating to the throne the sturdy little tsarevich Peter, then in
his fourth year. He purchased the allegiance of the stryeltsi, or musketeers,
and then, summoning the boyars of the council, earnestly represented to
them that Theodore, scarce able to live, was surely unable to reign, and
urged the substitution of little Peter. But the reactionary boyars, among
whom were the near kinsmen of Theodore, proclaimed him tsar and
Matvyeev was banished to Pustozersk, in northern Russia, where he
remained till Theodore’s death (April 27, 1682). Immediately afterwards
Peter was proclaimed tsar by the patriarch, and the first ukaz issued in
Peter’s name summoned Matvyeev to return to the capital and act as chief
adviser to the tsaritsa Natalia. He reached Moscow on the 15th of May,
prepared “to lay down his life for the tsar,” and at once proceeded to the
head of the Red Staircase to meet and argue with the assembled stryeltsi,
who had been instigated to rebel by the anti-Petrine faction. He had already
succeeded in partially pacifying them, when one of their colonels began to
abuse the still hesitating and suspicious musketeers. Infuriated, they seized
and flung Matvyeev into the square below, where he was hacked to pieces
by their comrades.
the tsar Alexius and Artamon were already on intimate terms, and on the
retirement of Orduin-Nashchokin Matvyeev became the tsar’s chief
counsellor. It was at his house, full of all the wondrous, half-forbidden
novelties of the west, that Alexius, after the death of his first consort,
Martha, met Matvyeev’s favourite pupil, the beautiful Natalia Naruishkina,
whom he married on the 21st of January 1672. At the end of the year
Matvyeev was raised to the rank of okolnichy, and on the 1st of September
1674 attained the still higher dignity of boyar. Matvyeev remained
paramount to the end of the reign and introduced play-acting and all sorts of
refining western novelties into Muscovy. The deplorable physical condition
of Alexius’s immediate successor, Theodore III. suggested to Matvyeev the
desirability of elevating to the throne the sturdy little tsarevich Peter, then in
his fourth year. He purchased the allegiance of the stryeltsi, or musketeers,
and then, summoning the boyars of the council, earnestly represented to
them that Theodore, scarce able to live, was surely unable to reign, and
urged the substitution of little Peter. But the reactionary boyars, among
whom were the near kinsmen of Theodore, proclaimed him tsar and
Matvyeev was banished to Pustozersk, in northern Russia, where he
remained till Theodore’s death (April 27, 1682). Immediately afterwards
Peter was proclaimed tsar by the patriarch, and the first ukaz issued in
Peter’s name summoned Matvyeev to return to the capital and act as chief
adviser to the tsaritsa Natalia. He reached Moscow on the 15th of May,
prepared “to lay down his life for the tsar,” and at once proceeded to the
head of the Red Staircase to meet and argue with the assembled stryeltsi,
who had been instigated to rebel by the anti-Petrine faction. He had already
succeeded in partially pacifying them, when one of their colonels began to
abuse the still hesitating and suspicious musketeers. Infuriated, they seized
and flung Matvyeev into the square below, where he was hacked to pieces
by their comrades.
Page 69
See R. Nisbet Bain, The First Romanovs (London, 1905); M. P.
Pogodin, The First Seventeen Years of the Life of Peter the Great
(Rus.), (Moscow, 1875); S. M. Solovev, History of Russia (Rus.), (vols.
12, 13, (St Petersburg, 1895, &c.); L. Shehepotev, A. S. Matvyeev as an
Educational and Political Reformer (Rus.), (St Petersburg, 1906).
(R. N. B.)
MAUBEUGE, a town of northern France, in the department of
Nord, situated on both banks of the Sambre, here canalized, 231⁄2 m. by rail
E. by S. of Valenciennes, and about 2 m. from the Belgian frontier. Pop.
(1906), town 13,569, commune 21,520. As a fortress Maubeuge has an old
enceinte of bastion trace which serves as the centre of an important
entrenched camp of 18 m. perimeter, constructed for the most part after the
war of 1870, but since modernized and augmented. The town has a board of
trade arbitration, a communal college, a commercial and industrial school;
and there are important foundries, forges and blast-furnaces, together with
manufactures of machine-tools, porcelain, &c. It is united by electric
tramway with Hautmont (pop. 12,473), also an important metallurgical
centre.
Maubeuge (Malbodium) owes its origin to a double monastery, for men
and women, founded in the 7th century by St Aldegonde relics of whom are
preserved in the church. It subsequently belonged to the territory of
Hainault. It was burnt by Louis XI., by Francis I., and by Henry II., and was
Pogodin, The First Seventeen Years of the Life of Peter the Great
(Rus.), (Moscow, 1875); S. M. Solovev, History of Russia (Rus.), (vols.
12, 13, (St Petersburg, 1895, &c.); L. Shehepotev, A. S. Matvyeev as an
Educational and Political Reformer (Rus.), (St Petersburg, 1906).
(R. N. B.)
MAUBEUGE, a town of northern France, in the department of
Nord, situated on both banks of the Sambre, here canalized, 231⁄2 m. by rail
E. by S. of Valenciennes, and about 2 m. from the Belgian frontier. Pop.
(1906), town 13,569, commune 21,520. As a fortress Maubeuge has an old
enceinte of bastion trace which serves as the centre of an important
entrenched camp of 18 m. perimeter, constructed for the most part after the
war of 1870, but since modernized and augmented. The town has a board of
trade arbitration, a communal college, a commercial and industrial school;
and there are important foundries, forges and blast-furnaces, together with
manufactures of machine-tools, porcelain, &c. It is united by electric
tramway with Hautmont (pop. 12,473), also an important metallurgical
centre.
Maubeuge (Malbodium) owes its origin to a double monastery, for men
and women, founded in the 7th century by St Aldegonde relics of whom are
preserved in the church. It subsequently belonged to the territory of
Hainault. It was burnt by Louis XI., by Francis I., and by Henry II., and was
Page 70
finally assigned to France by the Treaty of Nijmwegen. It was fortified at
Vauban by the command of Louis XIV., who under Turenne first saw
military service there. Besieged in 1793 by Prince Josias of Coburg, it was
relieved by the victory of Wattignies, which is commemorated by a
monument in the town. It was unsuccessfully besieged in 1814, but was
compelled to capitulate, after a vigorous resistance, in the Hundred Days.
MAUCH CHUNK, a borough and the county-seat of Carbon
county, Pennsylvania, U.S.A., on the W. bank of the Lehigh river and on the
Lehigh Coal and Navigation Company’s Canal, 46 m. by rail W.N.W. of
Easton. Pop. (1800), 4101; (1900), 4029 (571 foreign-born); (1910), 3952.
Mauch Chunk is served by the Central of New Jersey railway and, at East
Mauch Chunk, across the river, connected by electric railway, by the Lehigh
Valley railway. The borough lies in the valley of the Lehigh river, along
which runs one of its few streets and in another deeply cut valley at right
angles to the river; through this second valley east and west runs the main
street, on which is an electric railway; parallel to it on the south is High
Street, formerly an Irish settlement; half way up the steep hill, and on the
north at the top of the opposite hill is the ward of Upper Mauch Chunk,
reached by the electric railway. An incline railway, originally used to
transport coal from the mines to the river and named the “Switch-Back,”
now carries tourists up the steep slopes of Mount Pisgah and Mount
Jefferson, to Summit Hill, a rich anthracite coal region, with a famous
“burning mine,” which has been on fire since 1832, and then back. An
Vauban by the command of Louis XIV., who under Turenne first saw
military service there. Besieged in 1793 by Prince Josias of Coburg, it was
relieved by the victory of Wattignies, which is commemorated by a
monument in the town. It was unsuccessfully besieged in 1814, but was
compelled to capitulate, after a vigorous resistance, in the Hundred Days.
MAUCH CHUNK, a borough and the county-seat of Carbon
county, Pennsylvania, U.S.A., on the W. bank of the Lehigh river and on the
Lehigh Coal and Navigation Company’s Canal, 46 m. by rail W.N.W. of
Easton. Pop. (1800), 4101; (1900), 4029 (571 foreign-born); (1910), 3952.
Mauch Chunk is served by the Central of New Jersey railway and, at East
Mauch Chunk, across the river, connected by electric railway, by the Lehigh
Valley railway. The borough lies in the valley of the Lehigh river, along
which runs one of its few streets and in another deeply cut valley at right
angles to the river; through this second valley east and west runs the main
street, on which is an electric railway; parallel to it on the south is High
Street, formerly an Irish settlement; half way up the steep hill, and on the
north at the top of the opposite hill is the ward of Upper Mauch Chunk,
reached by the electric railway. An incline railway, originally used to
transport coal from the mines to the river and named the “Switch-Back,”
now carries tourists up the steep slopes of Mount Pisgah and Mount
Jefferson, to Summit Hill, a rich anthracite coal region, with a famous
“burning mine,” which has been on fire since 1832, and then back. An
Page 71
electric railway to the top of Flagstaff Mountain, built in 1900, was
completed in 1901 to Lehighton, 4 m. south-east of Mauch Chunk, where
coal is mined and silk and stoves are manufactured, and which had a
population in 1900 of 4629, and in 1910 of 5316. Immediately above
Mauch Chunk the river forms a horseshoe; on the opposite side, connected
by a bridge, is the borough of East Mauch Chunk (pop. 1900, 3458; 1910,
3548); and 2 m. up the river is Glen Onoko, with fine falls and cascades.
The principal buildings in Mauch Chunk are the county court house, a
county gaol, a Young Men’s Christian Association building, and the
Dimmick Memorial Library (1890). The borough was long a famous
shipping point for coal. It now has ironworks and foundries, and in East
Mauch Chunk there are silk mills. The name is Indian and means “Bear
Mountain,” this English name being used for a mountain on the east side of
the river. The borough was founded by the Lehigh Coal and Navigation
Company in 1818. This company began in 1827 the operation of the
“Switch-Back,” probably the first railway in the country to be used for
transporting coal. In 1831 the town was opened to individual enterprise, and
in 1850 it was incorporated as a borough. Mauch Chunk was for many
years the home of Asa Packer, the projector and builder of the Lehigh
Valley railroad from Mauch Chunk to Easton.
MAUCHLINE, a town in the division of Kyle, Ayrshire, Scotland.
Pop. (1901), 1767. It lies 8 m. E.S.E. of Kilmarnock and 11 m. E. by N. of
Ayr by the Glasgow and South-Western railway. It is situated on a gentle
completed in 1901 to Lehighton, 4 m. south-east of Mauch Chunk, where
coal is mined and silk and stoves are manufactured, and which had a
population in 1900 of 4629, and in 1910 of 5316. Immediately above
Mauch Chunk the river forms a horseshoe; on the opposite side, connected
by a bridge, is the borough of East Mauch Chunk (pop. 1900, 3458; 1910,
3548); and 2 m. up the river is Glen Onoko, with fine falls and cascades.
The principal buildings in Mauch Chunk are the county court house, a
county gaol, a Young Men’s Christian Association building, and the
Dimmick Memorial Library (1890). The borough was long a famous
shipping point for coal. It now has ironworks and foundries, and in East
Mauch Chunk there are silk mills. The name is Indian and means “Bear
Mountain,” this English name being used for a mountain on the east side of
the river. The borough was founded by the Lehigh Coal and Navigation
Company in 1818. This company began in 1827 the operation of the
“Switch-Back,” probably the first railway in the country to be used for
transporting coal. In 1831 the town was opened to individual enterprise, and
in 1850 it was incorporated as a borough. Mauch Chunk was for many
years the home of Asa Packer, the projector and builder of the Lehigh
Valley railroad from Mauch Chunk to Easton.
MAUCHLINE, a town in the division of Kyle, Ayrshire, Scotland.
Pop. (1901), 1767. It lies 8 m. E.S.E. of Kilmarnock and 11 m. E. by N. of
Ayr by the Glasgow and South-Western railway. It is situated on a gentle
Page 72
slope about 1 m. from the river Ayr, which flows through the south of the
parish of Mauchline. It is noted for its manufacture of snuff-boxes and
knick-knacks in wood, and of curling-stones. There is also some cabinet-
making, besides spinning and weaving, and its horse fairs and cattle
markets have more than local celebrity. The parish church, dating from
1829, stands in the middle of the village, and on the green a monument,
erected in 1830, marks the spot where five Covenanters were killed in 1685.
Robert Burns lived with his brother Gilbert on the farm of Mossgiel, about
a mile to the north, from 1784 to 1788. Mauchline kirkyard was the scene
of the “Holy Fair”; at “Poosie Nansie’s” (Agnes Gibson’s)—still, though
much altered, a popular inn—the “Jolly Beggars” held their high jinks; near
the church (in the poet’s day an old, barn-like structure) was the Whiteford
Arms inn, where on a pane of glass Burns wrote the epitaph on John Dove,
the landlord; “auld Nanse Tinnock’s” house, with the date of 1744 above
the door, nearly faces the entrance to the churchyard; the Rev. William Auld
was minister of Mauchline, and “Holy Willie,” whom the poet scourged in
the celebrated “Prayer,” was one of “Daddy Auld’s” elders; behind the
kirkyard stands the house of Gavin Hamilton, the lawyer and firm friend of
Burns, in which the poet was married. The braes of Ballochmyle, where he
met the heroine of his song, “The Lass o’ Ballochmyle,” lie about a mile to
the south-east. Adjoining them is the considerable manufacturing town of
Catrine (pop. 2340), with cotton factories, bleach fields and brewery, where
Dr Matthew Stewart (1717-1785), the father of Dugald Stewart—had a
mansion, and where there is a big water-wheel said to be inferior in size
only to that of Laxey in the Isle of Man. Barskimming House, 2 m. south by
west of Mauchline, the seat of Lord-President Miller (1717-1789), was
burned down in 1882. Near the confluence of the Fail and the Ayr was the
scene of Burns’s parting with Highland Mary.
parish of Mauchline. It is noted for its manufacture of snuff-boxes and
knick-knacks in wood, and of curling-stones. There is also some cabinet-
making, besides spinning and weaving, and its horse fairs and cattle
markets have more than local celebrity. The parish church, dating from
1829, stands in the middle of the village, and on the green a monument,
erected in 1830, marks the spot where five Covenanters were killed in 1685.
Robert Burns lived with his brother Gilbert on the farm of Mossgiel, about
a mile to the north, from 1784 to 1788. Mauchline kirkyard was the scene
of the “Holy Fair”; at “Poosie Nansie’s” (Agnes Gibson’s)—still, though
much altered, a popular inn—the “Jolly Beggars” held their high jinks; near
the church (in the poet’s day an old, barn-like structure) was the Whiteford
Arms inn, where on a pane of glass Burns wrote the epitaph on John Dove,
the landlord; “auld Nanse Tinnock’s” house, with the date of 1744 above
the door, nearly faces the entrance to the churchyard; the Rev. William Auld
was minister of Mauchline, and “Holy Willie,” whom the poet scourged in
the celebrated “Prayer,” was one of “Daddy Auld’s” elders; behind the
kirkyard stands the house of Gavin Hamilton, the lawyer and firm friend of
Burns, in which the poet was married. The braes of Ballochmyle, where he
met the heroine of his song, “The Lass o’ Ballochmyle,” lie about a mile to
the south-east. Adjoining them is the considerable manufacturing town of
Catrine (pop. 2340), with cotton factories, bleach fields and brewery, where
Dr Matthew Stewart (1717-1785), the father of Dugald Stewart—had a
mansion, and where there is a big water-wheel said to be inferior in size
only to that of Laxey in the Isle of Man. Barskimming House, 2 m. south by
west of Mauchline, the seat of Lord-President Miller (1717-1789), was
burned down in 1882. Near the confluence of the Fail and the Ayr was the
scene of Burns’s parting with Highland Mary.
Page 73
MAUDE, CYRIL (1862- ), English actor, was born in
London and educated at Charterhouse. He began his career as an actor in
1883 in America, and from 1896 to 1905 was co-manager with F. Harrison
of the Haymarket Theatre, London. There he became distinguished for his
quietly humorous acting in many parts. In 1906 he went into management
on his own account, and in 1907 opened his new theatre The Playhouse. In
1888 he married the actress Winifred Emery (b. 1862), who had made her
London début as a child in 1875, and acted with Irving at the Lyceum
between 1881 and 1887. She was a daughter of Samuel Anderson Emery
(1817-1881) and granddaughter of John Emery (1777-1822), both well-
known actors in their day.
MAULE, a coast province of central Chile, bounded N. by Talea, E.
by Linares and Nuble, and S. by Concepción, and lying between the rivers
Maule and Itata, which form its northern and southern boundaries. Pop.
(1895), 119,791; area, 2475 sq. m. Maule is traversed from north to south
by the coast range and its surfaces are much broken. The Buchupureo river
London and educated at Charterhouse. He began his career as an actor in
1883 in America, and from 1896 to 1905 was co-manager with F. Harrison
of the Haymarket Theatre, London. There he became distinguished for his
quietly humorous acting in many parts. In 1906 he went into management
on his own account, and in 1907 opened his new theatre The Playhouse. In
1888 he married the actress Winifred Emery (b. 1862), who had made her
London début as a child in 1875, and acted with Irving at the Lyceum
between 1881 and 1887. She was a daughter of Samuel Anderson Emery
(1817-1881) and granddaughter of John Emery (1777-1822), both well-
known actors in their day.
MAULE, a coast province of central Chile, bounded N. by Talea, E.
by Linares and Nuble, and S. by Concepción, and lying between the rivers
Maule and Itata, which form its northern and southern boundaries. Pop.
(1895), 119,791; area, 2475 sq. m. Maule is traversed from north to south
by the coast range and its surfaces are much broken. The Buchupureo river
Page 74
flows westward across the province. The climate is mild and healthy.
Agriculture and stock-raising are the principal occupations, and hides,
cattle, wheat and timber are exported. Transport facilities are afforded by
the Maule and the Itata, which are navigable, and by a branch of the
government railway from Cauquenes to Parral, an important town of
southern Linares. The provincial capital, Cauquenes (pop., in 1895, 8574;
1902 estimate, 9895), is centrally situated on the Buchupureo river, on the
eastern slopes of the coast cordilleras. The town and port of Constitución
(pop., in 1900, about 7000) on the south bank of the Maule, one mile above
its mouth, was formerly the capital of the province. The port suffers from a
dangerous bar at the mouth of the river, but is connected with Talca by rail
and has a considerable trade.
The Maule river, from which the province takes its name, is of historic
interest because it is said to have marked the southern limits of the Inca
Empire. It rises in the Laguna del Maule, an Andean lake near the Argentine
frontier, 7218 ft. above sea-level, and flows westward about 140 m. to the
Pacific, into which it discharges in 35° 18′ S. The upper part of its drainage
basin, to which the Anuario Hydrografico gives an area of 8000 sq. m.,
contains the volcanoes of San Pedro (11,800 ft.), the Descabezado (12,795
ft.), and others of the same group of lower elevations. The upper course and
tributaries of the Maule, principally in the province of Linares, are largely
used for irrigation.
Agriculture and stock-raising are the principal occupations, and hides,
cattle, wheat and timber are exported. Transport facilities are afforded by
the Maule and the Itata, which are navigable, and by a branch of the
government railway from Cauquenes to Parral, an important town of
southern Linares. The provincial capital, Cauquenes (pop., in 1895, 8574;
1902 estimate, 9895), is centrally situated on the Buchupureo river, on the
eastern slopes of the coast cordilleras. The town and port of Constitución
(pop., in 1900, about 7000) on the south bank of the Maule, one mile above
its mouth, was formerly the capital of the province. The port suffers from a
dangerous bar at the mouth of the river, but is connected with Talca by rail
and has a considerable trade.
The Maule river, from which the province takes its name, is of historic
interest because it is said to have marked the southern limits of the Inca
Empire. It rises in the Laguna del Maule, an Andean lake near the Argentine
frontier, 7218 ft. above sea-level, and flows westward about 140 m. to the
Pacific, into which it discharges in 35° 18′ S. The upper part of its drainage
basin, to which the Anuario Hydrografico gives an area of 8000 sq. m.,
contains the volcanoes of San Pedro (11,800 ft.), the Descabezado (12,795
ft.), and others of the same group of lower elevations. The upper course and
tributaries of the Maule, principally in the province of Linares, are largely
used for irrigation.
Page 75
MAULÉON, SAVARI DE (d. 1236), French soldier, was the
son of Raoul de Mauléon, vicomte de Thouars and lord of Mauléon (now
Châtillon-sur-Sèvre). Having espoused the cause of Arthur of Brittany, he
was captured at Mirebeau (1202), and imprisoned in the château of Corfe.
But John set him at liberty in 1204, gained him to his side and named him
seneschal of Poitou (1205). In 1211 Savari de Mauléon assisted Raymond
VI. count of Toulouse, and with him besieged Simon de Montfort in
Castelnaudary. Philip Augustus bought his services in 1212 and gave him
command of a fleet which was destroyed in the Flemish port of Damme.
Then Mauléon returned to John, whom he aided in his struggle with the
barons in 1215. He was one of those whom John designated on his deathbed
for a council of regency (1216). Then he went to Egypt (1219), and was
present at the taking of Damietta. Returning to Poitou he was a second time
seneschal for the king of England. He defended Saintonge against Louis
VIII. in 1224, but was accused of having given La Rochelle up to the king
of France, and the suspicions of the English again threw him back upon the
French. Louis VIII. then turned over to him the defence of La Rochelle and
the coast of Saintonge. In 1227 he took part in the rising of the barons of
Poitiers and Anjou against the young Louis IX. He enjoyed a certain
reputation for his poems in the langue d’oc.
See Chilhaud-Dumaine, “Savari de Mauléon,” in Positions des
Thèses des élèves de l’École des Chartes (1877); Histoire littéraire de
la France, xviii. 671-682.
son of Raoul de Mauléon, vicomte de Thouars and lord of Mauléon (now
Châtillon-sur-Sèvre). Having espoused the cause of Arthur of Brittany, he
was captured at Mirebeau (1202), and imprisoned in the château of Corfe.
But John set him at liberty in 1204, gained him to his side and named him
seneschal of Poitou (1205). In 1211 Savari de Mauléon assisted Raymond
VI. count of Toulouse, and with him besieged Simon de Montfort in
Castelnaudary. Philip Augustus bought his services in 1212 and gave him
command of a fleet which was destroyed in the Flemish port of Damme.
Then Mauléon returned to John, whom he aided in his struggle with the
barons in 1215. He was one of those whom John designated on his deathbed
for a council of regency (1216). Then he went to Egypt (1219), and was
present at the taking of Damietta. Returning to Poitou he was a second time
seneschal for the king of England. He defended Saintonge against Louis
VIII. in 1224, but was accused of having given La Rochelle up to the king
of France, and the suspicions of the English again threw him back upon the
French. Louis VIII. then turned over to him the defence of La Rochelle and
the coast of Saintonge. In 1227 he took part in the rising of the barons of
Poitiers and Anjou against the young Louis IX. He enjoyed a certain
reputation for his poems in the langue d’oc.
See Chilhaud-Dumaine, “Savari de Mauléon,” in Positions des
Thèses des élèves de l’École des Chartes (1877); Histoire littéraire de
la France, xviii. 671-682.
Page 76
MAULSTICK, or Mahlstick, a stick with a soft leather or padded
head, used by painters to support the hand that holds the brush. The word is
an adaptation of the Dutch maalstok, i.e. the painter’s stick, from malen, to
paint.
MAUNDY THURSDAY (through O.Fr. mandé from Lat.
mandatum, commandment, in allusion to Christ’s words: “A new
commandment give I unto you,” after he had washed the disciples’ feet at
the Last Supper), the Thursday before Easter. Maundy Thursday is
sometimes known as Sheer or Chare Thursday, either in allusion, it is
thought, to the “shearing” of heads and beards in preparation for Easter, or
more probably in the word’s Middle English sense of “pure,” in allusion to
the ablutions of the day. The chief ceremony, as kept from the early middle
ages onwards—the washing of the feet of twelve or more poor men or
beggars—was in the early Church almost unknown. Of Chrysostom and St
Augustine, who both speak of Maundy Thursday as being marked by a
solemn celebration of the Sacrament, the former does not mention the foot-
washing, and the latter merely alludes to it. Perhaps an indication of it may
be discerned as early as the 4th century in a custom, current in Spain,
northern Italy and elsewhere, of washing the feet of the catechumens
towards the end of Lent before their baptism. It was not, however,
universal, and in the 48th canon of the synod of Elvira (a.d. 306) it is
expressly prohibited (cf. Corp. Jur. Can., c. 104, caus. i. qu. 1). From the
4th century ceremonial foot-washing became yearly more common, till it
head, used by painters to support the hand that holds the brush. The word is
an adaptation of the Dutch maalstok, i.e. the painter’s stick, from malen, to
paint.
MAUNDY THURSDAY (through O.Fr. mandé from Lat.
mandatum, commandment, in allusion to Christ’s words: “A new
commandment give I unto you,” after he had washed the disciples’ feet at
the Last Supper), the Thursday before Easter. Maundy Thursday is
sometimes known as Sheer or Chare Thursday, either in allusion, it is
thought, to the “shearing” of heads and beards in preparation for Easter, or
more probably in the word’s Middle English sense of “pure,” in allusion to
the ablutions of the day. The chief ceremony, as kept from the early middle
ages onwards—the washing of the feet of twelve or more poor men or
beggars—was in the early Church almost unknown. Of Chrysostom and St
Augustine, who both speak of Maundy Thursday as being marked by a
solemn celebration of the Sacrament, the former does not mention the foot-
washing, and the latter merely alludes to it. Perhaps an indication of it may
be discerned as early as the 4th century in a custom, current in Spain,
northern Italy and elsewhere, of washing the feet of the catechumens
towards the end of Lent before their baptism. It was not, however,
universal, and in the 48th canon of the synod of Elvira (a.d. 306) it is
expressly prohibited (cf. Corp. Jur. Can., c. 104, caus. i. qu. 1). From the
4th century ceremonial foot-washing became yearly more common, till it
Page 77
was regarded as a necessary rite, to be performed by the pope, all Catholic
sovereigns, prelates, priests and nobles. In England the king washed the feet
of as many poor men as he was years old, and then distributed to them
meat, money and clothes. At Durham Cathedral, until the 16th century,
every charity-boy had a monk to wash his feet. At Peterborough Abbey, in
1530, Wolsey made “his maund in Our Lady’s Chapel, having fifty-nine
poor men whose feet he washed and kissed; and after he had wiped them he
gave every of the said poor men twelve pence in money, three ells of good
canvas to make them shirts, a pair of new shoes, a cast of red herrings and
three white herrings.” Queen Elizabeth performed the ceremony, the
paupers’ feet, however, being first washed by the yeomen of the laundry
with warm water and sweet herbs. James II. was the last English monarch to
perform the rite. William III. delegated the washing to his almoner, and this
was usual until the middle of the 18th century. Since 1754 the foot-washing
has been abandoned, and the ceremony now consists of the presentation of
Maundy money, officially called Maundy Pennies. These were first coined
in the reign of Charles II. They come straight from the Mint, and have their
edges unmilled. The service which formerly took place in the Chapel Royal,
Whitehall, is now held in Westminster Abbey. A procession is formed in the
nave, consisting of the lord high almoner representing the sovereign, the
clergy and the yeomen of the guard, the latter carrying white and red purses
in baskets. The clothes formerly given are now commuted for in cash. The
full ritual is gone through by the Roman Catholic archbishop of
Westminster, and abroad it survives in all Catholic countries, a notable
example being that of the Austrian emperor. In the Greek Church the rite
survives notably at Moscow, St Petersburg and Constantinople. It is on
Maundy Thursday that in the Church of Rome the sacred oil is blessed, and
the chrism prepared according to an elaborate ritual which is given in the
Pontificale.
sovereigns, prelates, priests and nobles. In England the king washed the feet
of as many poor men as he was years old, and then distributed to them
meat, money and clothes. At Durham Cathedral, until the 16th century,
every charity-boy had a monk to wash his feet. At Peterborough Abbey, in
1530, Wolsey made “his maund in Our Lady’s Chapel, having fifty-nine
poor men whose feet he washed and kissed; and after he had wiped them he
gave every of the said poor men twelve pence in money, three ells of good
canvas to make them shirts, a pair of new shoes, a cast of red herrings and
three white herrings.” Queen Elizabeth performed the ceremony, the
paupers’ feet, however, being first washed by the yeomen of the laundry
with warm water and sweet herbs. James II. was the last English monarch to
perform the rite. William III. delegated the washing to his almoner, and this
was usual until the middle of the 18th century. Since 1754 the foot-washing
has been abandoned, and the ceremony now consists of the presentation of
Maundy money, officially called Maundy Pennies. These were first coined
in the reign of Charles II. They come straight from the Mint, and have their
edges unmilled. The service which formerly took place in the Chapel Royal,
Whitehall, is now held in Westminster Abbey. A procession is formed in the
nave, consisting of the lord high almoner representing the sovereign, the
clergy and the yeomen of the guard, the latter carrying white and red purses
in baskets. The clothes formerly given are now commuted for in cash. The
full ritual is gone through by the Roman Catholic archbishop of
Westminster, and abroad it survives in all Catholic countries, a notable
example being that of the Austrian emperor. In the Greek Church the rite
survives notably at Moscow, St Petersburg and Constantinople. It is on
Maundy Thursday that in the Church of Rome the sacred oil is blessed, and
the chrism prepared according to an elaborate ritual which is given in the
Pontificale.
Page 78
MAUPASSANT, HENRI RENÉ ALBERT GUY
DE (1850-1893), French novelist and poet, was born at the Château of
Miromesnil in the department of Seine-Inférieure on the 5th August 1850.
His grandfather, a landed proprietor of a good Lorraine family, owned an
estate at Neuville-Champ-d’Oisel near Rouen, and bequeathed a moderate
fortune to his son, a Paris stockbroker, who married Mademoiselle Laure
Lepoitevin. Maupassant was educated at Yvetot and at the Rouen lycée. A
copy of verses entitled Le Dieu créateur, written during his year of
philosophy, has been preserved and printed. He entered the ministry of
marine, and was promoted by M. Bardoux to the Cabinet de l’Instruction
publique. A pleasant legend says that, in a report by his official chief,
Maupassant is mentioned as not reaching the standard of the department in
the matter of style. He may very well have been an unsatisfactory clerk, as
he divided his time between rowing expeditions and attending the literary
gatherings at the house of Gustave Flaubert, who was not, as he is often
alleged to be, connected with Maupassant by any blood tie. Flaubert was
not his uncle, nor his cousin, nor even his godfather, but merely an old
friend of Madame de Maupassant, whom he had known from childhood. At
the literary meetings Maupassant seldom shared in the conversation. Upon
those who met him—Tourgenieff, Alphonse Daudet, Catulle Mendès, José-
Maria de Heredia and Émile Zola—he left the impression of a simple young
athlete. Even Flaubert, to whom Maupassant submitted some sketches, was
not greatly struck by their talent, though he encouraged the youth to
DE (1850-1893), French novelist and poet, was born at the Château of
Miromesnil in the department of Seine-Inférieure on the 5th August 1850.
His grandfather, a landed proprietor of a good Lorraine family, owned an
estate at Neuville-Champ-d’Oisel near Rouen, and bequeathed a moderate
fortune to his son, a Paris stockbroker, who married Mademoiselle Laure
Lepoitevin. Maupassant was educated at Yvetot and at the Rouen lycée. A
copy of verses entitled Le Dieu créateur, written during his year of
philosophy, has been preserved and printed. He entered the ministry of
marine, and was promoted by M. Bardoux to the Cabinet de l’Instruction
publique. A pleasant legend says that, in a report by his official chief,
Maupassant is mentioned as not reaching the standard of the department in
the matter of style. He may very well have been an unsatisfactory clerk, as
he divided his time between rowing expeditions and attending the literary
gatherings at the house of Gustave Flaubert, who was not, as he is often
alleged to be, connected with Maupassant by any blood tie. Flaubert was
not his uncle, nor his cousin, nor even his godfather, but merely an old
friend of Madame de Maupassant, whom he had known from childhood. At
the literary meetings Maupassant seldom shared in the conversation. Upon
those who met him—Tourgenieff, Alphonse Daudet, Catulle Mendès, José-
Maria de Heredia and Émile Zola—he left the impression of a simple young
athlete. Even Flaubert, to whom Maupassant submitted some sketches, was
not greatly struck by their talent, though he encouraged the youth to
Page 79
persevere. Maupassant’s first essay was a dramatic piece twice given at
Étretat in 1873 before an audience which included Tourgenieff, Flaubert
and Meilhac. In this indecorous performance, of which nothing more is
heard, Maupassant played the part of a woman. During the next seven years
he served a severe apprenticeship to Flaubert, who by this time realized his
pupil’s exceptional gifts. In 1880 Maupassant published a volume of poems,
Des Vers, against which the public prosecutor of Etampes took proceedings
that were finally withdrawn through the influence of the senator Cordier.
From Flaubert, who had himself been prosecuted for his first book,
Madame Bovary, there came a letter congratulating the poet on the
similarity between their first literary experiences. Des Vers is an extremely
interesting experiment, which shows Maupassant to us still hesitating in his
choice of a medium; but he recognized that it was not wholly satisfactory,
and that its chief deficiency—the absence of verbal melody—was fatal.
Later in the same year he contributed to the Soirées de Médan, a collection
of short stories by MM. Zola, J.-K. Huysmans, Henry Céard, Léon
Hennique and Paul Alexis; and in Boule de suif the young unknown author
revealed himself to his amazed collaborators and to the public as an
admirable writer of prose and a consummate master of the conte. There is
perhaps no other instance in modern literary history of a writer beginning,
as a fully equipped artist, with a genuine masterpiece. This early success
was quickly followed by another. The volume entitled La Maison Tellier
(1881) confirmed the first impression, and vanquished even those who were
repelled by the author’s choice of subjects. In Mademoiselle Fifi (1883) he
repeated his previous triumphs as a conteur, and in this same year he, for
the first time, attempted to write on a larger scale. Choosing to portray the
life of a blameless girl, unfortunate in her marriage, unfortunate in her son,
consistently unfortunate in every circumstance of existence, he leaves her,
ruined and prematurely old, clinging to the tragic hope, which time, as one
feels, will belie, that she may find happiness in her grandson. This picture
Étretat in 1873 before an audience which included Tourgenieff, Flaubert
and Meilhac. In this indecorous performance, of which nothing more is
heard, Maupassant played the part of a woman. During the next seven years
he served a severe apprenticeship to Flaubert, who by this time realized his
pupil’s exceptional gifts. In 1880 Maupassant published a volume of poems,
Des Vers, against which the public prosecutor of Etampes took proceedings
that were finally withdrawn through the influence of the senator Cordier.
From Flaubert, who had himself been prosecuted for his first book,
Madame Bovary, there came a letter congratulating the poet on the
similarity between their first literary experiences. Des Vers is an extremely
interesting experiment, which shows Maupassant to us still hesitating in his
choice of a medium; but he recognized that it was not wholly satisfactory,
and that its chief deficiency—the absence of verbal melody—was fatal.
Later in the same year he contributed to the Soirées de Médan, a collection
of short stories by MM. Zola, J.-K. Huysmans, Henry Céard, Léon
Hennique and Paul Alexis; and in Boule de suif the young unknown author
revealed himself to his amazed collaborators and to the public as an
admirable writer of prose and a consummate master of the conte. There is
perhaps no other instance in modern literary history of a writer beginning,
as a fully equipped artist, with a genuine masterpiece. This early success
was quickly followed by another. The volume entitled La Maison Tellier
(1881) confirmed the first impression, and vanquished even those who were
repelled by the author’s choice of subjects. In Mademoiselle Fifi (1883) he
repeated his previous triumphs as a conteur, and in this same year he, for
the first time, attempted to write on a larger scale. Choosing to portray the
life of a blameless girl, unfortunate in her marriage, unfortunate in her son,
consistently unfortunate in every circumstance of existence, he leaves her,
ruined and prematurely old, clinging to the tragic hope, which time, as one
feels, will belie, that she may find happiness in her grandson. This picture
Page 80
of an average woman undergoing the constant agony of disillusion
Maupassant calls Une Vie (1883), and as in modern literature there is no
finer example of cruel observation, so there is no sadder book than this,
while the effect of extreme truthfulness which it conveys justifies its sub-
title—L’Humble vérité. Certain passages of Une Vie are of such a character
that the sale of the volume at railway bookstalls was forbidden throughout
France. The matter was brought before the chamber of deputies, with the
result of drawing still more attention to the book, and of advertising the
Contes de la bécasse (1883), a collection of stories as improper as they are
clever. Au soleil (1884), a book of travels which has the eminent qualities of
lucid observation and exact description, was less read than Clair de lune,
Miss Harriet, Les Sœurs Rondoli and Yvette, all published in 1883-1884
when Maupassant’s powers were at their highest level. Three further
collections of short tales, entitled Contes et nouvelles, Monsieur Parent, and
Contes du jour et de la nuit, issued in 1885, proved that while the author’s
vision was as incomparable as ever, his fecundity had not improved his
impeccable form. To 1885 also belongs an elaborate novel, Bel-ami, the
cynical history of a particularly detestable, brutal scoundrel who makes his
way in the world by means of his handsome face. Maupassant is here no
less vivid in realizing his literary men, financiers and frivolous women than
in dealing with his favourite peasants, boors and servants, to whom he
returned in Toine (1886) and in La Petite roque (1886). About this time
appeared the first symptoms of the malady which destroyed him; he wrote
less, and though the novel Mont-Oriol (1887) shows him apparently in
undiminished possession of his faculty, Le Horla (1887) suggests that he
was already subject to alarming hallucinations. Restored to some extent by
a sea-voyage, recorded in Sur l’eau (1888), he went back to short stories in
Le Rosier de Madame Husson (1888), a burst of Rabelaisian humour equal
to anything he had ever written. His novels Pierre et Jean (1888), Fort
comme la mort (1889), and Notre cœur (1890) are penetrating studies
Maupassant calls Une Vie (1883), and as in modern literature there is no
finer example of cruel observation, so there is no sadder book than this,
while the effect of extreme truthfulness which it conveys justifies its sub-
title—L’Humble vérité. Certain passages of Une Vie are of such a character
that the sale of the volume at railway bookstalls was forbidden throughout
France. The matter was brought before the chamber of deputies, with the
result of drawing still more attention to the book, and of advertising the
Contes de la bécasse (1883), a collection of stories as improper as they are
clever. Au soleil (1884), a book of travels which has the eminent qualities of
lucid observation and exact description, was less read than Clair de lune,
Miss Harriet, Les Sœurs Rondoli and Yvette, all published in 1883-1884
when Maupassant’s powers were at their highest level. Three further
collections of short tales, entitled Contes et nouvelles, Monsieur Parent, and
Contes du jour et de la nuit, issued in 1885, proved that while the author’s
vision was as incomparable as ever, his fecundity had not improved his
impeccable form. To 1885 also belongs an elaborate novel, Bel-ami, the
cynical history of a particularly detestable, brutal scoundrel who makes his
way in the world by means of his handsome face. Maupassant is here no
less vivid in realizing his literary men, financiers and frivolous women than
in dealing with his favourite peasants, boors and servants, to whom he
returned in Toine (1886) and in La Petite roque (1886). About this time
appeared the first symptoms of the malady which destroyed him; he wrote
less, and though the novel Mont-Oriol (1887) shows him apparently in
undiminished possession of his faculty, Le Horla (1887) suggests that he
was already subject to alarming hallucinations. Restored to some extent by
a sea-voyage, recorded in Sur l’eau (1888), he went back to short stories in
Le Rosier de Madame Husson (1888), a burst of Rabelaisian humour equal
to anything he had ever written. His novels Pierre et Jean (1888), Fort
comme la mort (1889), and Notre cœur (1890) are penetrating studies
Page 81
touched with a profounder sympathy than had hitherto distinguished him;
and this softening into pity for the tragedy of life is deepened in some of the
tales included in Inutile beauté (1890). One of these, Le Champ d’Oliviers,
is an unsurpassable example of poignant, emotional narrative. With La Vie
errante (1890), a volume of travels, Maupassant’s career practically closed.
Musotte, a theatrical piece written in collaboration with M. Jacques
Normand, was published in 1891. By this time inherited nervous maladies,
aggravated by excessive physical exercises and by the imprudent use of
drugs, had undermined his constitution. He began to take an interest in
religious problems, and for a while made the Imitation his handbook; but
his misanthropy deepened, and he suffered from curious delusions as to his
wealth and rank. A victim of general paralysis, of which La Folie des
grandeurs was one of the symptoms, he drank the waters at Aix-les-Bains
during the summer of 1891, and retired to Cannes, where he purposed
passing the winter. The singularities of conduct which had been observed at
Aix-les-Bains grew more and more marked. Maupassant’s reason slowly
gave way. On the 6th of January 1892 he attempted suicide, and was
removed to Paris, where he died in the most painful circumstances on the
6th of July 1893. He is buried in the cemetery of Montparnasse. The
opening chapters of two projected novels, L’Angélus and L’Ame étrangère,
were found among his papers; these, with La Paix du ménage, a comedy in
two acts, and two collections of tales, Le Père Milon (1898) and Le
Colporteur (1899), have been published posthumously. A correspondence,
called Amitié amoureuse (1897), and dedicated to his mother, is probably
unauthentic. Among the prefaces which he wrote for the works of others,
only one—an introduction to a French prose version of Mr Swinburne’s
Poems and Ballads—is likely to interest English readers.
Maupassant began as a follower of Flaubert and of M. Zola, but,
whatever the masters may have called themselves, they both remained
essentially romantiques. The pupil is the last of the “naturalists”: he even
and this softening into pity for the tragedy of life is deepened in some of the
tales included in Inutile beauté (1890). One of these, Le Champ d’Oliviers,
is an unsurpassable example of poignant, emotional narrative. With La Vie
errante (1890), a volume of travels, Maupassant’s career practically closed.
Musotte, a theatrical piece written in collaboration with M. Jacques
Normand, was published in 1891. By this time inherited nervous maladies,
aggravated by excessive physical exercises and by the imprudent use of
drugs, had undermined his constitution. He began to take an interest in
religious problems, and for a while made the Imitation his handbook; but
his misanthropy deepened, and he suffered from curious delusions as to his
wealth and rank. A victim of general paralysis, of which La Folie des
grandeurs was one of the symptoms, he drank the waters at Aix-les-Bains
during the summer of 1891, and retired to Cannes, where he purposed
passing the winter. The singularities of conduct which had been observed at
Aix-les-Bains grew more and more marked. Maupassant’s reason slowly
gave way. On the 6th of January 1892 he attempted suicide, and was
removed to Paris, where he died in the most painful circumstances on the
6th of July 1893. He is buried in the cemetery of Montparnasse. The
opening chapters of two projected novels, L’Angélus and L’Ame étrangère,
were found among his papers; these, with La Paix du ménage, a comedy in
two acts, and two collections of tales, Le Père Milon (1898) and Le
Colporteur (1899), have been published posthumously. A correspondence,
called Amitié amoureuse (1897), and dedicated to his mother, is probably
unauthentic. Among the prefaces which he wrote for the works of others,
only one—an introduction to a French prose version of Mr Swinburne’s
Poems and Ballads—is likely to interest English readers.
Maupassant began as a follower of Flaubert and of M. Zola, but,
whatever the masters may have called themselves, they both remained
essentially romantiques. The pupil is the last of the “naturalists”: he even
Page 82
destroyed naturalism, since he did all that can be done in that direction. He
had no psychology, no theories of art, no moral or strong social prejudices,
no disturbing imagination, no wealth of perplexing ideas. It is no paradox to
say that his marked limitations made him the incomparable artist that he
was. Undisturbed by any external influence, his marvellous vision enabled
him to become a supreme observer, and, given his literary sense, the rest
was simple. He prided himself in having no invention; he described nothing
that he had not seen. The peasants whom he had known as a boy figure in a
score of tales; what he saw in Government offices is set down in
L’Héritage; from Algiers he gathers the material for Maroca; he drinks the
waters and builds up Mont-Oriol; he enters journalism, constructs Bel-ami,
and, for the sake of precision, makes his brother, Hervé de Maupassant, sit
for the infamous hero’s portrait; he sees fashionable society, and, though it
wearied him intensely, he transcribes its life in Fort comme la mort and
Notre cœur. Fundamentally he finds all men alike. In every grade he finds
the same ferocious, cunning, animal instincts at work: it is not a gay world,
but he knows no other; he is possessed by the dread of growing old, of
ceasing to enjoy; the horror of death haunts him like a spectre. It is an
extremely simple outlook. Maupassant does not prefer good to bad, one
man to another; he never pauses to argue about the meaning of life, a
senseless thing which has the one advantage of yielding materials for art;
his one aim is to discover the hidden aspect of visible things, to relate what
he has observed, to give an objective rendering of it, and he has seen so
intensely and so serenely that he is the most exact transcriber in literature.
And as the substance is, so is the form: his style is exceedingly simple and
exceedingly strong; he uses no rare or superfluous word, and is content to
use the humblest word if only it conveys the exact picture of the thing seen.
In ten years he produced some thirty volumes. With the exception of Pierre
et Jean, his novels, excellent as they are, scarcely represent him at his best,
and of over two hundred contes a proportion must be rejected. But enough
had no psychology, no theories of art, no moral or strong social prejudices,
no disturbing imagination, no wealth of perplexing ideas. It is no paradox to
say that his marked limitations made him the incomparable artist that he
was. Undisturbed by any external influence, his marvellous vision enabled
him to become a supreme observer, and, given his literary sense, the rest
was simple. He prided himself in having no invention; he described nothing
that he had not seen. The peasants whom he had known as a boy figure in a
score of tales; what he saw in Government offices is set down in
L’Héritage; from Algiers he gathers the material for Maroca; he drinks the
waters and builds up Mont-Oriol; he enters journalism, constructs Bel-ami,
and, for the sake of precision, makes his brother, Hervé de Maupassant, sit
for the infamous hero’s portrait; he sees fashionable society, and, though it
wearied him intensely, he transcribes its life in Fort comme la mort and
Notre cœur. Fundamentally he finds all men alike. In every grade he finds
the same ferocious, cunning, animal instincts at work: it is not a gay world,
but he knows no other; he is possessed by the dread of growing old, of
ceasing to enjoy; the horror of death haunts him like a spectre. It is an
extremely simple outlook. Maupassant does not prefer good to bad, one
man to another; he never pauses to argue about the meaning of life, a
senseless thing which has the one advantage of yielding materials for art;
his one aim is to discover the hidden aspect of visible things, to relate what
he has observed, to give an objective rendering of it, and he has seen so
intensely and so serenely that he is the most exact transcriber in literature.
And as the substance is, so is the form: his style is exceedingly simple and
exceedingly strong; he uses no rare or superfluous word, and is content to
use the humblest word if only it conveys the exact picture of the thing seen.
In ten years he produced some thirty volumes. With the exception of Pierre
et Jean, his novels, excellent as they are, scarcely represent him at his best,
and of over two hundred contes a proportion must be rejected. But enough
Page 83
will remain to vindicate his claim to a permanent place in literature as an
unmatched observer and the most perfect master of the short story.
See also F. Brunetière, Le Roman naturaliste (1883); T. Lemaître,
Les Contemporains (vols. i. v. vi.); R. Doumic, Ecrivains d’aujourd’hui
(1894); an introduction by Henry James to The Odd Number ... (1891);
a critical preface by the earl of Crewe to Pierre and Jean (1902); A.
Symons, Studies in Prose and Verse (1904). There are many references
to Maupassant in the Journal des Goncourt, and some correspondence
with Marie Bashkirtseff was printed with Further Memoirs of that lady
in 1901.
(J. F. K.)
MAUPEOU, RENÉ NICOLAS CHARLES
AUGUSTIN (1714-1792), chancellor of France, was born on the 25th
of February 1714, being the eldest son of René Charles de Maupeou (1688-
1775), who was president of the parlement of Paris from 1743 to 1757. He
married in 1744 a rich heiress, Anne de Roncherolles, a cousin of Madame
d’Épinay. Entering public life, he was his father’s right hand in the conflicts
between the parlement and Christophe de Beaumont, archbishop of Paris,
who was supported by the court. Between 1763 and 1768, dates which
cover the revision of the case of Jean Calas and the trial of the comte de
Lally, Maupeou was himself president of the parlement. In 1768, through
the protection of Choiseul, whose fall two years later was in large measure
his work, he became chancellor in succession to his father, who had held the
unmatched observer and the most perfect master of the short story.
See also F. Brunetière, Le Roman naturaliste (1883); T. Lemaître,
Les Contemporains (vols. i. v. vi.); R. Doumic, Ecrivains d’aujourd’hui
(1894); an introduction by Henry James to The Odd Number ... (1891);
a critical preface by the earl of Crewe to Pierre and Jean (1902); A.
Symons, Studies in Prose and Verse (1904). There are many references
to Maupassant in the Journal des Goncourt, and some correspondence
with Marie Bashkirtseff was printed with Further Memoirs of that lady
in 1901.
(J. F. K.)
MAUPEOU, RENÉ NICOLAS CHARLES
AUGUSTIN (1714-1792), chancellor of France, was born on the 25th
of February 1714, being the eldest son of René Charles de Maupeou (1688-
1775), who was president of the parlement of Paris from 1743 to 1757. He
married in 1744 a rich heiress, Anne de Roncherolles, a cousin of Madame
d’Épinay. Entering public life, he was his father’s right hand in the conflicts
between the parlement and Christophe de Beaumont, archbishop of Paris,
who was supported by the court. Between 1763 and 1768, dates which
cover the revision of the case of Jean Calas and the trial of the comte de
Lally, Maupeou was himself president of the parlement. In 1768, through
the protection of Choiseul, whose fall two years later was in large measure
his work, he became chancellor in succession to his father, who had held the
Page 84
office for a few days only. He determined to support the royal authority
against the parlement, which in league with the provincial magistratures
was seeking to arrogate to itself the functions of the states-general. He
allied himself with the duc d’Aiguillon and Madame du Barry, and secured
for a creature of his own, the Abbé Terrai, the office of comptroller-general.
The struggle came over the trial of the case of the duc d’Aiguillon, ex-
governor of Brittany, and of La Chalotais, procureur-général of the
province, who had been imprisoned by the governor for accusations against
his administration. When the parlement showed signs of hostility against
Aiguillon, Maupeou read letters patent from Louis XV. annulling the
proceedings. Louis replied to remonstrances from the parlement by a lit de
justice, in which he demanded the surrender of the minutes of procedure.
On the 27th of November 1770 appeared the Édit de règlement et de
discipline, which was promulgated by the chancellor, forbidding the union
of the various branches of the parlement and correspondence with the
provincial magistratures. It also made a strike on the part of the parlement
punishable by confiscation of goods, and forbade further obstruction to the
registration of royal decrees after the royal reply had been given to a first
remonstrance. This edict the magistrates refused to register, and it was
registered in a lit de justice held at Versailles on the 7th of December,
whereupon the parlement suspended its functions. After five summonses to
return to their duties, the magistrates were surprised individually on the
night of the 19th of January 1771 by musketeers, who required them to sign
yes or no to a further request to return. Thirty-eight magistrates gave an
affirmative answer, but on the exile of their former colleagues by lettres de
cachet they retracted, and were also exiled. Maupeou installed the council
of state to administer justice pending the establishment of six superior
courts in the provinces, and of a new parlement in Paris. The cour des aides
was next suppressed.
against the parlement, which in league with the provincial magistratures
was seeking to arrogate to itself the functions of the states-general. He
allied himself with the duc d’Aiguillon and Madame du Barry, and secured
for a creature of his own, the Abbé Terrai, the office of comptroller-general.
The struggle came over the trial of the case of the duc d’Aiguillon, ex-
governor of Brittany, and of La Chalotais, procureur-général of the
province, who had been imprisoned by the governor for accusations against
his administration. When the parlement showed signs of hostility against
Aiguillon, Maupeou read letters patent from Louis XV. annulling the
proceedings. Louis replied to remonstrances from the parlement by a lit de
justice, in which he demanded the surrender of the minutes of procedure.
On the 27th of November 1770 appeared the Édit de règlement et de
discipline, which was promulgated by the chancellor, forbidding the union
of the various branches of the parlement and correspondence with the
provincial magistratures. It also made a strike on the part of the parlement
punishable by confiscation of goods, and forbade further obstruction to the
registration of royal decrees after the royal reply had been given to a first
remonstrance. This edict the magistrates refused to register, and it was
registered in a lit de justice held at Versailles on the 7th of December,
whereupon the parlement suspended its functions. After five summonses to
return to their duties, the magistrates were surprised individually on the
night of the 19th of January 1771 by musketeers, who required them to sign
yes or no to a further request to return. Thirty-eight magistrates gave an
affirmative answer, but on the exile of their former colleagues by lettres de
cachet they retracted, and were also exiled. Maupeou installed the council
of state to administer justice pending the establishment of six superior
courts in the provinces, and of a new parlement in Paris. The cour des aides
was next suppressed.
Page 85
Voltaire praised this revolution, applauding the suppression of the old
hereditary magistrature, but in general Maupeou’s policy was regarded as
the triumph of tyranny. The remonstrances of the princes, of the nobles, and
of the minor courts, were met by exile and suppression, but by the end of
1771 the new system was established, and the Bar, which had offered a
passive resistance, recommenced to plead. But the death of Louis XV. in
May 1774 ruined the chancellor. The restoration of the parlements was
followed by a renewal of the quarrels between the new king and the
magistrature. Maupeou and Terrai were replaced by Malesherbes and
Turgot. Maupeou lived in retreat until his death at Thuit on the 29th of July
1792, having lived to see the overthrow of the ancien régime. His work, in
so far as it was directed towards the separation of the judicial and political
functions and to the reform of the abuses attaching to a hereditary
magistrature, was subsequently endorsed by the Revolution; but no
justification of his violent methods or defence of his intriguing and
avaricious character is possible. He aimed at securing absolute power for
Louis XV., but his action was in reality a serious blow to the monarchy.
The chief authority for the administration of Maupeou is the compte
rendu in his own justification presented by him to Louis XVI. in 1789,
which included a dossier of his speeches and edicts, and is preserved in
the Bibliothèque nationale. These documents, in the hands of his
former secretary, C. F. Lebrun, duc de Plaisance, formed the basis of
the judicial system of France as established under the consulate (cf. C.
F. Lebrun, Opinions, rapports et choix d’écrits politiques, published
posthumously in 1829). See further Maupeouana (6 vols., Paris, 1775),
which contains the pamphlets directed against him; Journal hist. de la
révolution opérée ... par M. de Maupeou (7 vols., 1775); the official
correspondence of Mercy-Argenteau, the letters of Mme d’Épinay; and
Jules Flammermont, Le Chancelier Maupeou et les parlements (1883).
hereditary magistrature, but in general Maupeou’s policy was regarded as
the triumph of tyranny. The remonstrances of the princes, of the nobles, and
of the minor courts, were met by exile and suppression, but by the end of
1771 the new system was established, and the Bar, which had offered a
passive resistance, recommenced to plead. But the death of Louis XV. in
May 1774 ruined the chancellor. The restoration of the parlements was
followed by a renewal of the quarrels between the new king and the
magistrature. Maupeou and Terrai were replaced by Malesherbes and
Turgot. Maupeou lived in retreat until his death at Thuit on the 29th of July
1792, having lived to see the overthrow of the ancien régime. His work, in
so far as it was directed towards the separation of the judicial and political
functions and to the reform of the abuses attaching to a hereditary
magistrature, was subsequently endorsed by the Revolution; but no
justification of his violent methods or defence of his intriguing and
avaricious character is possible. He aimed at securing absolute power for
Louis XV., but his action was in reality a serious blow to the monarchy.
The chief authority for the administration of Maupeou is the compte
rendu in his own justification presented by him to Louis XVI. in 1789,
which included a dossier of his speeches and edicts, and is preserved in
the Bibliothèque nationale. These documents, in the hands of his
former secretary, C. F. Lebrun, duc de Plaisance, formed the basis of
the judicial system of France as established under the consulate (cf. C.
F. Lebrun, Opinions, rapports et choix d’écrits politiques, published
posthumously in 1829). See further Maupeouana (6 vols., Paris, 1775),
which contains the pamphlets directed against him; Journal hist. de la
révolution opérée ... par M. de Maupeou (7 vols., 1775); the official
correspondence of Mercy-Argenteau, the letters of Mme d’Épinay; and
Jules Flammermont, Le Chancelier Maupeou et les parlements (1883).
Page 86
MAUPERTUIS, PIERRE LOUIS MOREAU DE
(1698-1759), French mathematician and astronomer, was born at St Malo
on the 17th of July 1698. When twenty years of age he entered the army,
becoming lieutenant in a regiment of cavalry, and employing his leisure on
mathematical studies. After five years he quitted the army and was admitted
in 1723 a member of the Academy of Sciences. In 1728 he visited London,
and was elected a fellow of the Royal Society. In 1736 he acted as chief of
the expedition sent by Louis XV. into Lapland to measure the length of a
degree of the meridian (see Earth, Figure of), and on his return home he
became a member of almost all the scientific societies of Europe. In 1740
Maupertuis went to Berlin on the invitation of the king of Prussia, and took
part in the battle of Mollwitz, where he was taken prisoner by the Austrians.
On his release he returned to Berlin, and thence to Paris, where he was
elected director of the Academy of Sciences in 1742, and in the following
year was admitted into the Academy. Returning to Berlin in 1744, at the
desire of Frederick II., he was chosen president of the Royal Academy of
Sciences in 1746. Finding his health declining, he repaired in 1757 to the
south of France, but went in 1758 to Basel, where he died on the 27th of
July 1759. Maupertuis was unquestionably a man of considerable ability as
a mathematician, but his restless, gloomy disposition involved him in
constant quarrels, of which his controversies with König and Voltaire
during the latter part of his life furnish examples.
(1698-1759), French mathematician and astronomer, was born at St Malo
on the 17th of July 1698. When twenty years of age he entered the army,
becoming lieutenant in a regiment of cavalry, and employing his leisure on
mathematical studies. After five years he quitted the army and was admitted
in 1723 a member of the Academy of Sciences. In 1728 he visited London,
and was elected a fellow of the Royal Society. In 1736 he acted as chief of
the expedition sent by Louis XV. into Lapland to measure the length of a
degree of the meridian (see Earth, Figure of), and on his return home he
became a member of almost all the scientific societies of Europe. In 1740
Maupertuis went to Berlin on the invitation of the king of Prussia, and took
part in the battle of Mollwitz, where he was taken prisoner by the Austrians.
On his release he returned to Berlin, and thence to Paris, where he was
elected director of the Academy of Sciences in 1742, and in the following
year was admitted into the Academy. Returning to Berlin in 1744, at the
desire of Frederick II., he was chosen president of the Royal Academy of
Sciences in 1746. Finding his health declining, he repaired in 1757 to the
south of France, but went in 1758 to Basel, where he died on the 27th of
July 1759. Maupertuis was unquestionably a man of considerable ability as
a mathematician, but his restless, gloomy disposition involved him in
constant quarrels, of which his controversies with König and Voltaire
during the latter part of his life furnish examples.
Page 87
The following are his most important works: Sur la figure de la terre
(Paris, 1738); Discours sur la parallaxe de la lune (Paris, 1741);
Discours sur la figure des astres (Paris, 1742); Éléments de la
géographie (Paris, 1742); Lettre sur la comète de 1742 (Paris, 1742);
Astronomie nautique (Paris, 1745 and 1746); Vénus physique (Paris,
1745); Essai de cosmologie (Amsterdam, 1750). His Œuvres were
published in 1752 at Dresden and in 1756 at Lyons.
MAU RANIPUR, a town of British India in Jahnsi district, in the
United Provinces. Pop. (1901), 17,231. It contains a large community of
wealthy merchants and bankers. A special variety of red cotton cloth,
known as kharua, is manufactured and exported to all parts of India. Trees
line many of the streets, and handsome temples ornament the town.
MAUREL, ABDIAS (d. 1705), Camisard leader, became a
cavalry officer in the French army and gained distinction in Italy; here he
served under Marshal Catinat, and on this account he himself is sometimes
(Paris, 1738); Discours sur la parallaxe de la lune (Paris, 1741);
Discours sur la figure des astres (Paris, 1742); Éléments de la
géographie (Paris, 1742); Lettre sur la comète de 1742 (Paris, 1742);
Astronomie nautique (Paris, 1745 and 1746); Vénus physique (Paris,
1745); Essai de cosmologie (Amsterdam, 1750). His Œuvres were
published in 1752 at Dresden and in 1756 at Lyons.
MAU RANIPUR, a town of British India in Jahnsi district, in the
United Provinces. Pop. (1901), 17,231. It contains a large community of
wealthy merchants and bankers. A special variety of red cotton cloth,
known as kharua, is manufactured and exported to all parts of India. Trees
line many of the streets, and handsome temples ornament the town.
MAUREL, ABDIAS (d. 1705), Camisard leader, became a
cavalry officer in the French army and gained distinction in Italy; here he
served under Marshal Catinat, and on this account he himself is sometimes
Page 88
known as Catinat. In 1702, when the revolt in the Cévennes broke out, he
became one of the Camisard leaders, and in this capacity his name was soon
known and feared. He refused to accept the peace made by Jean Cavalier in
1704, and after passing a few weeks in Switzerland he returned to France
and became one of the chiefs of those Camisards who were still in arms. He
was deeply concerned in a plot to capture some French towns, a scheme
which, it was hoped, would be helped by England and Holland. But it
failed; Maurel was betrayed, and with three other leaders of the movement
was burned to death at Nîmes on the 22nd of April 1705. He was a man of
great physical strength; but he was very cruel, and boasted he had killed
200 Roman Catholics with his own hands.
MAUREL, VICTOR (1848- ), French singer, was born at
Marseilles, and educated in music at the Paris Conservatoire. He made his
début in opera at Paris in 1868, and in London in 1873, and from that time
onwards his admirable acting and vocal method established his reputation
as one of the finest of operatic baritones. He created the leading part in
Verdi’s Otello, and was equally fine in Wagnerian and Italian opera.
became one of the Camisard leaders, and in this capacity his name was soon
known and feared. He refused to accept the peace made by Jean Cavalier in
1704, and after passing a few weeks in Switzerland he returned to France
and became one of the chiefs of those Camisards who were still in arms. He
was deeply concerned in a plot to capture some French towns, a scheme
which, it was hoped, would be helped by England and Holland. But it
failed; Maurel was betrayed, and with three other leaders of the movement
was burned to death at Nîmes on the 22nd of April 1705. He was a man of
great physical strength; but he was very cruel, and boasted he had killed
200 Roman Catholics with his own hands.
MAUREL, VICTOR (1848- ), French singer, was born at
Marseilles, and educated in music at the Paris Conservatoire. He made his
début in opera at Paris in 1868, and in London in 1873, and from that time
onwards his admirable acting and vocal method established his reputation
as one of the finest of operatic baritones. He created the leading part in
Verdi’s Otello, and was equally fine in Wagnerian and Italian opera.
Page 89
MAURENBRECHER, KARL PETER WILHELM
(1838-1892), German historian, was born at Bonn on the 21st of December,
1838, and studied in Berlin and Munich under Ranke and Von Sybel, being
especially influenced by the latter historian. After doing some research
work at Simancas in Spain, he became professor of history at the university
of Dorpat in 1867; and was then in turn professor at Königsberg, Bonn and
Leipzig. He died at Leipzig on the 6th of November, 1892.
Many of Maurenbrecher’s works are concerned with the
Reformation, among them being England im Reformationszeitalter
(Düsseldorf, 1866); Karl V. und die deutschen Protestanten
(Düsseldorf, 1865); Studien und Skizzen zur Geschichte der
Reformationszeit (Leipzig, 1874); and the incomplete Geschichte der
Katholischen Reformation (Nördlingen, 1880). He also wrote Don
Karlos (Berlin, 1876); Gründung des deutschen Reiches 1859-1871
(Leipzig, 1892, and again 1902); and Geschichte der deutschen
Königswahlen (Leipzig, 1889). See G. Wolf, Wilhelm Maurenbrecher
(Berlin, 1893).
(1838-1892), German historian, was born at Bonn on the 21st of December,
1838, and studied in Berlin and Munich under Ranke and Von Sybel, being
especially influenced by the latter historian. After doing some research
work at Simancas in Spain, he became professor of history at the university
of Dorpat in 1867; and was then in turn professor at Königsberg, Bonn and
Leipzig. He died at Leipzig on the 6th of November, 1892.
Many of Maurenbrecher’s works are concerned with the
Reformation, among them being England im Reformationszeitalter
(Düsseldorf, 1866); Karl V. und die deutschen Protestanten
(Düsseldorf, 1865); Studien und Skizzen zur Geschichte der
Reformationszeit (Leipzig, 1874); and the incomplete Geschichte der
Katholischen Reformation (Nördlingen, 1880). He also wrote Don
Karlos (Berlin, 1876); Gründung des deutschen Reiches 1859-1871
(Leipzig, 1892, and again 1902); and Geschichte der deutschen
Königswahlen (Leipzig, 1889). See G. Wolf, Wilhelm Maurenbrecher
(Berlin, 1893).
Page 90
MAUREPAS, JEAN FRÉDÉRIC PHÉLYPEAUX,
Comte de (1701-1781), French statesman, was born on the 9th of July 1701
at Versailles, being the son of Jérôme de Pontchartrain, secretary of state for
the marine and the royal household. Maurepas succeeded to his father’s
charge at fourteen, and began his functions in the royal household at
seventeen, while in 1725 he undertook the actual administration of the navy.
Although essentially light and frivolous in character, Maurepas was
seriously interested in scientific matters, and he used the best brains of
France to apply science to questions of navigation and of naval
construction. He was disgraced in 1749, and exiled from Paris for an
epigram against Madame de Pompadour. On the accession of Louis XVI.,
twenty-five years later, he became a minister of state and Louis XVI.’s chief
adviser. He gave Turgot the direction of finance, placed Lamoignon-
Malesherbes over the royal household and made Vergennes minister for
foreign affairs. At the outset of his new career he showed his weakness by
recalling to their functions, in deference to popular clamour, the members of
the old parlement ousted by Maupeou, thus reconstituting the most
dangerous enemy of the royal power. This step, and his intervention on
behalf of the American states, helped to pave the way for the French
revolution. Jealous of his personal ascendancy over Louis XVI., he
intrigued against Turgot, whose disgrace in 1776 was followed after six
months of disorder by the appointment of Necker. In 1781 Maurepas
Comte de (1701-1781), French statesman, was born on the 9th of July 1701
at Versailles, being the son of Jérôme de Pontchartrain, secretary of state for
the marine and the royal household. Maurepas succeeded to his father’s
charge at fourteen, and began his functions in the royal household at
seventeen, while in 1725 he undertook the actual administration of the navy.
Although essentially light and frivolous in character, Maurepas was
seriously interested in scientific matters, and he used the best brains of
France to apply science to questions of navigation and of naval
construction. He was disgraced in 1749, and exiled from Paris for an
epigram against Madame de Pompadour. On the accession of Louis XVI.,
twenty-five years later, he became a minister of state and Louis XVI.’s chief
adviser. He gave Turgot the direction of finance, placed Lamoignon-
Malesherbes over the royal household and made Vergennes minister for
foreign affairs. At the outset of his new career he showed his weakness by
recalling to their functions, in deference to popular clamour, the members of
the old parlement ousted by Maupeou, thus reconstituting the most
dangerous enemy of the royal power. This step, and his intervention on
behalf of the American states, helped to pave the way for the French
revolution. Jealous of his personal ascendancy over Louis XVI., he
intrigued against Turgot, whose disgrace in 1776 was followed after six
months of disorder by the appointment of Necker. In 1781 Maurepas
Page 91
deserted Necker as he had done Turgot, and he died at Versailles on the 21st
of November 1781.
Maurepas is credited with contributions to the collection of facetiae
known as the Étrennes de la Saint Jean (2nd ed., 1742). Four volumes
of Mémoires de Maurepas, purporting to be collected by his secretary
and edited by J. L. G. Soulavie in 1792, must be regarded as
apocryphal. Some of his letters were published in 1896 by the Soc. de
l’hist. de Paris. His éloge in the Academy of Sciences was pronounced
by Condorcet.
MAURER, GEORG LUDWIG VON (1790-1872),
German statesman and historian, son of a Protestant pastor, was born at
Erpolzheim, near Dürkheim, in the Rhenish Palatinate, on the 2nd of
November 1790. Educated at Heidelberg, he went in 1812 to reside in Paris,
where he entered upon a systematic study of the ancient legal institutions of
the Germans. Returning to Germany in 1814, he received an appointment
under the Bavarian government, and afterwards filled several important
official positions. In 1824 he published at Heidelberg his Geschichte des
altgermanischen und namentlich altbayrischen öffentlich-mündlichen
Gerichtsverfahrens, which obtained the first prize of the academy of
Munich, and in 1826 he became professor in the university of Munich. In
1829 he returned to official life, and was soon offered an important post. In
1832, when Otto (Otho), son of Louis I., king of Bavaria, was chosen to fill
of November 1781.
Maurepas is credited with contributions to the collection of facetiae
known as the Étrennes de la Saint Jean (2nd ed., 1742). Four volumes
of Mémoires de Maurepas, purporting to be collected by his secretary
and edited by J. L. G. Soulavie in 1792, must be regarded as
apocryphal. Some of his letters were published in 1896 by the Soc. de
l’hist. de Paris. His éloge in the Academy of Sciences was pronounced
by Condorcet.
MAURER, GEORG LUDWIG VON (1790-1872),
German statesman and historian, son of a Protestant pastor, was born at
Erpolzheim, near Dürkheim, in the Rhenish Palatinate, on the 2nd of
November 1790. Educated at Heidelberg, he went in 1812 to reside in Paris,
where he entered upon a systematic study of the ancient legal institutions of
the Germans. Returning to Germany in 1814, he received an appointment
under the Bavarian government, and afterwards filled several important
official positions. In 1824 he published at Heidelberg his Geschichte des
altgermanischen und namentlich altbayrischen öffentlich-mündlichen
Gerichtsverfahrens, which obtained the first prize of the academy of
Munich, and in 1826 he became professor in the university of Munich. In
1829 he returned to official life, and was soon offered an important post. In
1832, when Otto (Otho), son of Louis I., king of Bavaria, was chosen to fill
Page 92
the throne of Greece, a council of regency was nominated during his
minority, and Maurer was appointed a member. He applied himself
energetically to the task of creating institutions adapted to the requirements
of a modern civilized community; but grave difficulties soon arose and
Maurer was recalled in 1834, when he returned to Munich. This loss was a
serious one for Greece. Maurer was the ablest, most energetic and most
liberal-minded member of the council, and it was through his enlightened
efforts that Greece obtained a revised penal code, regular tribunals and an
improved system of civil procedure. Soon after his recall he published Das
griechische Volk in öffentlicher, kirchlicher, und privatrechtlicher Beziehung
vor und nach dem Freiheitskampfe bis zum 31 Juli 1834 (Heidelberg, 1835-
1836), a useful source of information for the history of Greece before Otto
ascended the throne, and also for the labours of the council of regency to
the time of the author’s recall. After the fall of the ministry of Karl von Abel
(1788-1859) in 1847, he became chief Bavarian minister and head of the
departments of foreign affairs and of justice, but was overthrown in the
same year. He died at Munich on the 9th of May 1872. His only son,
Conrad von Maurer (1823-1902), was a Scandinavian scholar of some
repute, and like his father was a professor at the university of Munich.
Maurer’s most important contribution to history is a series of books
on the early institutions of the Germans. These are: Einleitung zur
Geschichte der Mark-, Hof-, Dorf-, und Stadtverfassung und der
öffentlichen Gewalt (Munich, 1854); Geschichte der Markenverfassung
in Deutschland (Erlangen, 1856); Geschichte der Fronhöfe, der
Bauernhöfe, und der Hofverfassung in Deutschland (Erlangen, 1862-
1863); Geschichte der Dorfverfassung in Deutschland (Erlangen,
1865-1866); and Geschichte der Slädteverfassung in Deutschland
(Erlangen, 1869-1871). These works are still important authorities for
the early history of the Germans. Among other works are, Das Stadt-
und Landrechtsbuch Ruprechts von Freising, ein Beitrag zur
minority, and Maurer was appointed a member. He applied himself
energetically to the task of creating institutions adapted to the requirements
of a modern civilized community; but grave difficulties soon arose and
Maurer was recalled in 1834, when he returned to Munich. This loss was a
serious one for Greece. Maurer was the ablest, most energetic and most
liberal-minded member of the council, and it was through his enlightened
efforts that Greece obtained a revised penal code, regular tribunals and an
improved system of civil procedure. Soon after his recall he published Das
griechische Volk in öffentlicher, kirchlicher, und privatrechtlicher Beziehung
vor und nach dem Freiheitskampfe bis zum 31 Juli 1834 (Heidelberg, 1835-
1836), a useful source of information for the history of Greece before Otto
ascended the throne, and also for the labours of the council of regency to
the time of the author’s recall. After the fall of the ministry of Karl von Abel
(1788-1859) in 1847, he became chief Bavarian minister and head of the
departments of foreign affairs and of justice, but was overthrown in the
same year. He died at Munich on the 9th of May 1872. His only son,
Conrad von Maurer (1823-1902), was a Scandinavian scholar of some
repute, and like his father was a professor at the university of Munich.
Maurer’s most important contribution to history is a series of books
on the early institutions of the Germans. These are: Einleitung zur
Geschichte der Mark-, Hof-, Dorf-, und Stadtverfassung und der
öffentlichen Gewalt (Munich, 1854); Geschichte der Markenverfassung
in Deutschland (Erlangen, 1856); Geschichte der Fronhöfe, der
Bauernhöfe, und der Hofverfassung in Deutschland (Erlangen, 1862-
1863); Geschichte der Dorfverfassung in Deutschland (Erlangen,
1865-1866); and Geschichte der Slädteverfassung in Deutschland
(Erlangen, 1869-1871). These works are still important authorities for
the early history of the Germans. Among other works are, Das Stadt-
und Landrechtsbuch Ruprechts von Freising, ein Beitrag zur
Page 93
Geschichte des Schwabenspiegels (Stuttgart, 1839); Über die
Freipflege (plegium liberale), und die Entstehung der grossen und
kleinen Jury in England (Munich, 1848); and Über die deutsche
Reichsterritorial- und Rechtsgeschichte (1830).
Sec K. T. von Heigel, Denkwürdigkeiten des bayrischen Staatsrats
G. L. von Maurer (Munich, 1903).
MAURETANIA, the ancient name of the north-western angle of
the African continent, and under the Roman Empire also of a large territory
eastward of that angle. The name had different significations at different
times; but before the Roman occupation, Mauretania comprised a
considerable part of the modern Morocco i.e. the northern portion bounded
on the east by Algiers. Towards the south we may suppose it bounded by
the Atlas range, and it seems to have been regarded by geographers as
extending along the coast to the Atlantic as far as the point where that chain
descends to the sea, in about 30 N. lat. (Strabo, p. 825). The magnificent
plateau in which the city of Morocco is situated seems to have been
unknown to ancient geographers, and was certainly never included in the
Roman Empire. On the other hand, the Gaetulians to the south of the Atlas
range, on the date-producing slopes towards the Sahara, seem to have
owned a precarious subjection to the kings of Mauretania, as afterwards to
the Roman government. A large part of the country is of great natural
fertility, and in ancient times produced large quantities of corn, while the
slopes of Atlas were clothed with forests, which, besides other kinds of
Freipflege (plegium liberale), und die Entstehung der grossen und
kleinen Jury in England (Munich, 1848); and Über die deutsche
Reichsterritorial- und Rechtsgeschichte (1830).
Sec K. T. von Heigel, Denkwürdigkeiten des bayrischen Staatsrats
G. L. von Maurer (Munich, 1903).
MAURETANIA, the ancient name of the north-western angle of
the African continent, and under the Roman Empire also of a large territory
eastward of that angle. The name had different significations at different
times; but before the Roman occupation, Mauretania comprised a
considerable part of the modern Morocco i.e. the northern portion bounded
on the east by Algiers. Towards the south we may suppose it bounded by
the Atlas range, and it seems to have been regarded by geographers as
extending along the coast to the Atlantic as far as the point where that chain
descends to the sea, in about 30 N. lat. (Strabo, p. 825). The magnificent
plateau in which the city of Morocco is situated seems to have been
unknown to ancient geographers, and was certainly never included in the
Roman Empire. On the other hand, the Gaetulians to the south of the Atlas
range, on the date-producing slopes towards the Sahara, seem to have
owned a precarious subjection to the kings of Mauretania, as afterwards to
the Roman government. A large part of the country is of great natural
fertility, and in ancient times produced large quantities of corn, while the
slopes of Atlas were clothed with forests, which, besides other kinds of
Page 94
timber, produced the celebrated ornamental wood called citrum (Plin. Hist.
Nat. 13-96), for tables of which the Romans gave fabulous prices. (For
physical geography, see Morocco.)
Mauretania, or Maurusia as it was called by Greek writers, signified
the land of the Mauri, a term still retained in the modern name of
Moors (q.v.). The origin and ethnical affinities of the race are uncertain;
but it is probable that all the inhabitants of this northern tract of Africa
were kindred races belonging to the great Berber family, possibly with
an intermingled fair-skinned race from Europe (see Tissot, Géographie
comparée de la province romaine d’Afrique, i. 400 seq.; also Berbers).
They first appear in history at the time of the Jugurthine War (110-106
b.c.), when Mauretania was under the government of Bocchus and
seems to have been recognized as organized state (Sallust, Jugurtha,
19). To this Bocchus was given, after the war, the western part of
Jugurtha’s kingdom of Numidia, perhaps as far east as Saldae (Bougie).
Sixty years later, at the time of the dictator Caesar, we find two
Mauretanian kingdoms, one to the west of the river Mulucha under
Bogud, and the other to the east under a Bocchus; as to the date or
cause of the division we are ignorant. Both these kings took Caesar’s
part in the civil wars, and had their territory enlarged by him (Appian,
b.c. 4, 54). In 25 b.c., after their deaths, Augustus gave the two
kingdoms to Juba II. of Numidia (see under Juba), with the river
Ampsaga as the eastern frontier (Plin. 5. 22; Ptol. 4. 3. 1). Juba and his
son Ptolemaeus after him reigned till a.d. 40, when the latter was put to
death by Caligula, and shortly afterwards Claudius incorporated the
kingdom into the Roman state as two provinces, viz. Mauretania
Tingitana to the west of the Mulucha and M. Caesariensis to the east of
that river, the latter taking its name from the city Caesarea (formerly
Iol), which Juba had thus named and adopted as his capital. Thus the
dividing line between the two provinces was the same as that which
Nat. 13-96), for tables of which the Romans gave fabulous prices. (For
physical geography, see Morocco.)
Mauretania, or Maurusia as it was called by Greek writers, signified
the land of the Mauri, a term still retained in the modern name of
Moors (q.v.). The origin and ethnical affinities of the race are uncertain;
but it is probable that all the inhabitants of this northern tract of Africa
were kindred races belonging to the great Berber family, possibly with
an intermingled fair-skinned race from Europe (see Tissot, Géographie
comparée de la province romaine d’Afrique, i. 400 seq.; also Berbers).
They first appear in history at the time of the Jugurthine War (110-106
b.c.), when Mauretania was under the government of Bocchus and
seems to have been recognized as organized state (Sallust, Jugurtha,
19). To this Bocchus was given, after the war, the western part of
Jugurtha’s kingdom of Numidia, perhaps as far east as Saldae (Bougie).
Sixty years later, at the time of the dictator Caesar, we find two
Mauretanian kingdoms, one to the west of the river Mulucha under
Bogud, and the other to the east under a Bocchus; as to the date or
cause of the division we are ignorant. Both these kings took Caesar’s
part in the civil wars, and had their territory enlarged by him (Appian,
b.c. 4, 54). In 25 b.c., after their deaths, Augustus gave the two
kingdoms to Juba II. of Numidia (see under Juba), with the river
Ampsaga as the eastern frontier (Plin. 5. 22; Ptol. 4. 3. 1). Juba and his
son Ptolemaeus after him reigned till a.d. 40, when the latter was put to
death by Caligula, and shortly afterwards Claudius incorporated the
kingdom into the Roman state as two provinces, viz. Mauretania
Tingitana to the west of the Mulucha and M. Caesariensis to the east of
that river, the latter taking its name from the city Caesarea (formerly
Iol), which Juba had thus named and adopted as his capital. Thus the
dividing line between the two provinces was the same as that which
Page 95
had originally separated Mauretania from Numidia (q.v.). These
provinces were governed until the time of Diocletian by imperial
procurators, and were occasionally united for military purposes. Under
and after Diocletian M. Tingitana was attached administratively to the
dioicesis of Spain, with which it was in all respects closely connected;
while M. Caesariensis was divided by making its eastern part into a
separate government, which was called M. Sitifensis from the Roman
colony Sitifis.
In the two provinces of Mauretania there were at the time of Pliny a
number of towns, including seven (possibly eight) Roman colonies in
M. Tingitana and eleven in M. Caesariensis; others were added later.
These were mostly military foundations, and served the purpose of
securing civilization against the inroads of the natives, who were not in
a condition to be used as material for town-life as in Gaul and Spain,
but were under the immediate government of the procurators, retaining
their own clan organization. Of these colonies the most important,
beginning from the west, were Lixus on the Atlantic, Tingis (Tangier),
Rusaddir (Melila, Melilla), Cartenna (Tenes), Iol or Caesarea
(Cherchel), Icosium (Algiers), Saldae (Bougie), Igilgili (Jijelli) and
Sitifis (Setif). All these were on the coast but the last, which was some
distance inland. Besides these there were many municipia or oppida
civium romanorum (Plin. 5. 19 seq.), but, as has been made clear by
French archaeologists who have explored these regions, Roman
settlements are less frequent the farther we go west, and M. Tingitana
has as yet yielded but scanty evidence of Roman civilization. On the
whole Mauretania was in a flourishing condition down to the irruption
of the Vandals in a.d. 429; in the Notitia nearly a hundred and seventy
episcopal sees are enumerated here, but we must remember that
numbers of these were mere villages.
provinces were governed until the time of Diocletian by imperial
procurators, and were occasionally united for military purposes. Under
and after Diocletian M. Tingitana was attached administratively to the
dioicesis of Spain, with which it was in all respects closely connected;
while M. Caesariensis was divided by making its eastern part into a
separate government, which was called M. Sitifensis from the Roman
colony Sitifis.
In the two provinces of Mauretania there were at the time of Pliny a
number of towns, including seven (possibly eight) Roman colonies in
M. Tingitana and eleven in M. Caesariensis; others were added later.
These were mostly military foundations, and served the purpose of
securing civilization against the inroads of the natives, who were not in
a condition to be used as material for town-life as in Gaul and Spain,
but were under the immediate government of the procurators, retaining
their own clan organization. Of these colonies the most important,
beginning from the west, were Lixus on the Atlantic, Tingis (Tangier),
Rusaddir (Melila, Melilla), Cartenna (Tenes), Iol or Caesarea
(Cherchel), Icosium (Algiers), Saldae (Bougie), Igilgili (Jijelli) and
Sitifis (Setif). All these were on the coast but the last, which was some
distance inland. Besides these there were many municipia or oppida
civium romanorum (Plin. 5. 19 seq.), but, as has been made clear by
French archaeologists who have explored these regions, Roman
settlements are less frequent the farther we go west, and M. Tingitana
has as yet yielded but scanty evidence of Roman civilization. On the
whole Mauretania was in a flourishing condition down to the irruption
of the Vandals in a.d. 429; in the Notitia nearly a hundred and seventy
episcopal sees are enumerated here, but we must remember that
numbers of these were mere villages.
Page 96
In 1904 the term Mauretania was revived as an official designation
by the French government, and applied to the territory north of the
lower Senegal under French protection (see Senegal).
To the authorities quoted under Africa, Roman, may be added here
Göbel, Die West-küste Afrikas im Alterthum. (W. W. F.*)
MAURIAC, a town of central France, capital of an arrondissement
in the department of Cantal, 39 m. N.N.W. of Aurillac by rail. Pop. (1906),
2558. Mauriac, built on the slope of a volcanic hill, has a church of the 12th
century, and the buildings of an old abbey now used as public offices and
dwellings; the town owes its origin to the abbey, founded during the 6th
century. It is the seat of a sub-prefect and has a tribunal of first instance and
a communal college. There are marble quarries in the vicinity.
MAURICE [or Mauritius], ST (d. c. 286), an early Christian
martyr, who, with his companions, is commemorated by the Roman
Catholic Church on the 22nd of September. The oldest form of his story is
by the French government, and applied to the territory north of the
lower Senegal under French protection (see Senegal).
To the authorities quoted under Africa, Roman, may be added here
Göbel, Die West-küste Afrikas im Alterthum. (W. W. F.*)
MAURIAC, a town of central France, capital of an arrondissement
in the department of Cantal, 39 m. N.N.W. of Aurillac by rail. Pop. (1906),
2558. Mauriac, built on the slope of a volcanic hill, has a church of the 12th
century, and the buildings of an old abbey now used as public offices and
dwellings; the town owes its origin to the abbey, founded during the 6th
century. It is the seat of a sub-prefect and has a tribunal of first instance and
a communal college. There are marble quarries in the vicinity.
MAURICE [or Mauritius], ST (d. c. 286), an early Christian
martyr, who, with his companions, is commemorated by the Roman
Catholic Church on the 22nd of September. The oldest form of his story is
Page 97
found in the Passio ascribed to Eucherius, bishop of Lyons, c. 450, who
relates how the “Theban” legion commanded by Mauritius was sent to north
Italy to reinforce the army of Maximinian. Maximinian wished to use them
in persecuting the Christians, but as they themselves were of this faith, they
refused, and for this, after having been twice decimated, the legion was
exterminated at Octodurum (Martigny) near Geneva. In late versions this
legend was expanded and varied, the martyrdom was connected with a
refusal to take part in a great sacrifice ordered at Octodurum and the name
of Exsuperius was added to that of Mauritius. Gregory of Tours (c. 539-
593) speaks of a company of the same legion which suffered at Cologne.
The Magdeburg Centuries, in spite of Mauritius being the patron
saint of Magdeburg, declared the whole legend fictitious; J. A. du
Bordien La Légion thébéenne (Amsterdam, 1705); J. J. Hottinger in
Helvetische Kirchengeschichte (Zürich, 1708); and F. W. Rettberg,
Kirchengeschichte Deutschlands (Göttingen, 1845-1848) have also
demonstrated its untrustworthiness, while the Bollandists, De Rivaz
and Joh. Friedrich uphold it. Apart from the a priori improbability of a
whole legion being martyred, the difficulties are that in 286 Christians
everywhere throughout the empire were not molested, that at no later
date have we evidence of the presence of Maximinian in the Valais, and
that none of the writers nearest to the event (Eusebius, Lactantius,
Orosius, Sulpicius Severus) know anything of it. It is of course quite
possible that isolated cases of officers being put to death for their faith
occurred during Maximinian’s reign, and on some such cases the
legend may have grown up during the century and a half between
Maximinian and Eucherius. The cult of St Maurice and the Theban
legion is found in Switzerland (where two places bear the name in
Valais, besides St Moritz in Grisons), along the Rhine, and in north
Italy. The foundation of the abbey of St Maurice (Agaunum) in the
relates how the “Theban” legion commanded by Mauritius was sent to north
Italy to reinforce the army of Maximinian. Maximinian wished to use them
in persecuting the Christians, but as they themselves were of this faith, they
refused, and for this, after having been twice decimated, the legion was
exterminated at Octodurum (Martigny) near Geneva. In late versions this
legend was expanded and varied, the martyrdom was connected with a
refusal to take part in a great sacrifice ordered at Octodurum and the name
of Exsuperius was added to that of Mauritius. Gregory of Tours (c. 539-
593) speaks of a company of the same legion which suffered at Cologne.
The Magdeburg Centuries, in spite of Mauritius being the patron
saint of Magdeburg, declared the whole legend fictitious; J. A. du
Bordien La Légion thébéenne (Amsterdam, 1705); J. J. Hottinger in
Helvetische Kirchengeschichte (Zürich, 1708); and F. W. Rettberg,
Kirchengeschichte Deutschlands (Göttingen, 1845-1848) have also
demonstrated its untrustworthiness, while the Bollandists, De Rivaz
and Joh. Friedrich uphold it. Apart from the a priori improbability of a
whole legion being martyred, the difficulties are that in 286 Christians
everywhere throughout the empire were not molested, that at no later
date have we evidence of the presence of Maximinian in the Valais, and
that none of the writers nearest to the event (Eusebius, Lactantius,
Orosius, Sulpicius Severus) know anything of it. It is of course quite
possible that isolated cases of officers being put to death for their faith
occurred during Maximinian’s reign, and on some such cases the
legend may have grown up during the century and a half between
Maximinian and Eucherius. The cult of St Maurice and the Theban
legion is found in Switzerland (where two places bear the name in
Valais, besides St Moritz in Grisons), along the Rhine, and in north
Italy. The foundation of the abbey of St Maurice (Agaunum) in the
Page 98
Valais is usually ascribed to Sigismund of Burgundy (515). Relics of
the saint are preserved here and at Brieg and Turin.
MAURICE (Mauricius Flavius Tiberius) (c. 539-602), East Roman
emperor from 582 to 602, was of Roman descent, but a native of Arabissus
in Cappadocia. He spent his youth at the court of Justin II., and, having
joined the army, fought with distinction in the Persian War (578-581). At
the age of forty-three he was declared Caesar by the dying emperor Tiberius
II., who bestowed upon him the hand of his daughter Constantina. Maurice
brought the Persian War to a successful close by the restoration of Chosroes
II. to the throne (591). On the northern frontier he at first bought off the
Avars by payments which compelled him to exercise strict economy in his
general administration, but after 595 inflicted several defeats upon them
through his general Crispus. By his strict discipline and his refusal to
ransom a captive corps he provoked to mutiny the army on the Danube. The
revolt spread to the popular factions in Constantinople, and Maurice
consented to abdicate. He withdrew to Chalcedon, but was hunted down
and put to death after witnessing the slaughter of his five sons.
The work on military art (στρατηγικά) ascribed to him is a
contemporary work of unknown authorship (ed. Scheffer, Arriani
tactica et Mauricii ars militaris, Upsala, 1664; see Max Jähns, Gesch.
d. Kriegswissensch., i. 152-156).
the saint are preserved here and at Brieg and Turin.
MAURICE (Mauricius Flavius Tiberius) (c. 539-602), East Roman
emperor from 582 to 602, was of Roman descent, but a native of Arabissus
in Cappadocia. He spent his youth at the court of Justin II., and, having
joined the army, fought with distinction in the Persian War (578-581). At
the age of forty-three he was declared Caesar by the dying emperor Tiberius
II., who bestowed upon him the hand of his daughter Constantina. Maurice
brought the Persian War to a successful close by the restoration of Chosroes
II. to the throne (591). On the northern frontier he at first bought off the
Avars by payments which compelled him to exercise strict economy in his
general administration, but after 595 inflicted several defeats upon them
through his general Crispus. By his strict discipline and his refusal to
ransom a captive corps he provoked to mutiny the army on the Danube. The
revolt spread to the popular factions in Constantinople, and Maurice
consented to abdicate. He withdrew to Chalcedon, but was hunted down
and put to death after witnessing the slaughter of his five sons.
The work on military art (στρατηγικά) ascribed to him is a
contemporary work of unknown authorship (ed. Scheffer, Arriani
tactica et Mauricii ars militaris, Upsala, 1664; see Max Jähns, Gesch.
d. Kriegswissensch., i. 152-156).
Page 99
See Theophylactus Simocatta, Vita Mauricii (ed. de Boor, 1887); E.
Gibbon, The Decline and Fall of the Roman Empire (ed. Bury, London,
1896, v. 19-21, 57); J. B. Bury, The Later Roman Empire (London,
1889, ii. 83-94); G. Finlay, History of Greece (ed. 1877, Oxford, i. 299-
306).
MAURICE (1521-1553), elector of Saxony, elder son of Henry,
duke of Saxony, belonging to the Albertine branch of the Wettin family, was
born at Freiberg on the 21st of March 1521. In January 1541 he married
Agnes, daughter of Philip, landgrave of Hesse. In that year he became duke
of Saxony by his father’s death, and he continued Henry’s work in
forwarding the progress of the Reformation. Duke Henry had decreed that
his lands should be divided between his two sons, but as a partition was
regarded as undesirable the whole of the duchy came to his elder son.
Maurice, however, made generous provision for his brother Augustus, and
the desire to compensate him still further was one of the minor threads of
his subsequent policy. In 1542 he assisted the emperor Charles V. against
the Turks, in 1543 against William, duke of Cleves, and in 1544 against the
French; but his ambition soon took a wider range. The harmonious relations
which subsisted between the two branches of the Wettins were disturbed by
the interference of Maurice in Cleves, a proceeding distasteful to the Saxon
elector, John Frederick; and a dispute over the bishopric of Meissen having
widened the breach, war was only averted by the mediation of Philip of
Hesse and Luther. About this time Maurice seized the idea of securing for
Gibbon, The Decline and Fall of the Roman Empire (ed. Bury, London,
1896, v. 19-21, 57); J. B. Bury, The Later Roman Empire (London,
1889, ii. 83-94); G. Finlay, History of Greece (ed. 1877, Oxford, i. 299-
306).
MAURICE (1521-1553), elector of Saxony, elder son of Henry,
duke of Saxony, belonging to the Albertine branch of the Wettin family, was
born at Freiberg on the 21st of March 1521. In January 1541 he married
Agnes, daughter of Philip, landgrave of Hesse. In that year he became duke
of Saxony by his father’s death, and he continued Henry’s work in
forwarding the progress of the Reformation. Duke Henry had decreed that
his lands should be divided between his two sons, but as a partition was
regarded as undesirable the whole of the duchy came to his elder son.
Maurice, however, made generous provision for his brother Augustus, and
the desire to compensate him still further was one of the minor threads of
his subsequent policy. In 1542 he assisted the emperor Charles V. against
the Turks, in 1543 against William, duke of Cleves, and in 1544 against the
French; but his ambition soon took a wider range. The harmonious relations
which subsisted between the two branches of the Wettins were disturbed by
the interference of Maurice in Cleves, a proceeding distasteful to the Saxon
elector, John Frederick; and a dispute over the bishopric of Meissen having
widened the breach, war was only averted by the mediation of Philip of
Hesse and Luther. About this time Maurice seized the idea of securing for
Page 100
himself the electoral dignity held by John Frederick, and his opportunity
came when Charles was preparing to attack the league of Schmalkalden.
Although educated as a Lutheran, religious questions had never seriously
appealed to Maurice. As a youth he had joined the league of Schmalkalden,
but this adhesion, as well as his subsequent declaration to stand by the
confession of Augsburg, cannot be regarded as the decision of his maturer
years. In June 1546 he took a decided step by making a secret agreement
with Charles at Regensburg. Maurice was promised some rights over the
archbishopric of Magdeburg and the bishopric of Halberstadt; immunity, in
part at least, for his subjects from the Tridentine decrees; and the question
of transferring the electoral dignity was discussed. In return the duke
probably agreed to aid Charles in his proposed attack on the league as soon
as he could gain the consent of the Saxon estates, or at all events to remain
neutral during the impending war. The struggle began in July 1546, and in
October Maurice declared war against John Frederick. He secured the
formal consent of Charles to the transfer of the electoral dignity and took
the field in November. He had gained a few successes when John Frederick
hastened from south Germany to defend his dominions. Maurice’s ally,
Albert Alcibiades, prince of Bayreuth, was taken prisoner at Rochlitz; and
the duke, driven from electoral Saxony, was unable to prevent his own
lands from being overrun. Salvation, however, was at hand. Marching
against John Frederick, Charles V., aided by Maurice, gained a decisive
victory at Mühlberg in April 1547, after which by the capitulation of
Wittenberg John Frederick renounced the electoral dignity in favour of
Maurice, who also obtained a large part of his kinsman’s lands. The formal
investiture of the new elector took place at Augsburg in February 1548.
The plans of Maurice soon took a form less agreeable to the emperor. The
continued imprisonment of his father-in-law, Philip of Hesse, whom he had
induced to surrender to Charles and whose freedom he had guaranteed, was
neither his greatest nor his only cause of complaint. The emperor had
came when Charles was preparing to attack the league of Schmalkalden.
Although educated as a Lutheran, religious questions had never seriously
appealed to Maurice. As a youth he had joined the league of Schmalkalden,
but this adhesion, as well as his subsequent declaration to stand by the
confession of Augsburg, cannot be regarded as the decision of his maturer
years. In June 1546 he took a decided step by making a secret agreement
with Charles at Regensburg. Maurice was promised some rights over the
archbishopric of Magdeburg and the bishopric of Halberstadt; immunity, in
part at least, for his subjects from the Tridentine decrees; and the question
of transferring the electoral dignity was discussed. In return the duke
probably agreed to aid Charles in his proposed attack on the league as soon
as he could gain the consent of the Saxon estates, or at all events to remain
neutral during the impending war. The struggle began in July 1546, and in
October Maurice declared war against John Frederick. He secured the
formal consent of Charles to the transfer of the electoral dignity and took
the field in November. He had gained a few successes when John Frederick
hastened from south Germany to defend his dominions. Maurice’s ally,
Albert Alcibiades, prince of Bayreuth, was taken prisoner at Rochlitz; and
the duke, driven from electoral Saxony, was unable to prevent his own
lands from being overrun. Salvation, however, was at hand. Marching
against John Frederick, Charles V., aided by Maurice, gained a decisive
victory at Mühlberg in April 1547, after which by the capitulation of
Wittenberg John Frederick renounced the electoral dignity in favour of
Maurice, who also obtained a large part of his kinsman’s lands. The formal
investiture of the new elector took place at Augsburg in February 1548.
The plans of Maurice soon took a form less agreeable to the emperor. The
continued imprisonment of his father-in-law, Philip of Hesse, whom he had
induced to surrender to Charles and whose freedom he had guaranteed, was
neither his greatest nor his only cause of complaint. The emperor had
Page 101
refused to complete the humiliation of the family of John Frederick; he had
embarked upon a course of action which boded danger to the elector’s
Lutheran subjects, and his increased power was a menace to the position of
Maurice. Assuring Charles of his continued loyalty, the elector entered into
negotiations with the discontented Protestant princes. An event happened
which gave him a base of operations, and enabled him to mask his schemes
against the emperor. In 1550 he had been entrusted with the execution of
the imperial ban against the city of Magdeburg, and under cover of these
operations he was able to collect troops and to concert measures with his
allies. Favourable terms were granted to Magdeburg, which surrendered and
remained in the power of Maurice, and in January 1552 a treaty was
concluded with Henry II. of France at Chambord. Meanwhile Maurice had
refused to recognize the Interim issued from Augsburg in May 1548 as
binding on Saxony; but a compromise was arranged on the basis of which
the Leipzig Interim was drawn up for his lands. It is uncertain how far
Charles was ignorant of the elector’s preparations, but certainly he was
unprepared for the attack made by Maurice and his allies in March 1552.
Augsburg was taken, the pass of Ehrenberg was forced, and in a few days
the emperor left Innsbruck as a fugitive. Ferdinand undertook to make
peace, and the Treaty of Passau, signed in August 1552, was the result.
Maurice obtained a general amnesty and freedom for Philip of Hesse, but
was unable to obtain a perpetual religious peace for the Lutherans. Charles
stubbornly insisted that this question must be referred to the Diet, and
Maurice was obliged to give way. He then fought against the Turks, and
renewed his communications with Henry of France. Returning from
Hungary the elector placed himself at the head of the princes who were
seeking to check the career of his former ally, Albert Alcibiades, whose
depredations were making him a curse to Germany. The rival armies met at
Sievershausen on the 9th of July 1553, where after a fierce encounter Albert
embarked upon a course of action which boded danger to the elector’s
Lutheran subjects, and his increased power was a menace to the position of
Maurice. Assuring Charles of his continued loyalty, the elector entered into
negotiations with the discontented Protestant princes. An event happened
which gave him a base of operations, and enabled him to mask his schemes
against the emperor. In 1550 he had been entrusted with the execution of
the imperial ban against the city of Magdeburg, and under cover of these
operations he was able to collect troops and to concert measures with his
allies. Favourable terms were granted to Magdeburg, which surrendered and
remained in the power of Maurice, and in January 1552 a treaty was
concluded with Henry II. of France at Chambord. Meanwhile Maurice had
refused to recognize the Interim issued from Augsburg in May 1548 as
binding on Saxony; but a compromise was arranged on the basis of which
the Leipzig Interim was drawn up for his lands. It is uncertain how far
Charles was ignorant of the elector’s preparations, but certainly he was
unprepared for the attack made by Maurice and his allies in March 1552.
Augsburg was taken, the pass of Ehrenberg was forced, and in a few days
the emperor left Innsbruck as a fugitive. Ferdinand undertook to make
peace, and the Treaty of Passau, signed in August 1552, was the result.
Maurice obtained a general amnesty and freedom for Philip of Hesse, but
was unable to obtain a perpetual religious peace for the Lutherans. Charles
stubbornly insisted that this question must be referred to the Diet, and
Maurice was obliged to give way. He then fought against the Turks, and
renewed his communications with Henry of France. Returning from
Hungary the elector placed himself at the head of the princes who were
seeking to check the career of his former ally, Albert Alcibiades, whose
depredations were making him a curse to Germany. The rival armies met at
Sievershausen on the 9th of July 1553, where after a fierce encounter Albert
Page 102
was defeated. The victor, however, was wounded during the fight and died
two days later.
Maurice was a friend to learning, and devoted some of the secularized
church property to the advancement of education. Very different estimates
have been formed of his character. He has been represented as the saviour
of German Protestantism on the one hand, and on the other as a traitor to his
faith and country. In all probability he was neither the one nor the other, but
a man of great ambition who, indifferent to religious considerations, made
good use of the exigencies of the time. He was generous and enlightened, a
good soldier and a clever diplomatist. He left an only daughter Anna (d.
1577), who became the second wife of William the Silent, prince of Orange.
The elector’s Politische Korrespondenz has been edited by E.
Brandenburg (Leipzig, 1900-1904); and a sketch of him is given by
Roger Ascham in A Report and Discourse of the Affairs and State of
Germany (London, 1864-1865). See also F. A. von Langenn, Moritz
Herzog und Churfürst zu Sachsen (Leipzig, 1841); G. Voigt, Moritz
von Sachsen (Leipzig, 1876); E. Brandenburg, Moritz von Sachsen
(Leipzig, 1898); S. Issleib, Moritz von Sachsen als protestantischer
Fürst (Hamburg, 1898); J. Witter, Die Beziehung und der Verkehr des
Kurfürsten Moritz mit König Ferdinand (Jena, 1886); L. von Ranke,
Deutsche Geschichte im Zeitalter der Reformation, Bde. IV. and V.
(Leipzig, 1882); and W. Maurenbrecher in the Allgemeine deutsche
Biographie, Bd. XXII. (Leipzig, 1885). For bibliography see
Maurenbrecher; and The Cambridge Modern History, vol. ii.
(Cambridge, 1903).
two days later.
Maurice was a friend to learning, and devoted some of the secularized
church property to the advancement of education. Very different estimates
have been formed of his character. He has been represented as the saviour
of German Protestantism on the one hand, and on the other as a traitor to his
faith and country. In all probability he was neither the one nor the other, but
a man of great ambition who, indifferent to religious considerations, made
good use of the exigencies of the time. He was generous and enlightened, a
good soldier and a clever diplomatist. He left an only daughter Anna (d.
1577), who became the second wife of William the Silent, prince of Orange.
The elector’s Politische Korrespondenz has been edited by E.
Brandenburg (Leipzig, 1900-1904); and a sketch of him is given by
Roger Ascham in A Report and Discourse of the Affairs and State of
Germany (London, 1864-1865). See also F. A. von Langenn, Moritz
Herzog und Churfürst zu Sachsen (Leipzig, 1841); G. Voigt, Moritz
von Sachsen (Leipzig, 1876); E. Brandenburg, Moritz von Sachsen
(Leipzig, 1898); S. Issleib, Moritz von Sachsen als protestantischer
Fürst (Hamburg, 1898); J. Witter, Die Beziehung und der Verkehr des
Kurfürsten Moritz mit König Ferdinand (Jena, 1886); L. von Ranke,
Deutsche Geschichte im Zeitalter der Reformation, Bde. IV. and V.
(Leipzig, 1882); and W. Maurenbrecher in the Allgemeine deutsche
Biographie, Bd. XXII. (Leipzig, 1885). For bibliography see
Maurenbrecher; and The Cambridge Modern History, vol. ii.
(Cambridge, 1903).
Page 103
MAURICE, JOHN FREDERICK DENISON (1805-
1872), English theologian, was born at Normanston, Suffolk, on the 29th of
August, 1805. He was the son of a Unitarian minister, and entered Trinity
College, Cambridge, in 1823, though it was then impossible for any but
members of the Established Church to obtain a degree. Together with John
Sterling (with whom he founded the Apostles’ Club) he migrated to Trinity
Hall, whence he obtained a first class in civil law in 1827; he then came to
London, and gave himself to literary work, writing a novel, Eustace
Conyers, and editing the London Literary Chronicle until 1830, and also for
a short time the Athenaeum. At this time he was much perplexed as to his
religious opinions, and he ultimately found relief in a decision to take a
further university course and to seek Anglican orders. Entering Exeter
College, Oxford, he took a second class in classics in 1831. He was
ordained in 1834, and after a short curacy at Bubbenhall in Warwickshire
was appointed chaplain of Guy’s Hospital, and became thenceforward a
sensible factor in the intellectual and social life of London. From 1839 to
1841 Maurice was editor of the Education Magazine. In 1840 he was
appointed professor of English history and literature in King’s College, and
to this post in 1846 was added the chair of divinity. In 1845 he was Boyle
lecturer and Warburton lecturer. These chairs he held till 1853. In that year
he published Theological Essays, wherein were stated opinions which
savoured to the principal, Dr R. W. Jelf, and to the council, of unsound
theology in regard to eternal punishment. He had previously been called on
1872), English theologian, was born at Normanston, Suffolk, on the 29th of
August, 1805. He was the son of a Unitarian minister, and entered Trinity
College, Cambridge, in 1823, though it was then impossible for any but
members of the Established Church to obtain a degree. Together with John
Sterling (with whom he founded the Apostles’ Club) he migrated to Trinity
Hall, whence he obtained a first class in civil law in 1827; he then came to
London, and gave himself to literary work, writing a novel, Eustace
Conyers, and editing the London Literary Chronicle until 1830, and also for
a short time the Athenaeum. At this time he was much perplexed as to his
religious opinions, and he ultimately found relief in a decision to take a
further university course and to seek Anglican orders. Entering Exeter
College, Oxford, he took a second class in classics in 1831. He was
ordained in 1834, and after a short curacy at Bubbenhall in Warwickshire
was appointed chaplain of Guy’s Hospital, and became thenceforward a
sensible factor in the intellectual and social life of London. From 1839 to
1841 Maurice was editor of the Education Magazine. In 1840 he was
appointed professor of English history and literature in King’s College, and
to this post in 1846 was added the chair of divinity. In 1845 he was Boyle
lecturer and Warburton lecturer. These chairs he held till 1853. In that year
he published Theological Essays, wherein were stated opinions which
savoured to the principal, Dr R. W. Jelf, and to the council, of unsound
theology in regard to eternal punishment. He had previously been called on
Page 104
to clear himself from charges of heterodoxy brought against him in the
Quarterly Review (1851), and had been acquitted by a committee of inquiry.
Now again he maintained with great warmth of conviction that his views
were in close accordance with Scripture and the Anglican standards, but the
council, without specifying any distinct “heresy” and declining to submit
the case to the judgment of competent theologians, ruled otherwise, and he
was deprived of his professorships. He held at the same time the chaplaincy
of Lincoln’s Inn, for which he had resigned Guy’s (1846-1860), but when
he offered to resign this the benchers refused. Nor was he assailed in the
incumbency of St. Peter’s, Vere Street, which he held for nine years (1860-
1869), and where he drew round him a circle of thoughtful people. During
the early years of this period he was engaged in a hot and bitter controversy
with H. L. Mansel (afterwards dean of St Paul’s), arising out of the latter’s
Bampton lecture upon reason and revelation.
During his residence in London Maurice was specially identified with
two important movements for education. He helped to found Queen’s
College for the education of women (1848), and the Working Men’s
College (1854), of which he was the first principal. He strongly advocated
the abolition of university tests (1853), and threw himself with great energy
into all that affected the social life of the people. Certain abortive attempts
at co-operation among working men, and the movement known as Christian
Socialism, were the immediate outcome of his teaching. In 1866 Maurice
was appointed professor of moral philosophy at Cambridge, and from 1870
to 1872 was incumbent of St Edward’s in that city. He died on the 1st of
April 1872.
He was twice married, first to Anna Barton, a sister of John Sterling’s
wife, secondly to a half-sister of his friend Archdeacon Hare. His son
Major-General Sir J. Frederick Maurice (b. 1841), became a distinguished
soldier and one of the most prominent military writers of his time.
Quarterly Review (1851), and had been acquitted by a committee of inquiry.
Now again he maintained with great warmth of conviction that his views
were in close accordance with Scripture and the Anglican standards, but the
council, without specifying any distinct “heresy” and declining to submit
the case to the judgment of competent theologians, ruled otherwise, and he
was deprived of his professorships. He held at the same time the chaplaincy
of Lincoln’s Inn, for which he had resigned Guy’s (1846-1860), but when
he offered to resign this the benchers refused. Nor was he assailed in the
incumbency of St. Peter’s, Vere Street, which he held for nine years (1860-
1869), and where he drew round him a circle of thoughtful people. During
the early years of this period he was engaged in a hot and bitter controversy
with H. L. Mansel (afterwards dean of St Paul’s), arising out of the latter’s
Bampton lecture upon reason and revelation.
During his residence in London Maurice was specially identified with
two important movements for education. He helped to found Queen’s
College for the education of women (1848), and the Working Men’s
College (1854), of which he was the first principal. He strongly advocated
the abolition of university tests (1853), and threw himself with great energy
into all that affected the social life of the people. Certain abortive attempts
at co-operation among working men, and the movement known as Christian
Socialism, were the immediate outcome of his teaching. In 1866 Maurice
was appointed professor of moral philosophy at Cambridge, and from 1870
to 1872 was incumbent of St Edward’s in that city. He died on the 1st of
April 1872.
He was twice married, first to Anna Barton, a sister of John Sterling’s
wife, secondly to a half-sister of his friend Archdeacon Hare. His son
Major-General Sir J. Frederick Maurice (b. 1841), became a distinguished
soldier and one of the most prominent military writers of his time.
Page 105
Those who knew Maurice best were deeply impressed with the
spirituality of his character. “Whenever he woke in the night,” says his wife,
“he was always praying.” Charles Kingsley called him “the most beautiful
human soul whom God has ever allowed me to meet with.” As regards his
intellectual attainments we may set Julius Hare’s verdict “the greatest mind
since Plato” over against Ruskin’s “by nature puzzle-headed and indeed
wrong-headed.” Such contradictory impressions bespeak a life made up of
contradictory elements. Maurice was a man of peace, yet his life was spent
in a series of conflicts; of deep humility, yet so polemical that he often
seemed biased; of large charity, yet bitter in his attack upon the religious
press of his time; a loyal churchman who detested the label “Broad,” yet
poured out criticism upon the leaders of the Church. With an intense
capacity for visualizing the unseen, and a kindly dignity, he combined a
large sense of humour. While most of the “Broad Churchmen” were
influenced by ethical and emotional considerations in their repudiation of
the dogma of everlasting torment, he was swayed by purely intellectual and
theological arguments, and in questions of a more general liberty he often
opposed the proposed Liberal theologians, though he as often took their
side if he saw them hard pressed. He had a wide metaphysical and
philosophical knowledge which he applied to the history of theology. He
was a strenuous advocate of ecclesiastical control in elementary education,
and an opponent of the new school of higher biblical criticism, though so
far an evolutionist as to believe in growth and development as applied to the
history of nations.
As a preacher, his message was apparently simple; his two great
convictions were the fatherhood of God, and that all religious systems
which had any stability lasted because of a portion of truth which had
to be disentangled from the error differentiating them from the
doctrines of the Church of England as understood by himself. His love
to God as his Father was a passionate adoration which filled his whole
spirituality of his character. “Whenever he woke in the night,” says his wife,
“he was always praying.” Charles Kingsley called him “the most beautiful
human soul whom God has ever allowed me to meet with.” As regards his
intellectual attainments we may set Julius Hare’s verdict “the greatest mind
since Plato” over against Ruskin’s “by nature puzzle-headed and indeed
wrong-headed.” Such contradictory impressions bespeak a life made up of
contradictory elements. Maurice was a man of peace, yet his life was spent
in a series of conflicts; of deep humility, yet so polemical that he often
seemed biased; of large charity, yet bitter in his attack upon the religious
press of his time; a loyal churchman who detested the label “Broad,” yet
poured out criticism upon the leaders of the Church. With an intense
capacity for visualizing the unseen, and a kindly dignity, he combined a
large sense of humour. While most of the “Broad Churchmen” were
influenced by ethical and emotional considerations in their repudiation of
the dogma of everlasting torment, he was swayed by purely intellectual and
theological arguments, and in questions of a more general liberty he often
opposed the proposed Liberal theologians, though he as often took their
side if he saw them hard pressed. He had a wide metaphysical and
philosophical knowledge which he applied to the history of theology. He
was a strenuous advocate of ecclesiastical control in elementary education,
and an opponent of the new school of higher biblical criticism, though so
far an evolutionist as to believe in growth and development as applied to the
history of nations.
As a preacher, his message was apparently simple; his two great
convictions were the fatherhood of God, and that all religious systems
which had any stability lasted because of a portion of truth which had
to be disentangled from the error differentiating them from the
doctrines of the Church of England as understood by himself. His love
to God as his Father was a passionate adoration which filled his whole
Page 106
heart. The prophetic, even apocalyptic, note of his preaching was
particularly impressive. He prophesied in London as Isaiah prophesied
to the little towns of Palestine and Syria, “often with dark foreboding,
but seeing through all unrest and convulsion the working out of a sure
divine purpose.” Both at King’s College and at Cambridge Maurice
gathered round him a band of earnest students, to whom he directly
taught much that was valuable drawn from wide stores of his own
reading, wide rather than deep, for he never was, strictly speaking, a
learned man. Still more did he encourage the habit of inquiry and
research, more valuable than his direct teaching. In his Socratic power
of convincing his pupils of their ignorance he did more than perhaps
any other man of his time to awaken in those who came under his sway
the desire for knowledge and the process of independent thought.
As a social reformer, Maurice was before his time, and gave his
eager support to schemes for which the world was not ready. From an
early period of his life in London the condition of the poor pressed
upon him with consuming force; the enormous magnitude of the social
questions involved was a burden which he could hardly bear. For many
years he was the clergyman whom working men of all opinions seemed
to trust even if their faith in other religious men and all religious
systems had faded, and he had a marvellous power of attracting the
zealot and the outcast.
His works cover nearly 40 volumes, often obscure, often
tautological, and with no great distinction of style. But their high
purpose and philosophical outlook give his writings a permanent place
in the history of the thought of his time. The following are the more
important works—some of them were rewritten and in a measure
recast, and the date given is not necessarily that of the first appearance
of the book, but of its more complete and abiding form: Eustace
particularly impressive. He prophesied in London as Isaiah prophesied
to the little towns of Palestine and Syria, “often with dark foreboding,
but seeing through all unrest and convulsion the working out of a sure
divine purpose.” Both at King’s College and at Cambridge Maurice
gathered round him a band of earnest students, to whom he directly
taught much that was valuable drawn from wide stores of his own
reading, wide rather than deep, for he never was, strictly speaking, a
learned man. Still more did he encourage the habit of inquiry and
research, more valuable than his direct teaching. In his Socratic power
of convincing his pupils of their ignorance he did more than perhaps
any other man of his time to awaken in those who came under his sway
the desire for knowledge and the process of independent thought.
As a social reformer, Maurice was before his time, and gave his
eager support to schemes for which the world was not ready. From an
early period of his life in London the condition of the poor pressed
upon him with consuming force; the enormous magnitude of the social
questions involved was a burden which he could hardly bear. For many
years he was the clergyman whom working men of all opinions seemed
to trust even if their faith in other religious men and all religious
systems had faded, and he had a marvellous power of attracting the
zealot and the outcast.
His works cover nearly 40 volumes, often obscure, often
tautological, and with no great distinction of style. But their high
purpose and philosophical outlook give his writings a permanent place
in the history of the thought of his time. The following are the more
important works—some of them were rewritten and in a measure
recast, and the date given is not necessarily that of the first appearance
of the book, but of its more complete and abiding form: Eustace
Page 107
Conway, or the Brother and Sister, a novel (1834); The Kingdom of
Christ (1842); Christmas Day and Other Sermons (1843); The Unity of
the New Testament (1844); The Epistle to the Hebrews (1846); The
Religions of the World (1847); Moral and Metaphysical Philosophy (at
first an article in the Encyclopaedia Metropolitana, 1848); The Church
a Family (1850); The Old Testament (1851); Theological Essays
(1853); The Prophets and Kings of the Old Testament (1853); Lectures
on Ecclesiastical History (1854); The Doctrine of Sacrifice (1854); The
Patriarchs and Lawgivers of the Old Testament (1855); The Epistles of
St John (1857); The Commandments as Instruments of National
Reformation (1866); On the Gospel of St Luke (1868); The Conscience:
Lectures on Casuistry (1868); The Lord’s Prayer, a Manual (1870). The
greater part of these works were first delivered as sermons or lectures.
Maurice also contributed many prefaces and introductions to the works
of friends, as to Archdeacon Hare’s Charges, Kingsley’s Saint’s
Tragedy, &c.
See Life by his son (2 vols., London, 1884), and a monograph by C.
F. G. Masterman (1907) in “Leader of the Church” series; W. E. Collins
in Typical English Churchmen, pp. 327-360 (1902), and T. Hughes in
The Friendship of Books (1873).
MAURICE OF NASSAU, prince of Orange (1567-1625), the
second son of William the Silent, by Anna, only daughter of the famous
Maurice, elector of Saxony, was born at Dillenburg. At the time of his
Christ (1842); Christmas Day and Other Sermons (1843); The Unity of
the New Testament (1844); The Epistle to the Hebrews (1846); The
Religions of the World (1847); Moral and Metaphysical Philosophy (at
first an article in the Encyclopaedia Metropolitana, 1848); The Church
a Family (1850); The Old Testament (1851); Theological Essays
(1853); The Prophets and Kings of the Old Testament (1853); Lectures
on Ecclesiastical History (1854); The Doctrine of Sacrifice (1854); The
Patriarchs and Lawgivers of the Old Testament (1855); The Epistles of
St John (1857); The Commandments as Instruments of National
Reformation (1866); On the Gospel of St Luke (1868); The Conscience:
Lectures on Casuistry (1868); The Lord’s Prayer, a Manual (1870). The
greater part of these works were first delivered as sermons or lectures.
Maurice also contributed many prefaces and introductions to the works
of friends, as to Archdeacon Hare’s Charges, Kingsley’s Saint’s
Tragedy, &c.
See Life by his son (2 vols., London, 1884), and a monograph by C.
F. G. Masterman (1907) in “Leader of the Church” series; W. E. Collins
in Typical English Churchmen, pp. 327-360 (1902), and T. Hughes in
The Friendship of Books (1873).
MAURICE OF NASSAU, prince of Orange (1567-1625), the
second son of William the Silent, by Anna, only daughter of the famous
Maurice, elector of Saxony, was born at Dillenburg. At the time of his
Page 108
father’s assassination in 1584 he was being educated at the university of
Leiden, at the expense of the states of Holland and Zeeland. Despite his
youth he was made stadtholder of those two provinces and president of the
council of state. During the period of Leicester’s governorship he remained
in the background, engaged in acquiring a thorough knowledge of the
military art, and in 1586 the States of Holland conferred upon him the title
of prince. On the withdrawal of Leicester from the Netherlands in August
1587, Johan van Oldenbarneveldt, the advocate of Holland, became the
leading statesman of the country, a position which he retained for upwards
of thirty years. He had been a devoted adherent of William the Silent and he
now used his influence to forward the interests of Maurice. In 1588 he was
appointed by the States-General captain and admiral-general of the Union,
in 1590 he was elected stadtholder of Utrecht and Overysel, and in 1591 of
Gelderland. From this time forward, Oldenbarneveldt at the head of the
civil government and Maurice in command of the armed forces of the
republic worked together in the task of rescuing the United Netherlands
from Spanish domination (for details see Holland). Maurice soon showed
himself to be a general second in skill to none of his contemporaries. He
was especially famed for his consummate knowledge of the science of
sieges. The twelve years’ truce on the 9th of April 1609 brought to an end
the cordial relations between Maurice and Oldenbarneveldt. Maurice was
opposed to the truce, but the advocate’s policy triumphed and henceforward
there was enmity between them. The theological disputes between the
Remonstrants and contra-Remonstrants found them on different sides; and
the theological quarrel soon became a political one. Oldenbarneveldt,
supported by the states of Holland, came forward as the champion of
provincial sovereignty against that of the states-general; Maurice threw the
weight of his sword on the side of the union. The struggle was a short one,
for the army obeyed the general who had so often led them to victory.
Oldenbarneveldt perished on the scaffold, and the share which Maurice had
Leiden, at the expense of the states of Holland and Zeeland. Despite his
youth he was made stadtholder of those two provinces and president of the
council of state. During the period of Leicester’s governorship he remained
in the background, engaged in acquiring a thorough knowledge of the
military art, and in 1586 the States of Holland conferred upon him the title
of prince. On the withdrawal of Leicester from the Netherlands in August
1587, Johan van Oldenbarneveldt, the advocate of Holland, became the
leading statesman of the country, a position which he retained for upwards
of thirty years. He had been a devoted adherent of William the Silent and he
now used his influence to forward the interests of Maurice. In 1588 he was
appointed by the States-General captain and admiral-general of the Union,
in 1590 he was elected stadtholder of Utrecht and Overysel, and in 1591 of
Gelderland. From this time forward, Oldenbarneveldt at the head of the
civil government and Maurice in command of the armed forces of the
republic worked together in the task of rescuing the United Netherlands
from Spanish domination (for details see Holland). Maurice soon showed
himself to be a general second in skill to none of his contemporaries. He
was especially famed for his consummate knowledge of the science of
sieges. The twelve years’ truce on the 9th of April 1609 brought to an end
the cordial relations between Maurice and Oldenbarneveldt. Maurice was
opposed to the truce, but the advocate’s policy triumphed and henceforward
there was enmity between them. The theological disputes between the
Remonstrants and contra-Remonstrants found them on different sides; and
the theological quarrel soon became a political one. Oldenbarneveldt,
supported by the states of Holland, came forward as the champion of
provincial sovereignty against that of the states-general; Maurice threw the
weight of his sword on the side of the union. The struggle was a short one,
for the army obeyed the general who had so often led them to victory.
Oldenbarneveldt perished on the scaffold, and the share which Maurice had
Page 109
in securing the illegal condemnation by a packed court of judges of the aged
patriot must ever remain a stain upon his memory.
Maurice, who had on the death of his elder brother Philip William, in
February 1618, become prince of Orange, was now supreme in the state, but
during the remainder of his life he sorely missed the wise counsels of the
experienced Oldenbarneveldt. War broke out again in 1621, but success had
ceased to accompany him on his campaigns. His health gave way, and he
died, a prematurely aged man, at the Hague on the 4th of April 1625. He
was buried by his father’s side at Delft.
Bibliography.—I. Commelin, Wilhelm en Maurits v. Nassau, pr. v.
Orangien, haer leven en bedrijf (Amsterdam, 1651); G. Groen van
Prinsterer, Archives ou correspondance de la maison d’Orange-
Nassau, 1e série, 9 vols. (Leiden, 1841-1861); G. Groen van Prinsterer,
Maurice et Barneveldt (Utrecht, 1875); J. L. Motley, Life and Death of
John of Barneveldt (2 vols., The Hague, 1894); C. M. Kemp, v.d.
Maurits v. Nassau, prins v. Oranje in zijn leven en verdiensten (4 vols.,
Rotterdam, 1845); M. O. Nutting, The Days of Prince Maurice (Boston
and Chicago, 1894).
MAURISTS, a congregation of French Benedictines called after St
Maurus (d. 565), a disciple of St Benedict and the legendary introducer of
the Benedictine rule and life into Gaul.1 At the end of the 16th century the
Benedictine monasteries of France had fallen into a state of disorganization
patriot must ever remain a stain upon his memory.
Maurice, who had on the death of his elder brother Philip William, in
February 1618, become prince of Orange, was now supreme in the state, but
during the remainder of his life he sorely missed the wise counsels of the
experienced Oldenbarneveldt. War broke out again in 1621, but success had
ceased to accompany him on his campaigns. His health gave way, and he
died, a prematurely aged man, at the Hague on the 4th of April 1625. He
was buried by his father’s side at Delft.
Bibliography.—I. Commelin, Wilhelm en Maurits v. Nassau, pr. v.
Orangien, haer leven en bedrijf (Amsterdam, 1651); G. Groen van
Prinsterer, Archives ou correspondance de la maison d’Orange-
Nassau, 1e série, 9 vols. (Leiden, 1841-1861); G. Groen van Prinsterer,
Maurice et Barneveldt (Utrecht, 1875); J. L. Motley, Life and Death of
John of Barneveldt (2 vols., The Hague, 1894); C. M. Kemp, v.d.
Maurits v. Nassau, prins v. Oranje in zijn leven en verdiensten (4 vols.,
Rotterdam, 1845); M. O. Nutting, The Days of Prince Maurice (Boston
and Chicago, 1894).
MAURISTS, a congregation of French Benedictines called after St
Maurus (d. 565), a disciple of St Benedict and the legendary introducer of
the Benedictine rule and life into Gaul.1 At the end of the 16th century the
Benedictine monasteries of France had fallen into a state of disorganization
Page 110
and relaxation. In the abbey of St Vaune near Verdun a reform was initiated
by Dom Didier de la Cour, which spread to other houses in Lorraine, and in
1604 the reformed congregation of St Vaune was established, the most
distinguished members of which were Ceillier and Calmet. A number of
French houses joined the new congregation; but as Lorraine was still
independent of the French crown, it was considered desirable to form on the
same lines a separate congregation for France. Thus in 1621 was established
the famous French congregation of St Maur. Most of the Benedictine
monasteries of France, except those belonging to Cluny, gradually joined
the new congregation, which eventually embraced nearly two hundred
houses. The chief house was Saint-Germain-des-Prés, Paris, the residence
of the superior-general and centre of the literary activity of the
congregation. The primary idea of the movement was not the undertaking of
literary and historical work, but the return to a strict monastic régime and
the faithful carrying out of Benedictine life; and throughout the most
glorious period of Maurist history the literary work was not allowed to
interfere with the due performance of the choral office and the other duties
of the monastic life. Towards the end of the 18th century a tendency crept
in, in some quarters, to relax the monastic observances in favour of study;
but the constitutions of 1770 show that a strict monastic régime was
maintained until the end. The course of Maurist history and work was
checkered by the ecclesiastical controversies that distracted the French
Church during the 17th and 18th centuries. Some of the members identified
themselves with the Jansenist cause; but the bulk, including nearly all the
greatest names, pursued a middle path, opposing the lax moral theology
condemned in 1679 by Pope Innocent XI., and adhering to those strong
views on grace and predestination associated with the Augustinian and
Thomist schools of Catholic theology; and like all the theological faculties
and schools on French soil, they were bound to teach the four Gallican
articles. It seems that towards the end of the 18th century a rationalistic and
by Dom Didier de la Cour, which spread to other houses in Lorraine, and in
1604 the reformed congregation of St Vaune was established, the most
distinguished members of which were Ceillier and Calmet. A number of
French houses joined the new congregation; but as Lorraine was still
independent of the French crown, it was considered desirable to form on the
same lines a separate congregation for France. Thus in 1621 was established
the famous French congregation of St Maur. Most of the Benedictine
monasteries of France, except those belonging to Cluny, gradually joined
the new congregation, which eventually embraced nearly two hundred
houses. The chief house was Saint-Germain-des-Prés, Paris, the residence
of the superior-general and centre of the literary activity of the
congregation. The primary idea of the movement was not the undertaking of
literary and historical work, but the return to a strict monastic régime and
the faithful carrying out of Benedictine life; and throughout the most
glorious period of Maurist history the literary work was not allowed to
interfere with the due performance of the choral office and the other duties
of the monastic life. Towards the end of the 18th century a tendency crept
in, in some quarters, to relax the monastic observances in favour of study;
but the constitutions of 1770 show that a strict monastic régime was
maintained until the end. The course of Maurist history and work was
checkered by the ecclesiastical controversies that distracted the French
Church during the 17th and 18th centuries. Some of the members identified
themselves with the Jansenist cause; but the bulk, including nearly all the
greatest names, pursued a middle path, opposing the lax moral theology
condemned in 1679 by Pope Innocent XI., and adhering to those strong
views on grace and predestination associated with the Augustinian and
Thomist schools of Catholic theology; and like all the theological faculties
and schools on French soil, they were bound to teach the four Gallican
articles. It seems that towards the end of the 18th century a rationalistic and
Page 111
free-thinking spirit invaded some of the houses. The congregation was
suppressed and the monks scattered at the revolution, the last superior-
general with forty of his monks dying on the scaffold in Paris. The present
French congregation of Benedictines initiated by Dom Guéranger in 1833 is
a new creation and has no continuity with the congregation of St Maur.
The great claim of the Maurists to the gratitude and admiration of
posterity is their historical and critical school, which stands quite alone in
history, and produced an extraordinary number of colossal works of
erudition which still are of permanent value. The foundations of this school
were laid by Dom Tarisse, the first superior-general, who in 1632 issued
instructions to the superiors of the monasteries to train the young monks in
the habits of research and of organized work. The pioneers in production
were Ménard and d’Achery.
The following tables give, divided into groups, the most important
Maurist works, along with such information as may be useful to
students. All works are folio when not otherwise noted:—
I.—The Editions of the Fathers
Epistle of Barnabas Ménard 1645 1 in
(editio princeps) 4to
Lanfranc d’Achery 1648 1
Guibert of Nogent d’Achery 1651 1
Robert Pulleyn and Peter Mathou 1655 1
of Poitiers
Bernard Mabillon 1667 2
Anselm Gerberon 1675 1
Cassiodorus Garet 1679 1
suppressed and the monks scattered at the revolution, the last superior-
general with forty of his monks dying on the scaffold in Paris. The present
French congregation of Benedictines initiated by Dom Guéranger in 1833 is
a new creation and has no continuity with the congregation of St Maur.
The great claim of the Maurists to the gratitude and admiration of
posterity is their historical and critical school, which stands quite alone in
history, and produced an extraordinary number of colossal works of
erudition which still are of permanent value. The foundations of this school
were laid by Dom Tarisse, the first superior-general, who in 1632 issued
instructions to the superiors of the monasteries to train the young monks in
the habits of research and of organized work. The pioneers in production
were Ménard and d’Achery.
The following tables give, divided into groups, the most important
Maurist works, along with such information as may be useful to
students. All works are folio when not otherwise noted:—
I.—The Editions of the Fathers
Epistle of Barnabas Ménard 1645 1 in
(editio princeps) 4to
Lanfranc d’Achery 1648 1
Guibert of Nogent d’Achery 1651 1
Robert Pulleyn and Peter Mathou 1655 1
of Poitiers
Bernard Mabillon 1667 2
Anselm Gerberon 1675 1
Cassiodorus Garet 1679 1
Page 112
Augustine (see Kukula, Delfau, Blampin, 1681- 11
Die Mauriner- Coustant, 1700
Ausgabe des Guesnie
Augustinus, 1898)
Ambrose du Frische 1686- 2
1690
Acta martyrum sincera Ruinart 1689 1
Hilary Coustant 1693 1
Jerome Martianay 1693- 5
1706
Athanasius Loppin and 1698 3
Montfaucon
Gregory of Tours Ruinart 1699 1
Gregory the Great Sainte-Marthe 1705 4
Hildebert of Tours Beaugendre 1708 1
Irenaeus Massuet 1710 1
Chrysostom Montfaucon 1718- 13
1738
Cyril of Jerusalem Touttée and 1720 1
Maran
Epistolae romanorum Coustant 1721 1
pontificum2
Basil Garnier and 1721- 3
Maran 1730
Cyprian (Baluze, not a 1726 1
Maurist)
finished by
Maran
Origen Ch. de la Rue (1, 1733- 4
2, 3) V. de la 1759
Die Mauriner- Coustant, 1700
Ausgabe des Guesnie
Augustinus, 1898)
Ambrose du Frische 1686- 2
1690
Acta martyrum sincera Ruinart 1689 1
Hilary Coustant 1693 1
Jerome Martianay 1693- 5
1706
Athanasius Loppin and 1698 3
Montfaucon
Gregory of Tours Ruinart 1699 1
Gregory the Great Sainte-Marthe 1705 4
Hildebert of Tours Beaugendre 1708 1
Irenaeus Massuet 1710 1
Chrysostom Montfaucon 1718- 13
1738
Cyril of Jerusalem Touttée and 1720 1
Maran
Epistolae romanorum Coustant 1721 1
pontificum2
Basil Garnier and 1721- 3
Maran 1730
Cyprian (Baluze, not a 1726 1
Maurist)
finished by
Maran
Origen Ch. de la Rue (1, 1733- 4
2, 3) V. de la 1759
Page 113
Rue (4)
Justin and the Apologists Maran 1742 1
Gregory Nazianzen3 Maran and 1778 1
Clémencet
II.—Biblical Works
St Jerome’s Latin Bible Martianay 1693 1
Origen’s Hexapla Montfaucon 1713 2
Old Latin versions Sabbathier 1743- 3
1749
III.—Great Collections of Documents
Spicilegium d’Achery 1655- 13
1677 in
4to
Veterae analecta Mabillon 1675- 4 in
1685 8vo
Musaeum italicum Mabillon 1687- 2 in
1689 4to
Collectio nova patrum Montfaucon 1706 2
graecorum
Thesaurus novus Martène and 1717 5
anecdotorum Durand
Veterum scriptorum Martène and 1724- 9
collectio Durand 1733
De antiquis Martène 1690-
ecclesiaeritibus 1706
(Final form) 1736- 4
1738
Justin and the Apologists Maran 1742 1
Gregory Nazianzen3 Maran and 1778 1
Clémencet
II.—Biblical Works
St Jerome’s Latin Bible Martianay 1693 1
Origen’s Hexapla Montfaucon 1713 2
Old Latin versions Sabbathier 1743- 3
1749
III.—Great Collections of Documents
Spicilegium d’Achery 1655- 13
1677 in
4to
Veterae analecta Mabillon 1675- 4 in
1685 8vo
Musaeum italicum Mabillon 1687- 2 in
1689 4to
Collectio nova patrum Montfaucon 1706 2
graecorum
Thesaurus novus Martène and 1717 5
anecdotorum Durand
Veterum scriptorum Martène and 1724- 9
collectio Durand 1733
De antiquis Martène 1690-
ecclesiaeritibus 1706
(Final form) 1736- 4
1738
Page 114
IV.—Monastic History
Acta of the Benedictine d’Achery, 1668- 9
Saints Mabillon and 1701
Ruinart
Benedictine Annals (to Mabillon (1-4), 1703- 6
1157) Massuet (5), 1739
Martène (6)
V.—Ecclesiastical History and Antiquities of France
A.—General.
Gallia Christiana (3 other Sainte-Marthe (1, 1715- 13
vols. were published 2, 3) 1785
1856-1865)
Monuments de la Montfaucon 1729- 5
monarchie française 1733
Histoire littéraire de la Rivet, 1733- 12
France (16 other vols. Clémencet, 1763 in
were published 1814- Clément 4to
1881)
Recueil des historiens de Bouquet (1-8), 1738- 19
la France (4 other Brial (12-19) 1833
vols. were published
1840-1876)
Concilia Galliae (the Labbat 1789 1
printing of vol. ii.
was interrupted by
the Revolution; there
Acta of the Benedictine d’Achery, 1668- 9
Saints Mabillon and 1701
Ruinart
Benedictine Annals (to Mabillon (1-4), 1703- 6
1157) Massuet (5), 1739
Martène (6)
V.—Ecclesiastical History and Antiquities of France
A.—General.
Gallia Christiana (3 other Sainte-Marthe (1, 1715- 13
vols. were published 2, 3) 1785
1856-1865)
Monuments de la Montfaucon 1729- 5
monarchie française 1733
Histoire littéraire de la Rivet, 1733- 12
France (16 other vols. Clémencet, 1763 in
were published 1814- Clément 4to
1881)
Recueil des historiens de Bouquet (1-8), 1738- 19
la France (4 other Brial (12-19) 1833
vols. were published
1840-1876)
Concilia Galliae (the Labbat 1789 1
printing of vol. ii.
was interrupted by
the Revolution; there
Page 115
were to have been 8
vols.)
B.—Histories of the Provinces.
Bretagne Lobineau 1707 2
Paris Félibien and 1725 5
Lobineau
Languedoc Vaissette and de 1730- 5
Vic 1745
Bourgogne Plancher (1-3), 1739- 4
Merle (4) 1748,
1781
Bretagne Morice 1742- 5
1756
VI.—Miscellaneous Works of Technical Erudition
De re diplomatica Mabillon 1681 1
Ditto Supplement Mabillon 1704 1
Nouveau traité de Toustain and 1750- 6 in
diplomatique Tassin 1765 4to
Paleographia graeca Montfaucon 1708 1
Bibliotheca coisliniana Montfaucon 1715 1
Bibliotheca Montfaucon 1739 2
bibliothecarum
manuscriptorum nova
L’Antiquité expliqué Montfaucon 1719- 15
1724
New ed. of Du Cange’s Dantine and 1733- 6
glossarium Carpentier 1736
Ditto Supplement Carpentier 1766 4
vols.)
B.—Histories of the Provinces.
Bretagne Lobineau 1707 2
Paris Félibien and 1725 5
Lobineau
Languedoc Vaissette and de 1730- 5
Vic 1745
Bourgogne Plancher (1-3), 1739- 4
Merle (4) 1748,
1781
Bretagne Morice 1742- 5
1756
VI.—Miscellaneous Works of Technical Erudition
De re diplomatica Mabillon 1681 1
Ditto Supplement Mabillon 1704 1
Nouveau traité de Toustain and 1750- 6 in
diplomatique Tassin 1765 4to
Paleographia graeca Montfaucon 1708 1
Bibliotheca coisliniana Montfaucon 1715 1
Bibliotheca Montfaucon 1739 2
bibliothecarum
manuscriptorum nova
L’Antiquité expliqué Montfaucon 1719- 15
1724
New ed. of Du Cange’s Dantine and 1733- 6
glossarium Carpentier 1736
Ditto Supplement Carpentier 1766 4
Page 116
Apparatus ad le Nourry 1703 2
bibliothecam
maximam patrum
L’Art de vérifier les dates Dantine, Durand, 1750 1 in
Clémencet 4to
Ed. 2 Clément 1770 1
Ed. 3 Clément 1783- 3
1787
The 58 works in the above list comprise 199 great folio volumes and
39 in 4to or 8vo. The full Maurist bibliography contains the names of
some 220 writers and more than 700 works. The lesser works in large
measure cover the same fields as those in the list, but the number of
works of purely religious character, of piety, devotion and edification,
is very striking. Perhaps the most wonderful phenomenon of Maurist
work is that what was produced was only a portion of what was
contemplated and prepared for. The French Revolution cut short many
gigantic undertakings, the collected materials for which fill hundreds of
manuscript volumes in the Bibliothèque nationale of Paris and other
libraries of France. There are at Paris 31 volumes of Berthereau’s
materials for the Historians of the Crusades, not only in Latin and
Greek, but in the oriental tongues; from them have been taken in great
measure the Recueil des historiens des croisades, whereof 15 folio
volumes have been published by the Académie des Inscriptions. There
exist also the preparations for an edition of Rufinus and one of
Eusebius, and for the continuation of the Papal Letters and of the
Concilia Galliae. Dom Caffiaux and Dom Villevielle left 236 volumes
of materials for a Trésor généalogique. There are Benedictine
Antiquities (37 vols.), a Monasticon Gallicanum and a Monasticon
bibliothecam
maximam patrum
L’Art de vérifier les dates Dantine, Durand, 1750 1 in
Clémencet 4to
Ed. 2 Clément 1770 1
Ed. 3 Clément 1783- 3
1787
The 58 works in the above list comprise 199 great folio volumes and
39 in 4to or 8vo. The full Maurist bibliography contains the names of
some 220 writers and more than 700 works. The lesser works in large
measure cover the same fields as those in the list, but the number of
works of purely religious character, of piety, devotion and edification,
is very striking. Perhaps the most wonderful phenomenon of Maurist
work is that what was produced was only a portion of what was
contemplated and prepared for. The French Revolution cut short many
gigantic undertakings, the collected materials for which fill hundreds of
manuscript volumes in the Bibliothèque nationale of Paris and other
libraries of France. There are at Paris 31 volumes of Berthereau’s
materials for the Historians of the Crusades, not only in Latin and
Greek, but in the oriental tongues; from them have been taken in great
measure the Recueil des historiens des croisades, whereof 15 folio
volumes have been published by the Académie des Inscriptions. There
exist also the preparations for an edition of Rufinus and one of
Eusebius, and for the continuation of the Papal Letters and of the
Concilia Galliae. Dom Caffiaux and Dom Villevielle left 236 volumes
of materials for a Trésor généalogique. There are Benedictine
Antiquities (37 vols.), a Monasticon Gallicanum and a Monasticon
Page 117
Benedictinum (54 vols.). Of the Histories of the Provinces of France
barely half a dozen were printed, but all were in hand, and the
collections for the others fill 800 volumes of MSS. The materials for a
geography of Gaul and France in 50 volumes perished in a fire during
the Revolution.
When these figures were considered, and when one contemplates the
vastness of the works in progress during any decade of the century
1680-1780; and still more, when not only the quantity but the quality of
the work, and the abiding value of most of it is realized, it will be
recognized that the output was prodigious and unique in the history of
letters, as coming from a single society. The qualities that have made
Maurist work proverbial for sound learning are its fine critical tact and
its thoroughness.
The chief source of information on the Maurists and their work is
Dom Tassin’s Histoire littéraire de la congregation de Saint-Maur
(1770); it has been reduced to a bare bibliography and completed by de
Lama, Bibliothèque des écrivains de la congr. de S.-M. (1882). The two
works of de Broglie, Mabillon (2 vols., 1888) and Montfaucon (2 vols.,
1891), give a charming picture of the inner life of the great Maurists of
the earlier generation in the midst of their work and their friends.
Sketches of the lives of a few of the chief Maurists will be found in
McCarthy’s Principal Writers of the Congr. of S. M. (1868). Useful
information about their literary undertakings will be found in De
Lisle’s Cabinet des MSS. de la Bibl. Nat. Fonds St Germain-des-Prés.
General information will be found in the standard authorities: Helyot,
Hist. des ordres religieux (1718), vi. c. 37; Heimbucher, Orden und
Kongregationen (1907) i. § 36; Wetzer und Welte, Kirchenlexicon (ed.
2) and Herzog-Hauck’s Realencyklopädie (ed. 3), the latter an
barely half a dozen were printed, but all were in hand, and the
collections for the others fill 800 volumes of MSS. The materials for a
geography of Gaul and France in 50 volumes perished in a fire during
the Revolution.
When these figures were considered, and when one contemplates the
vastness of the works in progress during any decade of the century
1680-1780; and still more, when not only the quantity but the quality of
the work, and the abiding value of most of it is realized, it will be
recognized that the output was prodigious and unique in the history of
letters, as coming from a single society. The qualities that have made
Maurist work proverbial for sound learning are its fine critical tact and
its thoroughness.
The chief source of information on the Maurists and their work is
Dom Tassin’s Histoire littéraire de la congregation de Saint-Maur
(1770); it has been reduced to a bare bibliography and completed by de
Lama, Bibliothèque des écrivains de la congr. de S.-M. (1882). The two
works of de Broglie, Mabillon (2 vols., 1888) and Montfaucon (2 vols.,
1891), give a charming picture of the inner life of the great Maurists of
the earlier generation in the midst of their work and their friends.
Sketches of the lives of a few of the chief Maurists will be found in
McCarthy’s Principal Writers of the Congr. of S. M. (1868). Useful
information about their literary undertakings will be found in De
Lisle’s Cabinet des MSS. de la Bibl. Nat. Fonds St Germain-des-Prés.
General information will be found in the standard authorities: Helyot,
Hist. des ordres religieux (1718), vi. c. 37; Heimbucher, Orden und
Kongregationen (1907) i. § 36; Wetzer und Welte, Kirchenlexicon (ed.
2) and Herzog-Hauck’s Realencyklopädie (ed. 3), the latter an
Page 118
interesting appreciation by the Protestant historian Otto Zöckler of the
spirit and the merits of the work of the Maurists. (E. C. B.)
1 His festival is kept on the 15th of January. He founded the monastery of Glanfeuil or
St Maur-sur-Loire.
2 14 vols. of materials collected for the continuation are at Paris.
3 The printing of vol. ii. was impeded by the Revolution.
MAURITIUS, an island and British colony in the Indian Ocean
(known whilst a French possession as the Île de France). It lies between 57°
18′ and 57° 49′ E., and 19° 58′ and 20° 32′ S., 550 m. E. of Madagascar,
2300 m. from the Cape of Good Hope, and 9500 m. from England via Suez.
The island is irregularly elliptical—somewhat triangular—in shape, and is
36 m. long from N.N.E. to S.S.W., and about 23 m. broad. It is 130 m. in
circumference, and its total area is about 710 sq. m. (For map see
Madagascar.) The island is surrounded by coral reefs, so that the ports are
difficult of access.
From its mountainous character Mauritius is a most picturesque island,
and its scenery is very varied and beautiful. It has been admirably described
by Bernardin de St Pierre, who lived in the island towards the close of the
18th century, in Paul et Virginie. The most level portions of the coast
districts are the north and north-east, all the rest being broken by hills,
which vary from 500 to 2700 ft. in height. The principal mountain masses
spirit and the merits of the work of the Maurists. (E. C. B.)
1 His festival is kept on the 15th of January. He founded the monastery of Glanfeuil or
St Maur-sur-Loire.
2 14 vols. of materials collected for the continuation are at Paris.
3 The printing of vol. ii. was impeded by the Revolution.
MAURITIUS, an island and British colony in the Indian Ocean
(known whilst a French possession as the Île de France). It lies between 57°
18′ and 57° 49′ E., and 19° 58′ and 20° 32′ S., 550 m. E. of Madagascar,
2300 m. from the Cape of Good Hope, and 9500 m. from England via Suez.
The island is irregularly elliptical—somewhat triangular—in shape, and is
36 m. long from N.N.E. to S.S.W., and about 23 m. broad. It is 130 m. in
circumference, and its total area is about 710 sq. m. (For map see
Madagascar.) The island is surrounded by coral reefs, so that the ports are
difficult of access.
From its mountainous character Mauritius is a most picturesque island,
and its scenery is very varied and beautiful. It has been admirably described
by Bernardin de St Pierre, who lived in the island towards the close of the
18th century, in Paul et Virginie. The most level portions of the coast
districts are the north and north-east, all the rest being broken by hills,
which vary from 500 to 2700 ft. in height. The principal mountain masses
Page 119
are the north-western or Pouce range, in the district of Port Louis; the
south-western, in the districts of Rivière Noire and Savanne; and the south-
eastern range, in the Grand Port district. In the first of these, which consists
of one principal ridge with several lateral spurs, overlooking Port Louis, are
the singular peak of the Pouce (2650 ft.), so called from its supposed
resemblance to the human thumb; and the still loftier Pieter Botte (2685 ft.),
a tall obelisk of bare rock, crowned with a globular mass of stone. The
highest summit in the island is in the south-western mass of hills, the Piton
de la Rivière Noire, which is 2711 ft. above the sea. The south-eastern
group of hills consists of the Montagne du Bambou, with several spurs
running down to the sea. In the interior are extensive fertile plains, some
1200 ft. in height, forming the districts of Moka, Vacois, and Plaines
Wilhelms; and from nearly the centre of the island an abrupt peak, the Piton
du Milieu de l’Île rises to a height of 1932 ft. Other prominent summits are
the Trois Mamelles, the Montagne du Corps de Garde, the Signal Mountain,
near Port Louis, and the Morne Brabant, at the south-west corner of the
island.
The rivers are small, and none is navigable beyond a few hundred yards
from the sea. In the dry season little more than brooks, they become raging
torrents in the wet season. The principal stream is the Grande Rivière, with
a course of about 10 m. There is a remarkable and very deep lake, called
Grand Bassin, in the south of the island, it is probably the extinct crater of
an ancient volcano; similar lakes are the Mare aux Vacois and the Mare aux
Joncs, and there are other deep hollows which have a like origin.
Geology.—The island is of volcanic origin, but has ceased to show
signs of volcanic activity. All the rocks are of basalt and greyish-tinted
lavas, excepting some beds of upraised coral. Columnar basalt is seen
in several places. The remains of ancient craters can be distinguished,
but their outlines have been greatly destroyed by denudation. There are
south-western, in the districts of Rivière Noire and Savanne; and the south-
eastern range, in the Grand Port district. In the first of these, which consists
of one principal ridge with several lateral spurs, overlooking Port Louis, are
the singular peak of the Pouce (2650 ft.), so called from its supposed
resemblance to the human thumb; and the still loftier Pieter Botte (2685 ft.),
a tall obelisk of bare rock, crowned with a globular mass of stone. The
highest summit in the island is in the south-western mass of hills, the Piton
de la Rivière Noire, which is 2711 ft. above the sea. The south-eastern
group of hills consists of the Montagne du Bambou, with several spurs
running down to the sea. In the interior are extensive fertile plains, some
1200 ft. in height, forming the districts of Moka, Vacois, and Plaines
Wilhelms; and from nearly the centre of the island an abrupt peak, the Piton
du Milieu de l’Île rises to a height of 1932 ft. Other prominent summits are
the Trois Mamelles, the Montagne du Corps de Garde, the Signal Mountain,
near Port Louis, and the Morne Brabant, at the south-west corner of the
island.
The rivers are small, and none is navigable beyond a few hundred yards
from the sea. In the dry season little more than brooks, they become raging
torrents in the wet season. The principal stream is the Grande Rivière, with
a course of about 10 m. There is a remarkable and very deep lake, called
Grand Bassin, in the south of the island, it is probably the extinct crater of
an ancient volcano; similar lakes are the Mare aux Vacois and the Mare aux
Joncs, and there are other deep hollows which have a like origin.
Geology.—The island is of volcanic origin, but has ceased to show
signs of volcanic activity. All the rocks are of basalt and greyish-tinted
lavas, excepting some beds of upraised coral. Columnar basalt is seen
in several places. The remains of ancient craters can be distinguished,
but their outlines have been greatly destroyed by denudation. There are
Page 120
many caverns and steep ravines, and from the character of the rocks the
ascents are rugged and precipitous. The island has few minerals,
although iron, lead and copper in very small quantities have in former
times been obtained. The greater part of the surface is composed of a
volcanic breccia, with here and there lava-streams exposed in ravines,
and sometimes on the surface. The commonest lavas are dolerites. In at
least two places sedimentary rocks are found at considerable
elevations. In the Black River Mountains, at a height of about 1200 ft.,
there is a clay-slate; and near Midlands, in the Grand Port group of
mountains, a chloritic schist occurs about 1700 ft. above the sea,
forming the hill of La Selle. This schist is much contorted, but seems to
have a general dip to the south or south-east. Evidence of recent
elevation of the island is furnished by masses of coral reef and beach
coral rock standing at heights of 40 ft. above sea-level in the south, 12
ft. in the north and 7 ft. on the islands situated on the bank extending to
the north-east.1
Climate.—The climate is pleasant during the cool season of the year,
but oppressively hot in summer (December to April), except in the
elevated plains of the interior, where the thermometer ranges from 70°
to 80° F., while in Port Louis and on the coast generally it ranges from
90° to 96°. The mean temperature for the year at Port Louis is 78.6°.
There are two seasons, the cool and comparatively dry season, from
April to November, and the hotter season, during the rest of the year.
The climate is now less healthy than it was, severe epidemics of
malarial fever having frequently occurred, so that malaria now appears
to be endemic among the non-European population. The rainfall varies
greatly in different parts of the island. Cluny in the Grand Port (south-
eastern) district has a mean annual rainfall of 145 in.; Albion on the
west coast is the driest station, with a mean annual rainfall of 31 in.
The mean monthly rainfall for the whole island varies from 12 in. in
ascents are rugged and precipitous. The island has few minerals,
although iron, lead and copper in very small quantities have in former
times been obtained. The greater part of the surface is composed of a
volcanic breccia, with here and there lava-streams exposed in ravines,
and sometimes on the surface. The commonest lavas are dolerites. In at
least two places sedimentary rocks are found at considerable
elevations. In the Black River Mountains, at a height of about 1200 ft.,
there is a clay-slate; and near Midlands, in the Grand Port group of
mountains, a chloritic schist occurs about 1700 ft. above the sea,
forming the hill of La Selle. This schist is much contorted, but seems to
have a general dip to the south or south-east. Evidence of recent
elevation of the island is furnished by masses of coral reef and beach
coral rock standing at heights of 40 ft. above sea-level in the south, 12
ft. in the north and 7 ft. on the islands situated on the bank extending to
the north-east.1
Climate.—The climate is pleasant during the cool season of the year,
but oppressively hot in summer (December to April), except in the
elevated plains of the interior, where the thermometer ranges from 70°
to 80° F., while in Port Louis and on the coast generally it ranges from
90° to 96°. The mean temperature for the year at Port Louis is 78.6°.
There are two seasons, the cool and comparatively dry season, from
April to November, and the hotter season, during the rest of the year.
The climate is now less healthy than it was, severe epidemics of
malarial fever having frequently occurred, so that malaria now appears
to be endemic among the non-European population. The rainfall varies
greatly in different parts of the island. Cluny in the Grand Port (south-
eastern) district has a mean annual rainfall of 145 in.; Albion on the
west coast is the driest station, with a mean annual rainfall of 31 in.
The mean monthly rainfall for the whole island varies from 12 in. in
Page 121
March to 2.6 in. in September and October. The Royal Alfred
Observatory is situated at Pamplemousses, on the north-west or dry
side of the island. From January to the middle of April, Mauritius, in
common with the neighbouring islands and the surrounding ocean from
8° to 30° of southern latitude is subject to severe cyclones,
accompanied by torrents of rain, which often cause great destruction to
houses and plantations. These hurricanes generally last about eight
hours, but they appear to be less frequent and violent than in former
times, owing, it is thought, to the destruction of the ancient forests and
the consequent drier condition of the atmosphere.
Fauna and Flora.—Mauritius being an oceanic island of small size,
its present fauna is very limited in extent. When first seen by
Europeans it contained no mammals except a large fruit-eating bat
(Pteropus vulgaris), which is plentiful in the woods; but several
mammals have been introduced, and are now numerous in the
uncultivated region. Among these are two monkeys of the genera
Macacus and Cercopithecus, a stag (Cervus hippelaphus), a small hare,
a shrew-mouse, and the ubiquitous rat. A lemur and one of the curious
hedgehog-like Insectivora of Madagascar (Centetes ecaudatus) have
probably both been brought from the larger island. The avifauna
resembles that of Madagascar; there are species of a peculiar genus of
caterpillar shrikes (Campephagidae), as well as of the genera
Pratincola, Hypsipetes, Phedina, Tchitrea, Zosterops, Foudia,
Collocalia and Coracopsis, and peculiar forms of doves and parakeets.
The living reptiles are small and few in number. The surrounding seas
contain great numbers of fish; the coral reefs abound with a great
variety of molluscs; and there are numerous land-shells. The extinct
fauna of Mauritius has considerable interest. In common with the other
Mascarene islands, it was the home of the dodo (Didus ineptus); there
were also Aphanapteryx, a species of rail, and a short-winged heron
Observatory is situated at Pamplemousses, on the north-west or dry
side of the island. From January to the middle of April, Mauritius, in
common with the neighbouring islands and the surrounding ocean from
8° to 30° of southern latitude is subject to severe cyclones,
accompanied by torrents of rain, which often cause great destruction to
houses and plantations. These hurricanes generally last about eight
hours, but they appear to be less frequent and violent than in former
times, owing, it is thought, to the destruction of the ancient forests and
the consequent drier condition of the atmosphere.
Fauna and Flora.—Mauritius being an oceanic island of small size,
its present fauna is very limited in extent. When first seen by
Europeans it contained no mammals except a large fruit-eating bat
(Pteropus vulgaris), which is plentiful in the woods; but several
mammals have been introduced, and are now numerous in the
uncultivated region. Among these are two monkeys of the genera
Macacus and Cercopithecus, a stag (Cervus hippelaphus), a small hare,
a shrew-mouse, and the ubiquitous rat. A lemur and one of the curious
hedgehog-like Insectivora of Madagascar (Centetes ecaudatus) have
probably both been brought from the larger island. The avifauna
resembles that of Madagascar; there are species of a peculiar genus of
caterpillar shrikes (Campephagidae), as well as of the genera
Pratincola, Hypsipetes, Phedina, Tchitrea, Zosterops, Foudia,
Collocalia and Coracopsis, and peculiar forms of doves and parakeets.
The living reptiles are small and few in number. The surrounding seas
contain great numbers of fish; the coral reefs abound with a great
variety of molluscs; and there are numerous land-shells. The extinct
fauna of Mauritius has considerable interest. In common with the other
Mascarene islands, it was the home of the dodo (Didus ineptus); there
were also Aphanapteryx, a species of rail, and a short-winged heron
Page 122
(Ardea megacephala), which probably seldom flew. The defenceless
condition of these birds led to their extinction after the island was
colonized. Considerable quantities of the bones of the dodo and other
extinct birds—a rail (Aphanapteryx), and a short-winged heron—have
been discovered in the beds of some of the ancient lakes (see Dodo).
Several species of large fossil tortoises have also been discovered; they
are quite different from the living ones of Aldabra, in the same
zoological region.
Owing to the destruction of the primeval forests for the formation of
sugar plantations, the indigenous flora is only seen in parts of the
interior plains, in the river valleys and on the hills; and it is not now
easy to distinguish between what is native and what has come from
abroad. The principal timber tree is the ebony (Diospyros ebeneum),
which grows to a considerable size. Besides this there are bois de
cannelle, olive-tree, benzoin (Croton Benzoe), colophane (Colophonia),
and iron-wood, all of which arc useful in carpentry; the coco-nut palm,
an importation, but a tree which has been so extensively planted during
the last hundred years that it is extremely plentiful; the palmiste (Palma
dactylifera latifolia), the latanier (Corypha umbraculifera) and the
date-palm. The vacoa or vacois, (Pandanus utilis) is largely grown, the
long tough leaves being manufactured into bags for the export of sugar,
and the roots being also made of use; and in the few remnants of the
original forests the traveller’s tree (Urania speciosa), grows
abundantly. A species of bamboo is very plentiful in the river valleys
and in marshy situations. A large variety of fruit is produced, including
the tamarind, mango, banana, pine-apple, guava, shaddock, fig,
avocado-pear, litchi, custard-apple and the mabolo (Diospyros
discolor), a fruit of exquisite flavour, but very disagreeable odour.
Many of the roots and vegetables of Europe have been introduced, as
well as some of those peculiar to the tropics, including maize, millet,
condition of these birds led to their extinction after the island was
colonized. Considerable quantities of the bones of the dodo and other
extinct birds—a rail (Aphanapteryx), and a short-winged heron—have
been discovered in the beds of some of the ancient lakes (see Dodo).
Several species of large fossil tortoises have also been discovered; they
are quite different from the living ones of Aldabra, in the same
zoological region.
Owing to the destruction of the primeval forests for the formation of
sugar plantations, the indigenous flora is only seen in parts of the
interior plains, in the river valleys and on the hills; and it is not now
easy to distinguish between what is native and what has come from
abroad. The principal timber tree is the ebony (Diospyros ebeneum),
which grows to a considerable size. Besides this there are bois de
cannelle, olive-tree, benzoin (Croton Benzoe), colophane (Colophonia),
and iron-wood, all of which arc useful in carpentry; the coco-nut palm,
an importation, but a tree which has been so extensively planted during
the last hundred years that it is extremely plentiful; the palmiste (Palma
dactylifera latifolia), the latanier (Corypha umbraculifera) and the
date-palm. The vacoa or vacois, (Pandanus utilis) is largely grown, the
long tough leaves being manufactured into bags for the export of sugar,
and the roots being also made of use; and in the few remnants of the
original forests the traveller’s tree (Urania speciosa), grows
abundantly. A species of bamboo is very plentiful in the river valleys
and in marshy situations. A large variety of fruit is produced, including
the tamarind, mango, banana, pine-apple, guava, shaddock, fig,
avocado-pear, litchi, custard-apple and the mabolo (Diospyros
discolor), a fruit of exquisite flavour, but very disagreeable odour.
Many of the roots and vegetables of Europe have been introduced, as
well as some of those peculiar to the tropics, including maize, millet,
Page 123
yams, manioc, dhol, gram, &c. Small quantities of tea, rice and sago,
have been grown, as well as many of the spices (cloves, nutmeg,
ginger, pepper and allspice), and also cotton, indigo, betel, camphor,
turmeric and vanilla. The Royal Botanical Gardens at Pamplemousses,
which date from the French occupation of the island, contain a rich
collection of tropical and extra-tropical species.
Inhabitants.—The inhabitants consist of two great divisions, those of
European blood, chiefly French and British, together with numerous half-
caste people, and those of Asiatic or African blood. The population of
European blood, which calls itself Creole, is greater than that of any other
tropical colony; many of the inhabitants trace their descent from ancient
French families, and the higher and middle classes are distinguished for
their intellectual culture. French is more commonly spoken than English.
The Creole class is, however, diminishing, though slowly, and the most
numerous section of the population is of Indian blood.
The introduction of Indian coolies to work the sugar plantations
dates from the period of the emancipation of the slaves in 1834-1839.
At that time the negroes who showed great unwillingness to work on
their late masters’ estates, numbered about 66,000. Immigration from
India began in 1834, and at a census taken in 1846, when the total
population was 158,462, there were already 56,245 Indians in the
island. In 1851 the total population had increased to 180,823, while in
1861 it was 310,050. This great increase was almost entirely due to
Indian immigration, the Indian population, 77,996 in 1851, being
192,634 in 1861. From that year the increase in the Indian population
has been more gradual but steady, while the non-Indian population has
decreased. From 102,827 in 1851 it rose to 117,416 in 1861 to sink to
99,784 in 1871. The figures for the three following census years were:
—
have been grown, as well as many of the spices (cloves, nutmeg,
ginger, pepper and allspice), and also cotton, indigo, betel, camphor,
turmeric and vanilla. The Royal Botanical Gardens at Pamplemousses,
which date from the French occupation of the island, contain a rich
collection of tropical and extra-tropical species.
Inhabitants.—The inhabitants consist of two great divisions, those of
European blood, chiefly French and British, together with numerous half-
caste people, and those of Asiatic or African blood. The population of
European blood, which calls itself Creole, is greater than that of any other
tropical colony; many of the inhabitants trace their descent from ancient
French families, and the higher and middle classes are distinguished for
their intellectual culture. French is more commonly spoken than English.
The Creole class is, however, diminishing, though slowly, and the most
numerous section of the population is of Indian blood.
The introduction of Indian coolies to work the sugar plantations
dates from the period of the emancipation of the slaves in 1834-1839.
At that time the negroes who showed great unwillingness to work on
their late masters’ estates, numbered about 66,000. Immigration from
India began in 1834, and at a census taken in 1846, when the total
population was 158,462, there were already 56,245 Indians in the
island. In 1851 the total population had increased to 180,823, while in
1861 it was 310,050. This great increase was almost entirely due to
Indian immigration, the Indian population, 77,996 in 1851, being
192,634 in 1861. From that year the increase in the Indian population
has been more gradual but steady, while the non-Indian population has
decreased. From 102,827 in 1851 it rose to 117,416 in 1861 to sink to
99,784 in 1871. The figures for the three following census years were:
—
Page 124
1881. 1891. 1901.
Indians 248,993 255,920 259,086
Others 110,881 114,668 111,937
——— ——— ———
Total 359,874 370,588 371,023
——— ——— ———
Including the military and crews of ships in harbour, the total
population in 1901 was 373,336.2 This total included 198,958 Indo-
Mauritians, i.e. persons of Indian descent born in Mauritius, and 62,022
other Indians. There were 3,509 Chinese, while the remaining 108,847
included persons of European, African or mixed descent, Malagasy,
Malays and Sinhalese. The Indian female population increased from
51,019 in 1861 to 115,986 in 1901. In the same period the non-Indian
female population but slightly varied, being 56,070 in 1861 and 55,485
in 1901. The Indo-Mauritians are now dominant in commercial,
agricultural and domestic callings, and much town and agricultural
land has been transferred from the Creole planters to Indians and
Chinese. The tendency to an Indian peasant proprietorship is marked.
Since 1864 real property to the value of over £1,250,000 has been
acquired by Asiatics. Between 1881 and 1901 the number of sugar
estates decreased from 171 to 115, those sold being held in small
parcels by Indians. The average death-rate for the period 1873-1901
was 32.6 per 1000. The average birth-rate in the Indian community is
37 per 1000; in the non-Indian community 34 per 1000. Many
Mauritian Creoles have emigrated to South Africa. The great increase
in the population since 1851 has made Mauritius one of the most
densely peopled regions of the world, having over 520 persons per
square mile.
Indians 248,993 255,920 259,086
Others 110,881 114,668 111,937
——— ——— ———
Total 359,874 370,588 371,023
——— ——— ———
Including the military and crews of ships in harbour, the total
population in 1901 was 373,336.2 This total included 198,958 Indo-
Mauritians, i.e. persons of Indian descent born in Mauritius, and 62,022
other Indians. There were 3,509 Chinese, while the remaining 108,847
included persons of European, African or mixed descent, Malagasy,
Malays and Sinhalese. The Indian female population increased from
51,019 in 1861 to 115,986 in 1901. In the same period the non-Indian
female population but slightly varied, being 56,070 in 1861 and 55,485
in 1901. The Indo-Mauritians are now dominant in commercial,
agricultural and domestic callings, and much town and agricultural
land has been transferred from the Creole planters to Indians and
Chinese. The tendency to an Indian peasant proprietorship is marked.
Since 1864 real property to the value of over £1,250,000 has been
acquired by Asiatics. Between 1881 and 1901 the number of sugar
estates decreased from 171 to 115, those sold being held in small
parcels by Indians. The average death-rate for the period 1873-1901
was 32.6 per 1000. The average birth-rate in the Indian community is
37 per 1000; in the non-Indian community 34 per 1000. Many
Mauritian Creoles have emigrated to South Africa. The great increase
in the population since 1851 has made Mauritius one of the most
densely peopled regions of the world, having over 520 persons per
square mile.
Page 125
Chief Towns.—The capital and seat of government, the city of Port
Louis, is on the north-western side of the island, in 20° 10′ S., 57° 30′
E. at the head of an excellent harbour, a deep inlet about a mile long,
available for ships of the deepest draught. This is protected by Fort
William and Fort George, as well as by the citadel (Fort Adelaide), and
it has three graving-docks connected with the inner harbour, the depths
alongside quays and berths being from 12 to 28 ft. The trade of the
island passes almost entirely through the port. Government House is a
three-storeyed structure with broad verandas, of no particular style of
architecture, while the Protestant cathedral was formerly a powder
magazine, to which a tower and spire have been added. The Roman
Catholic cathedral is more pretentious in style, but is tawdry in its
interior. There are, besides the town-hall, Royal College, public offices
and theatre, large barracks and military stores. Port Louis, which is
governed by an elective municipal council, is surrounded by lofty hills
and its unhealthy situation is aggravated by the difficulty of effective
drainage owing to the small amount of tide in the harbour. Though
much has been done to make the town sanitary, including the provision
of a good water-supply, the death-rate is generally over 44 per 1000.
Consequently all those who can make their homes in the cooler uplands
of the interior. As a result the population of the city decreased from
about 70,000 in 1891 to 53,000 in 1901. The favourite residential town
is Curepipe, where the climate resembles that of the south of France. It
is built on the central plateau about 20 m. distant from Port Louis by
rail and 1800 ft. above the sea. Curepipe was incorporated in 1888 and
had a population (1901) of 13,000. On the railway between Port Louis
and Curepipe are other residential towns—Beau Bassin, Rose Hill and
Quatre Bornes. Mahébourg, pop. (1901), 4810, is a town on the shores
of Grand Port on the south-east side of the island, Souillac a small
town on the south coast.
Louis, is on the north-western side of the island, in 20° 10′ S., 57° 30′
E. at the head of an excellent harbour, a deep inlet about a mile long,
available for ships of the deepest draught. This is protected by Fort
William and Fort George, as well as by the citadel (Fort Adelaide), and
it has three graving-docks connected with the inner harbour, the depths
alongside quays and berths being from 12 to 28 ft. The trade of the
island passes almost entirely through the port. Government House is a
three-storeyed structure with broad verandas, of no particular style of
architecture, while the Protestant cathedral was formerly a powder
magazine, to which a tower and spire have been added. The Roman
Catholic cathedral is more pretentious in style, but is tawdry in its
interior. There are, besides the town-hall, Royal College, public offices
and theatre, large barracks and military stores. Port Louis, which is
governed by an elective municipal council, is surrounded by lofty hills
and its unhealthy situation is aggravated by the difficulty of effective
drainage owing to the small amount of tide in the harbour. Though
much has been done to make the town sanitary, including the provision
of a good water-supply, the death-rate is generally over 44 per 1000.
Consequently all those who can make their homes in the cooler uplands
of the interior. As a result the population of the city decreased from
about 70,000 in 1891 to 53,000 in 1901. The favourite residential town
is Curepipe, where the climate resembles that of the south of France. It
is built on the central plateau about 20 m. distant from Port Louis by
rail and 1800 ft. above the sea. Curepipe was incorporated in 1888 and
had a population (1901) of 13,000. On the railway between Port Louis
and Curepipe are other residential towns—Beau Bassin, Rose Hill and
Quatre Bornes. Mahébourg, pop. (1901), 4810, is a town on the shores
of Grand Port on the south-east side of the island, Souillac a small
town on the south coast.
Page 126
Industries.—The Sugar Plantations: The soil of the island is of
considerable fertility; it is a ferruginous red clay, but so largely mingled
with stones of all sizes that no plough can be used, and the hoe has to
be employed to prepare the ground for cultivation. The greater portion
of the plains is now a vast sugar plantation. The bright green of the
sugar fields is a striking feature in a view of Mauritius from the sea,
and gives a peculiar beauty and freshness to the prospect. The soil is
suitable for the cultivation of almost all kinds of tropical produce, and
it is to be regretted that the prosperity of the colony depends almost
entirely on one article of production, for the consequences are serious
when there is a failure, more or less, of the sugar crop. Guano is
extensively imported as a manure, and by its use the natural fertility of
the soil has been increased to a wonderful extent. Since the beginning
of the 20th century some attention has been paid to the cultivation of
tea and cotton, with encouraging results. Of the exports, sugar amounts
on an average to about 95% of the total. The quantity of sugar exported
rose from 102,000 tons in 1854 to 189,164 tons in 1877. The
competition of beet-sugar and the effect of bounties granted by various
countries then began to tell on the production in Mauritius, the average
crop for the seven years ending 1900-1901 being only 150,449 tons.
The Brussels Sugar Convention of 1902 led to an increase in
production, the average annual weight of sugar exported for the three
years 1904-1906 being 182,000 tons. The value of the crop was
likewise seriously affected by the causes mentioned, and by various
diseases which attacked the canes. Thus in 1878 the value of the sugar
exported was £3,408,000; in 1888 it had sunk to £1,911,000, and in
1898 to £1,632,000. In 1900 the value was £1,922,000, and in 1905 it
had risen to £2,172,000. India and the South African colonies between
them take some two-thirds of the total produce. The remainder is taken
chiefly by Great Britain, Canada and Hong-Kong. Next to sugar, aloe-
considerable fertility; it is a ferruginous red clay, but so largely mingled
with stones of all sizes that no plough can be used, and the hoe has to
be employed to prepare the ground for cultivation. The greater portion
of the plains is now a vast sugar plantation. The bright green of the
sugar fields is a striking feature in a view of Mauritius from the sea,
and gives a peculiar beauty and freshness to the prospect. The soil is
suitable for the cultivation of almost all kinds of tropical produce, and
it is to be regretted that the prosperity of the colony depends almost
entirely on one article of production, for the consequences are serious
when there is a failure, more or less, of the sugar crop. Guano is
extensively imported as a manure, and by its use the natural fertility of
the soil has been increased to a wonderful extent. Since the beginning
of the 20th century some attention has been paid to the cultivation of
tea and cotton, with encouraging results. Of the exports, sugar amounts
on an average to about 95% of the total. The quantity of sugar exported
rose from 102,000 tons in 1854 to 189,164 tons in 1877. The
competition of beet-sugar and the effect of bounties granted by various
countries then began to tell on the production in Mauritius, the average
crop for the seven years ending 1900-1901 being only 150,449 tons.
The Brussels Sugar Convention of 1902 led to an increase in
production, the average annual weight of sugar exported for the three
years 1904-1906 being 182,000 tons. The value of the crop was
likewise seriously affected by the causes mentioned, and by various
diseases which attacked the canes. Thus in 1878 the value of the sugar
exported was £3,408,000; in 1888 it had sunk to £1,911,000, and in
1898 to £1,632,000. In 1900 the value was £1,922,000, and in 1905 it
had risen to £2,172,000. India and the South African colonies between
them take some two-thirds of the total produce. The remainder is taken
chiefly by Great Britain, Canada and Hong-Kong. Next to sugar, aloe-
Page 127
fibre is the most important export, the average annual export for the
five years ending 1906 being 1840 tons. In addition, a considerable
quantity of molasses and smaller quantities of rum, vanilla and coco-
nut oil are exported. The imports are mainly rice, wheat, cotton goods,
wine, coal, hardware and haberdashery, and guano. The rice comes
principally from India and Madagascar; cattle are imported from
Madagascar, sheep from South Africa and Australia, and frozen meat
from Australia. The average annual value of the exports for the ten
years 1896-1905 was £2,153,159; the average annual value of the
imports for the same period £1,453,089. These figures when compared
with those in years before the beet and bounty-fed sugar had entered
into severe competition with cane sugar, show how greatly the island
had thereby suffered. In 1864 the exports were valued at £2,249,000; in
1868 at £2,339,000; in 1877 at £4,201,000 and in 1880 at £3,634,000.
And in each of the years named the imports exceeded £2,000,000 in
value. Nearly all the aloe-fibre exported is taken by Great Britain, and
France, while the molasses goes to India. Among the minor exports is
that of bambara or sea-slugs, which are sent to Hong-Kong and
Singapore. This industry is chiefly in Chinese hands. The great
majority of the imports are from Great Britain or British possessions.
The currency of Mauritius is rupees and cents of a rupee, the Indian
rupee (= 16d.) being the standard unit. The metric system of weights
and measures has been in force since 1878.
Communications.—There is a regular fortnightly steamship service
between Marseilles and Port Louis by the Messageries Maritimes, a
four-weekly service with Southampton via Cape Town by the Union
Castle, and a four-weekly service with Colombo direct by the British
India Co.’s boats. There is also frequent communication with
Madagascar, Réunion and Natal. The average annual tonnage of ships
five years ending 1906 being 1840 tons. In addition, a considerable
quantity of molasses and smaller quantities of rum, vanilla and coco-
nut oil are exported. The imports are mainly rice, wheat, cotton goods,
wine, coal, hardware and haberdashery, and guano. The rice comes
principally from India and Madagascar; cattle are imported from
Madagascar, sheep from South Africa and Australia, and frozen meat
from Australia. The average annual value of the exports for the ten
years 1896-1905 was £2,153,159; the average annual value of the
imports for the same period £1,453,089. These figures when compared
with those in years before the beet and bounty-fed sugar had entered
into severe competition with cane sugar, show how greatly the island
had thereby suffered. In 1864 the exports were valued at £2,249,000; in
1868 at £2,339,000; in 1877 at £4,201,000 and in 1880 at £3,634,000.
And in each of the years named the imports exceeded £2,000,000 in
value. Nearly all the aloe-fibre exported is taken by Great Britain, and
France, while the molasses goes to India. Among the minor exports is
that of bambara or sea-slugs, which are sent to Hong-Kong and
Singapore. This industry is chiefly in Chinese hands. The great
majority of the imports are from Great Britain or British possessions.
The currency of Mauritius is rupees and cents of a rupee, the Indian
rupee (= 16d.) being the standard unit. The metric system of weights
and measures has been in force since 1878.
Communications.—There is a regular fortnightly steamship service
between Marseilles and Port Louis by the Messageries Maritimes, a
four-weekly service with Southampton via Cape Town by the Union
Castle, and a four-weekly service with Colombo direct by the British
India Co.’s boats. There is also frequent communication with
Madagascar, Réunion and Natal. The average annual tonnage of ships
Page 128
entering Port Louis is about 750,000 of which five-sevenths is British.
Cable communication with Europe, via the Seychelles, Zanzibar and
Aden, was established in 1893, and the Mauritius section of the Cape-
Australian cable, via Rodriguez, was completed in 1902.
Railways connect all the principal places and sugar estates on the
island, that known as the Midland line, 36 miles long, beginning at Port
Louis crosses the island to Mahébourg, passing through Curepipe,
where it is 1822 ft. above the sea. There are in all over 120 miles of
railway, all owned and worked by the government. The first railway
was opened in 1864. The roads are well kept and there is an extensive
system of tramways for bringing produce from the sugar estates to the
railway lines. Traction engines are also largely used. There is a
complete telegraphic and telephonic service.
Government and Revenue.—Mauritius is a crown colony. The governor
is assisted by an executive council of five official and two elected members,
and a legislative council of 27 members, 8 sitting ex officio, 9 being
nominated by the governor and 10 elected on a moderate franchise. Two of
the elected members represent St Louis, the 8 rural districts into which the
island is divided electing each one member. At least one-third of the
nominated members must be persons not holding any public office. The
number of registered electors in 1908 was 6186. The legislative session
usually lasts from April to December. Members may speak either in French
or English. The average annual revenue of the colony for the ten years
1896-1905, was £608,245, the average annual expenditure during the same
period £663,606. Up to 1854 there was a surplus in hand, but since that time
expenditure has on many occasions exceeded income, and the public debt in
1908 was £1,305,000, mainly incurred however on reproductive works.
The island has largely retained the old French laws, the codes civil, de
procédure, du commerce, and d’instruction criminelle being still in force,
Cable communication with Europe, via the Seychelles, Zanzibar and
Aden, was established in 1893, and the Mauritius section of the Cape-
Australian cable, via Rodriguez, was completed in 1902.
Railways connect all the principal places and sugar estates on the
island, that known as the Midland line, 36 miles long, beginning at Port
Louis crosses the island to Mahébourg, passing through Curepipe,
where it is 1822 ft. above the sea. There are in all over 120 miles of
railway, all owned and worked by the government. The first railway
was opened in 1864. The roads are well kept and there is an extensive
system of tramways for bringing produce from the sugar estates to the
railway lines. Traction engines are also largely used. There is a
complete telegraphic and telephonic service.
Government and Revenue.—Mauritius is a crown colony. The governor
is assisted by an executive council of five official and two elected members,
and a legislative council of 27 members, 8 sitting ex officio, 9 being
nominated by the governor and 10 elected on a moderate franchise. Two of
the elected members represent St Louis, the 8 rural districts into which the
island is divided electing each one member. At least one-third of the
nominated members must be persons not holding any public office. The
number of registered electors in 1908 was 6186. The legislative session
usually lasts from April to December. Members may speak either in French
or English. The average annual revenue of the colony for the ten years
1896-1905, was £608,245, the average annual expenditure during the same
period £663,606. Up to 1854 there was a surplus in hand, but since that time
expenditure has on many occasions exceeded income, and the public debt in
1908 was £1,305,000, mainly incurred however on reproductive works.
The island has largely retained the old French laws, the codes civil, de
procédure, du commerce, and d’instruction criminelle being still in force,
Page 129
except so far as altered by colonial ordinances. A supreme court of civil and
criminal justice was established in 1831 under a chief judge and three
puisne judges.
Religion and Education.—The majority of the European inhabitants
belong to the Roman Catholic faith. They numbered at the 1901 census
117,102, and the Protestants 6644. Anglicans, Roman Catholics and the
Church of Scotland are helped by state grants. At the head of the
Anglican community is the bishop of Mauritius; the chief Romanist
dignitary is styled bishop of Port Louis. The Mahommedans number
over 30,000, but the majority of the Indian coolies are Hindus.
The educational system, as brought into force in 1900, is under a
director of public instruction assisted by an advisory committee, and
consists of two branches (1) superior or secondary instruction, (2)
primary instruction. For primary instruction there are government
schools and schools maintained by the Roman Catholics, Protestants
and other faiths, to which the government gives grants in aid. In 1908
there were 67 government schools with 8400 scholars and 90 grant
schools with 10,200 scholars, besides Hindu schools receiving no
grant. The Roman Catholic scholars number 67.72%; the Protestants
3.80%; Mahommedans 8.37%; and Hindus and others 20.11%.
Secondary and higher education is given in the Royal College and
associated schools at Port Louis and Curepipe.
Defence.—Mauritius occupies an important strategic position on the
route between South Africa and India and in relation to Madagascar
and East Africa, while in Port Louis it possesses one of the finest
harbours in the Indian Ocean. A permanent garrison of some 3000 men
is maintained in the island at a cost of about £180,000 per annum. To
the cost of the troops Mauritius contributes 51⁄2% of its annual revenue
—about £30,000.
criminal justice was established in 1831 under a chief judge and three
puisne judges.
Religion and Education.—The majority of the European inhabitants
belong to the Roman Catholic faith. They numbered at the 1901 census
117,102, and the Protestants 6644. Anglicans, Roman Catholics and the
Church of Scotland are helped by state grants. At the head of the
Anglican community is the bishop of Mauritius; the chief Romanist
dignitary is styled bishop of Port Louis. The Mahommedans number
over 30,000, but the majority of the Indian coolies are Hindus.
The educational system, as brought into force in 1900, is under a
director of public instruction assisted by an advisory committee, and
consists of two branches (1) superior or secondary instruction, (2)
primary instruction. For primary instruction there are government
schools and schools maintained by the Roman Catholics, Protestants
and other faiths, to which the government gives grants in aid. In 1908
there were 67 government schools with 8400 scholars and 90 grant
schools with 10,200 scholars, besides Hindu schools receiving no
grant. The Roman Catholic scholars number 67.72%; the Protestants
3.80%; Mahommedans 8.37%; and Hindus and others 20.11%.
Secondary and higher education is given in the Royal College and
associated schools at Port Louis and Curepipe.
Defence.—Mauritius occupies an important strategic position on the
route between South Africa and India and in relation to Madagascar
and East Africa, while in Port Louis it possesses one of the finest
harbours in the Indian Ocean. A permanent garrison of some 3000 men
is maintained in the island at a cost of about £180,000 per annum. To
the cost of the troops Mauritius contributes 51⁄2% of its annual revenue
—about £30,000.
Page 130
History.—Mauritius appears to have been unknown to European nations,
if not to all other peoples, until the year 1505, when it was discovered by
Mascarenhas, a Portuguese navigator. It had then no inhabitants, and there
seem to be no traces of a previous occupation by any people. The island
was retained for most of the 16th century by its discoverers, but they made
no settlements in it. In 1598 the Dutch took possession, and named the
island “Mauritius,” in honour of their stadtholder, Count Maurice of
Nassau. It had been previously called by the Portuguese “Ilha do Cerné,”
from the belief that it was the island so named by Pliny. But though the
Dutch built a fort at Grand Port and introduced a number of slaves and
convicts, they made no permanent settlement in Mauritius, finally
abandoning the island in 1710. From 1715 to 1767 (when the French
government assumed direct control) the island was held by agents of the
French East India Company, by whom its name was again changed to “Île
de France.” The Company was fortunate in having several able men as
governors of its colony, especially the celebrated Mahé de Labourdonnais
(q.v.), who made sugar planting the main industry of the inhabitants.3 Under
his direction roads were made, forts built, and considerable portions of the
forest were cleared, and the present capital, Port Louis, was founded.
Labourdonnais also promoted the planting of cotton and indigo, and is
remembered as the most enlightened and best of all the French governors.
He also put down the maroons or runaway slaves who had long been the
pest of the island. The colony continued to rise in value during the time it
was held by the French crown, and to one of the intendants,4 Pierre Poivre,
was due the introduction of the clove, nutmeg and other spices. Another
governor was D’Entrecasteaux, whose name is kept in remembrance by a
group of islands east of New Guinea.
During the long war between France and England, at the commencement
of the 19th century, Mauritius was a continual source of much mischief to
English Indiamen and other merchant vessels; and at length the British
if not to all other peoples, until the year 1505, when it was discovered by
Mascarenhas, a Portuguese navigator. It had then no inhabitants, and there
seem to be no traces of a previous occupation by any people. The island
was retained for most of the 16th century by its discoverers, but they made
no settlements in it. In 1598 the Dutch took possession, and named the
island “Mauritius,” in honour of their stadtholder, Count Maurice of
Nassau. It had been previously called by the Portuguese “Ilha do Cerné,”
from the belief that it was the island so named by Pliny. But though the
Dutch built a fort at Grand Port and introduced a number of slaves and
convicts, they made no permanent settlement in Mauritius, finally
abandoning the island in 1710. From 1715 to 1767 (when the French
government assumed direct control) the island was held by agents of the
French East India Company, by whom its name was again changed to “Île
de France.” The Company was fortunate in having several able men as
governors of its colony, especially the celebrated Mahé de Labourdonnais
(q.v.), who made sugar planting the main industry of the inhabitants.3 Under
his direction roads were made, forts built, and considerable portions of the
forest were cleared, and the present capital, Port Louis, was founded.
Labourdonnais also promoted the planting of cotton and indigo, and is
remembered as the most enlightened and best of all the French governors.
He also put down the maroons or runaway slaves who had long been the
pest of the island. The colony continued to rise in value during the time it
was held by the French crown, and to one of the intendants,4 Pierre Poivre,
was due the introduction of the clove, nutmeg and other spices. Another
governor was D’Entrecasteaux, whose name is kept in remembrance by a
group of islands east of New Guinea.
During the long war between France and England, at the commencement
of the 19th century, Mauritius was a continual source of much mischief to
English Indiamen and other merchant vessels; and at length the British
Page 131
government determined upon an expedition for its capture. This was
effected in 1810; and upon the restoration of peace in 1814 the possession
of the island was confirmed to Britain by the Treaty of Paris. By the eighth
article of capitulation it was agreed that the inhabitants should retain their
own laws, customs, and religion; and thus the island is still largely French
in language, habits, and predilections; but its name has again been changed
to that given by the Dutch. One of the most distinguished of the British
governors was Sir Robert Farquhar (1810-1823), who did much to abolish
the Malagasy slave trade and to establish friendly relations with the rising
power of the Hova sovereign of Madagascar. Later governors of note were
Sir Henry Barkly (1863-1871), and Sir J. Pope Hennessy (1883-1886 and
1888).
The history of the colony since its acquisition by Great Britain has been
one of social and political evolution. At first all power was concentrated in
the hands of the governor, but in 1832 a legislative council was constituted
on which non-official nominated members served. In 1884-1885 this
council was transformed into a partly elected body. Of more importance
than the constitutional changes were the economic results which followed
the freeing of the slaves (1834-1839)—for the loss of whose labour the
planters received over £2,000,000 compensation. Coolies were introduced
to supply the place of the negroes, immigration being definitely sanctioned
by the government of India in 1842. Though under government control the
system of coolie labour led to many abuses. A royal commission
investigated the matter in 1871 and since that time the evils which were
attendant on the system have been gradually remedied. One result of the
introduction of free labour has been to reduce the descendants of the slave
population to a small and unimportant class—Mauritius in this respect
offering a striking contrast to the British colonies in the West Indies. The
last half of the 19th century was, however, chiefly notable in Mauritius for
the number of calamities which overtook the island. In 1854 cholera caused
effected in 1810; and upon the restoration of peace in 1814 the possession
of the island was confirmed to Britain by the Treaty of Paris. By the eighth
article of capitulation it was agreed that the inhabitants should retain their
own laws, customs, and religion; and thus the island is still largely French
in language, habits, and predilections; but its name has again been changed
to that given by the Dutch. One of the most distinguished of the British
governors was Sir Robert Farquhar (1810-1823), who did much to abolish
the Malagasy slave trade and to establish friendly relations with the rising
power of the Hova sovereign of Madagascar. Later governors of note were
Sir Henry Barkly (1863-1871), and Sir J. Pope Hennessy (1883-1886 and
1888).
The history of the colony since its acquisition by Great Britain has been
one of social and political evolution. At first all power was concentrated in
the hands of the governor, but in 1832 a legislative council was constituted
on which non-official nominated members served. In 1884-1885 this
council was transformed into a partly elected body. Of more importance
than the constitutional changes were the economic results which followed
the freeing of the slaves (1834-1839)—for the loss of whose labour the
planters received over £2,000,000 compensation. Coolies were introduced
to supply the place of the negroes, immigration being definitely sanctioned
by the government of India in 1842. Though under government control the
system of coolie labour led to many abuses. A royal commission
investigated the matter in 1871 and since that time the evils which were
attendant on the system have been gradually remedied. One result of the
introduction of free labour has been to reduce the descendants of the slave
population to a small and unimportant class—Mauritius in this respect
offering a striking contrast to the British colonies in the West Indies. The
last half of the 19th century was, however, chiefly notable in Mauritius for
the number of calamities which overtook the island. In 1854 cholera caused
Page 132
the death of 17,000 persons; in 1867 over 30,000 people died of malarial
fever; in 1892 a hurricane of terrific violence caused immense destruction
of property and serious loss of life; in 1893 a great part of Port Louis was
destroyed by fire. There were in addition several epidemics of small-pox
and plague, and from about 1880 onward the continual decline in the price
of sugar seriously affected the islanders, especially the Creole population.
During 1902-1905 an outbreak of surra, which caused great mortality
among draught animals, further tried the sugar planters and necessitated
government help. Notwithstanding all these calamities the Mauritians,
especially the Indo-Mauritians, have succeeded in maintaining the position
of the colony as an important sugar-producing country.
Dependencies.—Dependent upon Mauritius and forming part of the
colony are a number of small islands scattered over a large extent of
the Indian Ocean. Of these the chief is Rodriguez (q.v.), 375 m. east of
Mauritius. Considerably north-east of Rodriguez lie the Oil Islands or
Chagos archipelago, of which the chief is Diego Garcia (see Chagos).
The Cargados, Carayos or St Brandon islets, deeps and shoals, lie at the
south end of the Nazareth Bank about 250 m. N.N.E. of Mauritius.
Until 1903 the Seychelles, Amirantes, Aldabra and other islands lying
north of Madagascar were also part of the colony of Mauritius. In the
year named they were formed into a separate colony (see Seychelles).
Two islands, Farquhar and Coetivy, though geographically within the
Seychelles area, remained dependent on Mauritius, being owned by
residents in that island. In 1908, however, Coetivy was transferred to
the Seychelles administration. Amsterdam and St Paul, uninhabited
islands in the South Indian Ocean, included in an official list of the
dependencies of Mauritius drawn up in 1880, were in 1893 annexed by
France. The total population of the dependencies of Mauritius was
estimated in 1905 at 5400.
fever; in 1892 a hurricane of terrific violence caused immense destruction
of property and serious loss of life; in 1893 a great part of Port Louis was
destroyed by fire. There were in addition several epidemics of small-pox
and plague, and from about 1880 onward the continual decline in the price
of sugar seriously affected the islanders, especially the Creole population.
During 1902-1905 an outbreak of surra, which caused great mortality
among draught animals, further tried the sugar planters and necessitated
government help. Notwithstanding all these calamities the Mauritians,
especially the Indo-Mauritians, have succeeded in maintaining the position
of the colony as an important sugar-producing country.
Dependencies.—Dependent upon Mauritius and forming part of the
colony are a number of small islands scattered over a large extent of
the Indian Ocean. Of these the chief is Rodriguez (q.v.), 375 m. east of
Mauritius. Considerably north-east of Rodriguez lie the Oil Islands or
Chagos archipelago, of which the chief is Diego Garcia (see Chagos).
The Cargados, Carayos or St Brandon islets, deeps and shoals, lie at the
south end of the Nazareth Bank about 250 m. N.N.E. of Mauritius.
Until 1903 the Seychelles, Amirantes, Aldabra and other islands lying
north of Madagascar were also part of the colony of Mauritius. In the
year named they were formed into a separate colony (see Seychelles).
Two islands, Farquhar and Coetivy, though geographically within the
Seychelles area, remained dependent on Mauritius, being owned by
residents in that island. In 1908, however, Coetivy was transferred to
the Seychelles administration. Amsterdam and St Paul, uninhabited
islands in the South Indian Ocean, included in an official list of the
dependencies of Mauritius drawn up in 1880, were in 1893 annexed by
France. The total population of the dependencies of Mauritius was
estimated in 1905 at 5400.
Page 133
Authorities.—F. Leguat, Voyages et aventures en deux isles désertes
des Indes orientales (Eng. trans., A New Voyage to the East Indies;
London, 1708); Prudham, “England’s Colonial Empire,” vol. i., The
Mauritius and its Dependencies (1846); C. P. Lucas, A Historical
Geography of the British Colonies, vol. i. (Oxford, 1888); Ch. Grant,
History of Mauritius, or the Isle of France and Neighbouring Islands
(1801); J. Milbert, Voyage pittoresque à l’Île-de-France, &c., 4 vols.
(1812); Aug. Billiard, Voyage aux colonies orientales (1822); P.
Beaton, Creoles and Coolies, or Five Years in Mauritius (1859); Paul
Chasteau, Histoire et description de l’île Maurice (1860); F. P.
Flemyng, Mauritius, or the Isle of France (1862); Ch. J. Boyle, Far
Away, or Sketches of Scenery and Society in Mauritius (1867); L.
Simonin, Les Pays lointains, notes de voyage (Maurice, &c.) (1867);
N. Pike, Sub-Tropical Rambles in the Land of the Aphanapteryx
(1873); A. R. Wallace. “The Mascarene Islands,” in ch. xi. vol. i. of The
Geographical Distribution of Animals (1876); K. Möbius, F. Richter
and E. von Martens, Beiträge zur Meeresfauna der Insel Mauritius und
der Seychellen (Berlin, 1880); G. Clark, A Brief Notice of the Fauna of
Mauritius (1881); A. d’Épinay, Renseignements pour servir à l’histoire
de l’Île de France jusqu’à 1810 (Mauritius, 1890); N. Decotter,
Geography of Mauritius and its Dependencies (Mauritius, 1892); H. de
Haga Haig, “The Physical Features and Geology of Mauritius” in vol.
li., Q. J. Geol. Soc. (1895); the Annual Reports on Mauritius issued by
the Colonial Office, London; The Mauritius Almanack published yearly
at Port Louis. A map of the island in six sheets on the scale of one inch
to a mile was issued by the War Office in 1905. (J. Si.*)
1 See Geog. Journ. (June 1895), p. 597.
2 The total population of the colony (including dependencies) on the 1st of January
1907 was estimated at 383,206.
des Indes orientales (Eng. trans., A New Voyage to the East Indies;
London, 1708); Prudham, “England’s Colonial Empire,” vol. i., The
Mauritius and its Dependencies (1846); C. P. Lucas, A Historical
Geography of the British Colonies, vol. i. (Oxford, 1888); Ch. Grant,
History of Mauritius, or the Isle of France and Neighbouring Islands
(1801); J. Milbert, Voyage pittoresque à l’Île-de-France, &c., 4 vols.
(1812); Aug. Billiard, Voyage aux colonies orientales (1822); P.
Beaton, Creoles and Coolies, or Five Years in Mauritius (1859); Paul
Chasteau, Histoire et description de l’île Maurice (1860); F. P.
Flemyng, Mauritius, or the Isle of France (1862); Ch. J. Boyle, Far
Away, or Sketches of Scenery and Society in Mauritius (1867); L.
Simonin, Les Pays lointains, notes de voyage (Maurice, &c.) (1867);
N. Pike, Sub-Tropical Rambles in the Land of the Aphanapteryx
(1873); A. R. Wallace. “The Mascarene Islands,” in ch. xi. vol. i. of The
Geographical Distribution of Animals (1876); K. Möbius, F. Richter
and E. von Martens, Beiträge zur Meeresfauna der Insel Mauritius und
der Seychellen (Berlin, 1880); G. Clark, A Brief Notice of the Fauna of
Mauritius (1881); A. d’Épinay, Renseignements pour servir à l’histoire
de l’Île de France jusqu’à 1810 (Mauritius, 1890); N. Decotter,
Geography of Mauritius and its Dependencies (Mauritius, 1892); H. de
Haga Haig, “The Physical Features and Geology of Mauritius” in vol.
li., Q. J. Geol. Soc. (1895); the Annual Reports on Mauritius issued by
the Colonial Office, London; The Mauritius Almanack published yearly
at Port Louis. A map of the island in six sheets on the scale of one inch
to a mile was issued by the War Office in 1905. (J. Si.*)
1 See Geog. Journ. (June 1895), p. 597.
2 The total population of the colony (including dependencies) on the 1st of January
1907 was estimated at 383,206.
Page 134
3 Labourdonnais is credited by several writers with the introduction of the sugar cane
into the island. Leguat, however, mentions it as being cultivated during the Dutch
occupation.
4 The régime introduced in 1767 divided the administration between a governor,
primarily charged with military matters, and an intendant.
MAURY, JEAN SIFFREIN (1746-1817), French cardinal and
archbishop of Paris, the son of a poor cobbler, was born on the 26th of June
1746 at Valréas in the Comtat-Venaissin, the district in France which
belonged to the pope. His acuteness was observed by the priests of the
seminary at Avignon, where he was educated and took orders. He tried his
fortune by writing éloges of famous persons, then a favourite practice; and
in 1771 his éloge on Fénelon was pronounced next best to Laharpe’s by the
Academy. The real foundation of his fortunes was the success of a
panegyric on St Louis delivered before the Academy in 1772, which caused
him to be recommended for an abbacy. In 1777 he published under the title
of Discours choisis his panegyrics on Saint Louis, Saint Augustine and
Fénelon, his remarks on Bossuet and his Essai sur l’éloquence de la chaire,
a volume which contains much good criticism, and remains a French
classic. The book was often reprinted as Principes de l’éloquence. He
became a favourite preacher in Paris, and was Lent preacher at court in
1781, when King Louis XVI. said of his sermon: “If the abbé had only said
a few words on religion he would have discussed every possible subject.” In
1781 he obtained the rich priory of Lyons, near Péronne, and in 1785 he
was elected to the Academy, as successor of Lefranc de Pompignan. His
into the island. Leguat, however, mentions it as being cultivated during the Dutch
occupation.
4 The régime introduced in 1767 divided the administration between a governor,
primarily charged with military matters, and an intendant.
MAURY, JEAN SIFFREIN (1746-1817), French cardinal and
archbishop of Paris, the son of a poor cobbler, was born on the 26th of June
1746 at Valréas in the Comtat-Venaissin, the district in France which
belonged to the pope. His acuteness was observed by the priests of the
seminary at Avignon, where he was educated and took orders. He tried his
fortune by writing éloges of famous persons, then a favourite practice; and
in 1771 his éloge on Fénelon was pronounced next best to Laharpe’s by the
Academy. The real foundation of his fortunes was the success of a
panegyric on St Louis delivered before the Academy in 1772, which caused
him to be recommended for an abbacy. In 1777 he published under the title
of Discours choisis his panegyrics on Saint Louis, Saint Augustine and
Fénelon, his remarks on Bossuet and his Essai sur l’éloquence de la chaire,
a volume which contains much good criticism, and remains a French
classic. The book was often reprinted as Principes de l’éloquence. He
became a favourite preacher in Paris, and was Lent preacher at court in
1781, when King Louis XVI. said of his sermon: “If the abbé had only said
a few words on religion he would have discussed every possible subject.” In
1781 he obtained the rich priory of Lyons, near Péronne, and in 1785 he
was elected to the Academy, as successor of Lefranc de Pompignan. His
Page 135
morals were as loose as those of his great rival Mirabeau, but he was famed
in Paris for his wit and gaiety. In 1789 he was elected a member of the
states-general by the clergy of the bailliage of Péronne, and from the first
proved to be the most able and persevering defender of the ancien régime,
although he had drawn up the greater part of the cahier of the clergy of
Péronne, which contained a considerable programme of reform. It is said
that he attempted to emigrate both in July and in October 1789; but after
that time he held firmly to his place, when almost universally deserted by
his friends. In the Constituent Assembly he took an active part in every
important debate, combating with especial vigour the alienation of the
property of the clergy. His life was often in danger, but his ready wit always
saved it, and it was said that one bon mot would preserve him for a month.
When he did emigrate in 1792 he found himself regarded as a martyr to the
church and the king, and was at once named archbishop in partibus, and
extra nuncio to the diet at Frankfort, and in 1794 cardinal. He was finally
made bishop of Montefiascone, and settled down in that little Italian town
—but not for long, for in 1798 the French drove him from his retreat, and
he sought refuge in Venice and St Petersburg. Next year he returned to
Rome as ambassador of the exiled Louis XVIII. at the papal court. In 1804
he began to prepare his return to France by a well-turned letter to Napoleon,
congratulating him on restoring religion to France once more. In 1806 he
did return; in 1807 he was again received into the Academy; and in 1810,
on the refusal of Cardinal Fesch, was made archbishop of Paris. He was
presently ordered by the pope to surrender his functions as archbishop of
Paris. This he refused to do. On the restoration of the Bourbons he was
summarily expelled from the Academy and from the archiepiscopal palace.
He retired to Rome, where he was imprisoned in the castle of St Angelo for
six months for his disobedience to the papal orders, and died in 1817, a year
or two after his release, of disease contracted in prison and of chagrin. As a
critic he was a very able writer, and Sainte-Beuve gives him the credit of
in Paris for his wit and gaiety. In 1789 he was elected a member of the
states-general by the clergy of the bailliage of Péronne, and from the first
proved to be the most able and persevering defender of the ancien régime,
although he had drawn up the greater part of the cahier of the clergy of
Péronne, which contained a considerable programme of reform. It is said
that he attempted to emigrate both in July and in October 1789; but after
that time he held firmly to his place, when almost universally deserted by
his friends. In the Constituent Assembly he took an active part in every
important debate, combating with especial vigour the alienation of the
property of the clergy. His life was often in danger, but his ready wit always
saved it, and it was said that one bon mot would preserve him for a month.
When he did emigrate in 1792 he found himself regarded as a martyr to the
church and the king, and was at once named archbishop in partibus, and
extra nuncio to the diet at Frankfort, and in 1794 cardinal. He was finally
made bishop of Montefiascone, and settled down in that little Italian town
—but not for long, for in 1798 the French drove him from his retreat, and
he sought refuge in Venice and St Petersburg. Next year he returned to
Rome as ambassador of the exiled Louis XVIII. at the papal court. In 1804
he began to prepare his return to France by a well-turned letter to Napoleon,
congratulating him on restoring religion to France once more. In 1806 he
did return; in 1807 he was again received into the Academy; and in 1810,
on the refusal of Cardinal Fesch, was made archbishop of Paris. He was
presently ordered by the pope to surrender his functions as archbishop of
Paris. This he refused to do. On the restoration of the Bourbons he was
summarily expelled from the Academy and from the archiepiscopal palace.
He retired to Rome, where he was imprisoned in the castle of St Angelo for
six months for his disobedience to the papal orders, and died in 1817, a year
or two after his release, of disease contracted in prison and of chagrin. As a
critic he was a very able writer, and Sainte-Beuve gives him the credit of
Page 136
discovering Father Jacques Bridayne, and of giving Bossuet his rightful
place as a preacher above Massillon; as a politician, his wit and eloquence
make him a worthy rival of Mirabeau. He sacrificed too much to personal
ambition, yet it would have been a graceful act if Louis XVIII. had
remembered the courageous supporter of Louis XVI., and the pope the one
intrepid defender of the Church in the states-general.
The Œuvres choisies du Cardinal Maury (5 vols., 1827) contain
what is worth preserving. Mgr Ricard has published Maury’s
Correspondance diplomatique (2 vols., Lille, 1891). For his life and
character see Vie du Cardinal Maury, by Louis Siffrein Maury, his
nephew (1828); J. J. F. Poujoulat, Cardinal Maury, sa vie et ses œuvres
(1855); Sainte-Beuve, Causeries du lundi (vol. iv.); Mgr Ricard,
L’Abbé Maury (1746-1791), L’Abbé Maury avant 1789, L’Abbé Maury
et Mirabeau (1887); G. Bonet-Maury, Le Cardinal Maury d’après ses
mémoires et sa correspondance inédits (Paris, 1892); A. Aulard, Les
Orateurs de la constituante (Paris, 1882). Of the many libels written
against him during the Revolution the most noteworthy are the Petit
carême de l’abbé Maury, with a supplement called the Seconde année
(1790), and the Vie privée de l’abbé Maury (1790), claimed by J. R.
Hébert, but attributed by some writers to Restif de la Bretonne. For
further bibliographical details see J. M. Quérard, La France littéraire,
vol. v. (1833).
place as a preacher above Massillon; as a politician, his wit and eloquence
make him a worthy rival of Mirabeau. He sacrificed too much to personal
ambition, yet it would have been a graceful act if Louis XVIII. had
remembered the courageous supporter of Louis XVI., and the pope the one
intrepid defender of the Church in the states-general.
The Œuvres choisies du Cardinal Maury (5 vols., 1827) contain
what is worth preserving. Mgr Ricard has published Maury’s
Correspondance diplomatique (2 vols., Lille, 1891). For his life and
character see Vie du Cardinal Maury, by Louis Siffrein Maury, his
nephew (1828); J. J. F. Poujoulat, Cardinal Maury, sa vie et ses œuvres
(1855); Sainte-Beuve, Causeries du lundi (vol. iv.); Mgr Ricard,
L’Abbé Maury (1746-1791), L’Abbé Maury avant 1789, L’Abbé Maury
et Mirabeau (1887); G. Bonet-Maury, Le Cardinal Maury d’après ses
mémoires et sa correspondance inédits (Paris, 1892); A. Aulard, Les
Orateurs de la constituante (Paris, 1882). Of the many libels written
against him during the Revolution the most noteworthy are the Petit
carême de l’abbé Maury, with a supplement called the Seconde année
(1790), and the Vie privée de l’abbé Maury (1790), claimed by J. R.
Hébert, but attributed by some writers to Restif de la Bretonne. For
further bibliographical details see J. M. Quérard, La France littéraire,
vol. v. (1833).
Page 137
MAURY, LOUIS FERDINAND ALFRED (1817-
1892), French scholar, was born at Meaux on the 23rd of March 1817. In
1836, having completed his education, he entered the Bibliothèque
Nationale, and afterwards the Bibliothèque de l’Institut (1844), where he
devoted himself to the study of archaeology, ancient and modern languages,
medicine and law. Gifted with a great capacity for work, a remarkable
memory and an unbiassed and critical mind, he produced without great
effort a number of learned pamphlets and books on the most varied
subjects. He rendered great service to the Académie des Inscriptions et
Belles Lettres, of which he had been elected a member in 1857. Napoleon
III. employed him in research work connected with the Histoire de César,
and he was rewarded, proportionately to his active, if modest, part in this
work, with the positions of librarian of the Tuileries (1860), professor at the
College of France (1862) and director-general of the Archives (1868). It
was not, however, to the imperial favour that he owed these high positions.
He used his influence for the advancement of science and higher education,
and with Victor Duruy was one of the founders of the École des Hautes
Études. He died at Paris four years after his retirement from the last post, on
the 11th of February 1892.
Bibliography.—His works are numerous: Les Fées au moyen âge
and Histoire des légendes pieuses au moyen âge; two books filled with
ingenious ideas, which were published in 1843, and reprinted after the
death of the author, with numerous additions under the title Croyances
et légendes du moyen âge (1896); Histoire des grandes forêts de la
Gaule et de l’ancienne France (1850, a 3rd ed. revised appeared in
1867 under the title Les Forêts de la Gaule et de l’ancienne France);
La Terre et l’homme, a general historical sketch of geology, geography
and ethnology, being the introduction to the Histoire universelle, by
Victor Duruy (1854); Histoire des religions de la Grèce antique, (3
vols., 1857-1859); La Magie et l’astrologie dans l’antiquité et dans le
1892), French scholar, was born at Meaux on the 23rd of March 1817. In
1836, having completed his education, he entered the Bibliothèque
Nationale, and afterwards the Bibliothèque de l’Institut (1844), where he
devoted himself to the study of archaeology, ancient and modern languages,
medicine and law. Gifted with a great capacity for work, a remarkable
memory and an unbiassed and critical mind, he produced without great
effort a number of learned pamphlets and books on the most varied
subjects. He rendered great service to the Académie des Inscriptions et
Belles Lettres, of which he had been elected a member in 1857. Napoleon
III. employed him in research work connected with the Histoire de César,
and he was rewarded, proportionately to his active, if modest, part in this
work, with the positions of librarian of the Tuileries (1860), professor at the
College of France (1862) and director-general of the Archives (1868). It
was not, however, to the imperial favour that he owed these high positions.
He used his influence for the advancement of science and higher education,
and with Victor Duruy was one of the founders of the École des Hautes
Études. He died at Paris four years after his retirement from the last post, on
the 11th of February 1892.
Bibliography.—His works are numerous: Les Fées au moyen âge
and Histoire des légendes pieuses au moyen âge; two books filled with
ingenious ideas, which were published in 1843, and reprinted after the
death of the author, with numerous additions under the title Croyances
et légendes du moyen âge (1896); Histoire des grandes forêts de la
Gaule et de l’ancienne France (1850, a 3rd ed. revised appeared in
1867 under the title Les Forêts de la Gaule et de l’ancienne France);
La Terre et l’homme, a general historical sketch of geology, geography
and ethnology, being the introduction to the Histoire universelle, by
Victor Duruy (1854); Histoire des religions de la Grèce antique, (3
vols., 1857-1859); La Magie et l’astrologie dans l’antiquité et dans le
Page 138
moyen âge (1863); Histoire de l’ancienne académie des sciences
(1864); Histoire de l’Académie des Inscriptions et Belles Lettres
(1865); a learned paper on the reports of French archaeology, written
on the occasion of the universal exhibition (1867); a number of articles
in the Encyclopédie moderne (1846-1851), in Michaud’s Biographie
universelle (1858 and seq.), in the Journal des savants in the Revue des
deux mondes (1873, 1877, 1879-1880, &c.). A detailed bibliography of
his works has been placed by Auguste Longnon at the beginning of the
volume Les Croyances et légendes du moyen âge.
MAURY, MATTHEW FONTAINE (1806-1873),
American naval officer and hydrographer, was born near Fredericksburg in
Spottsylvania county, Virginia, on the 24th of January 1806. He was
educated at Harpeth academy, and in 1825 entered the navy as midshipman,
circumnavigating the globe in the “Vincennes,” during a cruise of four
years (1826-1830). In 1831 he was appointed master of the sloop
“Falmouth” on the Pacific station, and subsequently served in other vessels
before returning home in 1834, when he married his cousin, Ann Herndon.
In 1835-1836 he was actively engaged in producing for publication a
treatise on navigation, a remarkable achievement at so early a stage in his
career; he was at this time made lieutenant, and gazetted astronomer to a
South Sea exploring expedition, but resigned this position and was
appointed to the survey of southern harbours. In 1839 he met with an
accident which resulted in permanent lameness, and unfitted him for active
(1864); Histoire de l’Académie des Inscriptions et Belles Lettres
(1865); a learned paper on the reports of French archaeology, written
on the occasion of the universal exhibition (1867); a number of articles
in the Encyclopédie moderne (1846-1851), in Michaud’s Biographie
universelle (1858 and seq.), in the Journal des savants in the Revue des
deux mondes (1873, 1877, 1879-1880, &c.). A detailed bibliography of
his works has been placed by Auguste Longnon at the beginning of the
volume Les Croyances et légendes du moyen âge.
MAURY, MATTHEW FONTAINE (1806-1873),
American naval officer and hydrographer, was born near Fredericksburg in
Spottsylvania county, Virginia, on the 24th of January 1806. He was
educated at Harpeth academy, and in 1825 entered the navy as midshipman,
circumnavigating the globe in the “Vincennes,” during a cruise of four
years (1826-1830). In 1831 he was appointed master of the sloop
“Falmouth” on the Pacific station, and subsequently served in other vessels
before returning home in 1834, when he married his cousin, Ann Herndon.
In 1835-1836 he was actively engaged in producing for publication a
treatise on navigation, a remarkable achievement at so early a stage in his
career; he was at this time made lieutenant, and gazetted astronomer to a
South Sea exploring expedition, but resigned this position and was
appointed to the survey of southern harbours. In 1839 he met with an
accident which resulted in permanent lameness, and unfitted him for active
Page 139
service. In the same year, however, he began to write a series of articles on
naval reform and other subjects, under the title of Scraps from the Lucky-
Bag, which attracted much attention; and in 1841 he was placed in charge
of the Dépôt of Charts and Instruments, out of which grew the United States
Naval Observatory and the Hydrographie Office. He laboured assiduously
to obtain observations as to the winds and currents by distributing to
captains of vessels specially prepared log-books; and in the course of nine
years he had collected a sufficient number of logs to make two hundred
manuscript volumes, each with about two thousand five hundred days’
observations. One result was to show the necessity for combined action on
the part of maritime nations in regard to ocean meteorology. This led to an
international conference at Brussels in 1853, which produced the greatest
benefit to navigation as well as indirectly to meteorology. Maury attempted
to organize co-operative meteorological work on land, but the government
did not at this time take any steps in this direction. His oceanographical
work, however, received recognition in all parts of the civilized world, and
in 1855 it was proposed in the senate to remunerate him, but in the same
year the Naval Retiring Board, erected under an act to promote the
efficiency of the navy, placed him on the retired list. This action aroused
wide opposition, and in 1858 he was reinstated with the rank of commander
as from 1855. In 1853 Maury had published his Letters on the Amazon and
Atlantic Slopes of South America, and the most widely popular of his works,
the Physical Geography of the Sea, was published in London in 1855, and
in New York in 1856; it was translated into several European languages. On
the outbreak of the American Civil War in 1861, Maury threw in his lot with
the South, and became head of coast, harbour and river defences. He
invented an electric torpedo for harbour defence, and in 1862 was ordered
to England to purchase torpedo material, &c. Here he took active part in
organizing a petition for peace to the American people, which was
unsuccessful. Afterwards he became imperial commissioner of emigration
naval reform and other subjects, under the title of Scraps from the Lucky-
Bag, which attracted much attention; and in 1841 he was placed in charge
of the Dépôt of Charts and Instruments, out of which grew the United States
Naval Observatory and the Hydrographie Office. He laboured assiduously
to obtain observations as to the winds and currents by distributing to
captains of vessels specially prepared log-books; and in the course of nine
years he had collected a sufficient number of logs to make two hundred
manuscript volumes, each with about two thousand five hundred days’
observations. One result was to show the necessity for combined action on
the part of maritime nations in regard to ocean meteorology. This led to an
international conference at Brussels in 1853, which produced the greatest
benefit to navigation as well as indirectly to meteorology. Maury attempted
to organize co-operative meteorological work on land, but the government
did not at this time take any steps in this direction. His oceanographical
work, however, received recognition in all parts of the civilized world, and
in 1855 it was proposed in the senate to remunerate him, but in the same
year the Naval Retiring Board, erected under an act to promote the
efficiency of the navy, placed him on the retired list. This action aroused
wide opposition, and in 1858 he was reinstated with the rank of commander
as from 1855. In 1853 Maury had published his Letters on the Amazon and
Atlantic Slopes of South America, and the most widely popular of his works,
the Physical Geography of the Sea, was published in London in 1855, and
in New York in 1856; it was translated into several European languages. On
the outbreak of the American Civil War in 1861, Maury threw in his lot with
the South, and became head of coast, harbour and river defences. He
invented an electric torpedo for harbour defence, and in 1862 was ordered
to England to purchase torpedo material, &c. Here he took active part in
organizing a petition for peace to the American people, which was
unsuccessful. Afterwards he became imperial commissioner of emigration
Page 140
to the emperor Maximilian of Mexico, and attempted to form a Virginian
colony in that country. Incidentally he introduced there the cultivation of
cinchona. The scheme of colonization was abandoned by the emperor
(1866), and Maury, who had lost nearly his all during the war, settled for a
while in England, where he was presented with a testimonial raised by
public subscription, and among other honours received the degree of LL.D.
of Cambridge University (1868). In the same year, a general amnesty
admitting of his return to America, he accepted the professorship of
meteorology in the Virginia Military Institute, and settled at Lexington,
Virginia, where he died on the 1st of February 1873.
Among works published by Maury, in addition to those mentioned,
are the papers contributed by him to the Astronomical Observations of
the United States Observatory, Letter concerning Lanes for Steamers
crossing the Atlantic (1855); Physical Geography (1864) and Manual
of Geography (1871). In 1859 he began the publication of a series of
Nautical Monographs.
See Diana Fontaine Maury Corbin (his daughter), Life of Matthew
Fontaine Maury (London, 1888).
MAUSOLEUM, the term given to a monument erected to receive
the remains of a deceased person, which may sometimes take the form of a
sepulchral chapel. The term cenotaph (κενός, empty, τάφος, tomb) is
employed for a similar monument where the body is not buried in the
colony in that country. Incidentally he introduced there the cultivation of
cinchona. The scheme of colonization was abandoned by the emperor
(1866), and Maury, who had lost nearly his all during the war, settled for a
while in England, where he was presented with a testimonial raised by
public subscription, and among other honours received the degree of LL.D.
of Cambridge University (1868). In the same year, a general amnesty
admitting of his return to America, he accepted the professorship of
meteorology in the Virginia Military Institute, and settled at Lexington,
Virginia, where he died on the 1st of February 1873.
Among works published by Maury, in addition to those mentioned,
are the papers contributed by him to the Astronomical Observations of
the United States Observatory, Letter concerning Lanes for Steamers
crossing the Atlantic (1855); Physical Geography (1864) and Manual
of Geography (1871). In 1859 he began the publication of a series of
Nautical Monographs.
See Diana Fontaine Maury Corbin (his daughter), Life of Matthew
Fontaine Maury (London, 1888).
MAUSOLEUM, the term given to a monument erected to receive
the remains of a deceased person, which may sometimes take the form of a
sepulchral chapel. The term cenotaph (κενός, empty, τάφος, tomb) is
employed for a similar monument where the body is not buried in the
Page 141
structure. The term “mausoleum” originated with the magnificent
monument erected by Queen Artemisia in 353 b.c. in memory of her
husband King Mausolus, of which the remains were brought to England in
1859 by Sir Charles Newton and placed in the British Museum. The tombs
of Augustus and of Hadrian in Rome are perhaps the largest monuments of
the kind ever erected.
monument erected by Queen Artemisia in 353 b.c. in memory of her
husband King Mausolus, of which the remains were brought to England in
1859 by Sir Charles Newton and placed in the British Museum. The tombs
of Augustus and of Hadrian in Rome are perhaps the largest monuments of
the kind ever erected.
Page 142
MAUSOLUS (more correctly Maussollus), satrap and practically
ruler of Caria (377-353 b.c.). The part he took in the revolt against
Artaxerxes Mnemon, his conquest of a great part of Lycia, Ionia and of
several of the Greek islands, his co-operation with the Rhodians and their
allies in the war against Athens, and the removal of his capital from Mylasa,
the ancient seat of the Carian kings, to Halicarnassus are the leading facts of
his history. He is best known from the tomb erected for him by his widow
Artemisia. The architects Satyrus and Pythis, and the sculptors Scopas,
Leochares, Bryaxis and Timotheus, finished the work after her death. (See
Halicarnassus.) An inscription discovered at Mylasa (Böckh, Inscr. gr. ii.
2691 c.) details the punishment of certain conspirators who had made an
attempt upon his life at a festival in a temple at Labranda in 353.
See Diod. Sic. xv. 90, 3, xvi. 7, 4, 36, 2; Demosthenes, De
Rhodiorum libertate; J. B. Bury, Hist. of Greece (1902), ii. 271; W.
Judeich, Kleinasiatische Studien (Marburg, 1892), pp. 226-256, and
authorities under Halicarnassus.
ruler of Caria (377-353 b.c.). The part he took in the revolt against
Artaxerxes Mnemon, his conquest of a great part of Lycia, Ionia and of
several of the Greek islands, his co-operation with the Rhodians and their
allies in the war against Athens, and the removal of his capital from Mylasa,
the ancient seat of the Carian kings, to Halicarnassus are the leading facts of
his history. He is best known from the tomb erected for him by his widow
Artemisia. The architects Satyrus and Pythis, and the sculptors Scopas,
Leochares, Bryaxis and Timotheus, finished the work after her death. (See
Halicarnassus.) An inscription discovered at Mylasa (Böckh, Inscr. gr. ii.
2691 c.) details the punishment of certain conspirators who had made an
attempt upon his life at a festival in a temple at Labranda in 353.
See Diod. Sic. xv. 90, 3, xvi. 7, 4, 36, 2; Demosthenes, De
Rhodiorum libertate; J. B. Bury, Hist. of Greece (1902), ii. 271; W.
Judeich, Kleinasiatische Studien (Marburg, 1892), pp. 226-256, and
authorities under Halicarnassus.
Page 143
MAUVE, ANTON (1838-1888), Dutch landscape painter, was
born at Zaandam, the son of a Baptist minister. Much against the wish of his
parents he took up the study of art and entered the studio of Van Os, whose
dry academic manner had, however, but little attraction for him. He
benefited far more by his intimacy with his friends Jozef Israels and W.
Maris. Encouraged by their example he abandoned his early tight and
highly finished manner for a freer, looser method of painting, and the
brilliant palette of his youthful work for a tender lyric harmony which is
generally restricted to delicate greys, greens, and light blue. He excelled in
rendering the soft hazy atmosphere that lingers over the green meadows of
Holland, and devoted himself almost exclusively to depicting the peaceful
rural life of the fields and country lanes of Holland—especially of the
districts near Oosterbeck and Wolfhezen, the sand dunes of the coast at
Scheveningen, and the country near Laren, where he spent the last years of
his life. A little sad and melancholy, his pastoral scenes are nevertheless
conceived in a peaceful soothing lyrical mood, which is in marked contrast
to the epic power and almost tragic intensity of J. F. Millet. There are
fourteen of Mauve’s pictures at the Mesdag Museum at the Hague, and two
(“Milking Time” and “A Fishing Boat putting to Sea”) at the Ryks Museum
in Amsterdam. The Glasgow Corporation Gallery owns his painting of “A
Flock of Sheep.” The finest and most representative private collection of
pictures by Mauve was made by Mr J. C. J. Drucker, London.
born at Zaandam, the son of a Baptist minister. Much against the wish of his
parents he took up the study of art and entered the studio of Van Os, whose
dry academic manner had, however, but little attraction for him. He
benefited far more by his intimacy with his friends Jozef Israels and W.
Maris. Encouraged by their example he abandoned his early tight and
highly finished manner for a freer, looser method of painting, and the
brilliant palette of his youthful work for a tender lyric harmony which is
generally restricted to delicate greys, greens, and light blue. He excelled in
rendering the soft hazy atmosphere that lingers over the green meadows of
Holland, and devoted himself almost exclusively to depicting the peaceful
rural life of the fields and country lanes of Holland—especially of the
districts near Oosterbeck and Wolfhezen, the sand dunes of the coast at
Scheveningen, and the country near Laren, where he spent the last years of
his life. A little sad and melancholy, his pastoral scenes are nevertheless
conceived in a peaceful soothing lyrical mood, which is in marked contrast
to the epic power and almost tragic intensity of J. F. Millet. There are
fourteen of Mauve’s pictures at the Mesdag Museum at the Hague, and two
(“Milking Time” and “A Fishing Boat putting to Sea”) at the Ryks Museum
in Amsterdam. The Glasgow Corporation Gallery owns his painting of “A
Flock of Sheep.” The finest and most representative private collection of
pictures by Mauve was made by Mr J. C. J. Drucker, London.
Page 144
MAVROCORDATO, Mavrocordat or Mavrogordato, the name
of a family of Phanariot Greeks, distinguished in the history of Turkey,
Rumania and modern Greece. The family was founded by a merchant of
Chios, whose son Alexander Mavrocordato (c. 1636-1709), a doctor of
philosophy and medicine of Bologna, became dragoman to the sultan in
1673, and was much employed in negotiations with Austria. It was he who
drew up the treaty of Karlowitz (1699). He became a secretary of state, and
was created a count of the Holy Roman Empire. His authority, with that of
Hussein Kupruli and Rami Pasha, was supreme at the court of Mustapha II.,
and he did much to ameliorate the condition of the Christians in Turkey. He
was disgraced in 1703, but was recalled to court by Sultan Ahmed III. He
left some historical, grammatical, &c. treatises of little value.
His son Nicholas Mavrocordato (1670-1730) was grand dragoman to
the Divan (1697), and in 1708 was appointed hospodar (prince) of
Moldavia. Deposed, owing to the sultan’s suspicions, in favour of
Demetrius Cantacuzene, he was restored in 1711, and soon afterwards
became hospodar of Walachia. In 1716 he was deposed by the Austrians,
but was restored after the peace of Passarowitz. He was the first Greek set
to rule the Danubian principalities, and was responsible for establishing the
system which for a hundred years was to make the name of Greek hateful to
the Rumanians. He introduced Greek manners, the Greek language and
Greek costume, and set up a splendid court on the Byzantine model. For the
rest he was a man of enlightenment, founded libraries and was himself the
author of a curious work entitled Περὶ καθήκοντων (Bucharest, 1719). He
was succeeded as grand dragoman (1709) by his son John (Ioannes), who
was for a short while hospodar of Moldavia, and died in 1720.
Nicholas Mavrocordato was succeeded as prince of Walachia in 1730 by
his son Constantine. He was deprived in the same year, but again ruled the
principality from 1735 to 1741 and from 1744 to 1748; he was prince of
of a family of Phanariot Greeks, distinguished in the history of Turkey,
Rumania and modern Greece. The family was founded by a merchant of
Chios, whose son Alexander Mavrocordato (c. 1636-1709), a doctor of
philosophy and medicine of Bologna, became dragoman to the sultan in
1673, and was much employed in negotiations with Austria. It was he who
drew up the treaty of Karlowitz (1699). He became a secretary of state, and
was created a count of the Holy Roman Empire. His authority, with that of
Hussein Kupruli and Rami Pasha, was supreme at the court of Mustapha II.,
and he did much to ameliorate the condition of the Christians in Turkey. He
was disgraced in 1703, but was recalled to court by Sultan Ahmed III. He
left some historical, grammatical, &c. treatises of little value.
His son Nicholas Mavrocordato (1670-1730) was grand dragoman to
the Divan (1697), and in 1708 was appointed hospodar (prince) of
Moldavia. Deposed, owing to the sultan’s suspicions, in favour of
Demetrius Cantacuzene, he was restored in 1711, and soon afterwards
became hospodar of Walachia. In 1716 he was deposed by the Austrians,
but was restored after the peace of Passarowitz. He was the first Greek set
to rule the Danubian principalities, and was responsible for establishing the
system which for a hundred years was to make the name of Greek hateful to
the Rumanians. He introduced Greek manners, the Greek language and
Greek costume, and set up a splendid court on the Byzantine model. For the
rest he was a man of enlightenment, founded libraries and was himself the
author of a curious work entitled Περὶ καθήκοντων (Bucharest, 1719). He
was succeeded as grand dragoman (1709) by his son John (Ioannes), who
was for a short while hospodar of Moldavia, and died in 1720.
Nicholas Mavrocordato was succeeded as prince of Walachia in 1730 by
his son Constantine. He was deprived in the same year, but again ruled the
principality from 1735 to 1741 and from 1744 to 1748; he was prince of
Page 145
Moldavia from 1741 to 1744 and from 1748 to 1749. His rule was
distinguished by numerous tentative reforms in the fiscal and administrative
systems. He was wounded and taken prisoner in the affair of Galati during
the Russo-Turkish War, on the 5th of November 1769, and died in captivity.
Prince Alexander Mavrocordato (1791-1865), Greek statesman, a
descendant of the hospodars, was born at Constantinople on the 11th of
February 1791. In 1812 he went to the court of his uncle Ioannes Caradja,
hospodar of Walachia, with whom he passed into exile in Russia and Italy
(1817). He was a member of the Hetairia Philike and was among the
Phanariot Greeks who hastened to the Morea on the outbreak of the War of
Independence in 1821. He was active in endeavouring to establish a regular
government, and in January 1822 presided over the first Greek national
assembly at Epidaurus. He commanded the advance of the Greeks into
western Hellas the same year, and suffered a defeat at Peta on the 16th of
July, but retrieved this disaster somewhat by his successful resistance to the
first siege of Missolonghi (Nov. 1822 to Jan. 1823). His English sympathies
brought him, in the subsequent strife of factions, into opposition to the
“Russian” party headed by Demetrius Ypsilanti and Kolokotrones; and
though he held the portfolio of foreign affairs for a short while under the
presidency of Petrobey (Petros Mavromichales), he was compelled to
withdraw from affairs until February 1825, when he again became a
secretary of state. The landing of Ibrahim Pasha followed, and
Mavrocordato again joined the army, only escaping capture in the disaster
at Sphagia (Spakteria), on the 9th of May 1815, by swimming to Navarino.
After the fall of Missolonghi (April 22, 1826) he went into retirement, until
President Capo d’Istria made him a member of the committee for the
administration of war material, a position he resigned in 1828. After Capo
d’Istria’s murder (Oct. 9, 1831) and the resignation of his brother and
successor, Agostino Capo d’Istria (April 13, 1832), Mavrocordato became
minister of finance. He was vice-president of the National Assembly at
distinguished by numerous tentative reforms in the fiscal and administrative
systems. He was wounded and taken prisoner in the affair of Galati during
the Russo-Turkish War, on the 5th of November 1769, and died in captivity.
Prince Alexander Mavrocordato (1791-1865), Greek statesman, a
descendant of the hospodars, was born at Constantinople on the 11th of
February 1791. In 1812 he went to the court of his uncle Ioannes Caradja,
hospodar of Walachia, with whom he passed into exile in Russia and Italy
(1817). He was a member of the Hetairia Philike and was among the
Phanariot Greeks who hastened to the Morea on the outbreak of the War of
Independence in 1821. He was active in endeavouring to establish a regular
government, and in January 1822 presided over the first Greek national
assembly at Epidaurus. He commanded the advance of the Greeks into
western Hellas the same year, and suffered a defeat at Peta on the 16th of
July, but retrieved this disaster somewhat by his successful resistance to the
first siege of Missolonghi (Nov. 1822 to Jan. 1823). His English sympathies
brought him, in the subsequent strife of factions, into opposition to the
“Russian” party headed by Demetrius Ypsilanti and Kolokotrones; and
though he held the portfolio of foreign affairs for a short while under the
presidency of Petrobey (Petros Mavromichales), he was compelled to
withdraw from affairs until February 1825, when he again became a
secretary of state. The landing of Ibrahim Pasha followed, and
Mavrocordato again joined the army, only escaping capture in the disaster
at Sphagia (Spakteria), on the 9th of May 1815, by swimming to Navarino.
After the fall of Missolonghi (April 22, 1826) he went into retirement, until
President Capo d’Istria made him a member of the committee for the
administration of war material, a position he resigned in 1828. After Capo
d’Istria’s murder (Oct. 9, 1831) and the resignation of his brother and
successor, Agostino Capo d’Istria (April 13, 1832), Mavrocordato became
minister of finance. He was vice-president of the National Assembly at
Page 146
Argos (July, 1832), and was appointed by King Otto minister of finance,
and in 1833 premier. From 1834 onwards he was Greek envoy at Munich,
Berlin, London and—after a short interlude as premier in Greece in 1841—
Constantinople. In 1843, after the revolution of September, he returned to
Athens as minister without portfolio in the Metaxas cabinet, and from April
to August 1844 was head of the government formed after the fall of the
“Russian” party. Going into opposition, he distinguished himself by his
violent attacks on the Kolettis government. In 1854-1855 he was again head
of the government for a few months. He died in Aegina on the 18th of
August 1865.
See E. Legrand, Genealogie des Mavrocordato (Paris, 1886).
MAWKMAI (Burmese Maukmè), one of the largest states in the
eastern division of the southern Shan States of Burma. It lies approximately
between 19° 30′ and 20° 30′ N. and 97° 30′ and 98° 15′ E., and has an area
of 2,787 sq. m. The central portion of the state consists of a wide plain well
watered and under rice cultivation. The rest is chiefly hills in ranges
running north and south. There is a good deal of teak in the state, but it has
been ruinously worked. The sawbwa now works as contractor for
government, which takes one-third of the net profits. Rice is the chief crop,
but much tobacco of good quality is grown in the Langkö district on the
Têng river. There is also a great deal of cattle-breeding. The population in
1901 was 29,454, over two-thirds of whom were Shans and the remainder
and in 1833 premier. From 1834 onwards he was Greek envoy at Munich,
Berlin, London and—after a short interlude as premier in Greece in 1841—
Constantinople. In 1843, after the revolution of September, he returned to
Athens as minister without portfolio in the Metaxas cabinet, and from April
to August 1844 was head of the government formed after the fall of the
“Russian” party. Going into opposition, he distinguished himself by his
violent attacks on the Kolettis government. In 1854-1855 he was again head
of the government for a few months. He died in Aegina on the 18th of
August 1865.
See E. Legrand, Genealogie des Mavrocordato (Paris, 1886).
MAWKMAI (Burmese Maukmè), one of the largest states in the
eastern division of the southern Shan States of Burma. It lies approximately
between 19° 30′ and 20° 30′ N. and 97° 30′ and 98° 15′ E., and has an area
of 2,787 sq. m. The central portion of the state consists of a wide plain well
watered and under rice cultivation. The rest is chiefly hills in ranges
running north and south. There is a good deal of teak in the state, but it has
been ruinously worked. The sawbwa now works as contractor for
government, which takes one-third of the net profits. Rice is the chief crop,
but much tobacco of good quality is grown in the Langkö district on the
Têng river. There is also a great deal of cattle-breeding. The population in
1901 was 29,454, over two-thirds of whom were Shans and the remainder
Page 147
Taungthu, Burmese, Yangsek and Red Karens. The capital, Mawkmai,
stands in a fine rice plain in 20° 9′ N. and 97° 25′ E. It had about 150
houses when it first submitted in 1887, but was burnt out by the Red Karens
in the following year. It has since recovered. There are very fine orange
groves a few miles south of the town at Kantu-awn, called Kadugate by the
Burmese.
MAXENTIUS, MARCUS AURELIUS
VALERIUS, Roman emperor from a.d. 306 to 312, was the son of
Maximianus Herculius, and the son-in-law of Galerius. Owing to his vices
and incapacity he was left out of account in the division of the empire
which took place in 305. A variety of causes, however, had produced strong
dissatisfaction at Rome with many of the arrangements established by
Diocletian, and on the 28th of October 306, the public discontent found
expression in the massacre of those magistrates who remained loyal to
Flavius Valerius Severus and in the election of Maxentius to the imperial
dignity. With the help of his father, Maxentius was enabled to put Severus
to death and to repel the invasion of Galerius; his next steps were first to
banish Maximianus, and then, after achieving a military success in Africa
against the rebellious governor, L. Domitius Alexander, to declare war
against Constantine as having brought about the death of his father
Maximianus. His intention of carrying the war into Gaul was anticipated by
Constantine, who marched into Italy. Maxentius was defeated at Saxa
Rubra near Rome and drowned in the Tiber while attempting to make his
stands in a fine rice plain in 20° 9′ N. and 97° 25′ E. It had about 150
houses when it first submitted in 1887, but was burnt out by the Red Karens
in the following year. It has since recovered. There are very fine orange
groves a few miles south of the town at Kantu-awn, called Kadugate by the
Burmese.
MAXENTIUS, MARCUS AURELIUS
VALERIUS, Roman emperor from a.d. 306 to 312, was the son of
Maximianus Herculius, and the son-in-law of Galerius. Owing to his vices
and incapacity he was left out of account in the division of the empire
which took place in 305. A variety of causes, however, had produced strong
dissatisfaction at Rome with many of the arrangements established by
Diocletian, and on the 28th of October 306, the public discontent found
expression in the massacre of those magistrates who remained loyal to
Flavius Valerius Severus and in the election of Maxentius to the imperial
dignity. With the help of his father, Maxentius was enabled to put Severus
to death and to repel the invasion of Galerius; his next steps were first to
banish Maximianus, and then, after achieving a military success in Africa
against the rebellious governor, L. Domitius Alexander, to declare war
against Constantine as having brought about the death of his father
Maximianus. His intention of carrying the war into Gaul was anticipated by
Constantine, who marched into Italy. Maxentius was defeated at Saxa
Rubra near Rome and drowned in the Tiber while attempting to make his
Page 148
way across the Milvian bridge into Rome. He was a man of brutal and
worthless character; but although Gibbon’s statement that he was “just,
humane and even partial towards the afflicted Christians” may be
exaggerated, it is probable that he never exhibited any special hostility
towards them.
See De Broglie, L’Église et l’empire Romain au quatrième siècle
(1856-1866), and on the attitude of the Romans towards Christianity
generally, app. 8 in vol. ii. of J. B. Bury’s edition of Gibbon (Zosimus
ii. 9-18; Zonaras xii. 33, xiii. 1; Aurelius Victor, Epit. 40; Eutropius, x.
2).
MAXIM, SIR HIRAM STEVENS (1840- ), Anglo-
American engineer and inventor, was born at Sangerville, Maine, U.S.A.,
on the 5th of February 1840. After serving an apprenticeship with a
coachbuilder, he entered the machine works of his uncle, Levi Stevens, at
Fitchburg, Massachusetts, in 1864, and four years later he became a
draughtsman in the Novelty Iron Works and Shipbuilding Company in New
York City. About this period he produced several inventions connected with
illumination by gas; and from 1877 he was one of the numerous inventors
who were trying to solve the problem of making an efficient and durable
incandescent electric lamp, in this connexion introducing the widely-used
process of treating the carbon filaments by heating them in an atmosphere
of hydrocarbon vapour. In 1880 he came to Europe, and soon began to
worthless character; but although Gibbon’s statement that he was “just,
humane and even partial towards the afflicted Christians” may be
exaggerated, it is probable that he never exhibited any special hostility
towards them.
See De Broglie, L’Église et l’empire Romain au quatrième siècle
(1856-1866), and on the attitude of the Romans towards Christianity
generally, app. 8 in vol. ii. of J. B. Bury’s edition of Gibbon (Zosimus
ii. 9-18; Zonaras xii. 33, xiii. 1; Aurelius Victor, Epit. 40; Eutropius, x.
2).
MAXIM, SIR HIRAM STEVENS (1840- ), Anglo-
American engineer and inventor, was born at Sangerville, Maine, U.S.A.,
on the 5th of February 1840. After serving an apprenticeship with a
coachbuilder, he entered the machine works of his uncle, Levi Stevens, at
Fitchburg, Massachusetts, in 1864, and four years later he became a
draughtsman in the Novelty Iron Works and Shipbuilding Company in New
York City. About this period he produced several inventions connected with
illumination by gas; and from 1877 he was one of the numerous inventors
who were trying to solve the problem of making an efficient and durable
incandescent electric lamp, in this connexion introducing the widely-used
process of treating the carbon filaments by heating them in an atmosphere
of hydrocarbon vapour. In 1880 he came to Europe, and soon began to
Page 149
devote himself to the construction of a machine-gun which should be
automatically loaded and fired by the energy of the recoil (see Machine-
Gun). In order to realize the full usefulness of the weapon, which was first
exhibited in an underground range at Hatton Garden, London, in 1884, he
felt the necessity of employing a smokeless powder, and accordingly he
devised maximite, a mixture of trinitrocellulose, nitroglycerine and castor
oil, which was patented in 1889. He also undertook to make a flying
machine, and after numerous preliminary experiments constructed an
apparatus which was tried at Bexley Heath, Kent, in 1894. (See Flight.)
Having been naturalized as a British subject, he was knighted in 1901. His
younger brother, Hudson Maxim (b. 1853), took out numerous patents in
connexion with explosives.
MAXIMA AND MINIMA, in mathematics. By the maximum
or minimum value of an expression or quantity is meant primarily the
“greatest” or “least” value that it can receive. In general, however, there are
points at which its value ceases to increase and begins to decrease; its value
at such a point is called a maximum. So there are points at which its value
ceases to decrease and begins to increase; such a value is called a minimum.
There may be several maxima or minima, and a minimum is not necessarily
less than a maximum. For instance, the expression (x2 + x + 2)/(x − 1) can
take all values from −∞ to −1 and from +7 to +∞, but has, so long as x is
real, no value between -1 and +7. Here −1 is a maximum value, and +7 is a
automatically loaded and fired by the energy of the recoil (see Machine-
Gun). In order to realize the full usefulness of the weapon, which was first
exhibited in an underground range at Hatton Garden, London, in 1884, he
felt the necessity of employing a smokeless powder, and accordingly he
devised maximite, a mixture of trinitrocellulose, nitroglycerine and castor
oil, which was patented in 1889. He also undertook to make a flying
machine, and after numerous preliminary experiments constructed an
apparatus which was tried at Bexley Heath, Kent, in 1894. (See Flight.)
Having been naturalized as a British subject, he was knighted in 1901. His
younger brother, Hudson Maxim (b. 1853), took out numerous patents in
connexion with explosives.
MAXIMA AND MINIMA, in mathematics. By the maximum
or minimum value of an expression or quantity is meant primarily the
“greatest” or “least” value that it can receive. In general, however, there are
points at which its value ceases to increase and begins to decrease; its value
at such a point is called a maximum. So there are points at which its value
ceases to decrease and begins to increase; such a value is called a minimum.
There may be several maxima or minima, and a minimum is not necessarily
less than a maximum. For instance, the expression (x2 + x + 2)/(x − 1) can
take all values from −∞ to −1 and from +7 to +∞, but has, so long as x is
real, no value between -1 and +7. Here −1 is a maximum value, and +7 is a
Page 150
minimum value of the expression, though it can be made greater or less
than any assignable quantity.
The first general method of investigating maxima and minima seems to
have been published in a.d. 1629 by Pierre Fermat. Particular cases had
been discussed. Thus Euclid in book III. of the Elements finds the greatest
and least straight lines that can be drawn from a point to the circumference
of a circle, and in book VI. (in a proposition generally omitted from editions
of his works) finds the parallelogram of greatest area with a given
perimeter. Apollonius investigated the greatest and least distances of a point
from the perimeter of a conic section, and discovered them to be the
normals, and that their feet were the intersections of the conic with a
rectangular hyperbola. Some remarkable theorems on maximum areas are
attributed to Zenodorus, and preserved by Pappus and Theon of Alexandria.
The most noteworthy of them are the following:—
1. Of polygons of n sides with a given perimeter the regular polygon
encloses the greatest area.
2. Of two regular polygons of the same perimeter, that with the
greater number of sides encloses the greater area.
3. The circle encloses a greater area than any polygon of the same
perimeter.
4. The sum of the areas of two isosceles triangles on given bases, the
sum of whose perimeters is given, is greatest when the triangles are
similar.
5. Of segments of a circle of given perimeter, the semicircle encloses
the greatest area.
6. The sphere is the surface of given area which encloses the greatest
volume.
than any assignable quantity.
The first general method of investigating maxima and minima seems to
have been published in a.d. 1629 by Pierre Fermat. Particular cases had
been discussed. Thus Euclid in book III. of the Elements finds the greatest
and least straight lines that can be drawn from a point to the circumference
of a circle, and in book VI. (in a proposition generally omitted from editions
of his works) finds the parallelogram of greatest area with a given
perimeter. Apollonius investigated the greatest and least distances of a point
from the perimeter of a conic section, and discovered them to be the
normals, and that their feet were the intersections of the conic with a
rectangular hyperbola. Some remarkable theorems on maximum areas are
attributed to Zenodorus, and preserved by Pappus and Theon of Alexandria.
The most noteworthy of them are the following:—
1. Of polygons of n sides with a given perimeter the regular polygon
encloses the greatest area.
2. Of two regular polygons of the same perimeter, that with the
greater number of sides encloses the greater area.
3. The circle encloses a greater area than any polygon of the same
perimeter.
4. The sum of the areas of two isosceles triangles on given bases, the
sum of whose perimeters is given, is greatest when the triangles are
similar.
5. Of segments of a circle of given perimeter, the semicircle encloses
the greatest area.
6. The sphere is the surface of given area which encloses the greatest
volume.
Page 151
Serenus of Antissa investigated the somewhat trifling problem of finding
the triangle of greatest area whose sides are formed by the intersections
with the base and curved surface of a right circular cone of a plane drawn
through its vertex.
The next problem on maxima and minima of which there appears to be
any record occurs in a letter from Regiomontanus to Roder (July 4, 1471),
and is a particular numerical example of the problem of finding the point on
a given straight line at which two given points subtend a maximum angle.
N. Tartaglia in his General trattato de numeri et mesuri (c. 1556) gives,
without proof, a rule for dividing a number into two parts such that the
continued product of the numbers and their difference is a maximum.
Fermat investigated maxima and minima by means of the principle that
in the neighbourhood of a maximum or minimum the differences of the
values of a function are insensible, a method virtually the same as that of
the differential calculus, and of great use in dealing with geometrical
maxima and minima. His method was developed by Huygens, Leibnitz,
Newton and others, and in particular by John Hudde, who investigated
maxima and minima of functions of more than one independent variable,
and made some attempt to discriminate between maxima and minima, a
question first definitely settled, so far as one variable is concerned, by Colin
Maclaurin in his Treatise on Fluxions (1742). The method of the differential
calculus was perfected by Euler and Lagrange.
John Bernoulli’s famous problem of the “brachistochrone,” or curve of
quickest descent from one point to another under the action of gravity,
proposed in 1696, gave rise to a new kind of maximum and minimum
problem in which we have to find a curve and not points on a given curve.
From these problems arose the “Calculus of Variations.” (See Variations,
Calculus of.)
the triangle of greatest area whose sides are formed by the intersections
with the base and curved surface of a right circular cone of a plane drawn
through its vertex.
The next problem on maxima and minima of which there appears to be
any record occurs in a letter from Regiomontanus to Roder (July 4, 1471),
and is a particular numerical example of the problem of finding the point on
a given straight line at which two given points subtend a maximum angle.
N. Tartaglia in his General trattato de numeri et mesuri (c. 1556) gives,
without proof, a rule for dividing a number into two parts such that the
continued product of the numbers and their difference is a maximum.
Fermat investigated maxima and minima by means of the principle that
in the neighbourhood of a maximum or minimum the differences of the
values of a function are insensible, a method virtually the same as that of
the differential calculus, and of great use in dealing with geometrical
maxima and minima. His method was developed by Huygens, Leibnitz,
Newton and others, and in particular by John Hudde, who investigated
maxima and minima of functions of more than one independent variable,
and made some attempt to discriminate between maxima and minima, a
question first definitely settled, so far as one variable is concerned, by Colin
Maclaurin in his Treatise on Fluxions (1742). The method of the differential
calculus was perfected by Euler and Lagrange.
John Bernoulli’s famous problem of the “brachistochrone,” or curve of
quickest descent from one point to another under the action of gravity,
proposed in 1696, gave rise to a new kind of maximum and minimum
problem in which we have to find a curve and not points on a given curve.
From these problems arose the “Calculus of Variations.” (See Variations,
Calculus of.)
Page 152
The only general methods of attacking problems on maxima and minima
are those of the differential calculus or, in geometrical problems, what is
practically Fermat’s method. Some problems may be solved by algebra;
thus if y = ƒ(x) ÷ φ(x), where ƒ(x) and φ(x) are polynomials in x, the limits
to the values of yφ may be found from the consideration that the equation
yφ(x) − ƒ(x) = 0 must have real roots. This is a useful method in the case in
which φ(x) and ƒ(x) are quadratics, but scarcely ever in any other case. The
problem of finding the maximum product of n positive quantities whose
sum is given may also be found, algebraically, thus. If a and b are any two
real unequal quantities whatever {1⁄2(a + b)}2 > ab, so that we can increase
the product leaving the sum unaltered by replacing any two terms by half
their sum, and so long as any two of the quantities are unequal we can
increase the product. Now, the quantities being all positive, the product
cannot be increased without limit and must somewhere attain a maximum,
and no other form of the product than that in which they are all equal can be
the maximum, so that the product is a maximum when they are all equal. Its
minimum value is obviously zero. If the restriction that all the quantities
shall be positive is removed, the product can be made equal to any quantity,
positive or negative. So other theorems of algebra, which are stated as
theorems on inequalities, may be regarded as algebraic solutions of
problems on maxima and minima.
For purely geometrical questions the only general method available is
practically that employed by Fermat. If a quantity depends on the position
of some point P on a curve, and if its value is equal at two neighbouring
points P and P′, then at some position between P and P′ it attains a
maximum or minimum, and this position may be found by making P and P′
approach each other indefinitely. Take for instance the problem of
Regiomontanus “to find a point on a given straight line which subtends a
maximum angle at two given points A and B.” Let P and P′ be two near
points on the given straight line such that the angles APB and AP′B are
are those of the differential calculus or, in geometrical problems, what is
practically Fermat’s method. Some problems may be solved by algebra;
thus if y = ƒ(x) ÷ φ(x), where ƒ(x) and φ(x) are polynomials in x, the limits
to the values of yφ may be found from the consideration that the equation
yφ(x) − ƒ(x) = 0 must have real roots. This is a useful method in the case in
which φ(x) and ƒ(x) are quadratics, but scarcely ever in any other case. The
problem of finding the maximum product of n positive quantities whose
sum is given may also be found, algebraically, thus. If a and b are any two
real unequal quantities whatever {1⁄2(a + b)}2 > ab, so that we can increase
the product leaving the sum unaltered by replacing any two terms by half
their sum, and so long as any two of the quantities are unequal we can
increase the product. Now, the quantities being all positive, the product
cannot be increased without limit and must somewhere attain a maximum,
and no other form of the product than that in which they are all equal can be
the maximum, so that the product is a maximum when they are all equal. Its
minimum value is obviously zero. If the restriction that all the quantities
shall be positive is removed, the product can be made equal to any quantity,
positive or negative. So other theorems of algebra, which are stated as
theorems on inequalities, may be regarded as algebraic solutions of
problems on maxima and minima.
For purely geometrical questions the only general method available is
practically that employed by Fermat. If a quantity depends on the position
of some point P on a curve, and if its value is equal at two neighbouring
points P and P′, then at some position between P and P′ it attains a
maximum or minimum, and this position may be found by making P and P′
approach each other indefinitely. Take for instance the problem of
Regiomontanus “to find a point on a given straight line which subtends a
maximum angle at two given points A and B.” Let P and P′ be two near
points on the given straight line such that the angles APB and AP′B are
Page 153
equal. Then ABPP′ lie on a circle. By making P and P′ approach each other
we see that for a maximum or minimum value of the angle APB, P is a
point in which a circle drawn through AB touches the given straight line.
There are two such points, and unless the given straight line is at right
angles to AB the two angles obtained are not the same. It is easily seen that
both angles are maxima, one for points on the given straight line on one
side of its intersection with AB, the other for points on the other side. For
further examples of this method together with most other geometrical
problems on maxima and minima of any interest or importance the reader
may consult such a book as J. W. Russell’s A Sequel lo Elementary
Geometry (Oxford, 1907).
The method of the differential calculus is theoretically very simple.
Let u be a function of several variables x1, x2, x3 ... xn, supposed for the
present independent; if u is a maximum or minimum for the set of
values x1, x2, x3, ... xn, and u becomes u + δu, when x1, x2, x3 ... xn
receive small increments δx1, δx2, ... δxn; then δu must have the same
sign for all possible values of δx1, δ2 ... δxn.
Now
δu δ2u δ2u
δu = Σ
δx1 δx1 + 1⁄2 { Σ δx12 δx12 + 2Σ
δx1δx2 δx1δx2 ... } + ....
The sign of this expression in general is that of Σ(δu/δx1)δx1, which
cannot be one-signed when x1, x2, ... xn can take all possible values, for
a set of increments δx1, δx2 ... δxn, will give an opposite sign to the set
−δx1, −δx2, ... −δxn. Hence Σ(δu/δx1)δx1 must vanish for all sets of
increments δx1, ... δxn, and since these are independent, we must have
δu/δx1 = 0, δu/δx2 = 0, ... δu/δxn = 0. A value of u given by a set of
we see that for a maximum or minimum value of the angle APB, P is a
point in which a circle drawn through AB touches the given straight line.
There are two such points, and unless the given straight line is at right
angles to AB the two angles obtained are not the same. It is easily seen that
both angles are maxima, one for points on the given straight line on one
side of its intersection with AB, the other for points on the other side. For
further examples of this method together with most other geometrical
problems on maxima and minima of any interest or importance the reader
may consult such a book as J. W. Russell’s A Sequel lo Elementary
Geometry (Oxford, 1907).
The method of the differential calculus is theoretically very simple.
Let u be a function of several variables x1, x2, x3 ... xn, supposed for the
present independent; if u is a maximum or minimum for the set of
values x1, x2, x3, ... xn, and u becomes u + δu, when x1, x2, x3 ... xn
receive small increments δx1, δx2, ... δxn; then δu must have the same
sign for all possible values of δx1, δ2 ... δxn.
Now
δu δ2u δ2u
δu = Σ
δx1 δx1 + 1⁄2 { Σ δx12 δx12 + 2Σ
δx1δx2 δx1δx2 ... } + ....
The sign of this expression in general is that of Σ(δu/δx1)δx1, which
cannot be one-signed when x1, x2, ... xn can take all possible values, for
a set of increments δx1, δx2 ... δxn, will give an opposite sign to the set
−δx1, −δx2, ... −δxn. Hence Σ(δu/δx1)δx1 must vanish for all sets of
increments δx1, ... δxn, and since these are independent, we must have
δu/δx1 = 0, δu/δx2 = 0, ... δu/δxn = 0. A value of u given by a set of
Page 154
solutions of these equations is called a “critical value” of u. The value
of δu now becomes
δ2u δ2u
⁄
12 { Σ δx12 δx12 + 2 Σ
δx1δx2 δx1δx2 + ... };
for u to be a maximum or minimum this must have always the same
sign. For the case of a single variable x, corresponding to a value of x
given by the equation du/dx = 0, u is a maximum or minimum as
d2u/dx2 is negative or positive. If d2u/dx2 vanishes, then there is no
maximum or minimum unless d2u/dx2 vanishes, and there is a
maximum or minimum according as d4u/dx4 is negative or positive.
Generally, if the first differential coefficient which does not vanish is
even, there is a maximum or minimum according as this is negative or
positive. If it is odd, there is no maximum or minimum.
In the case of several variables, the quadratic
δ2u δ 2u
Σ δx12 + 2 Σ δx1δx2 + ...
δx12 δx1δx2
must be one-signed. The condition for this is that the series of
discriminants
a11 , a11 a12 , a11 a12 a13 , ...
a21 a22 a21 a22 a23
a31 a32 a33
where apq denotes δ2u/δapδaq should be all positive, if the quadratic is
always positive, and alternately negative and positive, if the quadratic
is always negative. If the first condition is satisfied the critical value is
of δu now becomes
δ2u δ2u
⁄
12 { Σ δx12 δx12 + 2 Σ
δx1δx2 δx1δx2 + ... };
for u to be a maximum or minimum this must have always the same
sign. For the case of a single variable x, corresponding to a value of x
given by the equation du/dx = 0, u is a maximum or minimum as
d2u/dx2 is negative or positive. If d2u/dx2 vanishes, then there is no
maximum or minimum unless d2u/dx2 vanishes, and there is a
maximum or minimum according as d4u/dx4 is negative or positive.
Generally, if the first differential coefficient which does not vanish is
even, there is a maximum or minimum according as this is negative or
positive. If it is odd, there is no maximum or minimum.
In the case of several variables, the quadratic
δ2u δ 2u
Σ δx12 + 2 Σ δx1δx2 + ...
δx12 δx1δx2
must be one-signed. The condition for this is that the series of
discriminants
a11 , a11 a12 , a11 a12 a13 , ...
a21 a22 a21 a22 a23
a31 a32 a33
where apq denotes δ2u/δapδaq should be all positive, if the quadratic is
always positive, and alternately negative and positive, if the quadratic
is always negative. If the first condition is satisfied the critical value is
Page 155
a minimum, if the second it is a maximum. For the case of two
variables the conditions are
δ2u δ2u δ 2u
δx12
·
δx22 > ( δx1δx2 )2
for a maximum or minimum at all and δ2u/δx12 and δ2u/δx22 both
negative for a maximum, and both positive for a minimum. It is
important to notice that by the quadratic being one-signed is meant that
it cannot be made to vanish except when δx1, δx2, ... δxn all vanish. If,
in the case of two variables,
δ2u δ2u δ 2u
δx12
·
δx22 = ( δx1δx2 )2
then the quadratic is one-signed unless it vanishes, but the value of u is
not necessarily a maximum or minimum, and the terms of the third and
possibly fourth order must be taken account of.
Take for instance the function u = x2 − xy2 + y2. Here the values x =
0, y = 0 satisfy the equations δu/δx = 0, δu/δy = 0, so that zero is a
critical value of u, but it is neither a maximum nor a minimum
although the terms of the second order are (δx)2, and are never
negative. Here δu = δx2 − δxδy2 + δy2, and by putting δx = 0 or an
infinitesimal of the same order as δy2, we can make the sign of δu
depend on that of δy2, and so be positive or negative as we please. On
the other hand, if we take the function u = x2 − xy2 + y4, x = 0, y = 0
make zero a critical value of u, and here δu = δx2 − δxδy2 + δy4, which
is always positive, because we can write it as the sum of two squares,
viz. (δx − 1⁄2δy2)2 + 3⁄4δy4; so that in this case zero is a minimum value
of u.
variables the conditions are
δ2u δ2u δ 2u
δx12
·
δx22 > ( δx1δx2 )2
for a maximum or minimum at all and δ2u/δx12 and δ2u/δx22 both
negative for a maximum, and both positive for a minimum. It is
important to notice that by the quadratic being one-signed is meant that
it cannot be made to vanish except when δx1, δx2, ... δxn all vanish. If,
in the case of two variables,
δ2u δ2u δ 2u
δx12
·
δx22 = ( δx1δx2 )2
then the quadratic is one-signed unless it vanishes, but the value of u is
not necessarily a maximum or minimum, and the terms of the third and
possibly fourth order must be taken account of.
Take for instance the function u = x2 − xy2 + y2. Here the values x =
0, y = 0 satisfy the equations δu/δx = 0, δu/δy = 0, so that zero is a
critical value of u, but it is neither a maximum nor a minimum
although the terms of the second order are (δx)2, and are never
negative. Here δu = δx2 − δxδy2 + δy2, and by putting δx = 0 or an
infinitesimal of the same order as δy2, we can make the sign of δu
depend on that of δy2, and so be positive or negative as we please. On
the other hand, if we take the function u = x2 − xy2 + y4, x = 0, y = 0
make zero a critical value of u, and here δu = δx2 − δxδy2 + δy4, which
is always positive, because we can write it as the sum of two squares,
viz. (δx − 1⁄2δy2)2 + 3⁄4δy4; so that in this case zero is a minimum value
of u.
Page 156
A critical value usually gives a maximum or minimum in the case of
a function of one variable, and often in the case of several independent
variables, but all maxima and minima, particularly absolutely greatest
and least values, are not necessarily critical values. If, for example, x is
restricted to lie between the values a and b and φ′(x) = 0 has no roots in
this interval, it follows that φ′(x) is one-signed as x increases from a to
b, so that φ(x) is increasing or diminishing all the time, and the greatest
and least values of φ(x) are φ(a) and φ(b), though neither of them is a
critical value. Consider the following example: A person in a boat a
miles from the nearest point of the beach wishes to reach as quickly as
possible a point b miles from that point along the shore. The ratio of his
rate of walking to his rate of rowing is cosec α. Where should he land?
Here let AB be the direction of the beach, A the nearest point to the
boat O, and B the point he wishes to reach. Clearly he must land, if at
all, between A and B. Suppose he lands at P. Let the angle AOP be θ, so
that OP = a secθ, and PB = b − a tan θ. If his rate of rowing is V miles
an hour his time will be a sec θ/V + (b − a tan θ) sin α/V hours. Call
this T. Then to the first power of δθ, δT = (a/V) sec2θ (sin θ − sin α)δθ,
so that if AOB > α, δT and δθ have opposite signs from θ = 0 to θ = α,
and the same signs from θ = α to θ = AOB. So that when AOB is > α, T
decreases from θ = 0 to θ = α, and then increases, so that he should
land at a point distant a tan α from A, unless a tan α > b. When this is
the case, δT and δθ have opposite signs throughout the whole range of
θ, so that T decreases as θ increases, and he should row direct to B. In
the first case the minimum value of T is also a critical value; in the
second case it is not.
The greatest and least values of the bending moments of loaded rods
are often at the extremities of the divisions of the rods and not at points
given by critical values.
a function of one variable, and often in the case of several independent
variables, but all maxima and minima, particularly absolutely greatest
and least values, are not necessarily critical values. If, for example, x is
restricted to lie between the values a and b and φ′(x) = 0 has no roots in
this interval, it follows that φ′(x) is one-signed as x increases from a to
b, so that φ(x) is increasing or diminishing all the time, and the greatest
and least values of φ(x) are φ(a) and φ(b), though neither of them is a
critical value. Consider the following example: A person in a boat a
miles from the nearest point of the beach wishes to reach as quickly as
possible a point b miles from that point along the shore. The ratio of his
rate of walking to his rate of rowing is cosec α. Where should he land?
Here let AB be the direction of the beach, A the nearest point to the
boat O, and B the point he wishes to reach. Clearly he must land, if at
all, between A and B. Suppose he lands at P. Let the angle AOP be θ, so
that OP = a secθ, and PB = b − a tan θ. If his rate of rowing is V miles
an hour his time will be a sec θ/V + (b − a tan θ) sin α/V hours. Call
this T. Then to the first power of δθ, δT = (a/V) sec2θ (sin θ − sin α)δθ,
so that if AOB > α, δT and δθ have opposite signs from θ = 0 to θ = α,
and the same signs from θ = α to θ = AOB. So that when AOB is > α, T
decreases from θ = 0 to θ = α, and then increases, so that he should
land at a point distant a tan α from A, unless a tan α > b. When this is
the case, δT and δθ have opposite signs throughout the whole range of
θ, so that T decreases as θ increases, and he should row direct to B. In
the first case the minimum value of T is also a critical value; in the
second case it is not.
The greatest and least values of the bending moments of loaded rods
are often at the extremities of the divisions of the rods and not at points
given by critical values.
Page 157
In the case of a function of several variables, X1, x2, ... xn, not
independent but connected by m functional relations u1 = 0, u2 = 0, ...,
um = 0, we might proceed to eliminate m of the variables; but
Lagrange’s “Method of undetermined Multipliers” is more elegant and
generally more useful.
We have δu1 = 0, δu2 = 0, ..., δum = 0. Consider instead of δu, what
is the same thing, viz., δu + λ1δu1 + λ2δu2 + ... + λmδum, where λ1, λ2,
... λm, are arbitrary multipliers. The terms of the first order in this
expression are
δu δu1 δum
Σ δx1 + λ1 Σ δx1 + ... + λm Σ δx1.
δx1 δx1 δx1
We can choose λ1, ... λm, to make the coefficients of δx1, δx2, ... δxm,
vanish, and the remaining δxm+1 to δxn may be regarded as
independent, so that, when u has a critical value, their coefficients must
also vanish. So that we put
δu δu1 δum
+ + ... + λm =0
δxr δxr δxr
for all values of r. These equations with the equations u1 = 0, ..., um = 0
are exactly enough to determine λ1, ..., λm, x1 x2, ..., xn, so that we find
critical values of u, and examine the terms of the second order to
decide whether we obtain a maximum or minimum.
To take a very simple illustration; consider the problem of
determining the maximum and minimum radii vectors of the ellipsoid
x2/a2 + y2/b2 + z2/c2 = 1, where a2 > b2 > c2. Here we require the
maximum and minimum values of x2 + y2 + z2 where x2/a2 + y2/b2 +
z2/c2 = 1.
independent but connected by m functional relations u1 = 0, u2 = 0, ...,
um = 0, we might proceed to eliminate m of the variables; but
Lagrange’s “Method of undetermined Multipliers” is more elegant and
generally more useful.
We have δu1 = 0, δu2 = 0, ..., δum = 0. Consider instead of δu, what
is the same thing, viz., δu + λ1δu1 + λ2δu2 + ... + λmδum, where λ1, λ2,
... λm, are arbitrary multipliers. The terms of the first order in this
expression are
δu δu1 δum
Σ δx1 + λ1 Σ δx1 + ... + λm Σ δx1.
δx1 δx1 δx1
We can choose λ1, ... λm, to make the coefficients of δx1, δx2, ... δxm,
vanish, and the remaining δxm+1 to δxn may be regarded as
independent, so that, when u has a critical value, their coefficients must
also vanish. So that we put
δu δu1 δum
+ + ... + λm =0
δxr δxr δxr
for all values of r. These equations with the equations u1 = 0, ..., um = 0
are exactly enough to determine λ1, ..., λm, x1 x2, ..., xn, so that we find
critical values of u, and examine the terms of the second order to
decide whether we obtain a maximum or minimum.
To take a very simple illustration; consider the problem of
determining the maximum and minimum radii vectors of the ellipsoid
x2/a2 + y2/b2 + z2/c2 = 1, where a2 > b2 > c2. Here we require the
maximum and minimum values of x2 + y2 + z2 where x2/a2 + y2/b2 +
z2/c2 = 1.
Page 158
We have
λ λ λ
δu = 2xδx ( 1 + a2 ) + 2yδy ( b2 ) + 2zδz ( c2 )
λ λ λ
+ δx2 ( 1 + a2 ) + δy2 ( b2 ) + δz2 ( c2 ).
To make the terms of the first order disappear, we have the three
equations:—
x (1 + λ/a2) = 0, y (1 + λ/b2) = 0, z (1 + λ/c2) = 0.
These have three sets of solutions consistent with the conditions x2/a2 +
y2/b2 + z2/c2 = 1, a2 > b2 > c2, viz.:—
(1) y = 0, z = 0, λ = −a2; (2) z = 0, x = 0, λ = −b2;
(3) x = 0, y = 0, λ = −c2.
In the case of (1) δu = δy2 (1 − a2/b2) + δz2 (1 − a2/c2), which is
always negative, so that u = a2 gives a maximum.
In the case of (3) δu = δx2 (1 − c2/a2) + δy2 (1 − c2/b2), which is
always positive, so that u = c2 gives a minimum.
In the case of (2) δu = δx2 (1 − b2/a2) − δz2(b2/c2 − 1), which can be
made either positive or negative, or even zero if we move in the planes
x2 (1 − b2/a2) = z2 (b2/c2 − 1), which are well known to be the central
planes of circular section. So that u = b2, though a critical value, is
neither a maximum nor minimum, and the central planes of circular
section divide the ellipsoid into four portions in two of which a2 > r2 >
b2, and in the other two b2 > r2 > c2. (A. E. J.)
λ λ λ
δu = 2xδx ( 1 + a2 ) + 2yδy ( b2 ) + 2zδz ( c2 )
λ λ λ
+ δx2 ( 1 + a2 ) + δy2 ( b2 ) + δz2 ( c2 ).
To make the terms of the first order disappear, we have the three
equations:—
x (1 + λ/a2) = 0, y (1 + λ/b2) = 0, z (1 + λ/c2) = 0.
These have three sets of solutions consistent with the conditions x2/a2 +
y2/b2 + z2/c2 = 1, a2 > b2 > c2, viz.:—
(1) y = 0, z = 0, λ = −a2; (2) z = 0, x = 0, λ = −b2;
(3) x = 0, y = 0, λ = −c2.
In the case of (1) δu = δy2 (1 − a2/b2) + δz2 (1 − a2/c2), which is
always negative, so that u = a2 gives a maximum.
In the case of (3) δu = δx2 (1 − c2/a2) + δy2 (1 − c2/b2), which is
always positive, so that u = c2 gives a minimum.
In the case of (2) δu = δx2 (1 − b2/a2) − δz2(b2/c2 − 1), which can be
made either positive or negative, or even zero if we move in the planes
x2 (1 − b2/a2) = z2 (b2/c2 − 1), which are well known to be the central
planes of circular section. So that u = b2, though a critical value, is
neither a maximum nor minimum, and the central planes of circular
section divide the ellipsoid into four portions in two of which a2 > r2 >
b2, and in the other two b2 > r2 > c2. (A. E. J.)
Page 159
MAXIMIANUS, a Latin elegiac poet who flourished during the
6th century a.d. He was an Etruscan by birth, and spent his youth at Rome,
where he enjoyed a great reputation as an orator. At an advanced age he was
sent on an important mission to the East, perhaps by Theodoric, if he is the
Maximianus to whom that monarch addressed a letter preserved in
Cassiodorus (Variarum, i. 21). The six elegies extant under his name,
written in old age, in which he laments the loss of his youth, contain
descriptions of various amours. They show the author’s familiarity with the
best writers of the Augustan age.
Editions by J. C. Wernsdorf, Poetae latini minores, vi.; E. Bährens,
Poetae latini minores, v.; M. Petschenig (1890), in C. F. Ascherson’s
Berliner Studien, xi.; R. Webster (Princeton, 1901; see Classical
Review, Oct. 1901), with introduction and commentary; see also
Robinson Ellis in American Journal of Philology, v. (1884) and
Teuffel-Schwabe, Hist. of Roman Literature (Eng. trans.), § 490. There
is an English version (as from Cornelius Gallus), by Hovenden Walker
(1689), under the title of The Impotent Lover.
6th century a.d. He was an Etruscan by birth, and spent his youth at Rome,
where he enjoyed a great reputation as an orator. At an advanced age he was
sent on an important mission to the East, perhaps by Theodoric, if he is the
Maximianus to whom that monarch addressed a letter preserved in
Cassiodorus (Variarum, i. 21). The six elegies extant under his name,
written in old age, in which he laments the loss of his youth, contain
descriptions of various amours. They show the author’s familiarity with the
best writers of the Augustan age.
Editions by J. C. Wernsdorf, Poetae latini minores, vi.; E. Bährens,
Poetae latini minores, v.; M. Petschenig (1890), in C. F. Ascherson’s
Berliner Studien, xi.; R. Webster (Princeton, 1901; see Classical
Review, Oct. 1901), with introduction and commentary; see also
Robinson Ellis in American Journal of Philology, v. (1884) and
Teuffel-Schwabe, Hist. of Roman Literature (Eng. trans.), § 490. There
is an English version (as from Cornelius Gallus), by Hovenden Walker
(1689), under the title of The Impotent Lover.
Page 160
MAXIMIANUS, MARCUS AURELIUS
VALERIUS, surnamed Herculius, Roman emperor from a.d. 286 to
305, was born of humble parents at Sirmium in Pannonia. He achieved
distinction during long service in the army, and having been made Caesar
by Diocletian in 285, received the title of Augustus in the following year
(April 1, 286). In 287 he suppressed the rising of the peasants (Bagaudae) in
Gaul, but in 289, after a three years’ struggle, his colleague and he were
compelled to acquiesce in the assumption by his lieutenant Carausius (who
had crossed over to Britain) of the title of Augustus. After 293 Maximianus
left the care of the Rhine frontier to Constantius Chlorus, who had been
designated Caesar in that year, but in 297 his arms achieved a rapid and
decisive victory over the barbarians of Mauretania, and in 302 he shared at
Rome the triumph of Diocletian, the last pageant of the kind ever witnessed
by that city. On the 1st of May 305, the day of Diocletian’s abdication, he
also, but without his colleague’s sincerity, divested himself of the imperial
dignity at Mediolanum (Milan), which had been his capital, and retired to a
villa in Lucania; in the following year, however, he was induced by his son
Maxentius to reassume the purple. In 307 he brought the emperor Flavius
Valerius Severus a captive to Rome, and also compelled Galerius to retreat,
but in 308 he was himself driven by Maxentius from Italy into Illyricum,
whence again he was compelled to seek refuge at Arelate (Arles), the court
of his son-in-law, Constantine. Here a false report was received, or
invented, of the death of Constantine, at that time absent on the Rhine.
VALERIUS, surnamed Herculius, Roman emperor from a.d. 286 to
305, was born of humble parents at Sirmium in Pannonia. He achieved
distinction during long service in the army, and having been made Caesar
by Diocletian in 285, received the title of Augustus in the following year
(April 1, 286). In 287 he suppressed the rising of the peasants (Bagaudae) in
Gaul, but in 289, after a three years’ struggle, his colleague and he were
compelled to acquiesce in the assumption by his lieutenant Carausius (who
had crossed over to Britain) of the title of Augustus. After 293 Maximianus
left the care of the Rhine frontier to Constantius Chlorus, who had been
designated Caesar in that year, but in 297 his arms achieved a rapid and
decisive victory over the barbarians of Mauretania, and in 302 he shared at
Rome the triumph of Diocletian, the last pageant of the kind ever witnessed
by that city. On the 1st of May 305, the day of Diocletian’s abdication, he
also, but without his colleague’s sincerity, divested himself of the imperial
dignity at Mediolanum (Milan), which had been his capital, and retired to a
villa in Lucania; in the following year, however, he was induced by his son
Maxentius to reassume the purple. In 307 he brought the emperor Flavius
Valerius Severus a captive to Rome, and also compelled Galerius to retreat,
but in 308 he was himself driven by Maxentius from Italy into Illyricum,
whence again he was compelled to seek refuge at Arelate (Arles), the court
of his son-in-law, Constantine. Here a false report was received, or
invented, of the death of Constantine, at that time absent on the Rhine.
Page 161
Maximianus at once grasped at the succession, but was soon driven to
Massilia (Marseilles), where, having been delivered up to his pursuers, he
strangled himself.
See Zosimus ii. 7-11; Zonaras xii. 31-33; Eutropius ix. 20, x. 2, 3;
Aurelius Victor p. 39. For the emperor Galerius Valerius Maximianus
see Galerius.
MAXIMILIAN I. (1573-1651), called “the Great,” elector and
duke of Bavaria, eldest son of William V. of Bavaria, was born at Munich
on the 17th of April 1573. He was educated by the Jesuits at the university
of Ingolstadt, and began to take part in the government in 1591. He married
in 1595 his cousin, Elizabeth, daughter of Charles II., duke of Lorraine, and
became duke of Bavaria upon his father’s abdication in 1597. He refrained
from any interference in German politics until 1607, when he was entrusted
with the duty of executing the imperial ban against the free city of
Donauwörth, a Protestant stronghold. In December 1607 his troops
occupied the city, and vigorous steps were taken to restore the supremacy of
the older faith. Some Protestant princes, alarmed at this action, formed a
union to defend their interests, which was answered in 1609 by the
establishment of a league, in the formation of which Maximilian took an
important part. Under his leadership an army was set on foot, but his policy
was strictly defensive and he refused to allow the league to become a tool in
the hands of the house of Habsburg. Dissensions among his colleagues led
Massilia (Marseilles), where, having been delivered up to his pursuers, he
strangled himself.
See Zosimus ii. 7-11; Zonaras xii. 31-33; Eutropius ix. 20, x. 2, 3;
Aurelius Victor p. 39. For the emperor Galerius Valerius Maximianus
see Galerius.
MAXIMILIAN I. (1573-1651), called “the Great,” elector and
duke of Bavaria, eldest son of William V. of Bavaria, was born at Munich
on the 17th of April 1573. He was educated by the Jesuits at the university
of Ingolstadt, and began to take part in the government in 1591. He married
in 1595 his cousin, Elizabeth, daughter of Charles II., duke of Lorraine, and
became duke of Bavaria upon his father’s abdication in 1597. He refrained
from any interference in German politics until 1607, when he was entrusted
with the duty of executing the imperial ban against the free city of
Donauwörth, a Protestant stronghold. In December 1607 his troops
occupied the city, and vigorous steps were taken to restore the supremacy of
the older faith. Some Protestant princes, alarmed at this action, formed a
union to defend their interests, which was answered in 1609 by the
establishment of a league, in the formation of which Maximilian took an
important part. Under his leadership an army was set on foot, but his policy
was strictly defensive and he refused to allow the league to become a tool in
the hands of the house of Habsburg. Dissensions among his colleagues led
Page 162
the duke to resign his office in 1616, but the approach of trouble brought
about his return to the league about two years later.
Having refused to become a candidate for the imperial throne in 1619,
Maximilian was faced with the complications arising from the outbreak of
war in Bohemia. After some delay he made a treaty with the emperor
Ferdinand II. in October 1619, and in return for large concessions placed
the forces of the league at the emperor’s service. Anxious to curtail the area
of the struggle, he made a treaty of neutrality with the Protestant Union, and
occupied Upper Austria as security for the expenses of the campaign. On
the 8th of November 1620 his troops under Count Tilly defeated the forces
of Frederick, king of Bohemia and count palatine of the Rhine, at the White
Hill near Prague. In spite of the arrangement with the union Tilly then
devastated the Rhenish Palatinate, and in February 1623 Maximilian was
formally invested with the electoral dignity and the attendant office of
imperial steward, which had been enjoyed since 1356 by the counts palatine
of the Rhine. After receiving the Upper Palatinate and restoring Upper
Austria to Ferdinand, Maximilian became leader of the party which sought
to bring about Wallenstein’s dismissal from the imperial service. At the diet
of Regensburg in 1630 Ferdinand was compelled to assent to this demand,
but the sequel was disastrous both for Bavaria and its ruler. Early in 1632
the Swedes marched into the duchy and occupied Munich, and Maximilian
could only obtain the assistance of the imperialists by placing himself under
the orders of Wallenstein, now restored to the command of the emperor’s
forces. The ravages of the Swedes and their French allies induced the
elector to enter into negotiations for peace with Gustavus Adolphus and
Cardinal Richelieu. He also proposed to disarm the Protestants by
modifying the Restitution edict of 1629; but these efforts were abortive. In
March 1647 he concluded an armistice with France and Sweden at Ulm, but
the entreaties of the emperor Ferdinand III. led him to disregard his
undertaking. Bavaria was again ravaged, and the elector’s forces defeated in
about his return to the league about two years later.
Having refused to become a candidate for the imperial throne in 1619,
Maximilian was faced with the complications arising from the outbreak of
war in Bohemia. After some delay he made a treaty with the emperor
Ferdinand II. in October 1619, and in return for large concessions placed
the forces of the league at the emperor’s service. Anxious to curtail the area
of the struggle, he made a treaty of neutrality with the Protestant Union, and
occupied Upper Austria as security for the expenses of the campaign. On
the 8th of November 1620 his troops under Count Tilly defeated the forces
of Frederick, king of Bohemia and count palatine of the Rhine, at the White
Hill near Prague. In spite of the arrangement with the union Tilly then
devastated the Rhenish Palatinate, and in February 1623 Maximilian was
formally invested with the electoral dignity and the attendant office of
imperial steward, which had been enjoyed since 1356 by the counts palatine
of the Rhine. After receiving the Upper Palatinate and restoring Upper
Austria to Ferdinand, Maximilian became leader of the party which sought
to bring about Wallenstein’s dismissal from the imperial service. At the diet
of Regensburg in 1630 Ferdinand was compelled to assent to this demand,
but the sequel was disastrous both for Bavaria and its ruler. Early in 1632
the Swedes marched into the duchy and occupied Munich, and Maximilian
could only obtain the assistance of the imperialists by placing himself under
the orders of Wallenstein, now restored to the command of the emperor’s
forces. The ravages of the Swedes and their French allies induced the
elector to enter into negotiations for peace with Gustavus Adolphus and
Cardinal Richelieu. He also proposed to disarm the Protestants by
modifying the Restitution edict of 1629; but these efforts were abortive. In
March 1647 he concluded an armistice with France and Sweden at Ulm, but
the entreaties of the emperor Ferdinand III. led him to disregard his
undertaking. Bavaria was again ravaged, and the elector’s forces defeated in
Page 163
May 1648 at Zusmarshausen. But the peace of Westphalia soon put an end
to the struggle. By this treaty it was agreed that Maximilian should retain
the electoral dignity, which was made hereditary in his family; and the
Upper Palatinate was incorporated with Bavaria. The elector died at
Ingolstadt on the 27th of September 1651. By his second wife, Maria Anne,
daughter of the emperor Ferdinand II., he left two sons, Ferdinand Maria,
who succeeded him, and Maximilian Philip. In 1839 a statue was erected to
his memory at Munich by Louis I., king of Bavaria. Weak in health and
feeble in frame, Maximilian had high ambitions both for himself and his
duchy, and was tenacious and resourceful in prosecuting his designs. As the
ablest prince of his age he sought to prevent Germany from becoming the
battleground of Europe, and although a rigid adherent of the Catholic faith,
was not always subservient to the priest.
See P. P. Wolf, Geschichte Kurfürst Maximilians I. und seiner Zeit
(Munich, 1807-1809); C. M. Freiherr von Aretin, Geschichte des
bayerschen Herzogs und Kurfürsten Maximilian des Ersten (Passau,
1842); M. Lossen, Die Reichstadt Donauwörth und Herzog Maximilian
(Munich, 1866); F. Stieve, Kurfürst Maximilian I. von Bayern (Munich,
1882); F. A. W. Schreiber, Maximilian I. der Katholische Kurfürst von
Bayern, und der dreissigjährige Krieg (Munich, 1868); M. Högl, Die
Bekehrung der Oberpfalz durch Kurfürst Maximilian I. (Regensburg,
1903).
to the struggle. By this treaty it was agreed that Maximilian should retain
the electoral dignity, which was made hereditary in his family; and the
Upper Palatinate was incorporated with Bavaria. The elector died at
Ingolstadt on the 27th of September 1651. By his second wife, Maria Anne,
daughter of the emperor Ferdinand II., he left two sons, Ferdinand Maria,
who succeeded him, and Maximilian Philip. In 1839 a statue was erected to
his memory at Munich by Louis I., king of Bavaria. Weak in health and
feeble in frame, Maximilian had high ambitions both for himself and his
duchy, and was tenacious and resourceful in prosecuting his designs. As the
ablest prince of his age he sought to prevent Germany from becoming the
battleground of Europe, and although a rigid adherent of the Catholic faith,
was not always subservient to the priest.
See P. P. Wolf, Geschichte Kurfürst Maximilians I. und seiner Zeit
(Munich, 1807-1809); C. M. Freiherr von Aretin, Geschichte des
bayerschen Herzogs und Kurfürsten Maximilian des Ersten (Passau,
1842); M. Lossen, Die Reichstadt Donauwörth und Herzog Maximilian
(Munich, 1866); F. Stieve, Kurfürst Maximilian I. von Bayern (Munich,
1882); F. A. W. Schreiber, Maximilian I. der Katholische Kurfürst von
Bayern, und der dreissigjährige Krieg (Munich, 1868); M. Högl, Die
Bekehrung der Oberpfalz durch Kurfürst Maximilian I. (Regensburg,
1903).
Page 164
MAXIMILIAN I. (Maximilian Joseph) (1756-1825), king of
Bavaria, was the son of the count palatine Frederick of Zweibrücken-
Birkenfeld, and was born on the 27th of May 1756. He was carefully
educated under the supervision of his uncle, Duke Christian IV. of
Zweibrücken, took service in 1777 as a colonel in the French army, and rose
rapidly to the rank of major-general. From 1782 to 1789 he was stationed at
Strassburg, but at the outbreak of the revolution he exchanged the French
for the Austrian service, taking part in the opening campaigns of the
revolutionary wars. On the 1st of April 1795 he succeeded his brother,
Charles II., as duke of Zweibrücken, and on the 16th of February 1799
became elector of Bavaria on the extinction of the Sulzbach line with the
death of the elector Charles Theodore.
The sympathy with France and with French ideas of enlightenment which
characterized his reign was at once manifested. In the newly organized
ministry Count Max Josef von Montgelas (q.v.), who, after falling into
disfavour with Charles Theodore, had acted for a time as Maximilian
Joseph’s private secretary, was the most potent influence, an influence
wholly “enlightened” and French. Agriculture and commerce were fostered,
the laws were ameliorated, a new criminal code drawn up, taxes and
imposts equalized without regard to traditional privileges, while a number
of religious houses were suppressed and their revenues used for educational
and other useful purposes. In foreign politics Maximilian Joseph’s attitude
was from the German point of view less commendable. With the growing
sentiment of German nationality he had from first to last no sympathy, and
his attitude throughout was dictated by wholly dynastic, or at least
Bavarian, considerations. Until 1813 he was the most faithful of Napoleon’s
German allies, the relation being cemented by the marriage of his daughter
to Eugène Beauharnais. His reward came with the treaty of Pressburg (Dec.
26, 1805), by the terms of which he was to receive the royal title and
Bavaria, was the son of the count palatine Frederick of Zweibrücken-
Birkenfeld, and was born on the 27th of May 1756. He was carefully
educated under the supervision of his uncle, Duke Christian IV. of
Zweibrücken, took service in 1777 as a colonel in the French army, and rose
rapidly to the rank of major-general. From 1782 to 1789 he was stationed at
Strassburg, but at the outbreak of the revolution he exchanged the French
for the Austrian service, taking part in the opening campaigns of the
revolutionary wars. On the 1st of April 1795 he succeeded his brother,
Charles II., as duke of Zweibrücken, and on the 16th of February 1799
became elector of Bavaria on the extinction of the Sulzbach line with the
death of the elector Charles Theodore.
The sympathy with France and with French ideas of enlightenment which
characterized his reign was at once manifested. In the newly organized
ministry Count Max Josef von Montgelas (q.v.), who, after falling into
disfavour with Charles Theodore, had acted for a time as Maximilian
Joseph’s private secretary, was the most potent influence, an influence
wholly “enlightened” and French. Agriculture and commerce were fostered,
the laws were ameliorated, a new criminal code drawn up, taxes and
imposts equalized without regard to traditional privileges, while a number
of religious houses were suppressed and their revenues used for educational
and other useful purposes. In foreign politics Maximilian Joseph’s attitude
was from the German point of view less commendable. With the growing
sentiment of German nationality he had from first to last no sympathy, and
his attitude throughout was dictated by wholly dynastic, or at least
Bavarian, considerations. Until 1813 he was the most faithful of Napoleon’s
German allies, the relation being cemented by the marriage of his daughter
to Eugène Beauharnais. His reward came with the treaty of Pressburg (Dec.
26, 1805), by the terms of which he was to receive the royal title and
Page 165
important territorial acquisitions in Swabia and Franconia to round off his
kingdom. The style of king he actually assumed on the 1st of January 1806.
The new king of Bavaria was the most important of the princes belonging
to the Confederation of the Rhine, and remained Napoleon’s ally until the
eve of the battle of Leipzig, when by the convention of Ried (Oct. 8, 1813)
he made the guarantee of the integrity of his kingdom the price of his
joining the Allies. By the first treaty of Paris (June 3, 1814), however, he
ceded Tirol to Austria in exchange for the former duchy of Würzburg. At
the congress of Vienna, too, which he attended in person, Maximilian had to
make further concessions to Austria, ceding the quarters of the Inn and
Hausruck in return for a part of the old Palatinate. The king fought hard to
maintain the contiguity of the Bavarian territories as guaranteed at Ried; but
the most he could obtain was an assurance from Metternich in the matter of
the Baden succession, in which he was also doomed to be disappointed (see
Baden: History, iii. 506).
At Vienna and afterwards Maximilian sturdily opposed any reconstitution
of Germany which should endanger the independence of Bavaria, and it
was his insistence on the principle of full sovereignty being left to the
German reigning princes that largely contributed to the loose and weak
organization of the new German Confederation. The Federal Act of the
Vienna congress was proclaimed in Bavaria, not as a law but as an
international treaty. It was partly to secure popular support in his resistance
to any interference of the federal diet in the internal affairs of Bavaria,
partly to give unity to his somewhat heterogeneous territories, that
Maximilian on the 26th of May 1818 granted a liberal constitution to his
people. Montgelas, who had opposed this concession, had fallen in the
previous year, and Maximilian had also reversed his ecclesiastical policy,
signing on the 24th of October 1817 a concordat with Rome by which the
powers of the clergy, largely curtailed under Montgelas’s administration,
kingdom. The style of king he actually assumed on the 1st of January 1806.
The new king of Bavaria was the most important of the princes belonging
to the Confederation of the Rhine, and remained Napoleon’s ally until the
eve of the battle of Leipzig, when by the convention of Ried (Oct. 8, 1813)
he made the guarantee of the integrity of his kingdom the price of his
joining the Allies. By the first treaty of Paris (June 3, 1814), however, he
ceded Tirol to Austria in exchange for the former duchy of Würzburg. At
the congress of Vienna, too, which he attended in person, Maximilian had to
make further concessions to Austria, ceding the quarters of the Inn and
Hausruck in return for a part of the old Palatinate. The king fought hard to
maintain the contiguity of the Bavarian territories as guaranteed at Ried; but
the most he could obtain was an assurance from Metternich in the matter of
the Baden succession, in which he was also doomed to be disappointed (see
Baden: History, iii. 506).
At Vienna and afterwards Maximilian sturdily opposed any reconstitution
of Germany which should endanger the independence of Bavaria, and it
was his insistence on the principle of full sovereignty being left to the
German reigning princes that largely contributed to the loose and weak
organization of the new German Confederation. The Federal Act of the
Vienna congress was proclaimed in Bavaria, not as a law but as an
international treaty. It was partly to secure popular support in his resistance
to any interference of the federal diet in the internal affairs of Bavaria,
partly to give unity to his somewhat heterogeneous territories, that
Maximilian on the 26th of May 1818 granted a liberal constitution to his
people. Montgelas, who had opposed this concession, had fallen in the
previous year, and Maximilian had also reversed his ecclesiastical policy,
signing on the 24th of October 1817 a concordat with Rome by which the
powers of the clergy, largely curtailed under Montgelas’s administration,
Page 166
were restored. The new parliament proved so intractable that in 1819
Maximilian was driven to appeal to the powers against his own creation;
but his Bavarian “particularism” and his genuine popular sympathies
prevented him from allowing the Carlsbad decrees to be strictly enforced
within his dominions. The suspects arrested by order of the Mainz
Commission he was accustomed to examine himself, with the result that in
many cases the whole proceedings were quashed, and in not a few the
accused dismissed with a present of money. Maximilian died on the 13th of
October 1825 and was succeeded by his son Louis I.
In private life Maximilian was kindly and simple. He loved to play the
part of Landesvater, walking about the streets of his capital en bourgeois
and entering into conversation with all ranks of his subjects, by whom he
was regarded with great affection. He was twice married: (1) in 1785 to
Princess Wilhelmine Auguste of Hesse-Darmstadt, (2) in 1797 to Princess
Caroline Friederike of Baden.
See G. Freiherr von Lerchenfeld, Gesch. Bayerns unter König
Maximilian Joseph I. (Berlin, 1854); J. M. Söltl, Max Joseph, König
von Bayern (Stuttgart, 1837); L. von Kobell, Unter den vier ersten
Königen Bayerns. Nach Briefen und eigenen Erinnerungen (Munich,
1894).
MAXIMILIAN II. (1811-1864), king of Bavaria, son of king
Louis I. and of his consort Theresa of Saxe-Hildburghausen, was born on
Maximilian was driven to appeal to the powers against his own creation;
but his Bavarian “particularism” and his genuine popular sympathies
prevented him from allowing the Carlsbad decrees to be strictly enforced
within his dominions. The suspects arrested by order of the Mainz
Commission he was accustomed to examine himself, with the result that in
many cases the whole proceedings were quashed, and in not a few the
accused dismissed with a present of money. Maximilian died on the 13th of
October 1825 and was succeeded by his son Louis I.
In private life Maximilian was kindly and simple. He loved to play the
part of Landesvater, walking about the streets of his capital en bourgeois
and entering into conversation with all ranks of his subjects, by whom he
was regarded with great affection. He was twice married: (1) in 1785 to
Princess Wilhelmine Auguste of Hesse-Darmstadt, (2) in 1797 to Princess
Caroline Friederike of Baden.
See G. Freiherr von Lerchenfeld, Gesch. Bayerns unter König
Maximilian Joseph I. (Berlin, 1854); J. M. Söltl, Max Joseph, König
von Bayern (Stuttgart, 1837); L. von Kobell, Unter den vier ersten
Königen Bayerns. Nach Briefen und eigenen Erinnerungen (Munich,
1894).
MAXIMILIAN II. (1811-1864), king of Bavaria, son of king
Louis I. and of his consort Theresa of Saxe-Hildburghausen, was born on
Page 167
the 28th of November 1811. After studying at Göttingen and Berlin and
travelling in Germany, Italy and Greece, he was introduced by his father
into the council of state (1836). From the first he showed a studious
disposition, declaring on one occasion that had he not been born in a royal
cradle his choice would have been to become a professor. As crown prince,
in the château of Hohenschwangau near Füssen, which he had rebuilt with
excellent taste, he gathered about him an intimate society of artists and men
of learning, and devoted his time to scientific and historical study. When the
abdication of Louis I. (March 28, 1848) called him suddenly to the throne,
his choice of ministers promised a liberal régime. The progress of the
revolution, however, gave him pause. He strenuously opposed the unionist
plans of the Frankfort parliament, refused to recognize the imperial
constitution devised by it, and assisted Austria in restoring the federal diet
and in carrying out the federal execution in Hesse and Holstein. Although,
however, from 1850 onwards his government tended in the direction of
absolutism, he refused to become the tool of the clerical reaction, and even
incurred the bitter criticism of the Ultramontanes by inviting a number of
celebrated men of learning and science (e.g. Liebig and Sybel) to Munich,
regardless of their religious views. Finally, in 1859, he dismissed the
reactionary ministry of von der Pfordten, and met the wishes of his people
for a moderate constitutional government. In his German policy he was
guided by the desire to maintain the union of the princes, and hoped to
attain this as against the perilous rivalry of Austria and Prussia by the
creation of a league of the “middle” and small states—the so-called Trias.
In 1863, however, seeing what he thought to be a better way, he supported
the project of reform proposed by Austria at the Fürstentag of Frankfort.
The failure of this proposal, and the attitude of Austria towards the
Confederation and in the Schleswig-Holstein question, undeceived him; but
before he could deal with the new situation created by the outbreak of the
war with Denmark he died suddenly at Munich, on the 10th of March 1864.
travelling in Germany, Italy and Greece, he was introduced by his father
into the council of state (1836). From the first he showed a studious
disposition, declaring on one occasion that had he not been born in a royal
cradle his choice would have been to become a professor. As crown prince,
in the château of Hohenschwangau near Füssen, which he had rebuilt with
excellent taste, he gathered about him an intimate society of artists and men
of learning, and devoted his time to scientific and historical study. When the
abdication of Louis I. (March 28, 1848) called him suddenly to the throne,
his choice of ministers promised a liberal régime. The progress of the
revolution, however, gave him pause. He strenuously opposed the unionist
plans of the Frankfort parliament, refused to recognize the imperial
constitution devised by it, and assisted Austria in restoring the federal diet
and in carrying out the federal execution in Hesse and Holstein. Although,
however, from 1850 onwards his government tended in the direction of
absolutism, he refused to become the tool of the clerical reaction, and even
incurred the bitter criticism of the Ultramontanes by inviting a number of
celebrated men of learning and science (e.g. Liebig and Sybel) to Munich,
regardless of their religious views. Finally, in 1859, he dismissed the
reactionary ministry of von der Pfordten, and met the wishes of his people
for a moderate constitutional government. In his German policy he was
guided by the desire to maintain the union of the princes, and hoped to
attain this as against the perilous rivalry of Austria and Prussia by the
creation of a league of the “middle” and small states—the so-called Trias.
In 1863, however, seeing what he thought to be a better way, he supported
the project of reform proposed by Austria at the Fürstentag of Frankfort.
The failure of this proposal, and the attitude of Austria towards the
Confederation and in the Schleswig-Holstein question, undeceived him; but
before he could deal with the new situation created by the outbreak of the
war with Denmark he died suddenly at Munich, on the 10th of March 1864.
Page 168
Maximilian was a man of amiable qualities and of intellectual
attainments far above the average, but as a king he was hampered by
constant ill-health, which compelled him to be often abroad, and when at
home to live much in the country. By his wife, Maria Hedwig, daughter of
Prince William of Prussia, whom he married in 1842, he had two sons,
Louis II., king of Bavaria, and Otto, king of Bavaria, both of whom lost
their reason.
See J. M. Söltl, Max der Zweite, König von Bayern (Munich, 1865);
biography by G. K. Heigel in Allgem. Deutsche Biographie, vol. xxi.
(Leipzig, 1885). Maximilian’s correspondence with Schlegel was
published at Stuttgart in 1890.
MAXIMILIAN I. (1459-1519), Roman emperor, son of the
emperor Frederick III. and Leonora, daughter of Edward, king of Portugal,
was born at Vienna Neustadt on the 22nd of March 1459. On the 18th of
August 1477, by his marriage at Ghent to Mary, who had just inherited
Burgundy and the Netherlands from her father Charles the Bold, duke of
Burgundy, he effected a union of great importance in the history of the
house of Habsburg. He at once undertook the defence of his wife’s
dominions from an attack by Louis XI., king of France, and defeated the
French forces at Guinegatte, the modern Enguinegatte, on the 7th of August
1479. But Maximilian was regarded with suspicion by the states of
Netherlands, and after suppressing a rising in Gelderland his position was
attainments far above the average, but as a king he was hampered by
constant ill-health, which compelled him to be often abroad, and when at
home to live much in the country. By his wife, Maria Hedwig, daughter of
Prince William of Prussia, whom he married in 1842, he had two sons,
Louis II., king of Bavaria, and Otto, king of Bavaria, both of whom lost
their reason.
See J. M. Söltl, Max der Zweite, König von Bayern (Munich, 1865);
biography by G. K. Heigel in Allgem. Deutsche Biographie, vol. xxi.
(Leipzig, 1885). Maximilian’s correspondence with Schlegel was
published at Stuttgart in 1890.
MAXIMILIAN I. (1459-1519), Roman emperor, son of the
emperor Frederick III. and Leonora, daughter of Edward, king of Portugal,
was born at Vienna Neustadt on the 22nd of March 1459. On the 18th of
August 1477, by his marriage at Ghent to Mary, who had just inherited
Burgundy and the Netherlands from her father Charles the Bold, duke of
Burgundy, he effected a union of great importance in the history of the
house of Habsburg. He at once undertook the defence of his wife’s
dominions from an attack by Louis XI., king of France, and defeated the
French forces at Guinegatte, the modern Enguinegatte, on the 7th of August
1479. But Maximilian was regarded with suspicion by the states of
Netherlands, and after suppressing a rising in Gelderland his position was
Page 169
further weakened by the death of his wife on the 27th of March 1482. He
claimed to be recognized as guardian of his young son Philip and as regent
of the Netherlands, but some of the states refused to agree to his demands
and disorder was general. Maximilian was compelled to assent to the treaty
of Arras in 1482 between the states of the Netherlands and Louis XI. This
treaty provided that Maximilian’s daughter Margaret should marry Charles,
the dauphin of France, and have for her dowry Artois and Franche-Comté,
two of the provinces in dispute, while the claim of Louis on the duchy of
Burgundy was tacitly admitted. Maximilian did not, however, abandon the
struggle in the Netherlands. Having crushed a rebellion at Utrecht, he
compelled the burghers of Ghent to restore Philip to him in 1485, and
returning to Germany was chosen king of the Romans, or German king, at
Frankfort on the 16th of February 1486, and crowned at Aix-la-Chapelle on
the 9th of the following April. Again in the Netherlands, he made a treaty
with Francis II., duke of Brittany, whose independence was threatened by
the French regent, Anne of Beaujeu, and the struggle with France was soon
renewed. This war was very unpopular with the trading cities of the
Netherlands, and early in 1488 Maximilian, having entered Bruges, was
detained there as a prisoner for nearly three months, and only set at liberty
on the approach of his father with a large force. On his release he had
promised he would maintain the treaty of Arras and withdraw from the
Netherlands; but he delayed his departure for nearly a year and took part in
a punitive campaign against his captors and their allies. On his return to
Germany he made peace with France at Frankfort in July 1489, and in
October several of the states of the Netherlands recognized him as their
ruler and as guardian of his son. In March 1490 the county of Tirol was
added to his possessions through the abdication of his kinsman, Count
Sigismund, and this district soon became his favourite residence.
Meanwhile the king had formed an alliance with Henry VII. king of
England, and Ferdinand II., king of Aragon, to defend the possessions of the
claimed to be recognized as guardian of his young son Philip and as regent
of the Netherlands, but some of the states refused to agree to his demands
and disorder was general. Maximilian was compelled to assent to the treaty
of Arras in 1482 between the states of the Netherlands and Louis XI. This
treaty provided that Maximilian’s daughter Margaret should marry Charles,
the dauphin of France, and have for her dowry Artois and Franche-Comté,
two of the provinces in dispute, while the claim of Louis on the duchy of
Burgundy was tacitly admitted. Maximilian did not, however, abandon the
struggle in the Netherlands. Having crushed a rebellion at Utrecht, he
compelled the burghers of Ghent to restore Philip to him in 1485, and
returning to Germany was chosen king of the Romans, or German king, at
Frankfort on the 16th of February 1486, and crowned at Aix-la-Chapelle on
the 9th of the following April. Again in the Netherlands, he made a treaty
with Francis II., duke of Brittany, whose independence was threatened by
the French regent, Anne of Beaujeu, and the struggle with France was soon
renewed. This war was very unpopular with the trading cities of the
Netherlands, and early in 1488 Maximilian, having entered Bruges, was
detained there as a prisoner for nearly three months, and only set at liberty
on the approach of his father with a large force. On his release he had
promised he would maintain the treaty of Arras and withdraw from the
Netherlands; but he delayed his departure for nearly a year and took part in
a punitive campaign against his captors and their allies. On his return to
Germany he made peace with France at Frankfort in July 1489, and in
October several of the states of the Netherlands recognized him as their
ruler and as guardian of his son. In March 1490 the county of Tirol was
added to his possessions through the abdication of his kinsman, Count
Sigismund, and this district soon became his favourite residence.
Meanwhile the king had formed an alliance with Henry VII. king of
England, and Ferdinand II., king of Aragon, to defend the possessions of the
Page 170
duchess Anne, daughter and successor of Francis, duke of Brittany. Early in
1490 he took a further step and was betrothed to the duchess, and later in
the same year the marriage was celebrated by proxy; but Brittany was still
occupied by French troops, and Maximilian was unable to go to the
assistance of his bride. The sequel was startling. In December 1491 Anne
was married to Charles VIII., king of France, and Maximilian’s daughter
Margaret, who had resided in France since her betrothal, was sent back to
her father. The inaction of Maximilian at this time is explained by the
condition of affairs in Hungary, where the death of king Matthias Corvinus
had brought about a struggle for this throne. The Roman king, who was an
unsuccessful candidate, took up arms, drove the Hungarians from Austria,
and regained Vienna, which had been in the possession of Matthias since
1485; but he was compelled by want of money to retreat, and on the 7th of
November 1491 signed the treaty of Pressburg with Ladislaus, king of
Bohemia, who had obtained the Hungarian throne. By this treaty it was
agreed that Maximilian should succeed to the crown in case Ladislaus left
no legitimate male issue. Having defeated the invading Turks at Villach in
1492, the king was eager to take revenge upon the king of France; but the
states of the Netherlands would afford him no assistance. The German diet
was indifferent, and in May 1493 he agreed to the peace of Senlis and
regained Artois and Franche-Comté.
In August 1493 the death of the emperor left Maximilian sole ruler of
Germany and head of the house of Habsburg; and on the 16th of March
1494 he married at Innsbruck Bianca Maria Sforza, daughter of Galeazzo
Sforza, duke of Milan (d. 1476). At this time Bianca’s uncle, Ludovico
Sforza, was invested with the duchy of Milan in return for the substantial
dowry which his niece brought to the king. Maximilian harboured the idea
of driving the Turks from Europe; but his appeal to all Christian sovereigns
was ineffectual. In 1494 he was again in the Netherlands, where he led an
expedition against the rebels of Gelderland, assisted Perkin Warbeck to
1490 he took a further step and was betrothed to the duchess, and later in
the same year the marriage was celebrated by proxy; but Brittany was still
occupied by French troops, and Maximilian was unable to go to the
assistance of his bride. The sequel was startling. In December 1491 Anne
was married to Charles VIII., king of France, and Maximilian’s daughter
Margaret, who had resided in France since her betrothal, was sent back to
her father. The inaction of Maximilian at this time is explained by the
condition of affairs in Hungary, where the death of king Matthias Corvinus
had brought about a struggle for this throne. The Roman king, who was an
unsuccessful candidate, took up arms, drove the Hungarians from Austria,
and regained Vienna, which had been in the possession of Matthias since
1485; but he was compelled by want of money to retreat, and on the 7th of
November 1491 signed the treaty of Pressburg with Ladislaus, king of
Bohemia, who had obtained the Hungarian throne. By this treaty it was
agreed that Maximilian should succeed to the crown in case Ladislaus left
no legitimate male issue. Having defeated the invading Turks at Villach in
1492, the king was eager to take revenge upon the king of France; but the
states of the Netherlands would afford him no assistance. The German diet
was indifferent, and in May 1493 he agreed to the peace of Senlis and
regained Artois and Franche-Comté.
In August 1493 the death of the emperor left Maximilian sole ruler of
Germany and head of the house of Habsburg; and on the 16th of March
1494 he married at Innsbruck Bianca Maria Sforza, daughter of Galeazzo
Sforza, duke of Milan (d. 1476). At this time Bianca’s uncle, Ludovico
Sforza, was invested with the duchy of Milan in return for the substantial
dowry which his niece brought to the king. Maximilian harboured the idea
of driving the Turks from Europe; but his appeal to all Christian sovereigns
was ineffectual. In 1494 he was again in the Netherlands, where he led an
expedition against the rebels of Gelderland, assisted Perkin Warbeck to
Page 171
make a descent upon England, and formally handed over the government of
the Low Countries to Philip. His attention was next turned to Italy, and,
alarmed at the progress of Charles VIII. in the peninsula, he signed the
league of Venice in March 1495, and about the same time arranged a
marriage between his son Philip and Joanna, daughter of Ferdinand and
Isabella, king and queen of Castile and Aragon. The need for help to
prosecute the war in Italy caused the king to call the diet to Worms in
March 1495, when he urged the necessity of checking the progress of
Charles. As during his father’s lifetime Maximilian had favoured the
reforming party among the princes, proposals for the better government of
the empire were brought forward at Worms as a necessary preliminary to
financial and military support. Some reforms were adopted, the public
peace was proclaimed without any limitation of time and a general tax was
levied. The three succeeding years were mainly occupied with quarrels with
the diet, with two invasions of France, and a war in Gelderland against
Charles, count of Egmont, who claimed that duchy, and was supported by
French troops. The reforms of 1495 were rendered abortive by the refusal of
Maximilian to attend the diets or to take any part in the working of the new
constitution, and in 1497 he strengthened his own authority by establishing
an Aulic Council (Reichshofrath), which he declared was competent to deal
with all business of the empire, and about the same time set up a court to
centralize the financial administration of Germany.
In February 1499 the king became involved in a war with the Swiss, who
had refused to pay the imperial taxes or to furnish a contribution for the
Italian expedition. Aided by France they defeated the German troops, and
the peace of Basel in September 1499 recognized them as virtually
independent of the empire. About this time Maximilian’s ally, Ludovico of
Milan, was taken prisoner by Louis XII., king of France, and Maximilian
was again compelled to ask the diet for help. An elaborate scheme for
raising an army was agreed to, and in return a council of regency
the Low Countries to Philip. His attention was next turned to Italy, and,
alarmed at the progress of Charles VIII. in the peninsula, he signed the
league of Venice in March 1495, and about the same time arranged a
marriage between his son Philip and Joanna, daughter of Ferdinand and
Isabella, king and queen of Castile and Aragon. The need for help to
prosecute the war in Italy caused the king to call the diet to Worms in
March 1495, when he urged the necessity of checking the progress of
Charles. As during his father’s lifetime Maximilian had favoured the
reforming party among the princes, proposals for the better government of
the empire were brought forward at Worms as a necessary preliminary to
financial and military support. Some reforms were adopted, the public
peace was proclaimed without any limitation of time and a general tax was
levied. The three succeeding years were mainly occupied with quarrels with
the diet, with two invasions of France, and a war in Gelderland against
Charles, count of Egmont, who claimed that duchy, and was supported by
French troops. The reforms of 1495 were rendered abortive by the refusal of
Maximilian to attend the diets or to take any part in the working of the new
constitution, and in 1497 he strengthened his own authority by establishing
an Aulic Council (Reichshofrath), which he declared was competent to deal
with all business of the empire, and about the same time set up a court to
centralize the financial administration of Germany.
In February 1499 the king became involved in a war with the Swiss, who
had refused to pay the imperial taxes or to furnish a contribution for the
Italian expedition. Aided by France they defeated the German troops, and
the peace of Basel in September 1499 recognized them as virtually
independent of the empire. About this time Maximilian’s ally, Ludovico of
Milan, was taken prisoner by Louis XII., king of France, and Maximilian
was again compelled to ask the diet for help. An elaborate scheme for
raising an army was agreed to, and in return a council of regency
Page 172
(Reichsregiment) was established, which amounted, in the words of a
Venetian envoy, to a deposition of the king. The relations were now very
strained between the reforming princes and Maximilian, who, unable to
raise an army, refused to attend the meetings of the council at Nuremberg,
while both parties treated for peace with France. The hostility of the king
rendered the council impotent. He was successful in winning the support of
many of the younger princes, and in establishing a new court of justice, the
members of which were named by himself. The negotiations with France
ended in the treaty of Blois, signed in September 1504, when Maximilian’s
grandson Charles was betrothed to Claude, daughter of Louis XII., and
Louis, invested with the duchy of Milan, agreed to aid the king of the
Romans to secure the imperial crown. A succession difficulty in Bavaria-
Landshut was only decided after Maximilian had taken up arms and
narrowly escaped with his life at Regensburg. In the settlement of this
question, made in 1505, he secured a considerable increase of territory, and
when the king met the diet at Cologne in 1505 he was at the height of his
power. His enemies at home were crushed, and their leader, Berthold,
elector of Mainz, was dead; while the outlook abroad was more favourable
than it had been since his accession.
It is at this period that Ranke believes Maximilian to have entertained the
idea of a universal monarchy; but whatever hopes he may have had were
shattered by the death of his son Philip and the rupture of the treaty of
Blois. The diet of Cologne discussed the question of reform in a halting
fashion, but afforded the king supplies for an expedition into Hungary, to
aid his ally Ladislaus, and to uphold his own influence in the East. Having
established his daughter Margaret as regent for Charles in the Netherlands,
Maximilian met the diet at Constance in 1507, when the imperial chamber
(Reichskammergericht) was revised and took a more permanent form, and
help was granted for an expedition to Italy. The king set out for Rome to
secure his coronation, but Venice refused to let him pass through her
Venetian envoy, to a deposition of the king. The relations were now very
strained between the reforming princes and Maximilian, who, unable to
raise an army, refused to attend the meetings of the council at Nuremberg,
while both parties treated for peace with France. The hostility of the king
rendered the council impotent. He was successful in winning the support of
many of the younger princes, and in establishing a new court of justice, the
members of which were named by himself. The negotiations with France
ended in the treaty of Blois, signed in September 1504, when Maximilian’s
grandson Charles was betrothed to Claude, daughter of Louis XII., and
Louis, invested with the duchy of Milan, agreed to aid the king of the
Romans to secure the imperial crown. A succession difficulty in Bavaria-
Landshut was only decided after Maximilian had taken up arms and
narrowly escaped with his life at Regensburg. In the settlement of this
question, made in 1505, he secured a considerable increase of territory, and
when the king met the diet at Cologne in 1505 he was at the height of his
power. His enemies at home were crushed, and their leader, Berthold,
elector of Mainz, was dead; while the outlook abroad was more favourable
than it had been since his accession.
It is at this period that Ranke believes Maximilian to have entertained the
idea of a universal monarchy; but whatever hopes he may have had were
shattered by the death of his son Philip and the rupture of the treaty of
Blois. The diet of Cologne discussed the question of reform in a halting
fashion, but afforded the king supplies for an expedition into Hungary, to
aid his ally Ladislaus, and to uphold his own influence in the East. Having
established his daughter Margaret as regent for Charles in the Netherlands,
Maximilian met the diet at Constance in 1507, when the imperial chamber
(Reichskammergericht) was revised and took a more permanent form, and
help was granted for an expedition to Italy. The king set out for Rome to
secure his coronation, but Venice refused to let him pass through her
Page 173
territories; and at Trant, on the 4th of February 1508, he took the important
step of assuming the title of Roman Emperor Elect, to which he soon
received the assent of pope Julius II. He attacked the Venetians, but finding
the war unpopular with the trading cities of southern Germany, made a truce
with the republic for three years. The treaty of Blois had contained a secret
article providing for an attack on Venice, and this ripened into the league of
Cambray, which was joined by the emperor in December 1509. He soon
took the field, but after his failure to capture Padua the league broke up; and
his sole ally, the French king, joined him in calling a general council at Pisa
to discuss the question of Church reform. A breach with pope Julius
followed, and at this time Maximilian appears to have entertained, perhaps
quite seriously, the idea of seating himself in the chair of St Peter. After a
period of vacillation he deserted Louis and joined the Holy League, which
had been formed to expel the French from Italy; but unable to raise troops,
he served with the English forces as a volunteer and shared in the victory
gained over the French at the battle of the Spurs near Thérouanne on the
16th of August 1513. In 1500 the diet had divided Germany into six circles,
for the maintenance of peace, to which the emperor at the diet of Cologne in
1512 added four others. Having made an alliance with Christian II., king of
Denmark, and interfered to protect the Teutonic Order against Sigismund I.,
king of Poland, Maximilian was again in Italy early in 1516 fighting the
French who had overrun Milan. His want of success compelled him on the
4th of December 1516 to sign the treaty of Brussels, which left Milan in the
hands of the French king, while Verona was soon afterwards transferred to
Venice. He attempted in vain to secure the election of his grandson Charles
as king of the Romans, and in spite of increasing infirmity was eager to lead
the imperial troops against the Turks. At the diet of Augsburg in 1518 the
emperor heard warnings of the Reformation in the shape of complaints
against papal exactions, and a repetition of the complaints preferred at the
diet of Mainz in 1517 about the administration of Germany. Leaving the
step of assuming the title of Roman Emperor Elect, to which he soon
received the assent of pope Julius II. He attacked the Venetians, but finding
the war unpopular with the trading cities of southern Germany, made a truce
with the republic for three years. The treaty of Blois had contained a secret
article providing for an attack on Venice, and this ripened into the league of
Cambray, which was joined by the emperor in December 1509. He soon
took the field, but after his failure to capture Padua the league broke up; and
his sole ally, the French king, joined him in calling a general council at Pisa
to discuss the question of Church reform. A breach with pope Julius
followed, and at this time Maximilian appears to have entertained, perhaps
quite seriously, the idea of seating himself in the chair of St Peter. After a
period of vacillation he deserted Louis and joined the Holy League, which
had been formed to expel the French from Italy; but unable to raise troops,
he served with the English forces as a volunteer and shared in the victory
gained over the French at the battle of the Spurs near Thérouanne on the
16th of August 1513. In 1500 the diet had divided Germany into six circles,
for the maintenance of peace, to which the emperor at the diet of Cologne in
1512 added four others. Having made an alliance with Christian II., king of
Denmark, and interfered to protect the Teutonic Order against Sigismund I.,
king of Poland, Maximilian was again in Italy early in 1516 fighting the
French who had overrun Milan. His want of success compelled him on the
4th of December 1516 to sign the treaty of Brussels, which left Milan in the
hands of the French king, while Verona was soon afterwards transferred to
Venice. He attempted in vain to secure the election of his grandson Charles
as king of the Romans, and in spite of increasing infirmity was eager to lead
the imperial troops against the Turks. At the diet of Augsburg in 1518 the
emperor heard warnings of the Reformation in the shape of complaints
against papal exactions, and a repetition of the complaints preferred at the
diet of Mainz in 1517 about the administration of Germany. Leaving the
Page 174
diet, he travelled to Wels in Upper Austria, where he died on the 12th of
January 1519. He was buried in the church of St George in Vienna
Neustadt, and a superb monument, which may still be seen, was raised to
his memory at Innsbruck.
Maximilian had many excellent personal qualities. He was not
handsome, but of a robust and well-proportioned frame. Simple in his
habits, conciliatory in his bearing, and catholic in his tastes, he enjoyed
great popularity and rarely made a personal enemy. He was a skilled
knight and a daring huntsman, and although not a great general, was
intrepid on the field of battle. His mental interests were extensive. He
knew something of six languages, and could discuss art, music,
literature or theology. He reorganized the university of Vienna and
encouraged the development of the universities of Ingolstadt and
Freiburg. He was the friend and patron of scholars, caused manuscripts
to be copied and medieval poems to be collected. He was the author of
military reforms, which included the establishment of standing troops,
called Landsknechte, the improvement of artillery by making cannon
portable, and some changes in the equipment of the cavalry. He was
continually devising plans for the better government of Austria, and
although they ended in failure, he established the unity of the Austrian
dominions. Maximilian has been called the second founder of the
house of Habsburg, and certainly by bringing about marriages between
Charles and Joanna and between his grandson Ferdinand and Anna,
daughter of Ladislaus, king of Hungary and Bohemia, he paved the
way for the vast empire of Charles V. and for the influence of the
Habsburgs in eastern Europe. But he had many qualities less desirable.
He was reckless and unstable, resorting often to lying and deceit, and
never pausing to count the cost of an enterprise or troubling to adapt
means to ends. For absurd and impracticable schemes in Italy and
elsewhere he neglected Germany, and sought to involve its princes in
January 1519. He was buried in the church of St George in Vienna
Neustadt, and a superb monument, which may still be seen, was raised to
his memory at Innsbruck.
Maximilian had many excellent personal qualities. He was not
handsome, but of a robust and well-proportioned frame. Simple in his
habits, conciliatory in his bearing, and catholic in his tastes, he enjoyed
great popularity and rarely made a personal enemy. He was a skilled
knight and a daring huntsman, and although not a great general, was
intrepid on the field of battle. His mental interests were extensive. He
knew something of six languages, and could discuss art, music,
literature or theology. He reorganized the university of Vienna and
encouraged the development of the universities of Ingolstadt and
Freiburg. He was the friend and patron of scholars, caused manuscripts
to be copied and medieval poems to be collected. He was the author of
military reforms, which included the establishment of standing troops,
called Landsknechte, the improvement of artillery by making cannon
portable, and some changes in the equipment of the cavalry. He was
continually devising plans for the better government of Austria, and
although they ended in failure, he established the unity of the Austrian
dominions. Maximilian has been called the second founder of the
house of Habsburg, and certainly by bringing about marriages between
Charles and Joanna and between his grandson Ferdinand and Anna,
daughter of Ladislaus, king of Hungary and Bohemia, he paved the
way for the vast empire of Charles V. and for the influence of the
Habsburgs in eastern Europe. But he had many qualities less desirable.
He was reckless and unstable, resorting often to lying and deceit, and
never pausing to count the cost of an enterprise or troubling to adapt
means to ends. For absurd and impracticable schemes in Italy and
elsewhere he neglected Germany, and sought to involve its princes in
Page 175
wars undertaken solely for private aggrandizement or personal
jealousy. Ignoring his responsibilities as ruler of Germany, he only
considered the question of its government when in need of money and
support from the princes. As the “last of the knights” he could not see
that the old order of society was passing away and a new order arising,
while he was fascinated by the glitter of the medieval empire and spent
the better part of his life in vague schemes for its revival. As “a gifted
amateur in politics” he increased the disorder of Germany and Italy and
exposed himself and the empire to the jeers of Europe.
Maximilian was also a writer of books, and his writings display his
inordinate vanity. His Geheimes Jagdbuch, containing about 2500
words, is a treatise purporting to teach his grandsons the art of hunting.
He inspired the production of The Dangers and Adventures of the
Famous Hero and Knight Sir Teuerdank, an allegorical poem
describing his adventures on his journey to marry Mary of Burgundy.
The emperor’s share in the work is not clear, but it seems certain that
the general scheme and many of the incidents are due to him. It was
first published at Nuremberg by Melchior Pfintzing in 1517, and was
adorned with woodcuts by Hans Leonhard Schäufelein. The Weisskunig
was long regarded as the work of the emperor’s secretary, Marx
Treitzsaurwein, but it is now believed that the greater part of the book
at least is the work of the emperor himself. It is an unfinished
autobiography containing an account of the achievements of
Maximilian, who is called “the young white king.” It was first
published at Vienna in 1775. He also is responsible for Freydal, an
allegorical account of the tournaments in which he took part during his
wooing of Mary of Burgundy; Ehrenpforten, Triumphwagen and Der
weisen könige Stammbaum, books concerning his own history and that
of the house of Habsburg, and works on various subjects, as Das
Stahlbuch, Die Baumeisterei and Die Gärtnerei. These works are all
jealousy. Ignoring his responsibilities as ruler of Germany, he only
considered the question of its government when in need of money and
support from the princes. As the “last of the knights” he could not see
that the old order of society was passing away and a new order arising,
while he was fascinated by the glitter of the medieval empire and spent
the better part of his life in vague schemes for its revival. As “a gifted
amateur in politics” he increased the disorder of Germany and Italy and
exposed himself and the empire to the jeers of Europe.
Maximilian was also a writer of books, and his writings display his
inordinate vanity. His Geheimes Jagdbuch, containing about 2500
words, is a treatise purporting to teach his grandsons the art of hunting.
He inspired the production of The Dangers and Adventures of the
Famous Hero and Knight Sir Teuerdank, an allegorical poem
describing his adventures on his journey to marry Mary of Burgundy.
The emperor’s share in the work is not clear, but it seems certain that
the general scheme and many of the incidents are due to him. It was
first published at Nuremberg by Melchior Pfintzing in 1517, and was
adorned with woodcuts by Hans Leonhard Schäufelein. The Weisskunig
was long regarded as the work of the emperor’s secretary, Marx
Treitzsaurwein, but it is now believed that the greater part of the book
at least is the work of the emperor himself. It is an unfinished
autobiography containing an account of the achievements of
Maximilian, who is called “the young white king.” It was first
published at Vienna in 1775. He also is responsible for Freydal, an
allegorical account of the tournaments in which he took part during his
wooing of Mary of Burgundy; Ehrenpforten, Triumphwagen and Der
weisen könige Stammbaum, books concerning his own history and that
of the house of Habsburg, and works on various subjects, as Das
Stahlbuch, Die Baumeisterei and Die Gärtnerei. These works are all
Page 176
profusely illustrated, some by Albrecht Dürer, and in the preparation of
the woodcuts Maximilian himself took the liveliest interest. A facsimile
of the original editions of Maximilian’s autobiographical and semi-
autobiographical works has been published in nine volumes in the
Jahrbücher der kunsthistorischen Sammlungen des Kaiserhauses
(Vienna, 1880-1888). For this edition S. Laschitzer wrote an
introduction to Sir Teuerdank, Q. von Leitner to Freydal, and N. A. von
Schultz to Der Weisskunig. The Holbein society issued a facsimile of
Sir Teuerdank (London, 1884) and Triumphwagen (London, 1883).
See Correspondance de l’empereur Maximilien I. et de Marguerite
d’Autriche, 1507-1519, edited by A. G. le Glay (Paris, 1839);
Maximilians I. vertraulicher Briefwechsel mit Sigmund Prüschenk,
edited by V. von Kraus (Innsbruck, 1875); J. Chmel, Urkunden, Briefe
und Aktenstücke zur Geschichte Maximilians I. und seiner Zeit.
(Stuttgart, 1845) and Aktenstücke und Briefe zur Geschichte des
Hauses Habsburg im Zeitalter Maximilians I. (Vienna, 1854-1858); K.
Klüpfel, Kaiser Maximilian I. (Berlin, 1864); H. Ulmann, Kaiser
Maximilian I. (Stuttgart, 1884); L. P. Gachard, Lettres inédites de
Maximilien I. sur les affaires des Pays Bas (Brussels, 1851-1852); L.
von Ranke, Geschichte der romanischen und germanischen Völker,
1494-1514 (Leipzig, 1874); R. W. S. Watson, Maximilian I. (London,
1902); A. Jäger, Über Kaiser Maximilians I. Verhältnis zum Papstthum
(Vienna, 1854); H. Ulmann, Kaiser Maximilians I. Absichten auf das
Papstthum (Stuttgart, 1888), and A. Schulte, Kaiser Maximilian I. als
Kandidat für den päpstlichen Stuhl (Leipzig, 1906). (A. W. H.*)
the woodcuts Maximilian himself took the liveliest interest. A facsimile
of the original editions of Maximilian’s autobiographical and semi-
autobiographical works has been published in nine volumes in the
Jahrbücher der kunsthistorischen Sammlungen des Kaiserhauses
(Vienna, 1880-1888). For this edition S. Laschitzer wrote an
introduction to Sir Teuerdank, Q. von Leitner to Freydal, and N. A. von
Schultz to Der Weisskunig. The Holbein society issued a facsimile of
Sir Teuerdank (London, 1884) and Triumphwagen (London, 1883).
See Correspondance de l’empereur Maximilien I. et de Marguerite
d’Autriche, 1507-1519, edited by A. G. le Glay (Paris, 1839);
Maximilians I. vertraulicher Briefwechsel mit Sigmund Prüschenk,
edited by V. von Kraus (Innsbruck, 1875); J. Chmel, Urkunden, Briefe
und Aktenstücke zur Geschichte Maximilians I. und seiner Zeit.
(Stuttgart, 1845) and Aktenstücke und Briefe zur Geschichte des
Hauses Habsburg im Zeitalter Maximilians I. (Vienna, 1854-1858); K.
Klüpfel, Kaiser Maximilian I. (Berlin, 1864); H. Ulmann, Kaiser
Maximilian I. (Stuttgart, 1884); L. P. Gachard, Lettres inédites de
Maximilien I. sur les affaires des Pays Bas (Brussels, 1851-1852); L.
von Ranke, Geschichte der romanischen und germanischen Völker,
1494-1514 (Leipzig, 1874); R. W. S. Watson, Maximilian I. (London,
1902); A. Jäger, Über Kaiser Maximilians I. Verhältnis zum Papstthum
(Vienna, 1854); H. Ulmann, Kaiser Maximilians I. Absichten auf das
Papstthum (Stuttgart, 1888), and A. Schulte, Kaiser Maximilian I. als
Kandidat für den päpstlichen Stuhl (Leipzig, 1906). (A. W. H.*)
Page 177
MAXIMILIAN II. (1527-1576), Roman emperor, was the eldest
son of the emperor Ferdinand I. by his wife Anne, daughter of Ladislaus,
king of Hungary and Bohemia, and was born in Vienna on the 31st of July
1527. Educated principally in Spain, he gained some experience of warfare
during the campaign of Charles V. against France in 1544, and also during
the war of the league of Schmalkalden, and soon began to take part in
imperial business. Having in September 1548 married his cousin Maria,
daughter of Charles V., he acted as the emperor’s representative in Spain
from 1548 to 1550, returning to Germany in December 1550 in order to
take part in the discussion over the imperial succession. Charles V. wished
his son Philip (afterwards king of Spain) to succeed him as emperor, but his
brother Ferdinand, who had already been designated as the next occupant of
the imperial throne, and Maximilian objected to this proposal. At length a
compromise was reached. Philip was to succeed Ferdinand, but during the
former’s reign Maximilian, as king of the Romans, was to govern Germany.
This arrangement was not carried out, and is only important because the
insistence of the emperor seriously disturbed the harmonious relations
which had hitherto existed between the two branches of the Habsburg
family; and the estrangement went so far that an illness which befell
Maximilian in 1552 was attributed to poison given to him in the interests of
his cousin and brother-in-law, Philip of Spain. About this time he took up
his residence in Vienna, and was engaged mainly in the government of the
Austrian dominions and in defending them against the Turks. The religious
son of the emperor Ferdinand I. by his wife Anne, daughter of Ladislaus,
king of Hungary and Bohemia, and was born in Vienna on the 31st of July
1527. Educated principally in Spain, he gained some experience of warfare
during the campaign of Charles V. against France in 1544, and also during
the war of the league of Schmalkalden, and soon began to take part in
imperial business. Having in September 1548 married his cousin Maria,
daughter of Charles V., he acted as the emperor’s representative in Spain
from 1548 to 1550, returning to Germany in December 1550 in order to
take part in the discussion over the imperial succession. Charles V. wished
his son Philip (afterwards king of Spain) to succeed him as emperor, but his
brother Ferdinand, who had already been designated as the next occupant of
the imperial throne, and Maximilian objected to this proposal. At length a
compromise was reached. Philip was to succeed Ferdinand, but during the
former’s reign Maximilian, as king of the Romans, was to govern Germany.
This arrangement was not carried out, and is only important because the
insistence of the emperor seriously disturbed the harmonious relations
which had hitherto existed between the two branches of the Habsburg
family; and the estrangement went so far that an illness which befell
Maximilian in 1552 was attributed to poison given to him in the interests of
his cousin and brother-in-law, Philip of Spain. About this time he took up
his residence in Vienna, and was engaged mainly in the government of the
Austrian dominions and in defending them against the Turks. The religious
Page 178
views of the king of Bohemia, as Maximilian had been called since his
recognition as the future ruler of that country in 1549, had always been
somewhat uncertain, and he had probably learned something of
Lutheranism in his youth; but his amicable relations with several Protestant
princes, which began about the time of the discussion over the succession,
were probably due more to political than to religious considerations.
However, in Vienna he became very intimate with Sebastian Pfauser (1520-
1569), a court preacher with strong leanings towards Lutheranism, and his
religious attitude caused some uneasiness to his father. Fears were freely
expressed that he would definitely leave the Catholic Church, and when
Ferdinand became emperor in 1558 he was prepared to assure Pope Paul IV.
that his son should not succeed him if he took this step. Eventually
Maximilian remained nominally an adherent of the older faith, although his
views were tinged with Lutheranism until the end of his life. After several
refusals he consented in 1560 to the banishment of Pfauser, and began again
to attend the services of the Catholic Church. This uneasiness having been
dispelled, in November 1562 Maximilian was chosen king of the Romans,
or German king, at Frankfort, where he was crowned a few days later, after
assuring the Catholic electors of his fidelity to their faith, and promising the
Protestant electors that he would publicly accept the confession of
Augsburg when he became emperor. He also took the usual oath to protect
the Church, and his election was afterwards confirmed by the papacy. In
September 1563 he was crowned king of Hungary, and on his father’s
death, in July 1564, succeeded to the empire and to the kingdoms of
Hungary and Bohemia.
The new emperor had already shown that he believed in the necessity for
a thorough reform of the Church. He was unable, however, to obtain the
consent of Pope Pius IV. to the marriage of the clergy, and in 1568 the
concession of communion in both kinds to the laity was withdrawn. On his
part Maximilian granted religious liberty to the Lutheran nobles and knights
recognition as the future ruler of that country in 1549, had always been
somewhat uncertain, and he had probably learned something of
Lutheranism in his youth; but his amicable relations with several Protestant
princes, which began about the time of the discussion over the succession,
were probably due more to political than to religious considerations.
However, in Vienna he became very intimate with Sebastian Pfauser (1520-
1569), a court preacher with strong leanings towards Lutheranism, and his
religious attitude caused some uneasiness to his father. Fears were freely
expressed that he would definitely leave the Catholic Church, and when
Ferdinand became emperor in 1558 he was prepared to assure Pope Paul IV.
that his son should not succeed him if he took this step. Eventually
Maximilian remained nominally an adherent of the older faith, although his
views were tinged with Lutheranism until the end of his life. After several
refusals he consented in 1560 to the banishment of Pfauser, and began again
to attend the services of the Catholic Church. This uneasiness having been
dispelled, in November 1562 Maximilian was chosen king of the Romans,
or German king, at Frankfort, where he was crowned a few days later, after
assuring the Catholic electors of his fidelity to their faith, and promising the
Protestant electors that he would publicly accept the confession of
Augsburg when he became emperor. He also took the usual oath to protect
the Church, and his election was afterwards confirmed by the papacy. In
September 1563 he was crowned king of Hungary, and on his father’s
death, in July 1564, succeeded to the empire and to the kingdoms of
Hungary and Bohemia.
The new emperor had already shown that he believed in the necessity for
a thorough reform of the Church. He was unable, however, to obtain the
consent of Pope Pius IV. to the marriage of the clergy, and in 1568 the
concession of communion in both kinds to the laity was withdrawn. On his
part Maximilian granted religious liberty to the Lutheran nobles and knights
Page 179
in Austria, and refused to allow the publication of the decrees of the council
of Trent. Amid general expectations on the part of the Protestants he met his
first Diet at Augsburg in March 1566. He refused to accede to the demands
of the Lutheran princes; on the other hand, although the increase of
sectarianism was discussed, no decisive steps were taken to suppress it, and
the only result of the meeting was a grant of assistance for the Turkish War,
which had just been renewed. Collecting a large and splendid army
Maximilian marched to defend his territories; but no decisive engagement
had taken place when a truce was made in 1568, and the emperor continued
to pay tribute to the sultan for Hungary. Meanwhile the relations between
Maximilian and Philip of Spain had improved; and the emperor’s
increasingly cautious and moderate attitude in religious matters was
doubtless due to the fact that the death of Philip’s son, Don Carlos, had
opened the way for the succession of Maximilian, or of one of his sons, to
the Spanish throne. Evidence of this friendly feeling was given in 1570,
when the emperor’s daughter, Anne, became the fourth wife of Philip; but
Maximilian was unable to moderate the harsh proceedings of the Spanish
king against the revolting inhabitants of the Netherlands. In 1570 the
emperor met the diet at Spires and asked for aid to place his eastern borders
in a state of defence, and also for power to repress the disorder caused by
troops in the service of foreign powers passing through Germany. He
proposed that his consent should be necessary before any soldiers for
foreign service were recruited in the empire; but the estates were unwilling
to strengthen the imperial authority, the Protestant princes regarded the
suggestion as an attempt to prevent them from assisting their coreligionists
in France and the Netherlands, and nothing was done in this direction,
although some assistance was voted for the defence of Austria. The
religious demands of the Protestants were still unsatisfied, while the policy
of toleration had failed to give peace to Austria. Maximilian’s power was
very limited; it was inability rather than unwillingness that prevented him
of Trent. Amid general expectations on the part of the Protestants he met his
first Diet at Augsburg in March 1566. He refused to accede to the demands
of the Lutheran princes; on the other hand, although the increase of
sectarianism was discussed, no decisive steps were taken to suppress it, and
the only result of the meeting was a grant of assistance for the Turkish War,
which had just been renewed. Collecting a large and splendid army
Maximilian marched to defend his territories; but no decisive engagement
had taken place when a truce was made in 1568, and the emperor continued
to pay tribute to the sultan for Hungary. Meanwhile the relations between
Maximilian and Philip of Spain had improved; and the emperor’s
increasingly cautious and moderate attitude in religious matters was
doubtless due to the fact that the death of Philip’s son, Don Carlos, had
opened the way for the succession of Maximilian, or of one of his sons, to
the Spanish throne. Evidence of this friendly feeling was given in 1570,
when the emperor’s daughter, Anne, became the fourth wife of Philip; but
Maximilian was unable to moderate the harsh proceedings of the Spanish
king against the revolting inhabitants of the Netherlands. In 1570 the
emperor met the diet at Spires and asked for aid to place his eastern borders
in a state of defence, and also for power to repress the disorder caused by
troops in the service of foreign powers passing through Germany. He
proposed that his consent should be necessary before any soldiers for
foreign service were recruited in the empire; but the estates were unwilling
to strengthen the imperial authority, the Protestant princes regarded the
suggestion as an attempt to prevent them from assisting their coreligionists
in France and the Netherlands, and nothing was done in this direction,
although some assistance was voted for the defence of Austria. The
religious demands of the Protestants were still unsatisfied, while the policy
of toleration had failed to give peace to Austria. Maximilian’s power was
very limited; it was inability rather than unwillingness that prevented him
Page 180
from yielding to the entreaties of Pope Pius V. to join in an attack on the
Turks both before and after the victory of Lepanto in 1571; and he remained
inert while the authority of the empire in north-eastern Europe was
threatened. His last important act was to make a bid for the throne of
Poland, either for himself or for his son Ernest. In December 1575 he was
elected by a powerful faction, but the diet which met at Regensburg was
loath to assist; and on the 12th of October 1576 the emperor died, refusing
on his deathbed to receive the last sacraments of the Church.
By his wife Maria he had a family of nine sons and six daughters. He was
succeeded by his eldest surviving son, Rudolph, who had been chosen king
of the Romans in October 1575. Another of his sons, Matthias, also became
emperor; three others, Ernest, Albert and Maximilian, took some part in the
government of the Habsburg territories or of the Netherlands, and a
daughter, Elizabeth, married Charles IX. king of France.
The religious attitude of Maximilian has given rise to much
discussion, and on this subject the writings of W. Maurenbrecher, W.
Goetz and E. Reimann in the Historische Zeitschrift, Bände VII., XV.,
XXXII. and LXXVII. (Munich, 1870 fol.) should be consulted, and
also O. H. Hopfen, Maximilian II. und der Kompromisskatholizismus
(Munich, 1895); C. Haupt, Melanchthons und seiner Lehrer Einfluss
auf Maximilian II. (Wittenberg, 1897); F. Walter, Die Wahl
Maximilians II. (Heidelberg, 1892); W. Goetz, Maximilians II. Wahl
zum römischen Könige (Würzburg, 1891), and T. J. Scherg, Über die
religiöse Entwickelung Kaiser Maximilians II. bis zu seiner Wahl zum
römischen Könige (Würzburg, 1903). For a more general account of his
life and work see Briefe und Akten zur Geschichte Maximilians II.,
edited by W. E. Schwarz (Paderborn, 1889-1891); M. Koch, Quellen
zur Geschichte des Kaisers Maximilian II. in Archiven gesammelt
(Leipzig, 1857-1861); R. Holtzmann, Kaiser Maximilian II. bis zu
Turks both before and after the victory of Lepanto in 1571; and he remained
inert while the authority of the empire in north-eastern Europe was
threatened. His last important act was to make a bid for the throne of
Poland, either for himself or for his son Ernest. In December 1575 he was
elected by a powerful faction, but the diet which met at Regensburg was
loath to assist; and on the 12th of October 1576 the emperor died, refusing
on his deathbed to receive the last sacraments of the Church.
By his wife Maria he had a family of nine sons and six daughters. He was
succeeded by his eldest surviving son, Rudolph, who had been chosen king
of the Romans in October 1575. Another of his sons, Matthias, also became
emperor; three others, Ernest, Albert and Maximilian, took some part in the
government of the Habsburg territories or of the Netherlands, and a
daughter, Elizabeth, married Charles IX. king of France.
The religious attitude of Maximilian has given rise to much
discussion, and on this subject the writings of W. Maurenbrecher, W.
Goetz and E. Reimann in the Historische Zeitschrift, Bände VII., XV.,
XXXII. and LXXVII. (Munich, 1870 fol.) should be consulted, and
also O. H. Hopfen, Maximilian II. und der Kompromisskatholizismus
(Munich, 1895); C. Haupt, Melanchthons und seiner Lehrer Einfluss
auf Maximilian II. (Wittenberg, 1897); F. Walter, Die Wahl
Maximilians II. (Heidelberg, 1892); W. Goetz, Maximilians II. Wahl
zum römischen Könige (Würzburg, 1891), and T. J. Scherg, Über die
religiöse Entwickelung Kaiser Maximilians II. bis zu seiner Wahl zum
römischen Könige (Würzburg, 1903). For a more general account of his
life and work see Briefe und Akten zur Geschichte Maximilians II.,
edited by W. E. Schwarz (Paderborn, 1889-1891); M. Koch, Quellen
zur Geschichte des Kaisers Maximilian II. in Archiven gesammelt
(Leipzig, 1857-1861); R. Holtzmann, Kaiser Maximilian II. bis zu
Page 181
seiner Thronbesteigung (Berlin, 1903); E. Wertheimer, Zur Geschichte
der Türkenkriege Maximilians II. (Vienna, 1875); L. von Ranke, Über
die Zeiten Ferdinands I. und Maximilians II. in Band VII. of his
Sämmtliche Werke (Leipzig, 1874), and J. Janssen, Geschichte des
deutschen Volkes seit dem Ausgang des Mittelalters, Bände IV. to VIII.
(Freiburg, 1885-1894), English translation by M. A. Mitchell and A. M.
Christie (London, 1896 fol.).
MAXIMILIAN (1832-1867), emperor of Mexico, second son of
the archduke Francis Charles of Austria, was born in the palace of
Schönbrunn, on the 6th of July 1832. He was a particularly clever boy,
showed considerable taste for the arts, and early displayed an interest in
science, especially botany. He was trained for the navy, and threw himself
into this career with so much zeal that he quickly rose to high command,
and was mainly instrumental in creating the naval port of Trieste and the
fleet with which Tegethoff won his victories in the Italian War. He had some
reputation as a Liberal, and this led, in February 1857, to his appointment as
viceroy of the Lombardo-Venetian kingdom; in the same year he married
the Princess Charlotte, daughter of Leopold I., king of the Belgians. On the
outbreak of the war of 1859 he retired into private life, chiefly at Trieste,
near which he built the beautiful chateau of Miramar. In this same year he
was first approached by Mexican exiles with the proposal to become the
candidate for the throne of Mexico. He did not at first accept, but sought to
satisfy his restless desire for adventure by a botanical expedition to the
der Türkenkriege Maximilians II. (Vienna, 1875); L. von Ranke, Über
die Zeiten Ferdinands I. und Maximilians II. in Band VII. of his
Sämmtliche Werke (Leipzig, 1874), and J. Janssen, Geschichte des
deutschen Volkes seit dem Ausgang des Mittelalters, Bände IV. to VIII.
(Freiburg, 1885-1894), English translation by M. A. Mitchell and A. M.
Christie (London, 1896 fol.).
MAXIMILIAN (1832-1867), emperor of Mexico, second son of
the archduke Francis Charles of Austria, was born in the palace of
Schönbrunn, on the 6th of July 1832. He was a particularly clever boy,
showed considerable taste for the arts, and early displayed an interest in
science, especially botany. He was trained for the navy, and threw himself
into this career with so much zeal that he quickly rose to high command,
and was mainly instrumental in creating the naval port of Trieste and the
fleet with which Tegethoff won his victories in the Italian War. He had some
reputation as a Liberal, and this led, in February 1857, to his appointment as
viceroy of the Lombardo-Venetian kingdom; in the same year he married
the Princess Charlotte, daughter of Leopold I., king of the Belgians. On the
outbreak of the war of 1859 he retired into private life, chiefly at Trieste,
near which he built the beautiful chateau of Miramar. In this same year he
was first approached by Mexican exiles with the proposal to become the
candidate for the throne of Mexico. He did not at first accept, but sought to
satisfy his restless desire for adventure by a botanical expedition to the
Page 182
tropical forests of Brazil. In 1863, however, under pressure from Napoleon
III., and after General Forey’s capture of the city of Mexico and the
plebiscite which confirmed his proclamation of the empire, he consented to
accept the crown. This decision was contrary to the advice of his brother,
the emperor Francis Joseph, and involved the loss of all his rights in
Austria. Maximilian landed at Vera Cruz on the 28th of May 1864; but from
the very outset he found himself involved in difficulties of the most serious
kind, which in 1866 made apparent to almost every one outside of Mexico
the necessity for his abdicating. Though urged to this course by Napoleon
himself, whose withdrawal from Mexico was the final blow to his cause,
Maximilian refused to desert his followers. Withdrawing, in February 1867,
to Querétaro, he there sustained a siege for several weeks, but on the 15th
of May resolved to attempt an escape through the enemy’s lines. He was,
however, arrested before he could carry out this resolution, and after trial by
court-martial was condemned to death. The sentence was carried out on the
19th of June 1867. His remains were conveyed to Vienna, where they were
buried in the imperial vault early in the following year. (See Mexico.)
Maximilian’s papers were published at Leipzig in 1867, in seven
volumes, under the title Aus meinem Leben, Reiseskizzen, Aphorismen,
Gedichte. See Pierre de la Gorce, Hist. du Second Empire, IV., liv. xxv.
ii. (Paris, 1904); article by von Hoffinger in Allgemeine Deutsche
Biographie, xxi. 70, where authorities are cited.
III., and after General Forey’s capture of the city of Mexico and the
plebiscite which confirmed his proclamation of the empire, he consented to
accept the crown. This decision was contrary to the advice of his brother,
the emperor Francis Joseph, and involved the loss of all his rights in
Austria. Maximilian landed at Vera Cruz on the 28th of May 1864; but from
the very outset he found himself involved in difficulties of the most serious
kind, which in 1866 made apparent to almost every one outside of Mexico
the necessity for his abdicating. Though urged to this course by Napoleon
himself, whose withdrawal from Mexico was the final blow to his cause,
Maximilian refused to desert his followers. Withdrawing, in February 1867,
to Querétaro, he there sustained a siege for several weeks, but on the 15th
of May resolved to attempt an escape through the enemy’s lines. He was,
however, arrested before he could carry out this resolution, and after trial by
court-martial was condemned to death. The sentence was carried out on the
19th of June 1867. His remains were conveyed to Vienna, where they were
buried in the imperial vault early in the following year. (See Mexico.)
Maximilian’s papers were published at Leipzig in 1867, in seven
volumes, under the title Aus meinem Leben, Reiseskizzen, Aphorismen,
Gedichte. See Pierre de la Gorce, Hist. du Second Empire, IV., liv. xxv.
ii. (Paris, 1904); article by von Hoffinger in Allgemeine Deutsche
Biographie, xxi. 70, where authorities are cited.
Page 183
MAXIMINUS, GAIUS JULIUS VERUS, Roman
emperor from a.d. 235 to 238, was born in a village on the confines of
Thrace. He was of barbarian parentage and was brought up as a shepherd.
His immense stature and enormous feats of strength attracted the attention
of the emperor Septimius Severus. He entered the army, and under
Caracalla rose to the rank of centurion. He carefully absented himself from
court during the reign of Heliogabalus, but under his successor Alexander
Severus, was appointed supreme commander of the Roman armies. After
the murder of Alexander in Gaul, hastened, it is said, by his instigation,
Maximinus was proclaimed emperor by the soldiers on the 19th of March
235. The three years of his reign, which were spent wholly in the camp,
were marked by great cruelty and oppression; the widespread discontent
thus produced culminated in a revolt in Africa and the assumption of the
purple by Gordian (q.v.). Maximinus, who was in Pannonia at the time,
marched against Rome, and passing over the Julian Alps descended on
Aquileia; while detained before that city he and his son were murdered in
their tent by a body of praetorians. Their heads were cut off and despatched
to Rome, where they were burnt on the Campus Martius by the exultant
crowd.
Capitolinus, Maximini duo; Herodian vi. 8, vii., viii. 1-5; Zosimus i.
13-15.
emperor from a.d. 235 to 238, was born in a village on the confines of
Thrace. He was of barbarian parentage and was brought up as a shepherd.
His immense stature and enormous feats of strength attracted the attention
of the emperor Septimius Severus. He entered the army, and under
Caracalla rose to the rank of centurion. He carefully absented himself from
court during the reign of Heliogabalus, but under his successor Alexander
Severus, was appointed supreme commander of the Roman armies. After
the murder of Alexander in Gaul, hastened, it is said, by his instigation,
Maximinus was proclaimed emperor by the soldiers on the 19th of March
235. The three years of his reign, which were spent wholly in the camp,
were marked by great cruelty and oppression; the widespread discontent
thus produced culminated in a revolt in Africa and the assumption of the
purple by Gordian (q.v.). Maximinus, who was in Pannonia at the time,
marched against Rome, and passing over the Julian Alps descended on
Aquileia; while detained before that city he and his son were murdered in
their tent by a body of praetorians. Their heads were cut off and despatched
to Rome, where they were burnt on the Campus Martius by the exultant
crowd.
Capitolinus, Maximini duo; Herodian vi. 8, vii., viii. 1-5; Zosimus i.
13-15.
Page 184
MAXIMINUS [Maximin], GALERIUS VALERIUS,
Roman emperor from a.d. 308 to 314, was originally an Illyrian shepherd
named Daia. He rose to high distinction after he had joined the army, and in
305 he was raised by his uncle, Galerius, to the rank of Caesar, with the
government of Syria and Egypt. In 308, after the elevation of Licinius, he
insisted on receiving the title of Augustus; on the death of Galerius, in 311,
he succeeded to the supreme command of the provinces of Asia, and when
Licinius and Constantine began to make common cause with one another
Maximinus entered into a secret alliance with Maxentius. He came to an
open rupture with Licinius in 313, sustained a crushing defeat in the
neighbourhood of Heraclea Pontica on the 30th of April, and fled, first to
Nicomedia and afterwards to Tarsus, where he died in August following.
His death was variously ascribed “to despair, to poison, and to the divine
justice.” Maximinus has a bad name in Christian annals, as having renewed
persecution after the publication of the toleration edict of Galerius, but it is
probable that he has been judged too harshly.
See Maxentius; Zosimus ii. 8; Aurelius Victor, Epit. 40.
MAXIMS, LEGAL. A maxim is an established principle or
proposition. The Latin term maxima is not to be found in Roman law with
any meaning exactly analogous to that of a legal maxim in the modern
sense of the word, but the treatises of many of the Roman jurists on Regulae
definitiones, and Sententiae juris are, in some measure, collections of
Roman emperor from a.d. 308 to 314, was originally an Illyrian shepherd
named Daia. He rose to high distinction after he had joined the army, and in
305 he was raised by his uncle, Galerius, to the rank of Caesar, with the
government of Syria and Egypt. In 308, after the elevation of Licinius, he
insisted on receiving the title of Augustus; on the death of Galerius, in 311,
he succeeded to the supreme command of the provinces of Asia, and when
Licinius and Constantine began to make common cause with one another
Maximinus entered into a secret alliance with Maxentius. He came to an
open rupture with Licinius in 313, sustained a crushing defeat in the
neighbourhood of Heraclea Pontica on the 30th of April, and fled, first to
Nicomedia and afterwards to Tarsus, where he died in August following.
His death was variously ascribed “to despair, to poison, and to the divine
justice.” Maximinus has a bad name in Christian annals, as having renewed
persecution after the publication of the toleration edict of Galerius, but it is
probable that he has been judged too harshly.
See Maxentius; Zosimus ii. 8; Aurelius Victor, Epit. 40.
MAXIMS, LEGAL. A maxim is an established principle or
proposition. The Latin term maxima is not to be found in Roman law with
any meaning exactly analogous to that of a legal maxim in the modern
sense of the word, but the treatises of many of the Roman jurists on Regulae
definitiones, and Sententiae juris are, in some measure, collections of
Page 185
maxims (see an article on “Latin Maxims in English Law” in Law Mag. and
Rev. xx. 285); Fortescue (De laudibus, c. 8) and Du Cange treat maxima and
regula as identical. The attitude of early English commentators towards the
maxims of the law was one of unmingled adulation. In Doctor and Student
(p. 26) they are described as “of the same strength and effect in the law as
statutes be.” Coke (Co. Litt. 11 A) says that a maxim is so called “Quia
maxima est ejus dignitas et certissima auctoritas, atque quod maxime
omnibus probetur.” “Not only,” observes Bacon in the Preface to his
Collection of Maxims, “will the use of maxims be in deciding doubt and
helping soundness of judgment, but, further, in gracing argument, in
correcting unprofitable subtlety, and reducing the same to a more sound and
substantial sense of law, in reclaiming vulgar errors, and, generally, in the
amendment in some measure of the very nature and complexion of the
whole law.” A similar note was sounded in Scotland; and it has been well
observed that “a glance at the pages of Morrison’s Dictionary or at other
early reports will show how frequently in the older Scots law questions
respecting the rights, remedies and liabilities of individuals were
determined by an immediate reference to legal maxims” (J. M. Irving,
Encyclo. Scots Law, s.v. “Maxims”). In later times less value has been
attached to the maxims of the law, as the development of civilization and
the increasing complexity of business relations have shown the necessity of
qualifying the propositions which they enunciate (see Stephen, Hist. Crim.
Law, ii. 94 n: Yarmouth v. France, 1887, 19 Q.B.D., per Lord Esher, at p.
653, and American authorities collected in Bouvier’s Law Dict. s.v.
“Maxim”). But both historically and practically they must always possess
interest and value.
A brief reference need only be made here, with examples by way of
illustration, to the field which the maxims of the law cover.
Rev. xx. 285); Fortescue (De laudibus, c. 8) and Du Cange treat maxima and
regula as identical. The attitude of early English commentators towards the
maxims of the law was one of unmingled adulation. In Doctor and Student
(p. 26) they are described as “of the same strength and effect in the law as
statutes be.” Coke (Co. Litt. 11 A) says that a maxim is so called “Quia
maxima est ejus dignitas et certissima auctoritas, atque quod maxime
omnibus probetur.” “Not only,” observes Bacon in the Preface to his
Collection of Maxims, “will the use of maxims be in deciding doubt and
helping soundness of judgment, but, further, in gracing argument, in
correcting unprofitable subtlety, and reducing the same to a more sound and
substantial sense of law, in reclaiming vulgar errors, and, generally, in the
amendment in some measure of the very nature and complexion of the
whole law.” A similar note was sounded in Scotland; and it has been well
observed that “a glance at the pages of Morrison’s Dictionary or at other
early reports will show how frequently in the older Scots law questions
respecting the rights, remedies and liabilities of individuals were
determined by an immediate reference to legal maxims” (J. M. Irving,
Encyclo. Scots Law, s.v. “Maxims”). In later times less value has been
attached to the maxims of the law, as the development of civilization and
the increasing complexity of business relations have shown the necessity of
qualifying the propositions which they enunciate (see Stephen, Hist. Crim.
Law, ii. 94 n: Yarmouth v. France, 1887, 19 Q.B.D., per Lord Esher, at p.
653, and American authorities collected in Bouvier’s Law Dict. s.v.
“Maxim”). But both historically and practically they must always possess
interest and value.
A brief reference need only be made here, with examples by way of
illustration, to the field which the maxims of the law cover.
Page 186
Commencing with rules founded on public policy, we may note the
famous principle—Salus populi suprema lex (xii. Tables: Bacon,
Maxims, reg. 12)—“the public welfare is the highest law.” It is on this
maxim that the coercive action of the State towards individual liberty
in a hundred matters is based. To the same category belong the maxims
—Summa ratio est quae pro religione facit (Co. Litt. 341 a)—“the best
rule is that which advances religion”—a maxim which finds its
application when the enforcement of foreign laws or judgments
supposed to violate our own laws or the principles of natural justice is
in question; and Dies dominicus non est juridicus, which exempts
Sunday from the lawful days for juridical acts. Among the maxims
relating to the crown, the most important are Rex non potest peccare (2
Rolle R. 304)—“The King can do no wrong”—which enshrines the
principle of ministerial responsibility, and Nullum tempus occurrit regi
(2 Co. Inst. 273)—“lapse of time does not bar the crown,” a maxim
qualified by various enactments in modern times. Passing to the
judicial office and the administration of justice, we may refer to the
rules—Audi alteram partem—a proposition too familiar to need either
translation or comment; Nemo debet esse judex in propriâ suâ causâ
(12 Co. Rep. 114)—“no man ought to be judge in his own cause”—a
maxim which French law, and the legal systems based upon or allied to
it, have embodied in an elaborate network of rules for judicial
challenge; and the maxim which defines the relative functions of judge
and jury, Ad quaestionem facti non respondent judices, ad quaestionem
legis non respondent juratores (8 Co. Rep. 155). The maxim Boni
judicis est ampliare jurisdictionem (Ch. Prec. 329) is certainly
erroneous as it stands, as a judge has no right to “extend his
jurisdiction.” If justitiam is substituted for jurisdictionem, as Lord
Mansfield said it should be (1 Burr. 304), the maxim is near the truth. A
group of maxims supposed to embody certain fundamental principles
famous principle—Salus populi suprema lex (xii. Tables: Bacon,
Maxims, reg. 12)—“the public welfare is the highest law.” It is on this
maxim that the coercive action of the State towards individual liberty
in a hundred matters is based. To the same category belong the maxims
—Summa ratio est quae pro religione facit (Co. Litt. 341 a)—“the best
rule is that which advances religion”—a maxim which finds its
application when the enforcement of foreign laws or judgments
supposed to violate our own laws or the principles of natural justice is
in question; and Dies dominicus non est juridicus, which exempts
Sunday from the lawful days for juridical acts. Among the maxims
relating to the crown, the most important are Rex non potest peccare (2
Rolle R. 304)—“The King can do no wrong”—which enshrines the
principle of ministerial responsibility, and Nullum tempus occurrit regi
(2 Co. Inst. 273)—“lapse of time does not bar the crown,” a maxim
qualified by various enactments in modern times. Passing to the
judicial office and the administration of justice, we may refer to the
rules—Audi alteram partem—a proposition too familiar to need either
translation or comment; Nemo debet esse judex in propriâ suâ causâ
(12 Co. Rep. 114)—“no man ought to be judge in his own cause”—a
maxim which French law, and the legal systems based upon or allied to
it, have embodied in an elaborate network of rules for judicial
challenge; and the maxim which defines the relative functions of judge
and jury, Ad quaestionem facti non respondent judices, ad quaestionem
legis non respondent juratores (8 Co. Rep. 155). The maxim Boni
judicis est ampliare jurisdictionem (Ch. Prec. 329) is certainly
erroneous as it stands, as a judge has no right to “extend his
jurisdiction.” If justitiam is substituted for jurisdictionem, as Lord
Mansfield said it should be (1 Burr. 304), the maxim is near the truth. A
group of maxims supposed to embody certain fundamental principles
Page 187
of legal right and obligations may next be referred to: (a) Ubi jus ibi
remedium (see Co. Litt. 197 b)—a maxim to which the evolution of the
flexible “action on the case,” by which wrongs unknown to the
“original writs” were dealt with, was historically due, but which must
be taken with the gloss Damnum absque injuria—“there are forms of
actual damage which do not constitute legal injury” for which the law
supplies no remedy; (b) Actus Dei nemini facit injuriam (2 Blackstone,
122)—and its allied maxim, Lex non cogit ad impossibilia (Co. Litt.
231 b)—on which the whole doctrine of vis major (force majeure) and
impossible conditions in the law of contract has been built up. In this
category may also be classed Volenti non fit injuria (Wingate, Maxims),
out of which sprang the theory—now profoundly modified by statute—
of “common employment” in the law of employers’ liability; see Smith
v. Baker, 1891, A.C. 325. Other maxims deal with rights of property—
Qui prior est tempore, potior est jure (Co. Litt. 14 a), which
consecrates the position of the beati possidentes alike in municipal and
in international law; Sic utere tuo ut alienum non laedas (9 Co. Rep.
59), which has played its part in the determination of the rights of
adjacent owners; and Domus sua cuique est tutissimum refugium (5 Co.
Rep. 92)—“a man’s house is his castle,” a doctrine which has imposed
limitations on the rights of execution creditors (see Execution). In the
laws of family relations there are the maxims Consensus non
concubitus facit matrimonium (Co. Litt. 33 a)—the canon law of
Europe prior to the council of Trent, and still law in Scotland, though
modified by legislation in England; and Pater is est quem nuptiae
demonstrant (see Co. Litt. 7 b), on which, in most civilized countries,
the presumption of legitimacy depends. In the interpretation of written
instruments, the maxim Noscitur a sociis (3 Term Reports, 87), which
proclaims the importance of the context, still applies. So do the rules
Expressio unius est exclusio alterius (Co. Litt. 210 a), and
remedium (see Co. Litt. 197 b)—a maxim to which the evolution of the
flexible “action on the case,” by which wrongs unknown to the
“original writs” were dealt with, was historically due, but which must
be taken with the gloss Damnum absque injuria—“there are forms of
actual damage which do not constitute legal injury” for which the law
supplies no remedy; (b) Actus Dei nemini facit injuriam (2 Blackstone,
122)—and its allied maxim, Lex non cogit ad impossibilia (Co. Litt.
231 b)—on which the whole doctrine of vis major (force majeure) and
impossible conditions in the law of contract has been built up. In this
category may also be classed Volenti non fit injuria (Wingate, Maxims),
out of which sprang the theory—now profoundly modified by statute—
of “common employment” in the law of employers’ liability; see Smith
v. Baker, 1891, A.C. 325. Other maxims deal with rights of property—
Qui prior est tempore, potior est jure (Co. Litt. 14 a), which
consecrates the position of the beati possidentes alike in municipal and
in international law; Sic utere tuo ut alienum non laedas (9 Co. Rep.
59), which has played its part in the determination of the rights of
adjacent owners; and Domus sua cuique est tutissimum refugium (5 Co.
Rep. 92)—“a man’s house is his castle,” a doctrine which has imposed
limitations on the rights of execution creditors (see Execution). In the
laws of family relations there are the maxims Consensus non
concubitus facit matrimonium (Co. Litt. 33 a)—the canon law of
Europe prior to the council of Trent, and still law in Scotland, though
modified by legislation in England; and Pater is est quem nuptiae
demonstrant (see Co. Litt. 7 b), on which, in most civilized countries,
the presumption of legitimacy depends. In the interpretation of written
instruments, the maxim Noscitur a sociis (3 Term Reports, 87), which
proclaims the importance of the context, still applies. So do the rules
Expressio unius est exclusio alterius (Co. Litt. 210 a), and
Page 188
Contemporanea expositio est optima et fortissima in lege (2 Co. Inst.
11), which lets in evidence of contemporaneous user as an aid to the
interpretation of statutes or documents; see Van Diemen’s Land Co. v.
Table Cape Marine Board, 1906, A.C. 92, 98. We may conclude this
sketch with a miscellaneous summary: Caveat emptor (Hob. 99)—“let
the purchaser beware”; Qui facit per alium facile per se, which affirms
the principal’s liability for the acts of his agent; Ignorantia juris
neminem excusat, on which rests the ordinary citizen’s obligation to
know the law; and Vigilantibus non dormientibus jura subveniunt (2
Co. Inst. 690), one of the maxims in accordance with which courts of
equity administer relief. Among other “maxims of equity” come the
rules that “he that seeks equity must do equity,” i.e. must act fairly, and
that “equity looks upon that as done which ought to be done”—a
principle from which the “conversion” into money of land directed to
be sold, and of money directed to be invested in the purchase of land, is
derived.
The principal collections of legal maxims are: English Law: Bacon,
Collection of Some Principal Rules and Maxims of the Common Law
(1630); Noy, Treatise of the principal Grounds and Maxims of the Law
of England (1641, 8th ed., 1824); Wingate, Maxims of Reason (1728);
Francis, Grounds and Rudiments of Law and Equity (2nd ed. 1751);
Lofft (annexed to his Reports, 1776); Broom, Legal Maxims (7th ed.
London, 1900). Scots Law: Lord Trayner, Latin Maxims and Phrases
(2nd ed., 1876); Stair, Institutions of the Law of Scotland, with Index
by More (Edinburgh, 1832). American Treatises: A. I. Morgan, English
Version of Legal Maxims (Cincinnati, 1878); S. S. Peloubet, Legal
Maxims in Law and Equity (New York, 1880). (A. W. R.)
11), which lets in evidence of contemporaneous user as an aid to the
interpretation of statutes or documents; see Van Diemen’s Land Co. v.
Table Cape Marine Board, 1906, A.C. 92, 98. We may conclude this
sketch with a miscellaneous summary: Caveat emptor (Hob. 99)—“let
the purchaser beware”; Qui facit per alium facile per se, which affirms
the principal’s liability for the acts of his agent; Ignorantia juris
neminem excusat, on which rests the ordinary citizen’s obligation to
know the law; and Vigilantibus non dormientibus jura subveniunt (2
Co. Inst. 690), one of the maxims in accordance with which courts of
equity administer relief. Among other “maxims of equity” come the
rules that “he that seeks equity must do equity,” i.e. must act fairly, and
that “equity looks upon that as done which ought to be done”—a
principle from which the “conversion” into money of land directed to
be sold, and of money directed to be invested in the purchase of land, is
derived.
The principal collections of legal maxims are: English Law: Bacon,
Collection of Some Principal Rules and Maxims of the Common Law
(1630); Noy, Treatise of the principal Grounds and Maxims of the Law
of England (1641, 8th ed., 1824); Wingate, Maxims of Reason (1728);
Francis, Grounds and Rudiments of Law and Equity (2nd ed. 1751);
Lofft (annexed to his Reports, 1776); Broom, Legal Maxims (7th ed.
London, 1900). Scots Law: Lord Trayner, Latin Maxims and Phrases
(2nd ed., 1876); Stair, Institutions of the Law of Scotland, with Index
by More (Edinburgh, 1832). American Treatises: A. I. Morgan, English
Version of Legal Maxims (Cincinnati, 1878); S. S. Peloubet, Legal
Maxims in Law and Equity (New York, 1880). (A. W. R.)
Page 189
MAXIMUS, the name of four Roman emperors.
I. M. Clodius Pupienus Maximus, joint emperor with D. Caelius Calvinus
Balbinus during a few months of the year a.d. 238. Pupienus was a
distinguished soldier, who had been proconsul of Bithynia, Achaea, and
Gallia Narbonensis. At the advanced age of seventy-four, he was chosen by
the senate with Balbinus to resist the barbarian Maximinus. Their complete
equality is shown by the fact that each assumed the titles of pontifex
maximus and princeps senatus. It was arranged that Pupienus should take
the field against Maximinus, while Balbinus remained at Rome to maintain
order, a task in which he signally failed. A revolt of the praetorians was not
repressed till much blood had been shed and a considerable part of the city
reduced to ashes. On his march, Pupienus, having received the news that
Maximinus had been assassinated by his own troops, returned in triumph to
Rome. Shortly afterwards, when both emperors were on the point of leaving
the city on an expedition—Pupienus against the Persians and Balbinus
against the Goths—the praetorians, who had always resented the
appointment of the senatorial emperors and cherished the memory of the
soldier-emperor Maximinus, seized the opportunity of revenge. When most
of the people were at the Capitoline games, they forced their way into the
palace, dragged Balbinus and Pupienus through the streets, and put them to
death.
I. M. Clodius Pupienus Maximus, joint emperor with D. Caelius Calvinus
Balbinus during a few months of the year a.d. 238. Pupienus was a
distinguished soldier, who had been proconsul of Bithynia, Achaea, and
Gallia Narbonensis. At the advanced age of seventy-four, he was chosen by
the senate with Balbinus to resist the barbarian Maximinus. Their complete
equality is shown by the fact that each assumed the titles of pontifex
maximus and princeps senatus. It was arranged that Pupienus should take
the field against Maximinus, while Balbinus remained at Rome to maintain
order, a task in which he signally failed. A revolt of the praetorians was not
repressed till much blood had been shed and a considerable part of the city
reduced to ashes. On his march, Pupienus, having received the news that
Maximinus had been assassinated by his own troops, returned in triumph to
Rome. Shortly afterwards, when both emperors were on the point of leaving
the city on an expedition—Pupienus against the Persians and Balbinus
against the Goths—the praetorians, who had always resented the
appointment of the senatorial emperors and cherished the memory of the
soldier-emperor Maximinus, seized the opportunity of revenge. When most
of the people were at the Capitoline games, they forced their way into the
palace, dragged Balbinus and Pupienus through the streets, and put them to
death.
Page 190
See Capitolinus, Life of Maximus and Balbinus; Herodian vii. 10,
viii. 6; Zonaras xii. 16; Orosius vii. 19; Eutropius ix. 2; Zosimus i. 14;
Aurelius Victor, Caesares, 26, epit. 26; H. Schiller, Geschichte der
römischen Kaiserzeit, i. 2; Gibbon, Decline and Fall, ch. 7 and (for the
chronology) appendix 12 (Bury’s edition).
II. Magnus Maximus, a native of Spain, who had accompanied
Theodosius on several expeditions and from 368 held high military rank in
Britain. The disaffected troops having proclaimed Maximus emperor, he
crossed over to Gaul, attacked Gratian (q.v.), and drove him from Paris to
Lyons, where he was murdered by a partisan of Maximus. Theodosius being
unable to avenge the death of his colleague, an agreement was made (384 or
385) by which Maximus was recognized as Augustus and sole emperor in
Gaul, Spain and Britain, while Valentinian II. was to remain unmolested in
Italy and Illyricum, Theodosius retaining his sovereignty in the East. In 387
Maximus crossed the Alps, Valentinian was speedily put to flight, while the
invader established himself in Milan and for the time became master of
Italy. Theodosius now took vigorous measures. Advancing with a powerful
army, he twice defeated the troops of Maximus—at Siscia on the Save, and
at Poetovio on the Danube. He then hurried on to Aquileia, where Maximus
had shut himself up, and had him beheaded. Under the name of Maxen
Wledig, Maximus appears in the list of Welsh royal heroes (see R.
Williams, Biog. Dict. of Eminent Welshmen, 1852; “The Dream of Maxen
Wledig,” in the Mabinogion).
Full account with classical references in H. Richter, Das
weströmische Reich, besonders unter den Kaisern Gratian, Valentinian
II. und Maximus (1865); see also H. Schiller, Geschichte der römischen
Kaiserzeit, ii. (1887); Gibbon, Decline and Fall, ch. 27; Tillemont,
Hist. des empereurs, v.
viii. 6; Zonaras xii. 16; Orosius vii. 19; Eutropius ix. 2; Zosimus i. 14;
Aurelius Victor, Caesares, 26, epit. 26; H. Schiller, Geschichte der
römischen Kaiserzeit, i. 2; Gibbon, Decline and Fall, ch. 7 and (for the
chronology) appendix 12 (Bury’s edition).
II. Magnus Maximus, a native of Spain, who had accompanied
Theodosius on several expeditions and from 368 held high military rank in
Britain. The disaffected troops having proclaimed Maximus emperor, he
crossed over to Gaul, attacked Gratian (q.v.), and drove him from Paris to
Lyons, where he was murdered by a partisan of Maximus. Theodosius being
unable to avenge the death of his colleague, an agreement was made (384 or
385) by which Maximus was recognized as Augustus and sole emperor in
Gaul, Spain and Britain, while Valentinian II. was to remain unmolested in
Italy and Illyricum, Theodosius retaining his sovereignty in the East. In 387
Maximus crossed the Alps, Valentinian was speedily put to flight, while the
invader established himself in Milan and for the time became master of
Italy. Theodosius now took vigorous measures. Advancing with a powerful
army, he twice defeated the troops of Maximus—at Siscia on the Save, and
at Poetovio on the Danube. He then hurried on to Aquileia, where Maximus
had shut himself up, and had him beheaded. Under the name of Maxen
Wledig, Maximus appears in the list of Welsh royal heroes (see R.
Williams, Biog. Dict. of Eminent Welshmen, 1852; “The Dream of Maxen
Wledig,” in the Mabinogion).
Full account with classical references in H. Richter, Das
weströmische Reich, besonders unter den Kaisern Gratian, Valentinian
II. und Maximus (1865); see also H. Schiller, Geschichte der römischen
Kaiserzeit, ii. (1887); Gibbon, Decline and Fall, ch. 27; Tillemont,
Hist. des empereurs, v.
Page 191
III. Maximus Tyrannus, made emperor in Spain by the Roman general,
Gerontius, who had rebelled against the usurper Constantine in 408. After
the defeat of Gerontius at Arelate (Arles) and his death in 411 Maximus
renounced the imperial title and was permitted by Constantine to retire into
private life. About 418 he rebelled again, but, failing in his attempt, was
seized, carried into Italy, and put to death at Ravenna in 422.
See Orosius vii. 42; Zosimus vi. 5; Sozomen ix. 3; E. A. Freeman,
“The Tyrants of Britain, Gaul and Spain, a.d. 406-411,” in English
Historical Review, i. (1886).
IV. Petronius Maximus, a member of the higher Roman nobility, had held
several court and public offices, including those of praefectus Romae (420)
and Italiae (439-441 and 445), and consul (433, 443). He was one of the
intimate associates of Valentinian III., whom he assisted in the palace
intrigues which led to the death of Aëtius in 454; but an outrage committed
on the wife of Maximus by the emperor turned his friendship into hatred.
Maximus was proclaimed emperor immediately after Valentinian’s murder
(March 16, 455), but after reigning less than three months, he was murdered
by some Burgundian mercenaries as he was fleeing before the troops of
Genseric, who, invited by Eudoxia, the widow of Valentinian, had landed at
the mouth of the Tiber (May or June 455).
See Procopius, Vand. i. 4; Sidonius Apollinaris, Panegyr. Aviti, ep. ii.
13; the various Chronicles; Gibbon, Decline and Fall, chs. 35, 36;
Tillemont, Hist. des empereurs, vi.
Gerontius, who had rebelled against the usurper Constantine in 408. After
the defeat of Gerontius at Arelate (Arles) and his death in 411 Maximus
renounced the imperial title and was permitted by Constantine to retire into
private life. About 418 he rebelled again, but, failing in his attempt, was
seized, carried into Italy, and put to death at Ravenna in 422.
See Orosius vii. 42; Zosimus vi. 5; Sozomen ix. 3; E. A. Freeman,
“The Tyrants of Britain, Gaul and Spain, a.d. 406-411,” in English
Historical Review, i. (1886).
IV. Petronius Maximus, a member of the higher Roman nobility, had held
several court and public offices, including those of praefectus Romae (420)
and Italiae (439-441 and 445), and consul (433, 443). He was one of the
intimate associates of Valentinian III., whom he assisted in the palace
intrigues which led to the death of Aëtius in 454; but an outrage committed
on the wife of Maximus by the emperor turned his friendship into hatred.
Maximus was proclaimed emperor immediately after Valentinian’s murder
(March 16, 455), but after reigning less than three months, he was murdered
by some Burgundian mercenaries as he was fleeing before the troops of
Genseric, who, invited by Eudoxia, the widow of Valentinian, had landed at
the mouth of the Tiber (May or June 455).
See Procopius, Vand. i. 4; Sidonius Apollinaris, Panegyr. Aviti, ep. ii.
13; the various Chronicles; Gibbon, Decline and Fall, chs. 35, 36;
Tillemont, Hist. des empereurs, vi.
Page 192
MAXIMUS, ST (c. 580-662), abbot of Chrysopolis, known as “the
Confessor” from his orthodox zeal in the Monothelite (q.v.) controversy, or
as “the monk,” was born of noble parentage at Constantinople about the
year 580. Educated with great care, he early became distinguished by his
talents and acquirements, and some time after the accession of the emperor
Heraclius in 610 was made his private secretary. In 630 he abandoned the
secular life and entered the monastery of Chrysopolis (Scutari), actuated, it
was believed, less by any longing for the life of a recluse than by the
dissatisfaction he felt with the Monothelite leanings of his master. The date
of his promotion to the abbacy is uncertain. In 633 he was one of the party
of Sophronius of Jerusalem (the chief original opponent of the
Monothelites) at the council of Alexandria; and in 645 he was again in
Africa, when he held in presence of the governor and a number of bishops
the disputation with Pyrrhus, the deposed and banished patriarch of
Constantinople, which resulted in the (temporary) conversion of his
interlocutor to the Dyothelite view. In the following year several African
synods, held under the influence of Maximus, declared for orthodoxy. In
649, after the accession of Martin I., he went to Rome, and did much to fan
the zeal of the new pope, who in October of that year held the (first)
Lateran synod, by which not only the Monothelite doctrine but also the
moderating ecthesis of Heraclius and typus of Constans II. were
anathematized. About 653 Maximus, for the part he had taken against the
latter document especially, was apprehended (together with the pope) by
Confessor” from his orthodox zeal in the Monothelite (q.v.) controversy, or
as “the monk,” was born of noble parentage at Constantinople about the
year 580. Educated with great care, he early became distinguished by his
talents and acquirements, and some time after the accession of the emperor
Heraclius in 610 was made his private secretary. In 630 he abandoned the
secular life and entered the monastery of Chrysopolis (Scutari), actuated, it
was believed, less by any longing for the life of a recluse than by the
dissatisfaction he felt with the Monothelite leanings of his master. The date
of his promotion to the abbacy is uncertain. In 633 he was one of the party
of Sophronius of Jerusalem (the chief original opponent of the
Monothelites) at the council of Alexandria; and in 645 he was again in
Africa, when he held in presence of the governor and a number of bishops
the disputation with Pyrrhus, the deposed and banished patriarch of
Constantinople, which resulted in the (temporary) conversion of his
interlocutor to the Dyothelite view. In the following year several African
synods, held under the influence of Maximus, declared for orthodoxy. In
649, after the accession of Martin I., he went to Rome, and did much to fan
the zeal of the new pope, who in October of that year held the (first)
Lateran synod, by which not only the Monothelite doctrine but also the
moderating ecthesis of Heraclius and typus of Constans II. were
anathematized. About 653 Maximus, for the part he had taken against the
latter document especially, was apprehended (together with the pope) by
Page 193
order of Constans and carried a prisoner to Constantinople. In 655, after
repeated examinations, in which he maintained his theological opinions
with memorable constancy, he was banished to Byzia in Thrace, and
afterwards to Perberis. In 662 he was again brought to Constantinople and
was condemned by a synod to be scourged, to have his tongue cut out by
the root, and to have his right hand chopped off. After this sentence had
been carried out he was again banished to Lazica, where he died on the 13th
of August 662. He is venerated as a saint both in the Greek and in the Latin
Churches. Maximus was not only a leader in the Monothelite struggle but a
mystic who zealously followed and advocated the system of Pseudo-
Dionysius, while adding to it an ethical element in the conception of the
freedom of the will. His works had considerable influence in shaping the
system of John Scotus Erigena.
The most important of the works of Maximus will be found in
Migne, Patrologia graeca, xc. xci., together with an anonymous life;
an exhaustive list in Wagenmann’s article in vol. xii. (1903) of Hauck-
Herzog’s Realencyklopädie where the following classification is
adopted: (a) exegetical, (b) scholia on the Fathers, (c) dogmatic and
controversial, (d) ethical and ascetic, (e) miscellaneous. The details of
the disputation with Pyrrhus and of the martyrdom are given very fully
and clearly in Hefele’s Conciliengeschichte, iii. For further literature
see H. Gelzer in C. Krumbacher’s Geschichte der byzantinischen
Litteratur (1897).
repeated examinations, in which he maintained his theological opinions
with memorable constancy, he was banished to Byzia in Thrace, and
afterwards to Perberis. In 662 he was again brought to Constantinople and
was condemned by a synod to be scourged, to have his tongue cut out by
the root, and to have his right hand chopped off. After this sentence had
been carried out he was again banished to Lazica, where he died on the 13th
of August 662. He is venerated as a saint both in the Greek and in the Latin
Churches. Maximus was not only a leader in the Monothelite struggle but a
mystic who zealously followed and advocated the system of Pseudo-
Dionysius, while adding to it an ethical element in the conception of the
freedom of the will. His works had considerable influence in shaping the
system of John Scotus Erigena.
The most important of the works of Maximus will be found in
Migne, Patrologia graeca, xc. xci., together with an anonymous life;
an exhaustive list in Wagenmann’s article in vol. xii. (1903) of Hauck-
Herzog’s Realencyklopädie where the following classification is
adopted: (a) exegetical, (b) scholia on the Fathers, (c) dogmatic and
controversial, (d) ethical and ascetic, (e) miscellaneous. The details of
the disputation with Pyrrhus and of the martyrdom are given very fully
and clearly in Hefele’s Conciliengeschichte, iii. For further literature
see H. Gelzer in C. Krumbacher’s Geschichte der byzantinischen
Litteratur (1897).
Page 194
MAXIMUS OF SMYRNA, a Greek philosopher of the Neo-
platonist school, who lived towards the end of the 4th century a.d. He was
perhaps the most important of the followers of Iamblichus. He is said to
have been of a rich and noble family, and exercised great influence over the
emperor Julian, who was commended to him by Aedesius. He pandered to
the emperor’s love of magic and theurgy, and by judicious administration of
the omens won a high position at court. His overbearing manner made him
numerous enemies, and, after being imprisoned on the death of Julian, he
was put to death by Valens. He is a representative of the least attractive side
of Neoplatonism. Attaching no value to logical proof and argument, he
enlarged on the wonders and mysteries of nature, and maintained his
position by the working of miracles. In logic he is reported to have agreed
with Eusebius, Iamblichus and Porphyry in asserting the validity of the
second and third figures of the syllogism.
MAXIMUS OF TYRE (Cassius Maximus Tyrius), a Greek
rhetorician and philosopher who flourished in the time of the Antonines and
Commodus (2nd century a.d.). After the manner of the sophists of his age,
he travelled extensively, delivering lectures on the way. His writings contain
many allusions to the history of Greece, while there is little reference to
Rome; hence it is inferred that he lived longer in Greece, perhaps as a
professor at Athens. Although nominally a Platonist, he is really an Eclectic
and one of the precursors of Neoplatonism. There are still extant by him
forty-one essays or discourses (διαλέξεις) on theological, ethical, and other
platonist school, who lived towards the end of the 4th century a.d. He was
perhaps the most important of the followers of Iamblichus. He is said to
have been of a rich and noble family, and exercised great influence over the
emperor Julian, who was commended to him by Aedesius. He pandered to
the emperor’s love of magic and theurgy, and by judicious administration of
the omens won a high position at court. His overbearing manner made him
numerous enemies, and, after being imprisoned on the death of Julian, he
was put to death by Valens. He is a representative of the least attractive side
of Neoplatonism. Attaching no value to logical proof and argument, he
enlarged on the wonders and mysteries of nature, and maintained his
position by the working of miracles. In logic he is reported to have agreed
with Eusebius, Iamblichus and Porphyry in asserting the validity of the
second and third figures of the syllogism.
MAXIMUS OF TYRE (Cassius Maximus Tyrius), a Greek
rhetorician and philosopher who flourished in the time of the Antonines and
Commodus (2nd century a.d.). After the manner of the sophists of his age,
he travelled extensively, delivering lectures on the way. His writings contain
many allusions to the history of Greece, while there is little reference to
Rome; hence it is inferred that he lived longer in Greece, perhaps as a
professor at Athens. Although nominally a Platonist, he is really an Eclectic
and one of the precursors of Neoplatonism. There are still extant by him
forty-one essays or discourses (διαλέξεις) on theological, ethical, and other
Page 195
philosophical commonplaces. With him God is the supreme being, one and
indivisible though called by many names, accessible to reason alone; but as
animals form the intermediate stage between plants and human beings, so
there exist intermediaries between God and man, viz. daemons, who dwell
on the confines of heaven and earth. The soul in many ways bears a great
resemblance to the divinity; it is partly mortal, partly immortal, and, when
freed from the fetters of the body, becomes a daemon. Life is the sleep of
the soul, from which it awakes at death. The style of Maximus is superior to
that of the ordinary sophistical rhetorician, but scholars differ widely as to
the merits of the essays themselves.
Maximus of Tyre must be distinguished from the Stoic Maximus, tutor of
Marcus Aurelius.
Editions by J. Davies, revised with valuable notes by J. Markland
(1740); J. J. Reiske (1774); F. Dübner (1840, with Theophrastus, &c.,
in the Didot series). Monographs by R. Rohdich (Beuthen, 1879); H.
Hobein, De Maximo Tyrio quaestiones philol. (Jena, 1895). There is an
English translation (1804) by Thomas Taylor, the Platonist.
MAX MÜLLER, FRIEDRICH (1823-1900), Anglo-German
orientalist and comparative philologist, was born at Dessau on the 6th of
December 1823, being the son of Wilhelm Müller (1794-1827), the German
poet, celebrated for his phil-Hellenic lyrics, who was ducal librarian at
Dessau. The elder Müller had endeared himself to the most intellectual
indivisible though called by many names, accessible to reason alone; but as
animals form the intermediate stage between plants and human beings, so
there exist intermediaries between God and man, viz. daemons, who dwell
on the confines of heaven and earth. The soul in many ways bears a great
resemblance to the divinity; it is partly mortal, partly immortal, and, when
freed from the fetters of the body, becomes a daemon. Life is the sleep of
the soul, from which it awakes at death. The style of Maximus is superior to
that of the ordinary sophistical rhetorician, but scholars differ widely as to
the merits of the essays themselves.
Maximus of Tyre must be distinguished from the Stoic Maximus, tutor of
Marcus Aurelius.
Editions by J. Davies, revised with valuable notes by J. Markland
(1740); J. J. Reiske (1774); F. Dübner (1840, with Theophrastus, &c.,
in the Didot series). Monographs by R. Rohdich (Beuthen, 1879); H.
Hobein, De Maximo Tyrio quaestiones philol. (Jena, 1895). There is an
English translation (1804) by Thomas Taylor, the Platonist.
MAX MÜLLER, FRIEDRICH (1823-1900), Anglo-German
orientalist and comparative philologist, was born at Dessau on the 6th of
December 1823, being the son of Wilhelm Müller (1794-1827), the German
poet, celebrated for his phil-Hellenic lyrics, who was ducal librarian at
Dessau. The elder Müller had endeared himself to the most intellectual
Page 196
circles in Germany by his amiable character and his genuine poetic gift; his
songs had been utilized by musical composers, notably Schubert; and it was
his son’s good fortune to meet in his youth with a succession of eminent
friends, who, already interested in him for his father’s sake, and charmed by
the qualities which they discovered in the young man himself, powerfully
aided him by advice and patronage. Mendelssohn, who was his godfather,
dissuaded him from indulging his natural bent to the study of music;
Professor Brockhaus of the University of Leipzig, where Max Müller
matriculated in 1841, induced him to take up Sanskrit; Bopp, at the
University of Berlin (1844), made the Sanskrit student a scientific
comparative philologist; Schelling at the same university, inspired him with
a love for metaphysical speculation, though failing to attract him to his own
philosophy; Burnouf, at Paris in the following year, by teaching him Zend,
started him on the track of inquiry into the science of comparative religion,
and impelled him to edit the Rig Veda; and when, in 1846, Max Müller
came to England upon this errand, Bunsen, in conjunction with Professor H.
H. Wilson, prevailed upon the East India Company to undertake the
expense of publication. Up to this time Max Müller had lived the life of a
poor student, supporting himself partly by copying manuscripts, but
Bunsen’s introductions to Queen Victoria and the prince consort, and to
Oxford University, laid the foundation for him of fame and fortune. In 1848
the printing of his Rig Veda at the University Press obliged him to settle in
Oxford, a step which decided his future career. He arrived at a favourable
conjuncture: the Tractarian strife, which had so long thrust learning into the
background, was just over, and Oxford was becoming accessible to modern
ideas. The young German excited curiosity and interest, and it was soon
discovered that, although a genuine scholar, he was no mere bookworm.
Part of his social success was due to his readiness to exert his musical
talents at private parties. Max Müller was speedily subjugated by the genius
loci. He was appointed deputy Taylorian professor of modern languages in
songs had been utilized by musical composers, notably Schubert; and it was
his son’s good fortune to meet in his youth with a succession of eminent
friends, who, already interested in him for his father’s sake, and charmed by
the qualities which they discovered in the young man himself, powerfully
aided him by advice and patronage. Mendelssohn, who was his godfather,
dissuaded him from indulging his natural bent to the study of music;
Professor Brockhaus of the University of Leipzig, where Max Müller
matriculated in 1841, induced him to take up Sanskrit; Bopp, at the
University of Berlin (1844), made the Sanskrit student a scientific
comparative philologist; Schelling at the same university, inspired him with
a love for metaphysical speculation, though failing to attract him to his own
philosophy; Burnouf, at Paris in the following year, by teaching him Zend,
started him on the track of inquiry into the science of comparative religion,
and impelled him to edit the Rig Veda; and when, in 1846, Max Müller
came to England upon this errand, Bunsen, in conjunction with Professor H.
H. Wilson, prevailed upon the East India Company to undertake the
expense of publication. Up to this time Max Müller had lived the life of a
poor student, supporting himself partly by copying manuscripts, but
Bunsen’s introductions to Queen Victoria and the prince consort, and to
Oxford University, laid the foundation for him of fame and fortune. In 1848
the printing of his Rig Veda at the University Press obliged him to settle in
Oxford, a step which decided his future career. He arrived at a favourable
conjuncture: the Tractarian strife, which had so long thrust learning into the
background, was just over, and Oxford was becoming accessible to modern
ideas. The young German excited curiosity and interest, and it was soon
discovered that, although a genuine scholar, he was no mere bookworm.
Part of his social success was due to his readiness to exert his musical
talents at private parties. Max Müller was speedily subjugated by the genius
loci. He was appointed deputy Taylorian professor of modern languages in
Page 197
1850, and the German government failed to tempt him back to Strassburg.
In the following year he was made M.A. and honorary fellow of Christ
Church, and in 1858 he was elected a fellow of All Souls. In 1854 the
Crimean War gave him the opportunity of utilizing his oriental learning in
vocabularies and schemes of transliteration. In 1857 he successfully
essayed another kind of literature in his beautiful story Deutsche Liebe,
written both in German and English. He had by this time become an
extensive contributor to English periodical literature, and had written
several of the essays subsequently collected as Chips from a German
Workshop. The most important of them was the fascinating essay on
“Comparative Mythology” in the Oxford Essays for 1856. His valuable
History of Ancient Sanskrit Literature, so far as it illustrates the primitive
religion of the Brahmans (and hence the Vedic period only), was published
in 1850.
Though Max Müller’s reputation was that of a comparative philologist
and orientalist, his professional duties at Oxford were long confined to
lecturing on modern languages, or at least their medieval forms. In 1860 the
death of Horace Hayman Wilson, professor of Sanskrit, seemed to open a
more congenial sphere to him. His claims to the succession seemed
incontestable, for his opponent, Monier Williams, though well qualified as a
Sanskritist, lacked Max Müller’s brilliant versatility, and although educated
at Oxford, had held no University office. But Max Müller was a Liberal,
and the friend of Liberals in university matters, in politics, and in theology,
and this consideration united with his foreign birth to bring the country
clergy in such hosts to the poll that the voice of resident Oxford was
overborne, and Monier Williams was elected by a large majority. It was the
one great disappointment of Max Müller’s life, and made a lasting
impression upon him. It was, nevertheless, serviceable to his influence and
reputation by permitting him to enter upon a wider field of subjects than
would have been possible otherwise. Directly, Sanskrit philology received
In the following year he was made M.A. and honorary fellow of Christ
Church, and in 1858 he was elected a fellow of All Souls. In 1854 the
Crimean War gave him the opportunity of utilizing his oriental learning in
vocabularies and schemes of transliteration. In 1857 he successfully
essayed another kind of literature in his beautiful story Deutsche Liebe,
written both in German and English. He had by this time become an
extensive contributor to English periodical literature, and had written
several of the essays subsequently collected as Chips from a German
Workshop. The most important of them was the fascinating essay on
“Comparative Mythology” in the Oxford Essays for 1856. His valuable
History of Ancient Sanskrit Literature, so far as it illustrates the primitive
religion of the Brahmans (and hence the Vedic period only), was published
in 1850.
Though Max Müller’s reputation was that of a comparative philologist
and orientalist, his professional duties at Oxford were long confined to
lecturing on modern languages, or at least their medieval forms. In 1860 the
death of Horace Hayman Wilson, professor of Sanskrit, seemed to open a
more congenial sphere to him. His claims to the succession seemed
incontestable, for his opponent, Monier Williams, though well qualified as a
Sanskritist, lacked Max Müller’s brilliant versatility, and although educated
at Oxford, had held no University office. But Max Müller was a Liberal,
and the friend of Liberals in university matters, in politics, and in theology,
and this consideration united with his foreign birth to bring the country
clergy in such hosts to the poll that the voice of resident Oxford was
overborne, and Monier Williams was elected by a large majority. It was the
one great disappointment of Max Müller’s life, and made a lasting
impression upon him. It was, nevertheless, serviceable to his influence and
reputation by permitting him to enter upon a wider field of subjects than
would have been possible otherwise. Directly, Sanskrit philology received
Page 198
little more from him, except in connexion with his later undertaking of The
Sacred Books of the East; but indirectly he exalted it more than any
predecessor by proclaiming its commanding position in the history of the
human intellect by his Science of Language, two courses of lectures
delivered at the Royal Institution in 1861 and 1863. Max Müller ought not
to be described as “the introducer of comparative philology into England.”
Prichard had proved the Aryan affinities of the Celtic languages by the
methods of comparative philology so long before as 1831; Winning’s
Manual of Comparative Philology had been published in 1838; the
discoveries of Bopp and Pott and Pictet had been recognized in brilliant
articles in the Quarterly Review, and had guided the researches of
Rawlinson. But Max Müller undoubtedly did far more to popularize the
subject than had been done, or could have been done, by any predecessor.
He was on less sure ground in another department of the study of language
—the problem of its origin. He wrote upon it as a disciple of Kant, whose
Critique of Pure Reason he translated. His essays on mythology are among
the most delightful of his writings, but their value is somewhat impaired by
a too uncompromising adherence to the seductive generalization of the solar
myth.
Max Müller’s studies in mythology led him to another field of activity in
which his influence was more durable and extensive, that of the
comparative science of religions. Here, so far as Great Britain is concerned,
he does deserve the fame of an originator, and his Introduction to the
Science of Religion (1873: the same year in which he lectured on the
subject, at Dean Stanley’s invitation, in Westminster Abbey, this being the
only occasion on which a layman had given an address there) marks an
epoch. It was followed by other works of importance, especially the four
volumes of Gifford lectures, delivered between 1888 and 1892; but the most
tangible result of the impulse he had given was the publication under his
editorship, from 1875 onwards, of The Sacred Books of the East, in fifty-
Sacred Books of the East; but indirectly he exalted it more than any
predecessor by proclaiming its commanding position in the history of the
human intellect by his Science of Language, two courses of lectures
delivered at the Royal Institution in 1861 and 1863. Max Müller ought not
to be described as “the introducer of comparative philology into England.”
Prichard had proved the Aryan affinities of the Celtic languages by the
methods of comparative philology so long before as 1831; Winning’s
Manual of Comparative Philology had been published in 1838; the
discoveries of Bopp and Pott and Pictet had been recognized in brilliant
articles in the Quarterly Review, and had guided the researches of
Rawlinson. But Max Müller undoubtedly did far more to popularize the
subject than had been done, or could have been done, by any predecessor.
He was on less sure ground in another department of the study of language
—the problem of its origin. He wrote upon it as a disciple of Kant, whose
Critique of Pure Reason he translated. His essays on mythology are among
the most delightful of his writings, but their value is somewhat impaired by
a too uncompromising adherence to the seductive generalization of the solar
myth.
Max Müller’s studies in mythology led him to another field of activity in
which his influence was more durable and extensive, that of the
comparative science of religions. Here, so far as Great Britain is concerned,
he does deserve the fame of an originator, and his Introduction to the
Science of Religion (1873: the same year in which he lectured on the
subject, at Dean Stanley’s invitation, in Westminster Abbey, this being the
only occasion on which a layman had given an address there) marks an
epoch. It was followed by other works of importance, especially the four
volumes of Gifford lectures, delivered between 1888 and 1892; but the most
tangible result of the impulse he had given was the publication under his
editorship, from 1875 onwards, of The Sacred Books of the East, in fifty-
Page 199
one volumes, including indexes, all but three of which appeared under his
superintendence during his lifetime. These comprise translations by the
most competent scholars of all the really important non-Christian scriptures
of Oriental nations, which can now be appreciated without a knowledge of
the original languages. Max Müller also wrote on Indian philosophy in his
latter years, and his exertions to stimulate search for Oriental manuscripts
and inscriptions were rewarded with important discoveries of early
Buddhist scriptures, in their Indian form, made in Japan. He was on
particularly friendly terms with native Japanese scholars, and after his death
his library was purchased by the university of Tôkyô.
In 1868 Max Müller had been indemnified for his disappointment over
the Sanskrit professorship by the establishment of a chair of Comparative
Philology to be filled by him. He retired, however, from the actual duties of
the post in 1875, when entering upon the editorship of The Sacred Books of
the East. The most remarkable external events of his latter years were his
delivery of lectures at the restored university of Strassburg in 1872, when
he devoted his honorarium to the endowment of a Sanskrit lectureship, and
his presidency over the International Congress of Orientalists in 1892. But
his days, if uneventful, were busy. He participated in every movement at
Oxford of which he could approve, and was intimate with nearly all its men
of light and leading; he was a curator of the Bodleian Library, and a
delegate of the University Press. He was acquainted with most of the
crowned heads
of Europe, and was an especial favourite with the English royal family.
His hospitality was ample, especially to visitors from India, where he was
far better known than any other European Orientalist. His distinctions,
conferred by foreign governments and learned societies, were innumerable,
and, having been naturalized shortly after his arrival in England, he
received the high honour of being made a privy councillor. In 1898 and
superintendence during his lifetime. These comprise translations by the
most competent scholars of all the really important non-Christian scriptures
of Oriental nations, which can now be appreciated without a knowledge of
the original languages. Max Müller also wrote on Indian philosophy in his
latter years, and his exertions to stimulate search for Oriental manuscripts
and inscriptions were rewarded with important discoveries of early
Buddhist scriptures, in their Indian form, made in Japan. He was on
particularly friendly terms with native Japanese scholars, and after his death
his library was purchased by the university of Tôkyô.
In 1868 Max Müller had been indemnified for his disappointment over
the Sanskrit professorship by the establishment of a chair of Comparative
Philology to be filled by him. He retired, however, from the actual duties of
the post in 1875, when entering upon the editorship of The Sacred Books of
the East. The most remarkable external events of his latter years were his
delivery of lectures at the restored university of Strassburg in 1872, when
he devoted his honorarium to the endowment of a Sanskrit lectureship, and
his presidency over the International Congress of Orientalists in 1892. But
his days, if uneventful, were busy. He participated in every movement at
Oxford of which he could approve, and was intimate with nearly all its men
of light and leading; he was a curator of the Bodleian Library, and a
delegate of the University Press. He was acquainted with most of the
crowned heads
of Europe, and was an especial favourite with the English royal family.
His hospitality was ample, especially to visitors from India, where he was
far better known than any other European Orientalist. His distinctions,
conferred by foreign governments and learned societies, were innumerable,
and, having been naturalized shortly after his arrival in England, he
received the high honour of being made a privy councillor. In 1898 and
Page 200
1899 he published autobiographical reminiscences under the title of Auld
Lang Syne. He was writing a more detailed autobiography when overtaken
by death on the 28th of October 1900. Max Müller married in 1859
Georgiana Adelaide Grenfell, sister of the wives of Charles Kingsley and J.
A. Froude. One of his daughters, Mrs Conybeare, distinguished herself by a
translation of Scherer’s History of German Literature.
Though undoubtedly a great scholar, Max Müller did not so much
represent scholarship pure and simple as her hybrid types—the scholar-
author and the scholar-courtier. In the former capacity, though manifesting
little of the originality of genius, he rendered vast service by popularizing
high truths among high minds. In his public and social character he
represented Oriental studies with a brilliancy, and conferred upon them a
distinction, which they had not previously enjoyed in Great Britain. There
were drawbacks in both respects: the author was too prone to build upon
insecure foundations, and the man of the world incurred censure for failings
which may perhaps be best indicated by the remark that he seemed too
much of a diplomatist. But the sum of foibles seems insignificant in
comparison with the life of intense labour dedicated to the service of culture
and humanity.
Max Müller’s Collected Works were published in 1903.
(R. G.)
MAXWELL, the name of a Scottish family, members of which
have held the titles of earl of Morton, earl of Nithsdale, Lord Maxwell, and
Lang Syne. He was writing a more detailed autobiography when overtaken
by death on the 28th of October 1900. Max Müller married in 1859
Georgiana Adelaide Grenfell, sister of the wives of Charles Kingsley and J.
A. Froude. One of his daughters, Mrs Conybeare, distinguished herself by a
translation of Scherer’s History of German Literature.
Though undoubtedly a great scholar, Max Müller did not so much
represent scholarship pure and simple as her hybrid types—the scholar-
author and the scholar-courtier. In the former capacity, though manifesting
little of the originality of genius, he rendered vast service by popularizing
high truths among high minds. In his public and social character he
represented Oriental studies with a brilliancy, and conferred upon them a
distinction, which they had not previously enjoyed in Great Britain. There
were drawbacks in both respects: the author was too prone to build upon
insecure foundations, and the man of the world incurred censure for failings
which may perhaps be best indicated by the remark that he seemed too
much of a diplomatist. But the sum of foibles seems insignificant in
comparison with the life of intense labour dedicated to the service of culture
and humanity.
Max Müller’s Collected Works were published in 1903.
(R. G.)
MAXWELL, the name of a Scottish family, members of which
have held the titles of earl of Morton, earl of Nithsdale, Lord Maxwell, and
Page 201
Lord Herries. The name is taken probably from Maccuswell, or Maxwell,
near Kelso, whither the family migrated from England about 1100. Sir
Herbert Maxwell won great fame by defending his castle of Carlaverock
against Edward I. in 1300; another Sir Herbert was made a lord of the
Scottish parliament before 1445; and his great-grandson John, 3rd Lord
Maxwell, was killed at Flodden in 1513. John’s son Robert, the 4th lord (d.
1546), was a member of the royal council under James V.; he was also an
extraordinary lord of session, high admiral, and warden of the west
marches, and was taken prisoner by the English at the rout of Solway Moss
in 1542. Robert’s grandson John, 7th Lord Maxwell (1553-1593), was the
second son of Robert, the 5th lord (d. 1552), and his wife Beatrix, daughter
of James Douglas, 3rd earl of Morton. After the execution of the regent
Morton, the 4th earl, in 1581 this earldom was bestowed upon Maxwell, but
in 1586 the attainder of the late earl was reversed and he was deprived of
his new title. He had helped in 1585 to drive the royal favourite James
Stewart, earl of Arran, from power, and he made active preparations to
assist the invading Spaniards in 1588. His son John, the 8th lord (c. 1586-
1613), was at feud with the Johnstones, who had killed his father in a
skirmish, and with the Douglases over the earldom of Morton, which he
regarded as his inheritance. After a life of exceptional and continuous
lawlessness he escaped from Scotland and in his absence was sentenced to
death; having returned to his native country he was seized and was
beheaded in Edinburgh. In 1618 John’s brother and heir Robert (d. 1646)
was restored to the lordship of Maxwell, and in 1620 was created earl of
Nithsdale, surrendering at this time his claim to the earldom of Morton. He
and his son Robert, afterwards the 2nd earl, fought under Montrose for
Charles I. during the Civil War. Robert died without sons in October 1667,
when a cousin John Maxwell, 7th Lord Herries (d. 1677), became third earl.
William, 5th earl of Nithsdale (1676-1744), a grandson of the third earl,
was like his ancestor a Roman Catholic and was attached to the cause of the
near Kelso, whither the family migrated from England about 1100. Sir
Herbert Maxwell won great fame by defending his castle of Carlaverock
against Edward I. in 1300; another Sir Herbert was made a lord of the
Scottish parliament before 1445; and his great-grandson John, 3rd Lord
Maxwell, was killed at Flodden in 1513. John’s son Robert, the 4th lord (d.
1546), was a member of the royal council under James V.; he was also an
extraordinary lord of session, high admiral, and warden of the west
marches, and was taken prisoner by the English at the rout of Solway Moss
in 1542. Robert’s grandson John, 7th Lord Maxwell (1553-1593), was the
second son of Robert, the 5th lord (d. 1552), and his wife Beatrix, daughter
of James Douglas, 3rd earl of Morton. After the execution of the regent
Morton, the 4th earl, in 1581 this earldom was bestowed upon Maxwell, but
in 1586 the attainder of the late earl was reversed and he was deprived of
his new title. He had helped in 1585 to drive the royal favourite James
Stewart, earl of Arran, from power, and he made active preparations to
assist the invading Spaniards in 1588. His son John, the 8th lord (c. 1586-
1613), was at feud with the Johnstones, who had killed his father in a
skirmish, and with the Douglases over the earldom of Morton, which he
regarded as his inheritance. After a life of exceptional and continuous
lawlessness he escaped from Scotland and in his absence was sentenced to
death; having returned to his native country he was seized and was
beheaded in Edinburgh. In 1618 John’s brother and heir Robert (d. 1646)
was restored to the lordship of Maxwell, and in 1620 was created earl of
Nithsdale, surrendering at this time his claim to the earldom of Morton. He
and his son Robert, afterwards the 2nd earl, fought under Montrose for
Charles I. during the Civil War. Robert died without sons in October 1667,
when a cousin John Maxwell, 7th Lord Herries (d. 1677), became third earl.
William, 5th earl of Nithsdale (1676-1744), a grandson of the third earl,
was like his ancestor a Roman Catholic and was attached to the cause of the
Page 202
exiled house of Stuart. In 1715 he joined the Jacobite insurgents, being
taken prisoner at the battle of Preston and sentenced to death. He escaped,
however, from the Tower of London through the courage and devotion of
his wife Winifred (d. 1749), daughter of William Herbert, 1st marquess of
Powis. He was attainted in 1716 and his titles became extinct, but his
estates passed to his son William (d. 1776), whose descendant, William
Constable-Maxwell, regained the title of Lord Herries in 1858. The
countess of Nithsdale wrote an account of her husband’s escape, which is
published in vol. i. of the Transactions of the Society of Antiquaries of
Scotland.
A few words may be added about other prominent members of the
Maxwell family. John Maxwell (c. 1590-1647), archbishop of Tuam,
was a Scottish ecclesiastic who took a leading part in helping
Archbishop Laud in his futile attempt to restore the liturgy in Scotland.
He was bishop of Ross from 1633 until 1638, when he was deposed by
the General Assembly; then crossing over to Ireland he was bishop of
Killala and Achonry from 1640 to 1645, and archbishop of Tuam from
1645 until his death. James Maxwell of Kirkconnell (c. 1708-1762), the
Jacobite, wrote the Narrative of Charles Prince of Wales’s Expedition
to Scotland in 1745, which was printed for the Maitland Club in 1841.
Robert Maxwell (1695-1765) was the author of Select Transactions of
the Society of Improvers and was a great benefactor to Scottish
agriculture. Sir Murray Maxwell (1775-1831), a naval officer, gained
much fame by his conduct when his ship the “Alceste” was wrecked in
Gaspar Strait in 1817. William Hamilton Maxwell (1792-1850), the
Irish novelist, wrote, in addition to several novels, a Life of the Duke of
Wellington (1839-1841 and again 1883), and a History of the Irish
Rebellion in 1798 (1845 and 1891). Sir Herbert Maxwell, 7th bart. (b.
1845), member of parliament for Wigtownshire from 1880 to 1906, and
president of the Society of Antiquaries of Scotland, became well
taken prisoner at the battle of Preston and sentenced to death. He escaped,
however, from the Tower of London through the courage and devotion of
his wife Winifred (d. 1749), daughter of William Herbert, 1st marquess of
Powis. He was attainted in 1716 and his titles became extinct, but his
estates passed to his son William (d. 1776), whose descendant, William
Constable-Maxwell, regained the title of Lord Herries in 1858. The
countess of Nithsdale wrote an account of her husband’s escape, which is
published in vol. i. of the Transactions of the Society of Antiquaries of
Scotland.
A few words may be added about other prominent members of the
Maxwell family. John Maxwell (c. 1590-1647), archbishop of Tuam,
was a Scottish ecclesiastic who took a leading part in helping
Archbishop Laud in his futile attempt to restore the liturgy in Scotland.
He was bishop of Ross from 1633 until 1638, when he was deposed by
the General Assembly; then crossing over to Ireland he was bishop of
Killala and Achonry from 1640 to 1645, and archbishop of Tuam from
1645 until his death. James Maxwell of Kirkconnell (c. 1708-1762), the
Jacobite, wrote the Narrative of Charles Prince of Wales’s Expedition
to Scotland in 1745, which was printed for the Maitland Club in 1841.
Robert Maxwell (1695-1765) was the author of Select Transactions of
the Society of Improvers and was a great benefactor to Scottish
agriculture. Sir Murray Maxwell (1775-1831), a naval officer, gained
much fame by his conduct when his ship the “Alceste” was wrecked in
Gaspar Strait in 1817. William Hamilton Maxwell (1792-1850), the
Irish novelist, wrote, in addition to several novels, a Life of the Duke of
Wellington (1839-1841 and again 1883), and a History of the Irish
Rebellion in 1798 (1845 and 1891). Sir Herbert Maxwell, 7th bart. (b.
1845), member of parliament for Wigtownshire from 1880 to 1906, and
president of the Society of Antiquaries of Scotland, became well
Page 203
known as a writer, his works including Life and Times of the Right
Hon. W. H. Smith (1893); Life of the Duke of Wellington (1899); The
House of Douglas (1902); Robert the Bruce (1897) and A Duke of
Britain (1895).
MAXWELL, JAMES CLERK (1831-1879), British
physicist, was the last representative of a younger branch of the well-known
Scottish family of Clerk of Penicuik, and was born at Edinburgh on the 13th
of November 1831. He was educated at the Edinburgh Academy (1840-
1847) and the university of Edinburgh (1847-1850). Entering at Cambridge
in 1850, he spent a term or two at Peterhouse, but afterwards migrated to
Trinity. In 1854 he took his degree as second wrangler, and was declared
equal with the senior wrangler of his year (E. J. Routh, q.v.) in the higher
ordeal of the Smith’s prize examination. He held the chair of Natural
Philosophy in Marischal College, Aberdeen, from 1856 till the fusion of the
two colleges there in 1860. For eight years subsequently he held the chair of
Physics and Astronomy in King’s College, London, but resigned in 1868
and retired to his estate of Glenlair in Kirkcudbrightshire. He was
summoned from his seclusion in 1871 to become the first holder of the
newly founded professorship of Experimental Physics in Cambridge; and it
was under his direction that the plans of the Cavendish Laboratory were
prepared. He superintended every step of the progress of the building and of
the purchase of the very valuable collection of apparatus with which it was
equipped at the expense of its munificent founder the seventh duke of
Hon. W. H. Smith (1893); Life of the Duke of Wellington (1899); The
House of Douglas (1902); Robert the Bruce (1897) and A Duke of
Britain (1895).
MAXWELL, JAMES CLERK (1831-1879), British
physicist, was the last representative of a younger branch of the well-known
Scottish family of Clerk of Penicuik, and was born at Edinburgh on the 13th
of November 1831. He was educated at the Edinburgh Academy (1840-
1847) and the university of Edinburgh (1847-1850). Entering at Cambridge
in 1850, he spent a term or two at Peterhouse, but afterwards migrated to
Trinity. In 1854 he took his degree as second wrangler, and was declared
equal with the senior wrangler of his year (E. J. Routh, q.v.) in the higher
ordeal of the Smith’s prize examination. He held the chair of Natural
Philosophy in Marischal College, Aberdeen, from 1856 till the fusion of the
two colleges there in 1860. For eight years subsequently he held the chair of
Physics and Astronomy in King’s College, London, but resigned in 1868
and retired to his estate of Glenlair in Kirkcudbrightshire. He was
summoned from his seclusion in 1871 to become the first holder of the
newly founded professorship of Experimental Physics in Cambridge; and it
was under his direction that the plans of the Cavendish Laboratory were
prepared. He superintended every step of the progress of the building and of
the purchase of the very valuable collection of apparatus with which it was
equipped at the expense of its munificent founder the seventh duke of
Page 204
Devonshire (chancellor of the university, and one of its most distinguished
alumni). He died at Cambridge on the 5th of November 1879.
For more than half of his brief life he held a prominent position in the
very foremost rank of natural philosophers. His contributions to scientific
societies began in his fifteenth year, when Professor J. D. Forbes
communicated to the Royal Society of Edinburgh a short paper of his on a
mechanical method of tracing Cartesian ovals. In his eighteenth year, while
still a student in Edinburgh, he contributed two valuable papers to the
Transactions of the same society—one of which, “On the Equilibrium of
Elastic Solids,” is remarkable, not only on account of its intrinsic power and
the youth of its author, but also because in it he laid the foundation of one
of the most singular discoveries of his later life, the temporary double
refraction produced in viscous liquids by shearing stress. Immediately after
taking his degree, he read to the Cambridge Philosophical Society a very
novel memoir, “On the Transformation of Surfaces by Bending.” This is
one of the few purely mathematical papers he published, and it exhibited at
once to experts the full genius of its author. About the same time appeared
his elaborate memoir, “On Faraday’s Lines of Force,” in which he gave the
first indication of some of those extraordinary electrical investigations
which culminated in the greatest work of his life. He obtained in 1859 the
Adams prize in Cambridge for a very original and powerful essay, “On the
Stability of Saturn’s Rings.” From 1855 to 1872 he published at intervals a
series of valuable investigations connected with the “Perception of Colour”
and “Colour-Blindness,” for the earlier of which he received the Rumford
medal from the Royal Society in 1860. The instruments which he devised
for these investigations were simple and convenient, but could not have
been thought of for the purpose except by a man whose knowledge was co-
extensive with his ingenuity. One of his greatest investigations bore on the
“Kinetic Theory of Gases.” Originating with D. Bernoulli, this theory was
advanced by the successive labours of John Herapath, J. P. Joule, and
alumni). He died at Cambridge on the 5th of November 1879.
For more than half of his brief life he held a prominent position in the
very foremost rank of natural philosophers. His contributions to scientific
societies began in his fifteenth year, when Professor J. D. Forbes
communicated to the Royal Society of Edinburgh a short paper of his on a
mechanical method of tracing Cartesian ovals. In his eighteenth year, while
still a student in Edinburgh, he contributed two valuable papers to the
Transactions of the same society—one of which, “On the Equilibrium of
Elastic Solids,” is remarkable, not only on account of its intrinsic power and
the youth of its author, but also because in it he laid the foundation of one
of the most singular discoveries of his later life, the temporary double
refraction produced in viscous liquids by shearing stress. Immediately after
taking his degree, he read to the Cambridge Philosophical Society a very
novel memoir, “On the Transformation of Surfaces by Bending.” This is
one of the few purely mathematical papers he published, and it exhibited at
once to experts the full genius of its author. About the same time appeared
his elaborate memoir, “On Faraday’s Lines of Force,” in which he gave the
first indication of some of those extraordinary electrical investigations
which culminated in the greatest work of his life. He obtained in 1859 the
Adams prize in Cambridge for a very original and powerful essay, “On the
Stability of Saturn’s Rings.” From 1855 to 1872 he published at intervals a
series of valuable investigations connected with the “Perception of Colour”
and “Colour-Blindness,” for the earlier of which he received the Rumford
medal from the Royal Society in 1860. The instruments which he devised
for these investigations were simple and convenient, but could not have
been thought of for the purpose except by a man whose knowledge was co-
extensive with his ingenuity. One of his greatest investigations bore on the
“Kinetic Theory of Gases.” Originating with D. Bernoulli, this theory was
advanced by the successive labours of John Herapath, J. P. Joule, and
Page 205
particularly R. Clausius, to such an extent as to put its general accuracy
beyond a doubt; but it received enormous developments from Maxwell,
who in this field appeared as an experimenter (on the laws of gaseous
friction) as well as a mathematician. He wrote an admirable textbook of the
Theory of Heat (1871), and a very excellent elementary treatise on Matter
and Motion (1876).
But the great work of his life was devoted to electricity. He began by
reading, with the most profound admiration and attention, the whole of
Faraday’s extraordinary self-revelations, and proceeded to translate the
ideas of that master into the succinct and expressive notation of the
mathematicians. A considerable part of this translation was accomplished
during his career as an undergraduate in Cambridge. The writer had the
opportunity of perusing the MS. of “On Faraday’s Lines of Force,” in a
form little different from the final one, a year before Maxwell took his
degree. His great object, as it was also the great object of Faraday, was to
overturn the idea of action at a distance. The splendid researches of S. D.
Poisson and K. F. Gauss had shown how to reduce all the phenomena of
statical electricity to mere attractions and repulsions exerted at a distance by
particles of an imponderable on one another. Lord Kelvin (Sir W. Thomson)
had, in 1846, shown that a totally different assumption, based upon other
analogies, led (by its own special mathematical methods) to precisely the
same results. He treated the resultant electric force at any point as
analogous to the flux of heat from sources distributed in the same manner as
the supposed electric particles. This paper of Thomson’s, whose ideas
Maxwell afterwards developed in an extraordinary manner, seems to have
given the first hint that there are at least two perfectly distinct methods of
arriving at the known formulae of statical electricity. The step to magnetic
phenomena was comparatively simple; but it was otherwise as regards
electro-magnetic phenomena, where current electricity is essentially
involved. An exceedingly ingenious, but highly artificial, theory had been
beyond a doubt; but it received enormous developments from Maxwell,
who in this field appeared as an experimenter (on the laws of gaseous
friction) as well as a mathematician. He wrote an admirable textbook of the
Theory of Heat (1871), and a very excellent elementary treatise on Matter
and Motion (1876).
But the great work of his life was devoted to electricity. He began by
reading, with the most profound admiration and attention, the whole of
Faraday’s extraordinary self-revelations, and proceeded to translate the
ideas of that master into the succinct and expressive notation of the
mathematicians. A considerable part of this translation was accomplished
during his career as an undergraduate in Cambridge. The writer had the
opportunity of perusing the MS. of “On Faraday’s Lines of Force,” in a
form little different from the final one, a year before Maxwell took his
degree. His great object, as it was also the great object of Faraday, was to
overturn the idea of action at a distance. The splendid researches of S. D.
Poisson and K. F. Gauss had shown how to reduce all the phenomena of
statical electricity to mere attractions and repulsions exerted at a distance by
particles of an imponderable on one another. Lord Kelvin (Sir W. Thomson)
had, in 1846, shown that a totally different assumption, based upon other
analogies, led (by its own special mathematical methods) to precisely the
same results. He treated the resultant electric force at any point as
analogous to the flux of heat from sources distributed in the same manner as
the supposed electric particles. This paper of Thomson’s, whose ideas
Maxwell afterwards developed in an extraordinary manner, seems to have
given the first hint that there are at least two perfectly distinct methods of
arriving at the known formulae of statical electricity. The step to magnetic
phenomena was comparatively simple; but it was otherwise as regards
electro-magnetic phenomena, where current electricity is essentially
involved. An exceedingly ingenious, but highly artificial, theory had been
Page 206
devised by W. E. Weber, which was found capable of explaining all the
phenomena investigated by Ampère as well as the induction currents of
Faraday. But this was based upon the assumption of a distance-action
between electric particles, the intensity of which depended on their relative
motion as well as on their position. This was, of course, even more
repugnant to Maxwell’s mind than the statical distance-action developed by
Poisson. The first paper of Maxwell’s in which an attempt at an admissible
physical theory of electromagnetism was made was communicated to the
Royal Society in 1867. But the theory, in a fully developed form, first
appeared in 1873 in his great treatise on Electricity and Magnetism. This
work was one of the most splendid monuments ever raised by the genius of
a single individual. Availing himself of the admirable generalized co-
ordinate system of Lagrange, Maxwell showed how to reduce all electric
and magnetic phenomena to stresses and motions of a material medium,
and, as one preliminary, but excessively severe, test of the truth of his
theory, he pointed out that (if the electro-magnetic medium be that which is
required for the explanation of the phenomena of light) the velocity of light
in vacuo should be numerically the same as the ratio of the electro-magnetic
and electrostatic units. In fact, the means of the best determinations of each
of these quantities separately agree with one another more closely than do
the various values of either.
One of Maxwell’s last great contributions to science was the editing (with
copious original notes) of the Electrical Researches of the Hon. Henry
Cavendish, from which it appeared that Cavendish, already famous by
many other researches (such as the mean density of the earth, the
composition of water, &c.), must be looked on as, in his day, a man of
Maxwell’s own stamp as a theorist and an experimenter of the very first
rank.
phenomena investigated by Ampère as well as the induction currents of
Faraday. But this was based upon the assumption of a distance-action
between electric particles, the intensity of which depended on their relative
motion as well as on their position. This was, of course, even more
repugnant to Maxwell’s mind than the statical distance-action developed by
Poisson. The first paper of Maxwell’s in which an attempt at an admissible
physical theory of electromagnetism was made was communicated to the
Royal Society in 1867. But the theory, in a fully developed form, first
appeared in 1873 in his great treatise on Electricity and Magnetism. This
work was one of the most splendid monuments ever raised by the genius of
a single individual. Availing himself of the admirable generalized co-
ordinate system of Lagrange, Maxwell showed how to reduce all electric
and magnetic phenomena to stresses and motions of a material medium,
and, as one preliminary, but excessively severe, test of the truth of his
theory, he pointed out that (if the electro-magnetic medium be that which is
required for the explanation of the phenomena of light) the velocity of light
in vacuo should be numerically the same as the ratio of the electro-magnetic
and electrostatic units. In fact, the means of the best determinations of each
of these quantities separately agree with one another more closely than do
the various values of either.
One of Maxwell’s last great contributions to science was the editing (with
copious original notes) of the Electrical Researches of the Hon. Henry
Cavendish, from which it appeared that Cavendish, already famous by
many other researches (such as the mean density of the earth, the
composition of water, &c.), must be looked on as, in his day, a man of
Maxwell’s own stamp as a theorist and an experimenter of the very first
rank.
Page 207
In private life Clerk Maxwell was one of the most lovable of men, a
sincere and unostentatious Christian. Though perfectly free from any trace
of envy or ill-will, he yet showed on fit occasion his contempt for that
pseudo-science which seeks for the applause of the ignorant by professing
to reduce the whole system of the universe to a fortuitous sequence of
uncaused events.
His collected works, including the series of articles on the properties
of matter, such as “Atom,” “Attraction,” “Capillary Action,”
“Diffusion,” “Ether,” &c., which he contributed to the 9th edition of
this encyclopaedia, were issued in two volumes by the Cambridge
University Press in 1890; and an extended biography, by his former
schoolfellow and lifelong friend Professor Lewis Campbell, was
published in 1882.
(P. G. T.)
MAXWELLTOWN, a burgh of barony and police burgh of
Kirkcudbrightshire, Scotland. Pop. (1901), 5796. It lies on the Nith,
opposite to Dumfries, with which it is connected by three bridges, being
united with it for parliamentary purposes. It has a station on the Glasgow &
South-Western line from Dumfries to Kirkcudbright. Its public buildings
include a court-house, the prison for the south-west of Scotland, and an
observatory and museum, housed in a disused windmill. The chief
manufactures are woollens and hosiery, besides dyeworks and sawmills. It
was a hamlet known as Bridgend up till 1810, in which year it was erected
sincere and unostentatious Christian. Though perfectly free from any trace
of envy or ill-will, he yet showed on fit occasion his contempt for that
pseudo-science which seeks for the applause of the ignorant by professing
to reduce the whole system of the universe to a fortuitous sequence of
uncaused events.
His collected works, including the series of articles on the properties
of matter, such as “Atom,” “Attraction,” “Capillary Action,”
“Diffusion,” “Ether,” &c., which he contributed to the 9th edition of
this encyclopaedia, were issued in two volumes by the Cambridge
University Press in 1890; and an extended biography, by his former
schoolfellow and lifelong friend Professor Lewis Campbell, was
published in 1882.
(P. G. T.)
MAXWELLTOWN, a burgh of barony and police burgh of
Kirkcudbrightshire, Scotland. Pop. (1901), 5796. It lies on the Nith,
opposite to Dumfries, with which it is connected by three bridges, being
united with it for parliamentary purposes. It has a station on the Glasgow &
South-Western line from Dumfries to Kirkcudbright. Its public buildings
include a court-house, the prison for the south-west of Scotland, and an
observatory and museum, housed in a disused windmill. The chief
manufactures are woollens and hosiery, besides dyeworks and sawmills. It
was a hamlet known as Bridgend up till 1810, in which year it was erected
Page 208
into a burgh of barony under its present name. To the north-west lies the
parish of Terregles, said to be a corruption of Tir-eglwys (terra ecclesia,
that is, “Kirk land”). The parish contains the beautiful ruin of Lincluden
Abbey (see Dumfries), and Terregles House, once the seat of William
Maxwell, last earl of Nithsdale. In the parish of Lochrutton, a few miles
south-west of Maxwelltown, there is a good example of a stone circle, the
“Seven Grey Sisters,” and an old peel-tower in the Mains of Hills.
MAY, PHIL (1864-1903), English caricaturist, was born at Wortley,
near Leeds, on the 22nd of April 1864, the son of an engineer. His father
died when the child was nine years old, and at twelve he had begun to earn
his living. Before he was fifteen he had acted as time-keeper at a foundry,
had tried to become a jockey, and had been on the stage at Scarborough and
Leeds. When he was about seventeen he went to London with a sovereign
in his pocket. He suffered extreme want, sleeping out in the parks and
streets, until he obtained employment as designer to a theatrical costumier.
He also drew posters and cartoons, and for about two years worked for the
St Stephen’s Review, until he was advised to go to Australia for his health.
During the three years he spent there he was attached to the Sydney
Bulletin, for which many of his best drawings were made. On his return to
Europe he went to Paris by way of Rome, where he worked hard for some
time before he appeared in 1892 in London to resume his interrupted
connexion with the St Stephen’s Review. His studies of the London
“guttersnipe” and the coster-girl rapidly made him famous. His overflowing
parish of Terregles, said to be a corruption of Tir-eglwys (terra ecclesia,
that is, “Kirk land”). The parish contains the beautiful ruin of Lincluden
Abbey (see Dumfries), and Terregles House, once the seat of William
Maxwell, last earl of Nithsdale. In the parish of Lochrutton, a few miles
south-west of Maxwelltown, there is a good example of a stone circle, the
“Seven Grey Sisters,” and an old peel-tower in the Mains of Hills.
MAY, PHIL (1864-1903), English caricaturist, was born at Wortley,
near Leeds, on the 22nd of April 1864, the son of an engineer. His father
died when the child was nine years old, and at twelve he had begun to earn
his living. Before he was fifteen he had acted as time-keeper at a foundry,
had tried to become a jockey, and had been on the stage at Scarborough and
Leeds. When he was about seventeen he went to London with a sovereign
in his pocket. He suffered extreme want, sleeping out in the parks and
streets, until he obtained employment as designer to a theatrical costumier.
He also drew posters and cartoons, and for about two years worked for the
St Stephen’s Review, until he was advised to go to Australia for his health.
During the three years he spent there he was attached to the Sydney
Bulletin, for which many of his best drawings were made. On his return to
Europe he went to Paris by way of Rome, where he worked hard for some
time before he appeared in 1892 in London to resume his interrupted
connexion with the St Stephen’s Review. His studies of the London
“guttersnipe” and the coster-girl rapidly made him famous. His overflowing
Page 209
sense of fun, his genuine sympathy with his subjects, and his kindly wit
were on a par with his artistic ability. It was often said that the extraordinary
economy of line which was a characteristic feature of his drawings had
been forced upon him by the deficiencies of the printing machines of the
Sydney Bulletin. It was in fact the result of a laborious process which
involved a number of preliminary sketches, and of a carefully considered
system of elimination. His later work included some excellent political
portraits. He became a regular member of the staff of Punch in 1896, and in
his later years his services were retained exclusively for Punch and the
Graphic. He died on the 5th of August 1903.
There was an exhibition of his drawings at the Fine Arts Society in
1895, and another at the Leicester Galleries in 1903. A selection of his
drawings contributed to the periodical press and from Phil May’s
Annual and Phil May’s Sketch Books, with a portrait and biography of
the artist, entitled The Phil May Folio, appeared in 1903.
MAY, THOMAS (1595-1650), English poet and historian, son of
Sir Thomas May of Mayfield, Sussex, was born in 1595. He entered Sidney
Sussex College, Cambridge, in 1609, and took his B.A. degree three years
later. His father having lost his fortune and sold the family estate, Thomas
May, who was hampered by an impediment in his speech, made literature
his profession. In 1620 he produced The Heir, an ingeniously constructed
comedy, and, probably about the same time, The Old Couple, which was
were on a par with his artistic ability. It was often said that the extraordinary
economy of line which was a characteristic feature of his drawings had
been forced upon him by the deficiencies of the printing machines of the
Sydney Bulletin. It was in fact the result of a laborious process which
involved a number of preliminary sketches, and of a carefully considered
system of elimination. His later work included some excellent political
portraits. He became a regular member of the staff of Punch in 1896, and in
his later years his services were retained exclusively for Punch and the
Graphic. He died on the 5th of August 1903.
There was an exhibition of his drawings at the Fine Arts Society in
1895, and another at the Leicester Galleries in 1903. A selection of his
drawings contributed to the periodical press and from Phil May’s
Annual and Phil May’s Sketch Books, with a portrait and biography of
the artist, entitled The Phil May Folio, appeared in 1903.
MAY, THOMAS (1595-1650), English poet and historian, son of
Sir Thomas May of Mayfield, Sussex, was born in 1595. He entered Sidney
Sussex College, Cambridge, in 1609, and took his B.A. degree three years
later. His father having lost his fortune and sold the family estate, Thomas
May, who was hampered by an impediment in his speech, made literature
his profession. In 1620 he produced The Heir, an ingeniously constructed
comedy, and, probably about the same time, The Old Couple, which was
Page 210
not printed until 1658. His other dramatic works are classical tragedies on
the subjects of Antigone, Cleopatra, and Agrippina. F. G. Fleay has
suggested that the more famous anonymous tragedy of Nero (printed 1624,
reprints in A. H. Bullen’s Old English Plays and the Mermaid Series)
should also be assigned to May. But his most important work in the
department of pure literature was his translation (1627) into heroic couplets
of the Pharsalia of Lucan. Its success led May to write a continuation of
Lucan’s narrative down to the death of Caesar. Charles I. became his patron,
and commanded him to write metrical histories of Henry II. and Edward
III., which were completed in 1635. When the earl of Pembroke, then lord
chamberlain, broke his staff across May’s shoulders at a masque, the king
took him under his protection as “my poet,” and Pembroke made him an
apology accompanied with a gift of £50. These marks of the royal favour
seem to have led May to expect the posts of poet-laureate and city
chronologer when they fell vacant on the death of Ben Jonson in 1637, but
he was disappointed, and he forsook the court and attached himself to the
party of the Parliament. In 1646 he is styled one of the “secretaries” of the
Parliament, and in 1647 he published his best known work, The History of
the Long Parliament. In this official apology for the moderate or
Presbyterian party, he professes to give an impartial statement of facts,
unaccompanied by any expression of party or personal opinion. If he
refrained from actual invective, he accomplished his purpose, according to
Guizot, by “omission, palliation and dissimulation.” Accusations of this
kind were foreseen by May, who says in his preface that if he gives more
information about the Parliament men than their opponents it is that he was
more conversant with them and their affairs. In 1650 he followed this with
another work written with a more definite bias, a Breviary of the History of
the Parliament of England, in Latin and English, in which he defended the
position of the Independents. He stopped short of the catastrophe of the
king’s execution, and it seems likely that his subservience to Cromwell was
the subjects of Antigone, Cleopatra, and Agrippina. F. G. Fleay has
suggested that the more famous anonymous tragedy of Nero (printed 1624,
reprints in A. H. Bullen’s Old English Plays and the Mermaid Series)
should also be assigned to May. But his most important work in the
department of pure literature was his translation (1627) into heroic couplets
of the Pharsalia of Lucan. Its success led May to write a continuation of
Lucan’s narrative down to the death of Caesar. Charles I. became his patron,
and commanded him to write metrical histories of Henry II. and Edward
III., which were completed in 1635. When the earl of Pembroke, then lord
chamberlain, broke his staff across May’s shoulders at a masque, the king
took him under his protection as “my poet,” and Pembroke made him an
apology accompanied with a gift of £50. These marks of the royal favour
seem to have led May to expect the posts of poet-laureate and city
chronologer when they fell vacant on the death of Ben Jonson in 1637, but
he was disappointed, and he forsook the court and attached himself to the
party of the Parliament. In 1646 he is styled one of the “secretaries” of the
Parliament, and in 1647 he published his best known work, The History of
the Long Parliament. In this official apology for the moderate or
Presbyterian party, he professes to give an impartial statement of facts,
unaccompanied by any expression of party or personal opinion. If he
refrained from actual invective, he accomplished his purpose, according to
Guizot, by “omission, palliation and dissimulation.” Accusations of this
kind were foreseen by May, who says in his preface that if he gives more
information about the Parliament men than their opponents it is that he was
more conversant with them and their affairs. In 1650 he followed this with
another work written with a more definite bias, a Breviary of the History of
the Parliament of England, in Latin and English, in which he defended the
position of the Independents. He stopped short of the catastrophe of the
king’s execution, and it seems likely that his subservience to Cromwell was
Page 211
not quite voluntary. In February 1650 he was brought to London from
Weymouth under a strong guard for having spread false reports of the
Parliament and of Cromwell. He died on the 13th of November in the same
year, and was buried in Westminster Abbey, but after the Restoration his
remains were exhumed and buried in a pit in the yard of St Margaret’s,
Westminster. May’s change of side made him many bitter enemies, and he
is the object of scathing condemnation from many of his contemporaries.
There is a long notice of May in the Biographia Britannica. See also
W. J. Courthope, Hist. of Eng. Poetry, vol. 3; and Guizot, Études
biographiques sur la révolution d’Angleterre (pp. 403-426, ed. 1851).
MAY, or Mey(e), WILLIAM (d. 1560), English divine, was the
brother of John May, bishop of Carlisle. He was educated at Cambridge,
where he was a fellow of Trinity Hall, and in 1537, president of Queen’s
College. May heartily supported the Reformation, signed the Ten Articles in
1536, and helped in the production of The Institution of a Christian Man.
He had close connexion with the diocese of Ely, being successively
chancellor, vicar-general and prebendary. In 1545 he was made a
prebendary of St Paul’s, and in the following year dean. His favourable
report on the Cambridge colleges saved them from dissolution. He was
dispossessed during the reign of Mary, but restored to the deanery on
Elizabeth’s accession. He died on the day of his election to the
archbishopric of York.
Weymouth under a strong guard for having spread false reports of the
Parliament and of Cromwell. He died on the 13th of November in the same
year, and was buried in Westminster Abbey, but after the Restoration his
remains were exhumed and buried in a pit in the yard of St Margaret’s,
Westminster. May’s change of side made him many bitter enemies, and he
is the object of scathing condemnation from many of his contemporaries.
There is a long notice of May in the Biographia Britannica. See also
W. J. Courthope, Hist. of Eng. Poetry, vol. 3; and Guizot, Études
biographiques sur la révolution d’Angleterre (pp. 403-426, ed. 1851).
MAY, or Mey(e), WILLIAM (d. 1560), English divine, was the
brother of John May, bishop of Carlisle. He was educated at Cambridge,
where he was a fellow of Trinity Hall, and in 1537, president of Queen’s
College. May heartily supported the Reformation, signed the Ten Articles in
1536, and helped in the production of The Institution of a Christian Man.
He had close connexion with the diocese of Ely, being successively
chancellor, vicar-general and prebendary. In 1545 he was made a
prebendary of St Paul’s, and in the following year dean. His favourable
report on the Cambridge colleges saved them from dissolution. He was
dispossessed during the reign of Mary, but restored to the deanery on
Elizabeth’s accession. He died on the day of his election to the
archbishopric of York.
Page 212
MAY, the fifth month of our modern year, the third of the old Roman
calendar. The origin of the name is disputed; the derivation from Maia, the
mother of Mercury, to whom the Romans were accustomed to sacrifice on
the first day of this month, is usually accepted. The ancient Romans used on
May Day to go in procession to the grotto of Egeria. From the 28th of April
to the 2nd of May was kept the festival in honour of Flora, goddess of
flowers. By the Romans the month was regarded as unlucky for marriages,
owing to the celebration on the 9th, 11th and 13th of the Lemuria, the
festival of the unhappy dead. This superstition has survived to the present
day.
In medieval and Tudor England, May Day was a great public holiday. All
classes of the people, young and old alike, were up with the dawn, and went
“a-Maying” in the woods. Branches of trees and flowers were borne back in
triumph to the towns and villages, the centre of the procession being
occupied by those who shouldered the maypole, glorious with ribbons and
wreaths. The maypole was usually of birch, and set up for the day only; but
in London and the larger towns the poles were of durable wood and
permanently erected. They were special eyesores to the Puritans. John
Stubbes in his Anatomy of Abuses (1583) speaks of them as those
“stinckyng idols,” about which the people “leape and daunce, as the heathen
did.” Maypoles were forbidden by the parliament in 1644, but came once
more into favour at the Restoration, the last to be erected in London being
that set up in 1661. This pole, which was of cedar, 134 ft. high, was set up
calendar. The origin of the name is disputed; the derivation from Maia, the
mother of Mercury, to whom the Romans were accustomed to sacrifice on
the first day of this month, is usually accepted. The ancient Romans used on
May Day to go in procession to the grotto of Egeria. From the 28th of April
to the 2nd of May was kept the festival in honour of Flora, goddess of
flowers. By the Romans the month was regarded as unlucky for marriages,
owing to the celebration on the 9th, 11th and 13th of the Lemuria, the
festival of the unhappy dead. This superstition has survived to the present
day.
In medieval and Tudor England, May Day was a great public holiday. All
classes of the people, young and old alike, were up with the dawn, and went
“a-Maying” in the woods. Branches of trees and flowers were borne back in
triumph to the towns and villages, the centre of the procession being
occupied by those who shouldered the maypole, glorious with ribbons and
wreaths. The maypole was usually of birch, and set up for the day only; but
in London and the larger towns the poles were of durable wood and
permanently erected. They were special eyesores to the Puritans. John
Stubbes in his Anatomy of Abuses (1583) speaks of them as those
“stinckyng idols,” about which the people “leape and daunce, as the heathen
did.” Maypoles were forbidden by the parliament in 1644, but came once
more into favour at the Restoration, the last to be erected in London being
that set up in 1661. This pole, which was of cedar, 134 ft. high, was set up
Page 213
by twelve British sailors under the personal supervision of James II., then
duke of York and lord high admiral, in the Strand on or about the site of the
present church of St Mary’s-in-the-Strand. Taken down in 1717, it was
conveyed to Wanstead Park in Essex, where it was fixed by Sir Isaac
Newton as part of the support of a large telescope, presented to the Royal
Society by a French astronomer.
For an account of the May Day survivals in rural England see P. H.
Ditchfield, Old English Customs extant at Present Times (1897).
MAY, ISLE OF, an island belonging to Fifeshire, Scotland, at the
entrance to the Firth of Forth, 5 m. S.E. of Crail and Anstruther. It has a
N.W. to S.E. trend, is more than 1 m. long, and measures at its widest about
1⁄3 m. St Adrian, who had settled here, was martyred by the Danes about the
middle of the 9th century. The ruins of the small chapel dedicated to him,
which was a favourite place of pilgrimage, still exist. The place where the
pilgrims—of whom James IV. was often one—landed is yet known as
Pilgrims’ Haven, and traces may yet be seen of the various wells of St
Andrew, St John, Our Lady, and the Pilgrims, though their waters have
become brackish. In 1499 Sir Andrew Wood of Largo, with the “Yellow
Carvel” and “Mayflower,” captured the English seaman Stephen Bull, and
three ships, after a fierce fight which took place between the island and the
Bass Rock. In 1636 a coal beacon was lighted on the May and maintained
duke of York and lord high admiral, in the Strand on or about the site of the
present church of St Mary’s-in-the-Strand. Taken down in 1717, it was
conveyed to Wanstead Park in Essex, where it was fixed by Sir Isaac
Newton as part of the support of a large telescope, presented to the Royal
Society by a French astronomer.
For an account of the May Day survivals in rural England see P. H.
Ditchfield, Old English Customs extant at Present Times (1897).
MAY, ISLE OF, an island belonging to Fifeshire, Scotland, at the
entrance to the Firth of Forth, 5 m. S.E. of Crail and Anstruther. It has a
N.W. to S.E. trend, is more than 1 m. long, and measures at its widest about
1⁄3 m. St Adrian, who had settled here, was martyred by the Danes about the
middle of the 9th century. The ruins of the small chapel dedicated to him,
which was a favourite place of pilgrimage, still exist. The place where the
pilgrims—of whom James IV. was often one—landed is yet known as
Pilgrims’ Haven, and traces may yet be seen of the various wells of St
Andrew, St John, Our Lady, and the Pilgrims, though their waters have
become brackish. In 1499 Sir Andrew Wood of Largo, with the “Yellow
Carvel” and “Mayflower,” captured the English seaman Stephen Bull, and
three ships, after a fierce fight which took place between the island and the
Bass Rock. In 1636 a coal beacon was lighted on the May and maintained
Page 214
by Alexander Cunningham of Barns. The oil light substituted for it in 1816
was replaced in 1888 by an electric light.
MAYA, an important tribe and stock of American Indians, the
dominant race of Yucatan and other states of Mexico and part of Central
America at the time of the Spanish conquest. They were then divided into
many nations, chief among them being the Maya proper, the Huastecs, the
Tzental, the Pokom, the Mame and the Cakchiquel and Quiché. They were
spread over Yucatan, Vera Cruz, Tabasco, Campeche, and Chiapas in
Mexico, and over the greater part of Guatemala and Salvador. In civilization
the Mayan peoples rivalled the Aztecs. Their traditions give as their place of
origin the extreme north; thence a migration took place, perhaps at the
beginning of the Christian era. They appear to have reached Yucatan as
early as the 5th century. From the evidence of the Quiché chronicles, which
are said to date back to about a.d. 700, Guatemala was shortly afterwards
overrun. Physically the Mayans are a dark-skinned, round-headed, short and
sturdy type. Although they were already decadent when the Spaniards
arrived they made a fierce resistance. They still form the bulk of the
inhabitants of Yucatan. For their culture, ruined cities, &c. see Central
America and Mexico.
was replaced in 1888 by an electric light.
MAYA, an important tribe and stock of American Indians, the
dominant race of Yucatan and other states of Mexico and part of Central
America at the time of the Spanish conquest. They were then divided into
many nations, chief among them being the Maya proper, the Huastecs, the
Tzental, the Pokom, the Mame and the Cakchiquel and Quiché. They were
spread over Yucatan, Vera Cruz, Tabasco, Campeche, and Chiapas in
Mexico, and over the greater part of Guatemala and Salvador. In civilization
the Mayan peoples rivalled the Aztecs. Their traditions give as their place of
origin the extreme north; thence a migration took place, perhaps at the
beginning of the Christian era. They appear to have reached Yucatan as
early as the 5th century. From the evidence of the Quiché chronicles, which
are said to date back to about a.d. 700, Guatemala was shortly afterwards
overrun. Physically the Mayans are a dark-skinned, round-headed, short and
sturdy type. Although they were already decadent when the Spaniards
arrived they made a fierce resistance. They still form the bulk of the
inhabitants of Yucatan. For their culture, ruined cities, &c. see Central
America and Mexico.
Page 215
MAYAGUEZ, the third largest city of Porto Rico, a seaport, and
the seat of government of the department of Mayaguez, on the west coast, at
the mouth of Rio Yaguez, about 72 m. W. by S. of San Juan. Pop. of the city
(1899), 15,187, including 1381 negroes and 4711 of mixed races; (1910),
16,591; of the municipal district, 35,700 (1899), of whom 2687 were
negroes and 9933 were of mixed races. Mayaguez is connected by the
American railroad of Porto Rico with San Juan and Ponce, and it is served
regularly by steamboats from San Juan, Ponce and New York, although its
harbour is not accessible to vessels drawing more than 16 ft. of water. It is
situated at the foot of Las Mesas mountains and commands picturesque
views. The climate is healthy and good water is obtained from the mountain
region. From the shipping district along the water-front a thoroughfare leads
to the main portion of the city, about 1 m. distant. There are four public
squares, in one of which is a statue of Columbus. Prominent among the
public buildings are the City Hall (containing a public library), San Antonio
Hospital, Roman Catholic churches, a Presbyterian church, the court-house
and a theatre. The United States has an agricultural experiment station here,
and the Insular Reform School is 1 m. south of the city. Coffee, sugar-cane
and tropical fruits are grown in the surrounding country; and the business of
the city consists chiefly in their export and the import of flour. Among the
manufactures are sugar, tobacco and chocolate. Mayaguez was founded
about the middle of the 18th century on the site of a hamlet which was first
the seat of government of the department of Mayaguez, on the west coast, at
the mouth of Rio Yaguez, about 72 m. W. by S. of San Juan. Pop. of the city
(1899), 15,187, including 1381 negroes and 4711 of mixed races; (1910),
16,591; of the municipal district, 35,700 (1899), of whom 2687 were
negroes and 9933 were of mixed races. Mayaguez is connected by the
American railroad of Porto Rico with San Juan and Ponce, and it is served
regularly by steamboats from San Juan, Ponce and New York, although its
harbour is not accessible to vessels drawing more than 16 ft. of water. It is
situated at the foot of Las Mesas mountains and commands picturesque
views. The climate is healthy and good water is obtained from the mountain
region. From the shipping district along the water-front a thoroughfare leads
to the main portion of the city, about 1 m. distant. There are four public
squares, in one of which is a statue of Columbus. Prominent among the
public buildings are the City Hall (containing a public library), San Antonio
Hospital, Roman Catholic churches, a Presbyterian church, the court-house
and a theatre. The United States has an agricultural experiment station here,
and the Insular Reform School is 1 m. south of the city. Coffee, sugar-cane
and tropical fruits are grown in the surrounding country; and the business of
the city consists chiefly in their export and the import of flour. Among the
manufactures are sugar, tobacco and chocolate. Mayaguez was founded
about the middle of the 18th century on the site of a hamlet which was first
Page 216
settled about 1680. It was incorporated as a town in 1836, and became a
city in 1873. In 1841 it was nearly all destroyed by fire.
MAYAVARAM, a town of British India, in the Tanjore district of
Madras, on the Cauvery river; junction on the South Indian railway, 174 m.
S.W. of Madras. Pop. (1901), 24,276. It possesses a speciality of fine cotton
and silk cloth, known as Kornad from the suburb in which the weavers live.
During October and November the town is the scene of a great pilgrimage
to the holy waters of the Cauvery.
MAYBOLE, a burgh of barony and police burgh of Ayrshire,
Scotland. Pop. (1901), 5892. It is situated 9 m. S. of Ayr and 501⁄4 m. S.W.
of Glasgow by the Glasgow & South-Western railway. It is an ancient place,
having received a charter from Duncan II. in 1193. In 1516 it was made a
burgh of regality, but for generations it remained under the subjection of the
Kennedys, afterwards earls of Cassillis and marquesses of Ailsa, the most
powerful family in Ayrshire. Of old Maybole was the capital of the district
city in 1873. In 1841 it was nearly all destroyed by fire.
MAYAVARAM, a town of British India, in the Tanjore district of
Madras, on the Cauvery river; junction on the South Indian railway, 174 m.
S.W. of Madras. Pop. (1901), 24,276. It possesses a speciality of fine cotton
and silk cloth, known as Kornad from the suburb in which the weavers live.
During October and November the town is the scene of a great pilgrimage
to the holy waters of the Cauvery.
MAYBOLE, a burgh of barony and police burgh of Ayrshire,
Scotland. Pop. (1901), 5892. It is situated 9 m. S. of Ayr and 501⁄4 m. S.W.
of Glasgow by the Glasgow & South-Western railway. It is an ancient place,
having received a charter from Duncan II. in 1193. In 1516 it was made a
burgh of regality, but for generations it remained under the subjection of the
Kennedys, afterwards earls of Cassillis and marquesses of Ailsa, the most
powerful family in Ayrshire. Of old Maybole was the capital of the district
Page 217
of Carrick, and for long its characteristic feature was the family mansions of
the barons of Carrick. The castle of the earls of Cassillis still remains. The
public buildings include the town-hall, the Ashgrove and the Lumsden
fresh-air fortnightly homes, and the Maybole combination poorhouse. The
leading manufactures are of boots and shoes and agricultural implements.
Two miles to the south-west are the ruins of Crossraguel (Cross of St
Regulus) Abbey, founded about 1240. Kirkoswald, where Burns spent his
seventeenth year, learning land-surveying, lies a little farther west. In the
parish churchyard lie “Tam o’ Shanter” (Douglas Graham) and “Souter
Johnnie” (John Davidson). Four miles to the west of Maybole on the coast
is Culzean Castle, the chief seat of the marquess of Ailsa, dating from 1777;
it stands on a basaltic cliff, beneath which are the Coves of Culzean, once
the retreat of outlaws and a resort of the fairies. Farther south are the ruins
of Turnberry Castle, where Robert Bruce is said to have been born. A few
miles to the north of Culzean are the ruins of Dunure Castle, an ancient
stronghold of the Kennedys.
MAYEN, a town of Germany, in the Prussian Rhine province, on the
northern declivity of the Eifel range, 16 m. W. from Coblenz, on the railway
Andernach-Gerolstein. Pop. (1905), 13,435. It is still partly surrounded by
medieval walls, and the ruins of a castle rise above the town. There are
some small industries, embracing textile manufactures, oil mills and
tanneries, and a trade in wine, while near the town are extensive quarries of
the barons of Carrick. The castle of the earls of Cassillis still remains. The
public buildings include the town-hall, the Ashgrove and the Lumsden
fresh-air fortnightly homes, and the Maybole combination poorhouse. The
leading manufactures are of boots and shoes and agricultural implements.
Two miles to the south-west are the ruins of Crossraguel (Cross of St
Regulus) Abbey, founded about 1240. Kirkoswald, where Burns spent his
seventeenth year, learning land-surveying, lies a little farther west. In the
parish churchyard lie “Tam o’ Shanter” (Douglas Graham) and “Souter
Johnnie” (John Davidson). Four miles to the west of Maybole on the coast
is Culzean Castle, the chief seat of the marquess of Ailsa, dating from 1777;
it stands on a basaltic cliff, beneath which are the Coves of Culzean, once
the retreat of outlaws and a resort of the fairies. Farther south are the ruins
of Turnberry Castle, where Robert Bruce is said to have been born. A few
miles to the north of Culzean are the ruins of Dunure Castle, an ancient
stronghold of the Kennedys.
MAYEN, a town of Germany, in the Prussian Rhine province, on the
northern declivity of the Eifel range, 16 m. W. from Coblenz, on the railway
Andernach-Gerolstein. Pop. (1905), 13,435. It is still partly surrounded by
medieval walls, and the ruins of a castle rise above the town. There are
some small industries, embracing textile manufactures, oil mills and
tanneries, and a trade in wine, while near the town are extensive quarries of
Page 218
basalt. Having been a Roman settlement, Mayen became a town in 1291. In
1689 it was destroyed by the French.
MAYENNE, CHARLES OF LORRAINE, Duke of
(1554-1611), second son of Francis of Lorraine, second duke of Guise, was
born on the 26th of March 1554. He was absent from France at the time of
the massacre of Saint Bartholomew, but took part in the siege of La
Rochelle in the following year, when he was created duke and peer of
France. He went with Henry of Valois, duke of Anjou (afterwards Henry
III.), on his election as king of Poland, but soon returned to France to
become the energetic supporter and lieutenant of his brother, the 3rd duke
of Guise. In 1577 he gained conspicuous successes over the Huguenot
forces in Poitou. As governor of Burgundy he raised his province in the
cause of the League in 1585. The assassination of his brothers at Blois on
the 23rd and 24th of December 1588 left him at the head of the Catholic
party. The Venetian ambassador, Mocenigo, states that Mayenne had
warned Henry III. that there was a plot afoot to seize his person and to send
him by force to Paris. At the time of the murder he was at Lyons, where he
received a letter from the king saying that he had acted on his warning, and
ordering him to retire to his government. Mayenne professed obedience, but
immediately made preparations for marching on Paris. After a vain attempt
to recover the persons of those of his relatives who had been arrested at
Blois he proceeded to recruit troops in his government of Burgundy and in
Champagne. Paris was devoted to the house of Guise and had been roused
1689 it was destroyed by the French.
MAYENNE, CHARLES OF LORRAINE, Duke of
(1554-1611), second son of Francis of Lorraine, second duke of Guise, was
born on the 26th of March 1554. He was absent from France at the time of
the massacre of Saint Bartholomew, but took part in the siege of La
Rochelle in the following year, when he was created duke and peer of
France. He went with Henry of Valois, duke of Anjou (afterwards Henry
III.), on his election as king of Poland, but soon returned to France to
become the energetic supporter and lieutenant of his brother, the 3rd duke
of Guise. In 1577 he gained conspicuous successes over the Huguenot
forces in Poitou. As governor of Burgundy he raised his province in the
cause of the League in 1585. The assassination of his brothers at Blois on
the 23rd and 24th of December 1588 left him at the head of the Catholic
party. The Venetian ambassador, Mocenigo, states that Mayenne had
warned Henry III. that there was a plot afoot to seize his person and to send
him by force to Paris. At the time of the murder he was at Lyons, where he
received a letter from the king saying that he had acted on his warning, and
ordering him to retire to his government. Mayenne professed obedience, but
immediately made preparations for marching on Paris. After a vain attempt
to recover the persons of those of his relatives who had been arrested at
Blois he proceeded to recruit troops in his government of Burgundy and in
Champagne. Paris was devoted to the house of Guise and had been roused
Page 219
to fury by the news of the murder. When Mayenne entered the city in
February 1589 he found it dominated by representatives of the sixteen
quarters of Paris, all fanatics of the League. He formed a council general to
direct the affairs of the city and to maintain relations with the other towns
faithful to the League. To this council each quarter sent four representatives,
and Mayenne added representatives of the various trades and professions of
Paris in order to counterbalance this revolutionary element. He constituted
himself “lieutenant-general of the state and crown of France,” taking his
oath before the parlement of Paris. In April he advanced on Tours. Henry
III. in his extremity sought an alliance with Henry of Navarre, and the allied
forces drove the leaguers back, and had laid siege to Paris, when the murder
of Henry III. by a Dominican fanatic changed the face of affairs and gave
new strength to the Catholic party.
Mayenne was urged to claim the crown for himself, but he was faithful to
the official programme of the League and proclaimed Charles, cardinal of
Bourbon, at that time a prisoner in the hands of Henry IV., as Charles X.
Henry IV. retired to Dieppe, followed by Mayenne, who joined his forces
with those of his cousin Charles, duke of Aumale, and Charles de Cossé,
comte de Brissac, and engaged the royal forces in a succession of fights in
the neighbourhood of Arques (September 1589). He was defeated and out-
marched by Henry IV., who moved on Paris, but retreated before Mayenne’s
forces. In 1590 Mayenne received additions to his army from the Spanish
Netherlands, and took the field again, only to suffer complete defeat at Ivry
(March 14, 1590). He then escaped to Mantes, and in September collected a
fresh army at Meaux, and with the assistance of Alexander Farnese, prince
of Parma, sent by Philip II., raised the siege of Paris, which was about to
surrender to Henry IV. Mayenne feared with reason the designs of Philip II.,
and his difficulties were increased by the death of Charles X., the “king of
the league.” The extreme section of the party, represented by the Sixteen,
urged him to proceed to the election of a Catholic king and to accept the
February 1589 he found it dominated by representatives of the sixteen
quarters of Paris, all fanatics of the League. He formed a council general to
direct the affairs of the city and to maintain relations with the other towns
faithful to the League. To this council each quarter sent four representatives,
and Mayenne added representatives of the various trades and professions of
Paris in order to counterbalance this revolutionary element. He constituted
himself “lieutenant-general of the state and crown of France,” taking his
oath before the parlement of Paris. In April he advanced on Tours. Henry
III. in his extremity sought an alliance with Henry of Navarre, and the allied
forces drove the leaguers back, and had laid siege to Paris, when the murder
of Henry III. by a Dominican fanatic changed the face of affairs and gave
new strength to the Catholic party.
Mayenne was urged to claim the crown for himself, but he was faithful to
the official programme of the League and proclaimed Charles, cardinal of
Bourbon, at that time a prisoner in the hands of Henry IV., as Charles X.
Henry IV. retired to Dieppe, followed by Mayenne, who joined his forces
with those of his cousin Charles, duke of Aumale, and Charles de Cossé,
comte de Brissac, and engaged the royal forces in a succession of fights in
the neighbourhood of Arques (September 1589). He was defeated and out-
marched by Henry IV., who moved on Paris, but retreated before Mayenne’s
forces. In 1590 Mayenne received additions to his army from the Spanish
Netherlands, and took the field again, only to suffer complete defeat at Ivry
(March 14, 1590). He then escaped to Mantes, and in September collected a
fresh army at Meaux, and with the assistance of Alexander Farnese, prince
of Parma, sent by Philip II., raised the siege of Paris, which was about to
surrender to Henry IV. Mayenne feared with reason the designs of Philip II.,
and his difficulties were increased by the death of Charles X., the “king of
the league.” The extreme section of the party, represented by the Sixteen,
urged him to proceed to the election of a Catholic king and to accept the
Page 220
help and the claims of their Spanish allies. But Mayenne, who had not the
popular gifts of his brother, the duke of Guise, had no sympathy with the
demagogues, and himself inclined to the moderate side of his party, which
began to urge reconciliation with Henry IV. He maintained the ancient
forms of the constitution against the revolutionary policy of the Sixteen,
who during his absence from Paris took the law into their own hands and in
November 1591 executed one of the leaders of the more moderate party,
Barnabé Brisson, president of the parlement. He returned to Paris and
executed four of the chief malcontents. The power of the Sixteen
diminished from that time, but with it the strength of the League.1
Mayenne entered into negotiations with Henry IV. while he was still
appearing to consider with Philip II. the succession to the French crown of
the Infanta Elizabeth, granddaughter, through her mother Elizabeth of
Valois, of Henry II. He demanded that Henry IV. should accomplish his
conversion to Catholicism before he was recognized by the leaguers. He
also desired the continuation to himself of the high offices which had
accumulated in his family and the reservation of their provinces to his
relatives among the leaguers. In 1593 he summoned the States General to
Paris and placed before them the claims of the Infanta, but they protested
against foreign intervention. Mayenne signed a truce at La Villette on the
31st of July 1593. The internal dissensions of the league continued to
increase, and the principal chiefs submitted. Mayenne finally made his
peace only in October 1595. Henry IV. allowed him the possession of
Chalon-sur-Saône, of Seurre and Soissons for three years, made him
governor of the Isle of France and paid a large indemnity. Mayenne died at
Soissons on the 3rd of October 1611.
A Histoire de la vie et de la mort du duc de Mayenne appeared at
Lyons in 1618. See also J. B. H. Capefigue, Hist. de la Réforme, de la
popular gifts of his brother, the duke of Guise, had no sympathy with the
demagogues, and himself inclined to the moderate side of his party, which
began to urge reconciliation with Henry IV. He maintained the ancient
forms of the constitution against the revolutionary policy of the Sixteen,
who during his absence from Paris took the law into their own hands and in
November 1591 executed one of the leaders of the more moderate party,
Barnabé Brisson, president of the parlement. He returned to Paris and
executed four of the chief malcontents. The power of the Sixteen
diminished from that time, but with it the strength of the League.1
Mayenne entered into negotiations with Henry IV. while he was still
appearing to consider with Philip II. the succession to the French crown of
the Infanta Elizabeth, granddaughter, through her mother Elizabeth of
Valois, of Henry II. He demanded that Henry IV. should accomplish his
conversion to Catholicism before he was recognized by the leaguers. He
also desired the continuation to himself of the high offices which had
accumulated in his family and the reservation of their provinces to his
relatives among the leaguers. In 1593 he summoned the States General to
Paris and placed before them the claims of the Infanta, but they protested
against foreign intervention. Mayenne signed a truce at La Villette on the
31st of July 1593. The internal dissensions of the league continued to
increase, and the principal chiefs submitted. Mayenne finally made his
peace only in October 1595. Henry IV. allowed him the possession of
Chalon-sur-Saône, of Seurre and Soissons for three years, made him
governor of the Isle of France and paid a large indemnity. Mayenne died at
Soissons on the 3rd of October 1611.
A Histoire de la vie et de la mort du duc de Mayenne appeared at
Lyons in 1618. See also J. B. H. Capefigue, Hist. de la Réforme, de la
Page 221
ligue et du règne de Henri IV. (8 vols., 1834-1835) and the literature
dealing with the house of Guise (q.v.).
1 The estates of the League in 1593 were the occasion of the famous Satire Ménippée,
circulated in MS. in that year, but only printed at Tours in 1594. It was the work of a circle
of men of letters who belonged to the politiques or party of the centre and ridiculed the
League. The authors were Pierre Le Roy, Jean Passerat, Florent Chrestien, Nicolas Rapin
and Pierre Pithou. It opened with “La vertu du catholicon,” in which a Spanish quack (the
cardinal of Plaisance) vaunts the virtues of his drug “catholicon composé,” manufactured in
the Escurial, while a Lorrainer rival (the cardinal of Pellevé) tries to sell a rival cure. A
mock account of the estates, with harangues delivered by Mayenne and the other chiefs of
the League, followed. Mayenne’s discourse is said to have been written by the jurist Pithou.
MAYENNE, a department of north-western France, three-fourths of
which formerly belonged to Lower Maine and the remainder to Anjou,
bounded on the N. by Manche and Orne, E. by Sarthe, S. by Maine-et-Loire
and W. by Ille-et-Vilaine. Area, 2012 sq. m. Pop. (1906), 305,457. Its
ancient geological formations connect it with Brittany. The surface is
agreeably undulating; forests are numerous, and the beauty of the cultivated
portions is enhanced by the hedgerows and lines of trees by which the farms
are divided. The highest point of the department, and indeed of the whole
north-west of France, is the Mont des Avaloirs (1368 ft.). Hydrographically
Mayenne belongs to the basins of the Loire, the Vilaine and the Sélune, the
first mentioned draining by far the larger part of the entire area. The
principal stream is the Mayenne, which passes successively from north to
south through Mayenne, Laval and Château-Gontier; by means of weirs and
dealing with the house of Guise (q.v.).
1 The estates of the League in 1593 were the occasion of the famous Satire Ménippée,
circulated in MS. in that year, but only printed at Tours in 1594. It was the work of a circle
of men of letters who belonged to the politiques or party of the centre and ridiculed the
League. The authors were Pierre Le Roy, Jean Passerat, Florent Chrestien, Nicolas Rapin
and Pierre Pithou. It opened with “La vertu du catholicon,” in which a Spanish quack (the
cardinal of Plaisance) vaunts the virtues of his drug “catholicon composé,” manufactured in
the Escurial, while a Lorrainer rival (the cardinal of Pellevé) tries to sell a rival cure. A
mock account of the estates, with harangues delivered by Mayenne and the other chiefs of
the League, followed. Mayenne’s discourse is said to have been written by the jurist Pithou.
MAYENNE, a department of north-western France, three-fourths of
which formerly belonged to Lower Maine and the remainder to Anjou,
bounded on the N. by Manche and Orne, E. by Sarthe, S. by Maine-et-Loire
and W. by Ille-et-Vilaine. Area, 2012 sq. m. Pop. (1906), 305,457. Its
ancient geological formations connect it with Brittany. The surface is
agreeably undulating; forests are numerous, and the beauty of the cultivated
portions is enhanced by the hedgerows and lines of trees by which the farms
are divided. The highest point of the department, and indeed of the whole
north-west of France, is the Mont des Avaloirs (1368 ft.). Hydrographically
Mayenne belongs to the basins of the Loire, the Vilaine and the Sélune, the
first mentioned draining by far the larger part of the entire area. The
principal stream is the Mayenne, which passes successively from north to
south through Mayenne, Laval and Château-Gontier; by means of weirs and
Page 222
sluices it is navigable below Mayenne, but traffic is inconsiderable. The
chief affluents are the Jouanne on the left, and on the right the Colmont, the
Ernée and the Oudon. A small area in the east of the department drains by
the Erve into the Sarthe; the Vilaine rises in the west, and in the north-west
two small rivers flow into the Sélune. The climate of Mayenne is generally
healthy except in the neighbourhood of the numerous marshes. The
temperature is lower and the moisture of the atmosphere greater than in the
neighbouring departments; the rainfall (about 32 in. annually) is above the
average for France.
Agriculture and stock-raising are prosperous. A large number of
horned cattle are reared, and in no other French department are so
many horses found within the same area; the breed, that of Craon, is
famed for its strength. Craon has also given its name to the most prized
breed of pigs in western France. Mayenne produces excellent butter
and poultry and a large quantity of honey. The cultivation of the vine is
very limited, and the most common beverage is cider. Wheat, oats,
barley and buckwheat, in the order named, are the most important
crops, and a large quantity of flax and hemp is produced. Game is
abundant. The timber grown is chiefly beech, oak, birch, elm and
chestnut. The department produces antimony, auriferous quartz and
coal. Marble, slate and other stone are quarried. There are several
chalybeate springs. The industries include flour-milling, brick and tile
making, brewing, cotton and wool spinning, and the production of
various textile fabrics (especially ticking) for which Laval and
Château-Gontier are the centres, agricultural implement making, wood
and marble sawing, tanning and dyeing. The exports include
agricultural produce, live-stock, stone and textiles; the chief imports
are coal, brandy, wine, furniture and clothing. The department is served
by the Western railway. It forms part of the circumscriptions of the IV.
army corps, the académie (educational division) of Rennes, and the
chief affluents are the Jouanne on the left, and on the right the Colmont, the
Ernée and the Oudon. A small area in the east of the department drains by
the Erve into the Sarthe; the Vilaine rises in the west, and in the north-west
two small rivers flow into the Sélune. The climate of Mayenne is generally
healthy except in the neighbourhood of the numerous marshes. The
temperature is lower and the moisture of the atmosphere greater than in the
neighbouring departments; the rainfall (about 32 in. annually) is above the
average for France.
Agriculture and stock-raising are prosperous. A large number of
horned cattle are reared, and in no other French department are so
many horses found within the same area; the breed, that of Craon, is
famed for its strength. Craon has also given its name to the most prized
breed of pigs in western France. Mayenne produces excellent butter
and poultry and a large quantity of honey. The cultivation of the vine is
very limited, and the most common beverage is cider. Wheat, oats,
barley and buckwheat, in the order named, are the most important
crops, and a large quantity of flax and hemp is produced. Game is
abundant. The timber grown is chiefly beech, oak, birch, elm and
chestnut. The department produces antimony, auriferous quartz and
coal. Marble, slate and other stone are quarried. There are several
chalybeate springs. The industries include flour-milling, brick and tile
making, brewing, cotton and wool spinning, and the production of
various textile fabrics (especially ticking) for which Laval and
Château-Gontier are the centres, agricultural implement making, wood
and marble sawing, tanning and dyeing. The exports include
agricultural produce, live-stock, stone and textiles; the chief imports
are coal, brandy, wine, furniture and clothing. The department is served
by the Western railway. It forms part of the circumscriptions of the IV.
army corps, the académie (educational division) of Rennes, and the
Page 223
court of appeal of Angers. It comprises three arrondissements (Laval,
Château-Gontier and Mayenne), with 27 cantons and 276 communes.
Laval, the capital, is the seat of a bishopric of the province of Tours.
The other principal towns are Château-Gontier and Mayenne, which
are treated under separate headings. The following places are also of
interest: Evron, which has a church of the 12th and 13th centuries;
Jublains, with a Roman fort and other Roman remains; Lassay, with a
fine château of the 14th and 16th centuries; and Ste Suzanne, which has
remains of medieval ramparts and a fortress with a keep of the
Romanesque period.
MAYENNE, a town of north-western France, capital of an
arrondissement in the department of Mayenne, 19 m. N.N.E. of Laval by
rail. Pop., town 7003, commune 10,020. Mayenne is an old feudal town,
irregularly built on hills on both sides of the river Mayenne. Of the old
castle overlooking the river several towers remain, one of which has
retained its conical roof; the vaulted chambers and chapel are ornamented in
the style of the 13th century; the building is now used as a prison. The
church of Notre-Dame, beside which there is a statue of Joan of Arc, dates
partly from the 12th century; the choir was rebuilt in the 19th century. In the
Place de Cheverus is a statue, by David of Angers, to Cardinal Jean de
Cheverus (1768-1836), who was born in Mayenne. Mayenne has a
subprefecture, tribunals of first instance and of commerce, a chamber of arts
and manufactures, and a board of trade-arbitration. There is a school of
Château-Gontier and Mayenne), with 27 cantons and 276 communes.
Laval, the capital, is the seat of a bishopric of the province of Tours.
The other principal towns are Château-Gontier and Mayenne, which
are treated under separate headings. The following places are also of
interest: Evron, which has a church of the 12th and 13th centuries;
Jublains, with a Roman fort and other Roman remains; Lassay, with a
fine château of the 14th and 16th centuries; and Ste Suzanne, which has
remains of medieval ramparts and a fortress with a keep of the
Romanesque period.
MAYENNE, a town of north-western France, capital of an
arrondissement in the department of Mayenne, 19 m. N.N.E. of Laval by
rail. Pop., town 7003, commune 10,020. Mayenne is an old feudal town,
irregularly built on hills on both sides of the river Mayenne. Of the old
castle overlooking the river several towers remain, one of which has
retained its conical roof; the vaulted chambers and chapel are ornamented in
the style of the 13th century; the building is now used as a prison. The
church of Notre-Dame, beside which there is a statue of Joan of Arc, dates
partly from the 12th century; the choir was rebuilt in the 19th century. In the
Place de Cheverus is a statue, by David of Angers, to Cardinal Jean de
Cheverus (1768-1836), who was born in Mayenne. Mayenne has a
subprefecture, tribunals of first instance and of commerce, a chamber of arts
and manufactures, and a board of trade-arbitration. There is a school of
Page 224
agriculture in the vicinity. The chief industry of the place is the manufacture
of tickings, linen, handkerchiefs and calicoes.
Mayenne had its origin in the castle built here by Juhel, baron of
Mayenne, the son of Geoffrey of Maine, in the beginning of the 11th
century. It was taken by the English in 1424, and several times suffered
capture by the opposing parties in the wars of religion and the Vendée. At
the beginning of the 16th century the territory passed to the family of Guise,
and in 1573 was made a duchy in favour of Charles of Mayenne, leader of
the League.
MAYER, JOHANN TOBIAS (1723-1762), German
astronomer, was born at Marbach, in Würtemberg, on the 17th of February
1723, and brought up at Esslingen in poor circumstances. A self-taught
mathematician, he had already published two original geometrical works
when, in 1746, he entered J. B. Homann’s cartographic establishment at
Nuremberg. Here he introduced many improvements in map-making, and
gained a scientific reputation which led (in 1751) to his election to the chair
of economy and mathematics in the university of Göttingen. In 1754 he
became superintendent of the observatory, where he laboured with great
zeal and success until his death, on the 20th of February 1762. His first
important astronomical work was a careful investigation of the libration of
the moon (Kosmographische Nachrichten, Nuremberg, 1750), and his chart
of the full moon (published in 1775) was unsurpassed for half a century.
of tickings, linen, handkerchiefs and calicoes.
Mayenne had its origin in the castle built here by Juhel, baron of
Mayenne, the son of Geoffrey of Maine, in the beginning of the 11th
century. It was taken by the English in 1424, and several times suffered
capture by the opposing parties in the wars of religion and the Vendée. At
the beginning of the 16th century the territory passed to the family of Guise,
and in 1573 was made a duchy in favour of Charles of Mayenne, leader of
the League.
MAYER, JOHANN TOBIAS (1723-1762), German
astronomer, was born at Marbach, in Würtemberg, on the 17th of February
1723, and brought up at Esslingen in poor circumstances. A self-taught
mathematician, he had already published two original geometrical works
when, in 1746, he entered J. B. Homann’s cartographic establishment at
Nuremberg. Here he introduced many improvements in map-making, and
gained a scientific reputation which led (in 1751) to his election to the chair
of economy and mathematics in the university of Göttingen. In 1754 he
became superintendent of the observatory, where he laboured with great
zeal and success until his death, on the 20th of February 1762. His first
important astronomical work was a careful investigation of the libration of
the moon (Kosmographische Nachrichten, Nuremberg, 1750), and his chart
of the full moon (published in 1775) was unsurpassed for half a century.
Page 225
But his fame rests chiefly on his lunar tables, communicated in 1752, with
new solar tables, to the Royal Society of Göttingen, and published in their
Transactions (vol. ii.). In 1755 he submitted to the English government an
amended body of MS. tables, which James Bradley compared with the
Greenwich observations, and found to be sufficiently accurate to determine
the moon’s place to 75″, and consequently the longitude at sea to about half
a degree. An improved set was afterwards published in London (1770), as
also the theory (Theoria lunae juxta systema Newtonianum, 1767) upon
which the tables are based. His widow, by whom they were sent to England,
received in consideration from the British government a grant of £3000.
Appended to the London edition of the solar and lunar tables are two short
tracts—the one on determining longitude by lunar distances, together with a
description of the repeating circle (invented by Mayer in 1752), the other on
a formula for atmospheric refraction, which applies a remarkably accurate
correction for temperature.
Mayer left behind him a considerable quantity of manuscript, part of
which was collected by G. C. Lichtenberg and published in one volume
(Opera inedita, Göttingen, 1775). It contains an easy and accurate method
for calculating eclipses; an essay on colour, in which three primary colours
are recognized; a catalogue of 998 zodiacal stars; and a memoir, the earliest
of any real value, on the proper motion of eighty stars, originally
communicated to the Göttingen Royal Society in 1760. The manuscript
residue includes papers on atmospheric refraction (dated 1755), on the
motion of Mars as affected by the perturbations of Jupiter and the Earth
(1756), and on terrestrial magnetism (1760 and 1762). In these last Mayer
sought to explain the magnetic action of the earth by a modification of
Euler’s hypothesis, and made the first really definite attempt to establish a
mathematical theory of magnetic action (C. Hansteen, Magnetismus der
Erde, i. 283). E. Klinkerfuss published in 1881 photo-lithographic
reproductions of Mayer’s local charts and general map of the moon; and his
new solar tables, to the Royal Society of Göttingen, and published in their
Transactions (vol. ii.). In 1755 he submitted to the English government an
amended body of MS. tables, which James Bradley compared with the
Greenwich observations, and found to be sufficiently accurate to determine
the moon’s place to 75″, and consequently the longitude at sea to about half
a degree. An improved set was afterwards published in London (1770), as
also the theory (Theoria lunae juxta systema Newtonianum, 1767) upon
which the tables are based. His widow, by whom they were sent to England,
received in consideration from the British government a grant of £3000.
Appended to the London edition of the solar and lunar tables are two short
tracts—the one on determining longitude by lunar distances, together with a
description of the repeating circle (invented by Mayer in 1752), the other on
a formula for atmospheric refraction, which applies a remarkably accurate
correction for temperature.
Mayer left behind him a considerable quantity of manuscript, part of
which was collected by G. C. Lichtenberg and published in one volume
(Opera inedita, Göttingen, 1775). It contains an easy and accurate method
for calculating eclipses; an essay on colour, in which three primary colours
are recognized; a catalogue of 998 zodiacal stars; and a memoir, the earliest
of any real value, on the proper motion of eighty stars, originally
communicated to the Göttingen Royal Society in 1760. The manuscript
residue includes papers on atmospheric refraction (dated 1755), on the
motion of Mars as affected by the perturbations of Jupiter and the Earth
(1756), and on terrestrial magnetism (1760 and 1762). In these last Mayer
sought to explain the magnetic action of the earth by a modification of
Euler’s hypothesis, and made the first really definite attempt to establish a
mathematical theory of magnetic action (C. Hansteen, Magnetismus der
Erde, i. 283). E. Klinkerfuss published in 1881 photo-lithographic
reproductions of Mayer’s local charts and general map of the moon; and his
Page 226
star-catalogue was re-edited by F. Baily in 1830 (Memoirs Roy. Astr. Soc. iv.
391) and by G. F. J. A. Auvers in 1894.
Authorities.—A. G. Kästner, Elogium Tobiae Mayeri (Göttingen,
1762); Connaissance des temps, 1767, p. 187 (J. Lalande); Monatliche
Correspondenz viii. 257, ix. 45, 415, 487, xi. 462; Allg. Geographische
Ephemeriden iii. 116, 1799 (portrait); Berliner Astr. Jahrbuch, Suppl.
Bd. iii. 209, 1797 (A. G. Kästner); J. B. J. Delambre, Hist. de l’Astr. au
XVIIIe siècle, p. 429; R. Grant, Hist. of Phys. Astr. pp. 46, 488, 555; A.
Berry, Short Hist. of Astr. p. 282; J. S. Pütter, Geschichte von der
Universität zu Göttingen, i. 68; J. Gehler, Physik. Wörterbuch neu
bearbeitet, vi. 746, 1039; Allg. Deutsche Biographie (S. Günther).
(A. M. C.)
MAYER, JULIUS ROBERT (1814-1878), German physicist,
was born at Heilbronn on the 25th of November 1814, studied medicine at
Tübingen, Munich and Paris, and after a journey to Java in 1840 as surgeon
of a Dutch vessel obtained a medical post in his native town. He claims
recognition as an independent a priori propounder of the “First Law of
Thermodynamics,” but more especially as having early and ably applied
that law to the explanation of many remarkable phenomena, both cosmical
and terrestrial. His first little paper on the subject, “Bemerkungen über die
Kräfte der unbelebten Natur,” appeared in 1842 in Liebig’s Annalen, five
years after the republication, in the same journal, of an extract from K. F.
391) and by G. F. J. A. Auvers in 1894.
Authorities.—A. G. Kästner, Elogium Tobiae Mayeri (Göttingen,
1762); Connaissance des temps, 1767, p. 187 (J. Lalande); Monatliche
Correspondenz viii. 257, ix. 45, 415, 487, xi. 462; Allg. Geographische
Ephemeriden iii. 116, 1799 (portrait); Berliner Astr. Jahrbuch, Suppl.
Bd. iii. 209, 1797 (A. G. Kästner); J. B. J. Delambre, Hist. de l’Astr. au
XVIIIe siècle, p. 429; R. Grant, Hist. of Phys. Astr. pp. 46, 488, 555; A.
Berry, Short Hist. of Astr. p. 282; J. S. Pütter, Geschichte von der
Universität zu Göttingen, i. 68; J. Gehler, Physik. Wörterbuch neu
bearbeitet, vi. 746, 1039; Allg. Deutsche Biographie (S. Günther).
(A. M. C.)
MAYER, JULIUS ROBERT (1814-1878), German physicist,
was born at Heilbronn on the 25th of November 1814, studied medicine at
Tübingen, Munich and Paris, and after a journey to Java in 1840 as surgeon
of a Dutch vessel obtained a medical post in his native town. He claims
recognition as an independent a priori propounder of the “First Law of
Thermodynamics,” but more especially as having early and ably applied
that law to the explanation of many remarkable phenomena, both cosmical
and terrestrial. His first little paper on the subject, “Bemerkungen über die
Kräfte der unbelebten Natur,” appeared in 1842 in Liebig’s Annalen, five
years after the republication, in the same journal, of an extract from K. F.
Page 227
Mohr’s paper on the nature of heat, and three years later he published Die
organische Bewegung in ihren Zusammenhange mit dem Stoffwechsel.
It has been repeatedly claimed for Mayer that he calculated the value
of the dynamical equivalent of heat, indirectly, no doubt, but in a
manner altogether free from error, and with a result according almost
exactly with that obtained by J. P. Joule after years of patient labour in
direct experimenting. This claim on Mayer’s behalf was first shown to
be baseless by W. Thomson (Lord Kelvin) and P. G. Tait in an article
on “Energy,” published in Good Words in 1862, which gave rise to a
long but lively discussion. A calm and judicial annihilation of the claim
is to be found in a brief article by Sir G. G. Stokes, Proc. Roy. Soc.,
1871, p. 54. See also Maxwell’s Theory of Heat, chap. xiii. Mayer
entirely ignored the grand fundamental principle laid down by Sadi
Carnot—that nothing can be concluded as to the relation between heat
and work from an experiment in which the working substance is left at
the end of an operation in a different physical state from that in which
it was at the commencement. Mayer has also been styled the discoverer
of the fact that heat consists in (the energy of) motion, a matter settled
at the very end of the 18th century by Count Rumford and Sir H. Davy;
but in the teeth of this statement we have Mayer’s own words, “We
might much rather assume the contrary—that in order to become heat
motion must cease to be motion.”
Mayer’s real merit consists in the fact that, having for himself made
out, on inadequate and even questionable grounds, the conservation of
energy, and having obtained (though by inaccurate reasoning) a
numerical result correct so far as his data permitted, he applied the
principle with great power and insight to the explanation of numerous
physical phenomena. His papers, which were republished in a single
volume with the title Die Mechanik der Wärme (3rd ed., 1893), are of
organische Bewegung in ihren Zusammenhange mit dem Stoffwechsel.
It has been repeatedly claimed for Mayer that he calculated the value
of the dynamical equivalent of heat, indirectly, no doubt, but in a
manner altogether free from error, and with a result according almost
exactly with that obtained by J. P. Joule after years of patient labour in
direct experimenting. This claim on Mayer’s behalf was first shown to
be baseless by W. Thomson (Lord Kelvin) and P. G. Tait in an article
on “Energy,” published in Good Words in 1862, which gave rise to a
long but lively discussion. A calm and judicial annihilation of the claim
is to be found in a brief article by Sir G. G. Stokes, Proc. Roy. Soc.,
1871, p. 54. See also Maxwell’s Theory of Heat, chap. xiii. Mayer
entirely ignored the grand fundamental principle laid down by Sadi
Carnot—that nothing can be concluded as to the relation between heat
and work from an experiment in which the working substance is left at
the end of an operation in a different physical state from that in which
it was at the commencement. Mayer has also been styled the discoverer
of the fact that heat consists in (the energy of) motion, a matter settled
at the very end of the 18th century by Count Rumford and Sir H. Davy;
but in the teeth of this statement we have Mayer’s own words, “We
might much rather assume the contrary—that in order to become heat
motion must cease to be motion.”
Mayer’s real merit consists in the fact that, having for himself made
out, on inadequate and even questionable grounds, the conservation of
energy, and having obtained (though by inaccurate reasoning) a
numerical result correct so far as his data permitted, he applied the
principle with great power and insight to the explanation of numerous
physical phenomena. His papers, which were republished in a single
volume with the title Die Mechanik der Wärme (3rd ed., 1893), are of
Page 228
unequal merit. But some, especially those on Celestial Dynamics and
Organic Motion, are admirable examples of what really valuable work
may be effected by a man of high intellectual powers, in spite of
imperfect information and defective logic.
Different, and it would appear exaggerated, estimates of Mayer are
given in John Tyndall’s papers in the Phil. Mag., 1863-1864 (whose
avowed object was “to raise a noble and a suffering man to the position
which his labours entitled him to occupy”), and in E. Dühring’s Robert
Mayer, der Galilei des neunzehnten Jahrhunderts, Chemnitz, 1880.
Some of the simpler facts of the case are summarized by Tait in the
Phil. Mag., 1864, ii. 289.
MAYFLOWER, the vessel which carried from Southampton,
England, to Plymouth, Massachusetts, the Pilgrims who established the first
permanent colony in New England. It was of about 180 tons burden, and in
company with the “Speedwell” sailed from Southampton on the 5th of
August 1620, the two having on board 120 Pilgrims. After two trials the
“Speedwell” was pronounced unseaworthy, and the “Mayflower” sailed
alone from Plymouth, England, on the 6th of September with the 100 (or
102) passengers, some 41 of whom on the 11th of November (O.S.) signed
the famous “Mayflower Compact” in Provincetown Harbor, and a small
party of whom, including William Bradford, sent to choose a place for
settlement, landed at what is now Plymouth, Massachusetts, on the 11th of
December (21st N.S.), an event which is celebrated, as Forefathers’ Day, on
Organic Motion, are admirable examples of what really valuable work
may be effected by a man of high intellectual powers, in spite of
imperfect information and defective logic.
Different, and it would appear exaggerated, estimates of Mayer are
given in John Tyndall’s papers in the Phil. Mag., 1863-1864 (whose
avowed object was “to raise a noble and a suffering man to the position
which his labours entitled him to occupy”), and in E. Dühring’s Robert
Mayer, der Galilei des neunzehnten Jahrhunderts, Chemnitz, 1880.
Some of the simpler facts of the case are summarized by Tait in the
Phil. Mag., 1864, ii. 289.
MAYFLOWER, the vessel which carried from Southampton,
England, to Plymouth, Massachusetts, the Pilgrims who established the first
permanent colony in New England. It was of about 180 tons burden, and in
company with the “Speedwell” sailed from Southampton on the 5th of
August 1620, the two having on board 120 Pilgrims. After two trials the
“Speedwell” was pronounced unseaworthy, and the “Mayflower” sailed
alone from Plymouth, England, on the 6th of September with the 100 (or
102) passengers, some 41 of whom on the 11th of November (O.S.) signed
the famous “Mayflower Compact” in Provincetown Harbor, and a small
party of whom, including William Bradford, sent to choose a place for
settlement, landed at what is now Plymouth, Massachusetts, on the 11th of
December (21st N.S.), an event which is celebrated, as Forefathers’ Day, on
Page 229
the 22nd of December. A “General Society of Mayflower Descendants” was
organized in 1894 by lineal descendants of passengers of the “Mayflower”
to “preserve their memory, their records, their history, and all facts relating
to them, their ancestors and their posterity.” Every lineal descendant, over
eighteen years of age, of any passenger of the “Mayflower” is eligible to
membership. Branch societies have since been organized in several of the
states and in the District of Columbia, and a triennial congress is held in
Plymouth.
See Azel Ames, The May-Flower and Her Log (Boston, 1901);
Blanche McManus, The Voyage of the Mayflower (New York, 1897);
The General Society of Mayflower: Meetings, Officers and Members,
arranged in State Societies, Ancestors and their Descendants (New
York, 1901). Also the articles Plymouth, Mass.; Massachusetts, §
History; Pilgrim; and Provincetown, Mass.
MAY-FLY. The Mayflies belong to the Ephemeridae, a remarkable
family of winged insects, included by Linnaeus in his order Neuroptera,
which derive their scientific name from ἐφήμερος, in allusion to their very
short lives. In some species it is possible that they have scarcely more than
one day’s existence, but others are far longer lived, though the extreme limit
is probably rarely more than a week. The family has very sharply defined
characters, which separate its members at once from all other neuropterous
(or pseudo-neuropterous) groups.
organized in 1894 by lineal descendants of passengers of the “Mayflower”
to “preserve their memory, their records, their history, and all facts relating
to them, their ancestors and their posterity.” Every lineal descendant, over
eighteen years of age, of any passenger of the “Mayflower” is eligible to
membership. Branch societies have since been organized in several of the
states and in the District of Columbia, and a triennial congress is held in
Plymouth.
See Azel Ames, The May-Flower and Her Log (Boston, 1901);
Blanche McManus, The Voyage of the Mayflower (New York, 1897);
The General Society of Mayflower: Meetings, Officers and Members,
arranged in State Societies, Ancestors and their Descendants (New
York, 1901). Also the articles Plymouth, Mass.; Massachusetts, §
History; Pilgrim; and Provincetown, Mass.
MAY-FLY. The Mayflies belong to the Ephemeridae, a remarkable
family of winged insects, included by Linnaeus in his order Neuroptera,
which derive their scientific name from ἐφήμερος, in allusion to their very
short lives. In some species it is possible that they have scarcely more than
one day’s existence, but others are far longer lived, though the extreme limit
is probably rarely more than a week. The family has very sharply defined
characters, which separate its members at once from all other neuropterous
(or pseudo-neuropterous) groups.
Page 230
These insects are universally aquatic in their preparatory states. The eggs
are dropped into the water by the female in large masses, resembling, in
some species, bunches of grapes in miniature. Probably several months
elapse before the young larvae are excluded. The sub-aquatic condition lasts
a considerable time: in Cloeon, a genus of small and delicate species, Sir J.
Lubbock (Lord Avebury) proved it to extend over more than six months; but
in larger and more robust genera (e.g. Palingenia) there appears reason to
believe that the greater part of three years is occupied in preparatory
conditions.
The larva is elongate and campodeiform. The head is rather large,
and is furnished at first with five simple eyes of nearly equal size; but
as it increases in size the homologues of the facetted eyes of the imago
become larger, whereas those equivalent to the ocelli remain small. The
antennae are long and thread-like, composed at first of few joints, but
the number of these latter apparently increases at each moult. The
mouth parts are well developed, consisting of an upper lip, powerful
mandibles, maxillae with three-jointed palpi, and a deeply quadrifid
labium or lower lip with three-jointed labial palpi. Distinct and
conspicuous maxillulae are associated with the tongue or hypopharynx.
There are three distinct and large thoracic segments, whereof the
prothorax is narrower than the others; the legs are much shorter and
stouter than in the winged insect, with monomerous tarsi terminated by
a single claw. The abdomen consists of ten segments, the tenth
furnished with long and slender multi-articulate tails, which appear to
be only two in number at first, but an intermediate one gradually
develops itself (though this latter is often lost in the winged insect).
Respiration is effected by means of external gills placed along both
sides of the dorsum of the abdomen and hinder segments of the thorax.
These vary in form: in some species they are entire plates, in others
they are cut up into numerous divisions, in all cases traversed by
are dropped into the water by the female in large masses, resembling, in
some species, bunches of grapes in miniature. Probably several months
elapse before the young larvae are excluded. The sub-aquatic condition lasts
a considerable time: in Cloeon, a genus of small and delicate species, Sir J.
Lubbock (Lord Avebury) proved it to extend over more than six months; but
in larger and more robust genera (e.g. Palingenia) there appears reason to
believe that the greater part of three years is occupied in preparatory
conditions.
The larva is elongate and campodeiform. The head is rather large,
and is furnished at first with five simple eyes of nearly equal size; but
as it increases in size the homologues of the facetted eyes of the imago
become larger, whereas those equivalent to the ocelli remain small. The
antennae are long and thread-like, composed at first of few joints, but
the number of these latter apparently increases at each moult. The
mouth parts are well developed, consisting of an upper lip, powerful
mandibles, maxillae with three-jointed palpi, and a deeply quadrifid
labium or lower lip with three-jointed labial palpi. Distinct and
conspicuous maxillulae are associated with the tongue or hypopharynx.
There are three distinct and large thoracic segments, whereof the
prothorax is narrower than the others; the legs are much shorter and
stouter than in the winged insect, with monomerous tarsi terminated by
a single claw. The abdomen consists of ten segments, the tenth
furnished with long and slender multi-articulate tails, which appear to
be only two in number at first, but an intermediate one gradually
develops itself (though this latter is often lost in the winged insect).
Respiration is effected by means of external gills placed along both
sides of the dorsum of the abdomen and hinder segments of the thorax.
These vary in form: in some species they are entire plates, in others
they are cut up into numerous divisions, in all cases traversed by
Page 231
numerous tracheal ramifications. According to the researches of
Lubbock and of E. Joly, the very young larvae have no breathing
organs, and respiration is effected through the skin. Lubbock traced at
least twenty moults in Cloeon; at about the tenth rudiments of the
wing-cases began to appear. These gradually become larger, and when
so the creature may be said to have entered its “nymph” stage; but there
is no condition analogous to the pupa-stage of insects with complete
metamorphoses.
There may be said to be three or four different modes of life in these
larvae: some are fossorial, and form tubes in the mud or clay in which
they live; others are found on or beneath stones; while others again
swim and crawl freely among water plants. It is probable that some are
carnivorous, either attacking other larvae or subsisting on more minute
forms of animal life; but others perhaps feed more exclusively on
vegetable matters of a low type, such as diatoms.
The most aberrant type of larva is that of the genus Prosopistoma,
which was originally described as an entomostracous crustacean on
account of the presence of a large carapace overlapping the greater part
of the body. The dorsal skeletal elements of the thorax and of the
anterior six abdominal segments unite with the wing-cases to form a
large respiratory chamber, containing five pairs of tracheal gills, with
lateral slits for the inflow and a posterior orifice for the outflow of
water. Species of this genus occur in Europe, Africa and Madagascar.
When the aquatic insect has reached its full growth it emerges from the
water or seeks its surface; the thorax splits down the back and the winged
form appears. But this is not yet perfect, although it has all the form of a
perfect insect and is capable of flight; it is what is variously termed a
“pseud-imago,” “sub-imago” or “pro-imago.” Contrary to the habits of all
other insects, there yet remains a pellicle that has to be shed, covering every
Lubbock and of E. Joly, the very young larvae have no breathing
organs, and respiration is effected through the skin. Lubbock traced at
least twenty moults in Cloeon; at about the tenth rudiments of the
wing-cases began to appear. These gradually become larger, and when
so the creature may be said to have entered its “nymph” stage; but there
is no condition analogous to the pupa-stage of insects with complete
metamorphoses.
There may be said to be three or four different modes of life in these
larvae: some are fossorial, and form tubes in the mud or clay in which
they live; others are found on or beneath stones; while others again
swim and crawl freely among water plants. It is probable that some are
carnivorous, either attacking other larvae or subsisting on more minute
forms of animal life; but others perhaps feed more exclusively on
vegetable matters of a low type, such as diatoms.
The most aberrant type of larva is that of the genus Prosopistoma,
which was originally described as an entomostracous crustacean on
account of the presence of a large carapace overlapping the greater part
of the body. The dorsal skeletal elements of the thorax and of the
anterior six abdominal segments unite with the wing-cases to form a
large respiratory chamber, containing five pairs of tracheal gills, with
lateral slits for the inflow and a posterior orifice for the outflow of
water. Species of this genus occur in Europe, Africa and Madagascar.
When the aquatic insect has reached its full growth it emerges from the
water or seeks its surface; the thorax splits down the back and the winged
form appears. But this is not yet perfect, although it has all the form of a
perfect insect and is capable of flight; it is what is variously termed a
“pseud-imago,” “sub-imago” or “pro-imago.” Contrary to the habits of all
other insects, there yet remains a pellicle that has to be shed, covering every
Page 232
part of the body. This final moult is effected soon after the insect’s
appearance in the winged form; the creature seeks a temporary resting-
place, the pellicle splits down the back, and the now perfect insect comes
forth, often differing very greatly in colours and markings from the
condition in which it was only a few moments before. If the observer takes
up a suitable position near water, his coat is often seen to be covered with
the cast sub-imaginal skins of these insects, which had chosen him as a
convenient object upon which to undergo their final change. In some few
genera of very low type it appears probable that, at any rate in the female,
this final change is never effected and that the creature dies a sub-imago.
The winged insect differs considerably in form from its sub-aquatic
condition. The head is smaller, often occupied almost entirely above in
the male by the very large eyes, which in some species are curiously
double in that sex, one portion being pillared, and forming what is
termed a “turban,” the mouth parts are aborted, for the creature is now
incapable of taking nutriment either solid or fluid; the antennae are
mere short bristles, consisting of two rather large basal joints and a
multi-articulate thread. The prothorax is much narrowed, whereas the
other segments (especially the mesothorax) are greatly enlarged; the
legs long and slender, the anterior pair often very much longer in the
male than in the female; the tarsi four- or five-jointed; but in some
genera (e.g. Oligoneuria and allies) the legs are aborted, and the
creatures are driven helplessly about by the wind. The wings are
carried erect: the anterior pair large, with numerous longitudinal
nervures, and usually abundant transverse reticulation; the posterior
pair very much smaller, often lanceolate, and frequently wanting
absolutely. The abdomen consists of ten segments; at the end are either
two or three long multi-articulate tails; in the male the ninth joint bears
forcipated appendages; in the female the oviducts terminate at the
junction of the seventh and eighth ventral segments. The independent
appearance in the winged form; the creature seeks a temporary resting-
place, the pellicle splits down the back, and the now perfect insect comes
forth, often differing very greatly in colours and markings from the
condition in which it was only a few moments before. If the observer takes
up a suitable position near water, his coat is often seen to be covered with
the cast sub-imaginal skins of these insects, which had chosen him as a
convenient object upon which to undergo their final change. In some few
genera of very low type it appears probable that, at any rate in the female,
this final change is never effected and that the creature dies a sub-imago.
The winged insect differs considerably in form from its sub-aquatic
condition. The head is smaller, often occupied almost entirely above in
the male by the very large eyes, which in some species are curiously
double in that sex, one portion being pillared, and forming what is
termed a “turban,” the mouth parts are aborted, for the creature is now
incapable of taking nutriment either solid or fluid; the antennae are
mere short bristles, consisting of two rather large basal joints and a
multi-articulate thread. The prothorax is much narrowed, whereas the
other segments (especially the mesothorax) are greatly enlarged; the
legs long and slender, the anterior pair often very much longer in the
male than in the female; the tarsi four- or five-jointed; but in some
genera (e.g. Oligoneuria and allies) the legs are aborted, and the
creatures are driven helplessly about by the wind. The wings are
carried erect: the anterior pair large, with numerous longitudinal
nervures, and usually abundant transverse reticulation; the posterior
pair very much smaller, often lanceolate, and frequently wanting
absolutely. The abdomen consists of ten segments; at the end are either
two or three long multi-articulate tails; in the male the ninth joint bears
forcipated appendages; in the female the oviducts terminate at the
junction of the seventh and eighth ventral segments. The independent
Page 233
opening of the genital ducts and the absence of an ectodermal vagina
and ejaculatory duct are remarkable archaic features of these insects, as
has been pointed out by J. A. Palmén. The sexual act takes place in the
air, and is of very short duration, but is apparently repeated several
times, at any rate in some cases.
Ephemeridae are found all over the world, even up to high northern
latitudes. F. J. Pictet, A. E. Eaton and others have given us valuable works
or monographs on the family; but the subject still remains little understood,
partly owing to the great difficulty of preserving such delicate insects; and it
appears probable they can only be satisfactorily investigated as moist
preparations. The number of described species is less than 200, spread over
many genera.
From the earliest times attention has been drawn to the enormous
abundance of species of the family in certain localities. Johann Anton
Scopoli, writing in the 18th century, speaks of them as so abundant in one
place in Carniola that in June twenty cartloads were carried away for
manure! Polymitarcys virgo, which, though not found in England, occurs in
many parts of Europe (and is common at Paris), emerges from the water
soon after sunset, and continues for several hours in such myriads as to
resemble snow showers, putting out lights, and causing inconvenience to
man, and annoyance to horses by entering their nostrils. In other parts of the
world they have been recorded in multitudes that obscured passers-by on
the other side of the street. And similar records might be multiplied almost
to any extent. In Britain, although they are often very abundant, we have
scarcely anything analogous.
Fish, as is well known, devour them greedily, and enjoy a veritable feast
during the short period in which any particular species appears. By anglers
the common English species of Ephemera (vulgata and danica, but more
especially the latter, which is more abundant) is known as the “may-fly,”
and ejaculatory duct are remarkable archaic features of these insects, as
has been pointed out by J. A. Palmén. The sexual act takes place in the
air, and is of very short duration, but is apparently repeated several
times, at any rate in some cases.
Ephemeridae are found all over the world, even up to high northern
latitudes. F. J. Pictet, A. E. Eaton and others have given us valuable works
or monographs on the family; but the subject still remains little understood,
partly owing to the great difficulty of preserving such delicate insects; and it
appears probable they can only be satisfactorily investigated as moist
preparations. The number of described species is less than 200, spread over
many genera.
From the earliest times attention has been drawn to the enormous
abundance of species of the family in certain localities. Johann Anton
Scopoli, writing in the 18th century, speaks of them as so abundant in one
place in Carniola that in June twenty cartloads were carried away for
manure! Polymitarcys virgo, which, though not found in England, occurs in
many parts of Europe (and is common at Paris), emerges from the water
soon after sunset, and continues for several hours in such myriads as to
resemble snow showers, putting out lights, and causing inconvenience to
man, and annoyance to horses by entering their nostrils. In other parts of the
world they have been recorded in multitudes that obscured passers-by on
the other side of the street. And similar records might be multiplied almost
to any extent. In Britain, although they are often very abundant, we have
scarcely anything analogous.
Fish, as is well known, devour them greedily, and enjoy a veritable feast
during the short period in which any particular species appears. By anglers
the common English species of Ephemera (vulgata and danica, but more
especially the latter, which is more abundant) is known as the “may-fly,”
Page 234
but the terms “green drake” and “bastard drake” are applied to conditions of
the same species. Useful information on this point will be found in
Ronalds’s Fly-Fisher’s Entomology, edited by Westwood.
Ephemeridae belong to a very ancient type of insects, and fossil imprints
of allied forms occur even in the Devonian and Carboniferous formations.
There is much to be said in favour of the view entertained by some
entomologists that the structural and developmental characteristics of may-
flies are sufficiently peculiar to warrant the formation for them of a special
order of insects, for which the names Agnatha, Plectoptera and
Ephemeroptera have been proposed. (See Hexapoda, Neuroptera.)
Bibliography.—Of especial value to students of these insects are A.
E. Eaton’s monograph (Trans. Linn. Soc. (2) iii. 1883-1885) and A.
Vayssière’s “Recherches sur l’organisation des larves” (Ann. Sci. Nat.
Zool. (6) xiii. 1882 (7) ix. 1890). J. A. Palmén’s memoirs Zur
Morphologie des Tracheensystems (Leipzig, 1877) and Über paarige
Ausführungsgänge der Geschlechtsorgane bei Insekten (Helsingfors,
1884), contain important observations on may-flies. See also L. C.
Miall, Nat. Hist. Aquatic Insects (London, 1895); J. G. Needham and
others (New York State Museum, Bull. 86, 1905).
(R. M’L.; G. H. C.)
MAYHEM (for derivation see Maiming), an old Anglo-French term
of the law signifying an assault whereby the injured person is deprived of a
the same species. Useful information on this point will be found in
Ronalds’s Fly-Fisher’s Entomology, edited by Westwood.
Ephemeridae belong to a very ancient type of insects, and fossil imprints
of allied forms occur even in the Devonian and Carboniferous formations.
There is much to be said in favour of the view entertained by some
entomologists that the structural and developmental characteristics of may-
flies are sufficiently peculiar to warrant the formation for them of a special
order of insects, for which the names Agnatha, Plectoptera and
Ephemeroptera have been proposed. (See Hexapoda, Neuroptera.)
Bibliography.—Of especial value to students of these insects are A.
E. Eaton’s monograph (Trans. Linn. Soc. (2) iii. 1883-1885) and A.
Vayssière’s “Recherches sur l’organisation des larves” (Ann. Sci. Nat.
Zool. (6) xiii. 1882 (7) ix. 1890). J. A. Palmén’s memoirs Zur
Morphologie des Tracheensystems (Leipzig, 1877) and Über paarige
Ausführungsgänge der Geschlechtsorgane bei Insekten (Helsingfors,
1884), contain important observations on may-flies. See also L. C.
Miall, Nat. Hist. Aquatic Insects (London, 1895); J. G. Needham and
others (New York State Museum, Bull. 86, 1905).
(R. M’L.; G. H. C.)
MAYHEM (for derivation see Maiming), an old Anglo-French term
of the law signifying an assault whereby the injured person is deprived of a
Page 235
member proper for his defence in fight, e.g. an arm, a leg, a fore tooth, &c.
The loss of an ear, jaw tooth, &c., was not mayhem. The most ancient
punishment in English law was retaliative—membrum pro membro, but
ultimately at common law fine and imprisonment. Various statutes were
passed aimed at the offence of maiming and disfiguring, which is now dealt
with by section 18 of the Offences against the Person Act 1861. Mayhem
may also be the ground of a civil action, which had this peculiarity that the
court on sight of the wound might increase the damages awarded by the
jury.
MAYHEW, HENRY (1812-1887), English author and journalist,
son of a London solicitor, was born in 1812. He was sent to Westminster
school, but ran away to sea. He sailed to India, and on his return studied law
for a short time under his father. He began his journalistic career by
founding, with Gilbert à Beckett, in 1831, a weekly paper, Figaro in
London. This was followed in 1832 by a short-lived paper called The Thief;
and he produced one or two successful farces. His brothers Horace (1816-
1872) and Augustus Septimus (1826-1875) were also journalists, and with
them Henry occasionally collaborated, notably with the younger in The
Greatest Plague of Life (1847) and in Acting Charades (1850). In 1841
Henry Mayhew was one of the leading spirits in the foundation of Punch, of
which he was for the first two years joint-editor with Mark Lemon. He
afterwards wrote on all kinds of subjects, and published a number of
volumes of no permanent reputation—humorous stories, travel and practical
The loss of an ear, jaw tooth, &c., was not mayhem. The most ancient
punishment in English law was retaliative—membrum pro membro, but
ultimately at common law fine and imprisonment. Various statutes were
passed aimed at the offence of maiming and disfiguring, which is now dealt
with by section 18 of the Offences against the Person Act 1861. Mayhem
may also be the ground of a civil action, which had this peculiarity that the
court on sight of the wound might increase the damages awarded by the
jury.
MAYHEW, HENRY (1812-1887), English author and journalist,
son of a London solicitor, was born in 1812. He was sent to Westminster
school, but ran away to sea. He sailed to India, and on his return studied law
for a short time under his father. He began his journalistic career by
founding, with Gilbert à Beckett, in 1831, a weekly paper, Figaro in
London. This was followed in 1832 by a short-lived paper called The Thief;
and he produced one or two successful farces. His brothers Horace (1816-
1872) and Augustus Septimus (1826-1875) were also journalists, and with
them Henry occasionally collaborated, notably with the younger in The
Greatest Plague of Life (1847) and in Acting Charades (1850). In 1841
Henry Mayhew was one of the leading spirits in the foundation of Punch, of
which he was for the first two years joint-editor with Mark Lemon. He
afterwards wrote on all kinds of subjects, and published a number of
volumes of no permanent reputation—humorous stories, travel and practical
Page 236
handbooks. He is credited with being the first to “write up” the poverty side
of London life from a philanthropic point of view; with the collaboration of
John Binny and others he published London Labour and London Poor
(1851; completed 1864) and other works on social and economic questions.
He died in London, on the 25th of July 1887. Horace Mayhew was for some
years sub-editor of Punch, and was the author of several humorous
publications and plays. The books of Horace and Augustus Mayhew owe
their survival chiefly to Cruikshank’s illustrations.
MAYHEW, JONATHAN (1720-1766), American clergyman,
was born at Martha’s Vineyard on the 8th of October 1720, being fifth in
descent from Thomas Mayhew (1592-1682), an early settler and the grantee
(1641) of Martha’s Vineyard. Thomas Mayhew (c. 1616-1657), the
younger, his son John (d. 1689) and John’s son, Experience (1673-1758),
were active missionaries among the Indians of Martha’s Vineyard and the
vicinity. Jonathan, the son of Experience, graduated at Harvard in 1744. So
liberal were his theological views that when he was to be ordained minister
of the West Church in Boston in 1747 only two ministers attended the first
council called for the ordination, and it was necessary to summon a second
council. Mayhew’s preaching made his church practically the first
“Unitarian” Congregational church in New England, though it was never
officially Unitarian. In 1763 he published Observations on the Charter and
Conduct of the Society for Propagating the Gospel in Foreign Parts, an
attack on the policy of the society in sending missionaries to New England
of London life from a philanthropic point of view; with the collaboration of
John Binny and others he published London Labour and London Poor
(1851; completed 1864) and other works on social and economic questions.
He died in London, on the 25th of July 1887. Horace Mayhew was for some
years sub-editor of Punch, and was the author of several humorous
publications and plays. The books of Horace and Augustus Mayhew owe
their survival chiefly to Cruikshank’s illustrations.
MAYHEW, JONATHAN (1720-1766), American clergyman,
was born at Martha’s Vineyard on the 8th of October 1720, being fifth in
descent from Thomas Mayhew (1592-1682), an early settler and the grantee
(1641) of Martha’s Vineyard. Thomas Mayhew (c. 1616-1657), the
younger, his son John (d. 1689) and John’s son, Experience (1673-1758),
were active missionaries among the Indians of Martha’s Vineyard and the
vicinity. Jonathan, the son of Experience, graduated at Harvard in 1744. So
liberal were his theological views that when he was to be ordained minister
of the West Church in Boston in 1747 only two ministers attended the first
council called for the ordination, and it was necessary to summon a second
council. Mayhew’s preaching made his church practically the first
“Unitarian” Congregational church in New England, though it was never
officially Unitarian. In 1763 he published Observations on the Charter and
Conduct of the Society for Propagating the Gospel in Foreign Parts, an
attack on the policy of the society in sending missionaries to New England
Page 237
contrary to its original purpose of “Maintaining Ministers of the Gospel” in
places “wholly destitute and unprovided with means for the maintenance of
ministers and for the public worship of God;” the Observations marked him
as a leader among those in New England who feared, as Mayhew said
(1762), “that there is a scheme forming for sending a bishop into this part of
the country, and that our Governor,1 a true churchman, is deeply in the
plot.” To an American reply to the Observations, entitled A Candid
Examination (1763), Mayhew wrote a Defense; and after the publication of
an Answer, anonymously published in London in 1764 and written by
Thomas Seeker, archbishop of Canterbury, he wrote a Second Defense. He
bitterly opposed the Stamp Act, and urged the necessity of colonial union
(or “communion”) to secure colonial liberties. He died on the 9th of July
1766. Mayhew was Dudleian lecturer at Harvard in 1765, and in 1749 had
received the degree of D.D. from the University of Aberdeen.
See Alden Bradford, Memoir of the Life and Writings of Rev.
Jonathan Mayhew (Boston, 1838), and “An Early Pulpit Champion of
Colonial Rights,” chapter vi., in vol. i. of M. C. Tyler’s Literary
History of the American Revolution (2 vols., New York, 1897).
1 Francis Bernard, whose project for a college at Northampton seemed to Mayhew and
others a move to strengthen Anglicanism.
MAYHEW, THOMAS, English 18th century cabinet-maker.
Mayhew was the less distinguished partner of William Ince (q.v.). The chief
places “wholly destitute and unprovided with means for the maintenance of
ministers and for the public worship of God;” the Observations marked him
as a leader among those in New England who feared, as Mayhew said
(1762), “that there is a scheme forming for sending a bishop into this part of
the country, and that our Governor,1 a true churchman, is deeply in the
plot.” To an American reply to the Observations, entitled A Candid
Examination (1763), Mayhew wrote a Defense; and after the publication of
an Answer, anonymously published in London in 1764 and written by
Thomas Seeker, archbishop of Canterbury, he wrote a Second Defense. He
bitterly opposed the Stamp Act, and urged the necessity of colonial union
(or “communion”) to secure colonial liberties. He died on the 9th of July
1766. Mayhew was Dudleian lecturer at Harvard in 1765, and in 1749 had
received the degree of D.D. from the University of Aberdeen.
See Alden Bradford, Memoir of the Life and Writings of Rev.
Jonathan Mayhew (Boston, 1838), and “An Early Pulpit Champion of
Colonial Rights,” chapter vi., in vol. i. of M. C. Tyler’s Literary
History of the American Revolution (2 vols., New York, 1897).
1 Francis Bernard, whose project for a college at Northampton seemed to Mayhew and
others a move to strengthen Anglicanism.
MAYHEW, THOMAS, English 18th century cabinet-maker.
Mayhew was the less distinguished partner of William Ince (q.v.). The chief
Page 238
source of information as to his work is supplied by his own drawings in the
volume of designs, The universal system of household furniture, which he
published in collaboration with his partner. The name of the firm appears to
have been Mayhew and Ince, but on the title page of this book the names
are reversed, perhaps as an indication that Ince was the more extensive
contributor. In the main Mayhew’s designs are heavy and clumsy, and often
downright extravagant, but he had a certain lightness of accomplishment in
his applications of the bizarre Chinese style. Of original talent he possessed
little, yet it is certain that much of his Chinese work has been attributed to
Chippendale. It is indeed often only by reference to books of design that the
respective work of the English cabinet-makers of the second half of the
18th century can be correctly attributed.
MAYMYO, a hill sanatorium in India, in the Mandalay district of
Upper Burma, 3500 ft. above the sea, with a station on the Mandalay-
Lashio railway 422 m. from Rangoon. Pop. (1901), 6223. It consists of an
undulating plateau, surrounded by hills, which are covered with thin oak
forest and bracken. Though not entirely free from malaria, it has been
chosen for the summer residence of the lieutenant-governor; and it is also
the permanent headquarters of the lieutenant-general commanding the
Burma division, and of other officials.
volume of designs, The universal system of household furniture, which he
published in collaboration with his partner. The name of the firm appears to
have been Mayhew and Ince, but on the title page of this book the names
are reversed, perhaps as an indication that Ince was the more extensive
contributor. In the main Mayhew’s designs are heavy and clumsy, and often
downright extravagant, but he had a certain lightness of accomplishment in
his applications of the bizarre Chinese style. Of original talent he possessed
little, yet it is certain that much of his Chinese work has been attributed to
Chippendale. It is indeed often only by reference to books of design that the
respective work of the English cabinet-makers of the second half of the
18th century can be correctly attributed.
MAYMYO, a hill sanatorium in India, in the Mandalay district of
Upper Burma, 3500 ft. above the sea, with a station on the Mandalay-
Lashio railway 422 m. from Rangoon. Pop. (1901), 6223. It consists of an
undulating plateau, surrounded by hills, which are covered with thin oak
forest and bracken. Though not entirely free from malaria, it has been
chosen for the summer residence of the lieutenant-governor; and it is also
the permanent headquarters of the lieutenant-general commanding the
Burma division, and of other officials.
Page 239
MAYNARD, FRANÇOIS DE (1582-1646), French poet,
was born at Toulouse in 1582. His father was conseiller in the parlement of
the town, and François was also trained for the law, becoming eventually
president of Aurillac. He became secretary to Margaret of Valois, wife of
Henry IV., for whom his early poems are written. He was a disciple of
Malherbe, who said that in the workmanship of his lines he excelled Racan,
but lacked his rival’s energy. In 1634 he accompanied the Cardinal de
Noailles to Rome and spent about two years in Italy. On his return to France
he made many unsuccessful efforts to obtain the favour of Richelieu, but
was obliged to retire to Toulouse. He never ceased to lament his exile from
Paris and his inability to be present at the meetings of the Academy, of
which he was one of the earliest members. The best of his poems is in
imitation of Horace, “Alcippe, reviens dans nos bois.” He died at Toulouse
on the 23rd of December 1646.
His works consist of odes, epigrams, songs and letters, and were
published in 1646 by Marin le Roy de Gomberville.
was born at Toulouse in 1582. His father was conseiller in the parlement of
the town, and François was also trained for the law, becoming eventually
president of Aurillac. He became secretary to Margaret of Valois, wife of
Henry IV., for whom his early poems are written. He was a disciple of
Malherbe, who said that in the workmanship of his lines he excelled Racan,
but lacked his rival’s energy. In 1634 he accompanied the Cardinal de
Noailles to Rome and spent about two years in Italy. On his return to France
he made many unsuccessful efforts to obtain the favour of Richelieu, but
was obliged to retire to Toulouse. He never ceased to lament his exile from
Paris and his inability to be present at the meetings of the Academy, of
which he was one of the earliest members. The best of his poems is in
imitation of Horace, “Alcippe, reviens dans nos bois.” He died at Toulouse
on the 23rd of December 1646.
His works consist of odes, epigrams, songs and letters, and were
published in 1646 by Marin le Roy de Gomberville.
Page 240
MAYNE, JASPER (1604-1672), English author, was baptized at
Hatherleigh, Devonshire, on the 23rd of November 1604. He was educated
at Westminster School and at Christ Church, Oxford, where he had a
distinguished career. He was presented to two college livings in
Oxfordshire, and was made D.D. in 1646. During the Commonwealth he
was dispossessed, and became chaplain to the duke of Devonshire. At the
Restoration he was made canon of Christ Church, archdeacon of Chichester
and chaplain in ordinary to the king. He wrote a farcical domestic comedy,
The City Match (1639), which is reprinted in vol. xiii. of Hazlitt’s edition of
Dodsley’s Old Plays, and a fantastic tragi-comedy entitled The Amorous
War (printed 1648). After receiving ecclesiastical preferment he gave up
poetry as unbefitting his profession. His other works comprise some
occasional gems, a translation of Lucian’s Dialogues (printed 1664) and a
number of sermons. He died on the 6th of December 1672 at Oxford.
MAYNOOTH, a small town of county Kildare, Ireland, on the
Midland Great Western railway and the Royal Canal, 15 m. W. by N. of
Dublin. Pop. (1901), 948. The Royal Catholic College of Maynooth,
founded by an Act of the Irish parliament in 1795, is the chief seminary for
the education of the Roman Catholic clergy of Ireland. The building is a
fine Gothic structure by A. W. Pugin, erected by a parliamentary grant
obtained in 1846. The chapel, with fine oak choir-stalls, mosaic pavements,
marble altars and stained glass, and with adjoining cloisters, was dedicated
in 1890. The average number of students is about 500—the number
Hatherleigh, Devonshire, on the 23rd of November 1604. He was educated
at Westminster School and at Christ Church, Oxford, where he had a
distinguished career. He was presented to two college livings in
Oxfordshire, and was made D.D. in 1646. During the Commonwealth he
was dispossessed, and became chaplain to the duke of Devonshire. At the
Restoration he was made canon of Christ Church, archdeacon of Chichester
and chaplain in ordinary to the king. He wrote a farcical domestic comedy,
The City Match (1639), which is reprinted in vol. xiii. of Hazlitt’s edition of
Dodsley’s Old Plays, and a fantastic tragi-comedy entitled The Amorous
War (printed 1648). After receiving ecclesiastical preferment he gave up
poetry as unbefitting his profession. His other works comprise some
occasional gems, a translation of Lucian’s Dialogues (printed 1664) and a
number of sermons. He died on the 6th of December 1672 at Oxford.
MAYNOOTH, a small town of county Kildare, Ireland, on the
Midland Great Western railway and the Royal Canal, 15 m. W. by N. of
Dublin. Pop. (1901), 948. The Royal Catholic College of Maynooth,
founded by an Act of the Irish parliament in 1795, is the chief seminary for
the education of the Roman Catholic clergy of Ireland. The building is a
fine Gothic structure by A. W. Pugin, erected by a parliamentary grant
obtained in 1846. The chapel, with fine oak choir-stalls, mosaic pavements,
marble altars and stained glass, and with adjoining cloisters, was dedicated
in 1890. The average number of students is about 500—the number
Page 241
specified under the act of 1845—and the full course of instruction is eight
years. Near the college stand the ruins of Maynooth Castle, probably built
in 1176, but subsequently extended, and formerly the residence of the
Fitzgerald family. It was besieged in the reigns of Henry VIII. and Edward
VI., and during the Cromwellian Wars, when it was demolished. The
beautiful mansion of Carton is about a mile from the town.
MAYO, RICHARD SOUTHWELL BOURKE, 6th
Earl of (1822-1872), British statesman, son of Robert Bourke, the 5th earl
(1797-1867), was born in Dublin on the 21st of February, 1822, and was
educated at Trinity College, Dublin. After travelling in Russia he entered
parliament, and sat successively for Kildare, Coleraine and Cockermouth.
He was chief secretary for Ireland in three administrations, in 1852, 1858
and 1866, and was appointed viceroy of India in January 1869. He
consolidated the frontiers of India and met Shere Ali, amir of Afghanistan,
in durbar at Umballa in March 1869. His reorganization of the finances of
the country put India on a paying basis; and he did much to promote
irrigation, railways, forests and other useful public works. Visiting the
convict settlement at Port Blair in the Andaman Islands, for the purpose of
inspection, the viceroy was assassinated by a convict on the 8th of February
1872. His successor was his son, Dermot Robert Wyndham Bourke (b.
1851) who became 7th earl of Mayo.
years. Near the college stand the ruins of Maynooth Castle, probably built
in 1176, but subsequently extended, and formerly the residence of the
Fitzgerald family. It was besieged in the reigns of Henry VIII. and Edward
VI., and during the Cromwellian Wars, when it was demolished. The
beautiful mansion of Carton is about a mile from the town.
MAYO, RICHARD SOUTHWELL BOURKE, 6th
Earl of (1822-1872), British statesman, son of Robert Bourke, the 5th earl
(1797-1867), was born in Dublin on the 21st of February, 1822, and was
educated at Trinity College, Dublin. After travelling in Russia he entered
parliament, and sat successively for Kildare, Coleraine and Cockermouth.
He was chief secretary for Ireland in three administrations, in 1852, 1858
and 1866, and was appointed viceroy of India in January 1869. He
consolidated the frontiers of India and met Shere Ali, amir of Afghanistan,
in durbar at Umballa in March 1869. His reorganization of the finances of
the country put India on a paying basis; and he did much to promote
irrigation, railways, forests and other useful public works. Visiting the
convict settlement at Port Blair in the Andaman Islands, for the purpose of
inspection, the viceroy was assassinated by a convict on the 8th of February
1872. His successor was his son, Dermot Robert Wyndham Bourke (b.
1851) who became 7th earl of Mayo.
Page 242
See Sir W. W. Hunter, Life of the Earl of Mayo, (1876), and The Earl
of Mayo in the Rulers of India Series (1891).
MAYO, a western county of Ireland, in the province of Connaught,
bounded N. and W. by the Atlantic Ocean, N.E. by Sligo, E. by
Roscommon, S.E. and S. by Galway. The area is 1,380,390 acres, or about
2157 sq. m., the county being the largest in Ireland after Cork and Galway.
About two-thirds of the boundary of Mayo is formed by sea, and the coast
is very much indented, and abounds in picturesque scenery. The principal
inlets are Killary Harbour between Mayo and Galway; Clew Bay, in which
are the harbours of Westport and Newport; Blacksod Bay and Broad Haven,
which form the peninsula of the Mullet; and Killala Bay between Mayo and
Sligo. The islands are very numerous, the principal being Inishturk, near
Killary Harbour; Clare Island, at the mouth of Clew Bay, where there are
many islets, all formed of drift; and Achill, the largest island off Ireland.
The coast scenery is not surpassed by that of Donegal northward and
Connemara southward, and there are several small coast-towns, among
which may be named Killala on the north coast, Belmullet on the isthmus
between Blacksod Bay and Broad Haven, Newport and Westport on Clew
Bay, with the watering-place of Mallaranny. The majestic cliffs of the north
coast, however, which reach an extreme height in Benwee Head (892 ft.),
are difficult of access and rarely visited. In the eastern half of the county the
surface is comparatively level, with occasional hills; the western half is
mountainous. Mweelrea (2688 ft.) is included in a mountain range lying
of Mayo in the Rulers of India Series (1891).
MAYO, a western county of Ireland, in the province of Connaught,
bounded N. and W. by the Atlantic Ocean, N.E. by Sligo, E. by
Roscommon, S.E. and S. by Galway. The area is 1,380,390 acres, or about
2157 sq. m., the county being the largest in Ireland after Cork and Galway.
About two-thirds of the boundary of Mayo is formed by sea, and the coast
is very much indented, and abounds in picturesque scenery. The principal
inlets are Killary Harbour between Mayo and Galway; Clew Bay, in which
are the harbours of Westport and Newport; Blacksod Bay and Broad Haven,
which form the peninsula of the Mullet; and Killala Bay between Mayo and
Sligo. The islands are very numerous, the principal being Inishturk, near
Killary Harbour; Clare Island, at the mouth of Clew Bay, where there are
many islets, all formed of drift; and Achill, the largest island off Ireland.
The coast scenery is not surpassed by that of Donegal northward and
Connemara southward, and there are several small coast-towns, among
which may be named Killala on the north coast, Belmullet on the isthmus
between Blacksod Bay and Broad Haven, Newport and Westport on Clew
Bay, with the watering-place of Mallaranny. The majestic cliffs of the north
coast, however, which reach an extreme height in Benwee Head (892 ft.),
are difficult of access and rarely visited. In the eastern half of the county the
surface is comparatively level, with occasional hills; the western half is
mountainous. Mweelrea (2688 ft.) is included in a mountain range lying
Page 243
between Killary Harbour and Lough Mask. The next highest summits are
Nephin (2646 ft.), to the west of Lough Conn, and Croagh Patrick (2510
ft.), to the south of Clew Bay. The river Moy flows northwards, forming
part of the boundary of the county with Sligo, and falls into Killala Bay.
The courses of the other streams are short, and except when swollen by
rains their volume is small. The principal lakes are Lough Mask and Lough
Corrib, on the borders of the county with Galway, and Loughs Conn in the
east, Carrowmore in the north-west, Beltra in the west, and Carra adjoining
Lough Mask. These loughs and the smaller loughs, with the streams
generally, afford admirable sport with salmon, sea-trout and brown trout,
and Ballina is a favourite centre.
Geology.—The wild and barren west of this county, including the
great hills on Achill Island, is formed of “Dalradian” rocks, schists and
quartzites, highly folded and metamorphosed, with intrusions of granite
near Belmullet. At Blacksod Bay the granite has been quarried as an
ornamental stone. Nephin Beg, Nephin and Croagh Patrick are typical
quartzite summits, the last named belonging possibly to a Silurian
horizon but rising from a metamorphosed area on the south side of
Clew Bay. The schists and gneisses of the Ox Mountain axis also enter
the county north of Castlebar. The Muilrea and Ben Gorm range,
bounding the fine fjord of Killary Harbour, is formed of terraced
Silurian rocks, from Bala to Ludlow age. These beds, with intercalated
lavas, form the mountainous west shore of Lough Mask, the east, like
that of Lough Corrib, being formed of low Carboniferous Limestone
ground. Silurian rocks, with Old Red Sandstone over them, come out at
the west end of the Curlew range at Ballaghaderreen. Clew Bay, with
its islets capped by glacial drift, is a submerged part of a synclinal of
Carboniferous strata, and Old Red Sandstone comes out on the north
side of this, from near Achill to Lough Conn. The country from Lough
Nephin (2646 ft.), to the west of Lough Conn, and Croagh Patrick (2510
ft.), to the south of Clew Bay. The river Moy flows northwards, forming
part of the boundary of the county with Sligo, and falls into Killala Bay.
The courses of the other streams are short, and except when swollen by
rains their volume is small. The principal lakes are Lough Mask and Lough
Corrib, on the borders of the county with Galway, and Loughs Conn in the
east, Carrowmore in the north-west, Beltra in the west, and Carra adjoining
Lough Mask. These loughs and the smaller loughs, with the streams
generally, afford admirable sport with salmon, sea-trout and brown trout,
and Ballina is a favourite centre.
Geology.—The wild and barren west of this county, including the
great hills on Achill Island, is formed of “Dalradian” rocks, schists and
quartzites, highly folded and metamorphosed, with intrusions of granite
near Belmullet. At Blacksod Bay the granite has been quarried as an
ornamental stone. Nephin Beg, Nephin and Croagh Patrick are typical
quartzite summits, the last named belonging possibly to a Silurian
horizon but rising from a metamorphosed area on the south side of
Clew Bay. The schists and gneisses of the Ox Mountain axis also enter
the county north of Castlebar. The Muilrea and Ben Gorm range,
bounding the fine fjord of Killary Harbour, is formed of terraced
Silurian rocks, from Bala to Ludlow age. These beds, with intercalated
lavas, form the mountainous west shore of Lough Mask, the east, like
that of Lough Corrib, being formed of low Carboniferous Limestone
ground. Silurian rocks, with Old Red Sandstone over them, come out at
the west end of the Curlew range at Ballaghaderreen. Clew Bay, with
its islets capped by glacial drift, is a submerged part of a synclinal of
Carboniferous strata, and Old Red Sandstone comes out on the north
side of this, from near Achill to Lough Conn. The country from Lough
Page 244
Conn northward to the sea is a lowland of Carboniferous Limestone,
with L. Carboniferous Sandstone against the Dalradian on the west.
Industries.—There are some very fertile regions in the level portions
of the county, but in the mountainous districts the soil is poor, the
holdings are subdivided beyond the possibility of affording proper
sustenance to their occupiers, and, except where fishing is combined
with agricultural operations, the circumstances of the peasantry are
among the most wretched of any district of Ireland. The proportion of
tillage to pasturage is roughly as 1 to 31⁄2. Oats and potatoes are the
principal crops. Cattle, sheep, pigs and poultry are reared. Coarse linen
and woollen cloths are manufactured to a small extent. At Foxford
woollen-mills are established at a nunnery, in connexion with a scheme
of technical instruction. Keel, Belmullet and Ballycastle are the
headquarters of sea and coast fishing districts, and Ballina of a salmon-
fishing district, and these fisheries are of some value to the poor
inhabitants. A branch of the Midland Great Western railway enters the
county from Athlone, in the south-east, and runs north to Ballina and
Killala on the coast, branches diverging from Claremorris to
Ballinrobe, and from Manulla to Westport and Achill on the west coast.
The Limerick and Sligo line of the Great Southern and Western passes
from south to north-east by way of Claremorris.
Population and Administration.—The population was 218,698 in 1891,
and 199,166 in 1901. The decrease of population and the number of
emigrants are slightly below the average of the Irish counties. Of the total
population about 97% are rural, and about the same percentage are Roman
Catholics. The chief towns are Ballina (pop. 4505), Westport (3892) and
Castlebar (3585), the county town. Ballaghaderreen, Claremorris (Clare),
Crossmolina and Swineford are lesser market towns; and Newport and
Westport are small seaports on Clew Bay. The county includes nine
with L. Carboniferous Sandstone against the Dalradian on the west.
Industries.—There are some very fertile regions in the level portions
of the county, but in the mountainous districts the soil is poor, the
holdings are subdivided beyond the possibility of affording proper
sustenance to their occupiers, and, except where fishing is combined
with agricultural operations, the circumstances of the peasantry are
among the most wretched of any district of Ireland. The proportion of
tillage to pasturage is roughly as 1 to 31⁄2. Oats and potatoes are the
principal crops. Cattle, sheep, pigs and poultry are reared. Coarse linen
and woollen cloths are manufactured to a small extent. At Foxford
woollen-mills are established at a nunnery, in connexion with a scheme
of technical instruction. Keel, Belmullet and Ballycastle are the
headquarters of sea and coast fishing districts, and Ballina of a salmon-
fishing district, and these fisheries are of some value to the poor
inhabitants. A branch of the Midland Great Western railway enters the
county from Athlone, in the south-east, and runs north to Ballina and
Killala on the coast, branches diverging from Claremorris to
Ballinrobe, and from Manulla to Westport and Achill on the west coast.
The Limerick and Sligo line of the Great Southern and Western passes
from south to north-east by way of Claremorris.
Population and Administration.—The population was 218,698 in 1891,
and 199,166 in 1901. The decrease of population and the number of
emigrants are slightly below the average of the Irish counties. Of the total
population about 97% are rural, and about the same percentage are Roman
Catholics. The chief towns are Ballina (pop. 4505), Westport (3892) and
Castlebar (3585), the county town. Ballaghaderreen, Claremorris (Clare),
Crossmolina and Swineford are lesser market towns; and Newport and
Westport are small seaports on Clew Bay. The county includes nine
Page 245
baronies. Assizes are held at Castlebar, and quarter sessions at Ballina,
Ballinrobe, Belmullet, Castlebar, Claremorris, Swineford and Westport. In
the Irish parliament two members were returned for the county, and two for
the borough of Castlebar, but at the union Castlebar was disfranchised. The
division since 1885 is into north, south, east and west parliamentary
divisions, each returning one member. The county is in the Protestant
diocese of Tuam and the Roman Catholic dioceses of Taum, Achonry,
Galway and Kilmacduagh, and Killala.
History and Antiquities.—Erris in Mayo was the scene of the landing of
the chief colony of the Firbolgs, and the battle which is said to have resulted
in the overthrow and almost annihilation of this tribe took place also in this
county, at Moytura near Cong. At the close of the 12th century what is now
the county of Mayo was granted, with other lands, by king John to William,
brother of Hubert de Burgh. After the murder of William de Burgh, 3rd earl
of Ulster (1333), the Bourkes (de Burghs) of the collateral male line,
rejecting the claim of William’s heiress (the wife of Lionel, son of King
Edward III.) to the succession, succeeded in holding the bulk of the De
Burgh possessions, what is now Mayo falling to the branch known by the
name of “MacWilliam Oughter,” who maintained their virtual independence
till the time of Elizabeth. Sir Henry Sydney, during his first viceroyalty,
after making efforts to improve communications between Dublin and
Connaught in 1566, arranged for the shiring of that province, and Mayo was
made shire ground, taking its name from the monastery of Maio or Mageo,
which was the seat of a bishop. Even after this period the MacWilliams
continued to exercise very great authority, which was regularized in 1603,
when “the MacWilliam Oughter,” Theobald Bourke, surrendered his lands
and received them back, to hold them by English tenure, with the title of
Viscount Mayo (see Burgh, De). Large confiscations of the estates in the
county were made in 1586, and on the termination of the wars of 1641; and
in 1666 the restoration of his estates to the 4th Viscount Mayo involved
Ballinrobe, Belmullet, Castlebar, Claremorris, Swineford and Westport. In
the Irish parliament two members were returned for the county, and two for
the borough of Castlebar, but at the union Castlebar was disfranchised. The
division since 1885 is into north, south, east and west parliamentary
divisions, each returning one member. The county is in the Protestant
diocese of Tuam and the Roman Catholic dioceses of Taum, Achonry,
Galway and Kilmacduagh, and Killala.
History and Antiquities.—Erris in Mayo was the scene of the landing of
the chief colony of the Firbolgs, and the battle which is said to have resulted
in the overthrow and almost annihilation of this tribe took place also in this
county, at Moytura near Cong. At the close of the 12th century what is now
the county of Mayo was granted, with other lands, by king John to William,
brother of Hubert de Burgh. After the murder of William de Burgh, 3rd earl
of Ulster (1333), the Bourkes (de Burghs) of the collateral male line,
rejecting the claim of William’s heiress (the wife of Lionel, son of King
Edward III.) to the succession, succeeded in holding the bulk of the De
Burgh possessions, what is now Mayo falling to the branch known by the
name of “MacWilliam Oughter,” who maintained their virtual independence
till the time of Elizabeth. Sir Henry Sydney, during his first viceroyalty,
after making efforts to improve communications between Dublin and
Connaught in 1566, arranged for the shiring of that province, and Mayo was
made shire ground, taking its name from the monastery of Maio or Mageo,
which was the seat of a bishop. Even after this period the MacWilliams
continued to exercise very great authority, which was regularized in 1603,
when “the MacWilliam Oughter,” Theobald Bourke, surrendered his lands
and received them back, to hold them by English tenure, with the title of
Viscount Mayo (see Burgh, De). Large confiscations of the estates in the
county were made in 1586, and on the termination of the wars of 1641; and
in 1666 the restoration of his estates to the 4th Viscount Mayo involved
Page 246
another confiscation, at the expense of Cromwell’s settlers. Killala was the
scene of the landing of a French squadron in connexion with the rebellion
of 1798. In 1879 the village of Knock in the south-east acquired notoriety
from a story that the Virgin Mary had appeared in the church, which
became the resort of many pilgrims.
There are round towers at Killala, Turlough, Meelick and Balla, and an
imperfect one at Aughagower. Killala was formerly a bishopric. The
monasteries were numerous, and many of them of considerable importance:
the principal being those at Mayo, Ballyhaunis, Cong, Ballinrobe,
Ballintober, Burrishoole, Cross or Holycross in the peninsula of Mullet,
Moyne, Roserk or Rosserick and Templemore or Strade. Of the old castles
the most notable are Carrigahooly near Newport, said to have been built by
the celebrated Grace O’Malley, and Deel Castle near Ballina, at one time
the residence of the earls of Arran.
See Hubert Thomas Knox, History of the County of Mayo (1908).
MAYOR, JOHN EYTON BICKERSTETH
(1825- ), English classical scholar, was born at Baddegama, Ceylon, on
the 28th of January 1825, and educated in England at Shrewsbury School
and St John’s College, Cambridge. From 1863 to 1867 he was librarian of
the university, and in 1872 succeeded H. A. J. Munro in the professorship of
Latin. His best-known work, an edition of thirteen satires of Juvenal, is
marked by an extraordinary wealth of illustrative quotations. His
scene of the landing of a French squadron in connexion with the rebellion
of 1798. In 1879 the village of Knock in the south-east acquired notoriety
from a story that the Virgin Mary had appeared in the church, which
became the resort of many pilgrims.
There are round towers at Killala, Turlough, Meelick and Balla, and an
imperfect one at Aughagower. Killala was formerly a bishopric. The
monasteries were numerous, and many of them of considerable importance:
the principal being those at Mayo, Ballyhaunis, Cong, Ballinrobe,
Ballintober, Burrishoole, Cross or Holycross in the peninsula of Mullet,
Moyne, Roserk or Rosserick and Templemore or Strade. Of the old castles
the most notable are Carrigahooly near Newport, said to have been built by
the celebrated Grace O’Malley, and Deel Castle near Ballina, at one time
the residence of the earls of Arran.
See Hubert Thomas Knox, History of the County of Mayo (1908).
MAYOR, JOHN EYTON BICKERSTETH
(1825- ), English classical scholar, was born at Baddegama, Ceylon, on
the 28th of January 1825, and educated in England at Shrewsbury School
and St John’s College, Cambridge. From 1863 to 1867 he was librarian of
the university, and in 1872 succeeded H. A. J. Munro in the professorship of
Latin. His best-known work, an edition of thirteen satires of Juvenal, is
marked by an extraordinary wealth of illustrative quotations. His
Page 247
Bibliographical Clue to Latin Literature (1873), based on E. Hübner’s
Grundriss zu Vorlesungen über die römische Litteraturgeschichte is a
valuable aid to the student, and his edition of Cicero’s Second Philippic is
widely used. He also edited the English works of J. Fisher, bishop of
Rochester, i. (1876); Thomas Baker’s History of St John’s College,
Cambridge (1869); Richard of Cirencester’s Speculum historiale de gestis
regum Angliae 447-1066 (1863-1869); Roger Ascham’s Schoolmaster (new
ed., 1883); the Latin Heptateuch (1889); and the Journal of Philology.
His brother, Joseph Bickersteth Mayor (1828- ), classical scholar
and theologian, was educated at Rugby and St John’s College, Cambridge,
and from 1870 to 1879 was professor of classics at King’s College, London.
His most important classical works are an edition of Cicero’s De natura
deorum (3 vols., 1880-1885) and Guide to the Choice of Classical Books
(3rd ed., 1885, with supplement, 1896). He also devoted attention to
theological literature and edited the epistles of St James (2nd ed., 1892), St
Jude and St Peter (1907), and the Miscellanies of Clement of Alexandria
(with F. J. A. Hort, 1902). From 1887 to 1893 he was editor of the Classical
Review. His Chapters on English Metre (1886) reached a second edition in
1901.
MAYOR (Lat. major, greater), in modern times the title of a
municipal officer who discharges judicial and administrative functions. The
French form of the word is maire. In Germany the corresponding title is
Grundriss zu Vorlesungen über die römische Litteraturgeschichte is a
valuable aid to the student, and his edition of Cicero’s Second Philippic is
widely used. He also edited the English works of J. Fisher, bishop of
Rochester, i. (1876); Thomas Baker’s History of St John’s College,
Cambridge (1869); Richard of Cirencester’s Speculum historiale de gestis
regum Angliae 447-1066 (1863-1869); Roger Ascham’s Schoolmaster (new
ed., 1883); the Latin Heptateuch (1889); and the Journal of Philology.
His brother, Joseph Bickersteth Mayor (1828- ), classical scholar
and theologian, was educated at Rugby and St John’s College, Cambridge,
and from 1870 to 1879 was professor of classics at King’s College, London.
His most important classical works are an edition of Cicero’s De natura
deorum (3 vols., 1880-1885) and Guide to the Choice of Classical Books
(3rd ed., 1885, with supplement, 1896). He also devoted attention to
theological literature and edited the epistles of St James (2nd ed., 1892), St
Jude and St Peter (1907), and the Miscellanies of Clement of Alexandria
(with F. J. A. Hort, 1902). From 1887 to 1893 he was editor of the Classical
Review. His Chapters on English Metre (1886) reached a second edition in
1901.
MAYOR (Lat. major, greater), in modern times the title of a
municipal officer who discharges judicial and administrative functions. The
French form of the word is maire. In Germany the corresponding title is
Page 248
Bürgermeister, in Italy sindico, and in Spain alcalde. “Mayor” had
originally a much wider significance. Among the nations which arose on the
ruins of the Roman empire of the West, and which made use of the Latin
spoken by their “Roman” subjects as their official and legal language,
major and the Low Latin feminine majorissa were found to be very
convenient terms to describe important officials of both sexes who had the
superintendence of others. Any female servant or slave in the household of
a barbarian, whose business it was to overlook other female servants or
slaves, would be quite naturally called a majorissa. So the male officer who
governed the king’s household would be the major domus. In the
households of the Frankish kings of the Merovingian line, the major domus,
who was also variously known as the gubernator, rector, moderator or
praefectus palatii, was so great an officer that he ended by evicting his
master. He was the “mayor of the palace” (q.v.). The fact that his office
became hereditary in the family of Pippin of Heristal made the fortune of
the Carolingian line. But besides the major domus (the major-domo), there
were other officers who were majores, the major cubiculi, mayor of the
bedchamber, and major equorum, mayor of the horse. In fact a word which
could be applied so easily and with accuracy in so many circumstances was
certain to be widely used by itself, or in its derivatives. The post-Augustine
majorinus, “one of the larger kind,” was the origin of the medieval Spanish
merinus, who in Castillian is the merino, and sometimes the merino mayor,
or chief merino. He was a judicial and administrative officer of the king’s.
The gregum merinus was the superintendent of the flocks of the corporation
of sheep-owners called the mesta. From him the sheep, and then the wool,
have come to be known as merinos—a word identical in origin with the
municipal title of mayor. The latter came directly from the heads of gilds,
and other associations of freemen, who had their banner and formed a group
on the populations of the towns, the majores baneriae or vexilli.
originally a much wider significance. Among the nations which arose on the
ruins of the Roman empire of the West, and which made use of the Latin
spoken by their “Roman” subjects as their official and legal language,
major and the Low Latin feminine majorissa were found to be very
convenient terms to describe important officials of both sexes who had the
superintendence of others. Any female servant or slave in the household of
a barbarian, whose business it was to overlook other female servants or
slaves, would be quite naturally called a majorissa. So the male officer who
governed the king’s household would be the major domus. In the
households of the Frankish kings of the Merovingian line, the major domus,
who was also variously known as the gubernator, rector, moderator or
praefectus palatii, was so great an officer that he ended by evicting his
master. He was the “mayor of the palace” (q.v.). The fact that his office
became hereditary in the family of Pippin of Heristal made the fortune of
the Carolingian line. But besides the major domus (the major-domo), there
were other officers who were majores, the major cubiculi, mayor of the
bedchamber, and major equorum, mayor of the horse. In fact a word which
could be applied so easily and with accuracy in so many circumstances was
certain to be widely used by itself, or in its derivatives. The post-Augustine
majorinus, “one of the larger kind,” was the origin of the medieval Spanish
merinus, who in Castillian is the merino, and sometimes the merino mayor,
or chief merino. He was a judicial and administrative officer of the king’s.
The gregum merinus was the superintendent of the flocks of the corporation
of sheep-owners called the mesta. From him the sheep, and then the wool,
have come to be known as merinos—a word identical in origin with the
municipal title of mayor. The latter came directly from the heads of gilds,
and other associations of freemen, who had their banner and formed a group
on the populations of the towns, the majores baneriae or vexilli.
Page 249
In England the major is the modern representative of the lord’s bailiff or
reeve (see Borough). We find the chief magistrate of London bearing the
title of portreeve for considerably more than a century after the Conquest.
This official was elected by popular choice, a privilege secured from king
John. By the beginning of the 11th century the title of portreeve1 gave way
to that of mayor as the designation of the chief officer of London,2 and the
adoption of the title by other boroughs followed at various intervals.
A mayor is now in England and America the official head of a
municipal government. In the United Kingdom the Municipal
Corporations Act, 1882, s. 15, regulates the election of mayors. He is to
be a fit person elected annually on the 9th of November by the council
of the borough from among the aldermen or councillors or persons
qualified to be such. His term of office is one year, but he is eligible for
re-election. He may appoint a deputy to act during illness or absence,
and such deputy must be either an alderman or councillor. A mayor
who is absent from the borough for more than two months becomes
disqualified and vacates his office. A mayor is ex officio during his year
of office and the next year a justice of the peace for the borough. He
receives such remuneration as the council thinks reasonable. The office
of mayor in an English borough does not entail any important
administrative duties. It is generally regarded as an honour conferred
for past services. The mayor is expected to devote much of his time to
ornamental functions and to preside over meetings which have for their
object the advancement of the public welfare. His administrative duties
are merely to act as returning officer at municipal elections, and as
chairman of the meetings of the council.
The position and power of an English mayor contrast very strongly
with those of the similar official in the United States. The latter is
reeve (see Borough). We find the chief magistrate of London bearing the
title of portreeve for considerably more than a century after the Conquest.
This official was elected by popular choice, a privilege secured from king
John. By the beginning of the 11th century the title of portreeve1 gave way
to that of mayor as the designation of the chief officer of London,2 and the
adoption of the title by other boroughs followed at various intervals.
A mayor is now in England and America the official head of a
municipal government. In the United Kingdom the Municipal
Corporations Act, 1882, s. 15, regulates the election of mayors. He is to
be a fit person elected annually on the 9th of November by the council
of the borough from among the aldermen or councillors or persons
qualified to be such. His term of office is one year, but he is eligible for
re-election. He may appoint a deputy to act during illness or absence,
and such deputy must be either an alderman or councillor. A mayor
who is absent from the borough for more than two months becomes
disqualified and vacates his office. A mayor is ex officio during his year
of office and the next year a justice of the peace for the borough. He
receives such remuneration as the council thinks reasonable. The office
of mayor in an English borough does not entail any important
administrative duties. It is generally regarded as an honour conferred
for past services. The mayor is expected to devote much of his time to
ornamental functions and to preside over meetings which have for their
object the advancement of the public welfare. His administrative duties
are merely to act as returning officer at municipal elections, and as
chairman of the meetings of the council.
The position and power of an English mayor contrast very strongly
with those of the similar official in the United States. The latter is
Page 250
elected directly by the voters within the city, usually for several years;
and he has extensive administrative powers.
The English method of selecting a mayor by the council is followed
for the corresponding functionaries in France (except Paris), the more
important cities of Italy, and in Germany, where, however, the central
government must confirm the choice of the council. Direct
appointment by the central government exists in Belgium, Holland,
Denmark, Norway, Sweden and the smaller towns of Italy and Spain.
As a rule, too, the term of office is longer in other countries than in the
United Kingdom. In France election is for four years, in Holland for
six, in Belgium for an indefinite period, and in Germany usually for
twelve years, but in some cases for life. In Germany the post may be
said to be a professional one, the burgomaster being the head of the city
magistracy, and requiring, in order to be eligible, a training in
administration. German burgomasters are most frequently elected by
promotion from another city. In France the maire, and a number of
experienced members termed “adjuncts,” who assist him as an
executive committee, are elected directly by the municipal council
from among their own number. Most of the administrative work is left
in the hands of the maire and his adjuncts, the full council meeting
comparatively seldom. The maire and the adjuncts receive no salary.
Further information will be found in the sections on local
government in the articles on the various countries; see also A. Shaw,
Municipal Government in Continental Europe; J. A. Fairlie, Municipal
Administration; S. and B. Webb, English Local Government; Redlich
and Hirst, Local Government in England; A. L. Lowell, The
Government of England.
1 If a place was of mercantile importance it was called a port (from porta, the city gate),
and the reeve or bailiff, a “portreeve.”
and he has extensive administrative powers.
The English method of selecting a mayor by the council is followed
for the corresponding functionaries in France (except Paris), the more
important cities of Italy, and in Germany, where, however, the central
government must confirm the choice of the council. Direct
appointment by the central government exists in Belgium, Holland,
Denmark, Norway, Sweden and the smaller towns of Italy and Spain.
As a rule, too, the term of office is longer in other countries than in the
United Kingdom. In France election is for four years, in Holland for
six, in Belgium for an indefinite period, and in Germany usually for
twelve years, but in some cases for life. In Germany the post may be
said to be a professional one, the burgomaster being the head of the city
magistracy, and requiring, in order to be eligible, a training in
administration. German burgomasters are most frequently elected by
promotion from another city. In France the maire, and a number of
experienced members termed “adjuncts,” who assist him as an
executive committee, are elected directly by the municipal council
from among their own number. Most of the administrative work is left
in the hands of the maire and his adjuncts, the full council meeting
comparatively seldom. The maire and the adjuncts receive no salary.
Further information will be found in the sections on local
government in the articles on the various countries; see also A. Shaw,
Municipal Government in Continental Europe; J. A. Fairlie, Municipal
Administration; S. and B. Webb, English Local Government; Redlich
and Hirst, Local Government in England; A. L. Lowell, The
Government of England.
1 If a place was of mercantile importance it was called a port (from porta, the city gate),
and the reeve or bailiff, a “portreeve.”
Page 251
2 The mayors of certain cities in the United Kingdom (London, York, Dublin) have
acquired by prescription the prefix of “lord.” In the case of London it seems to date from
1540. It has also been conferred during the closing years of the 19th century by letters
patent on other cities—Birmingham, Liverpool, Manchester, Bristol, Sheffield, Leeds,
Cardiff, Bradford, Newcastle-on-Tyne, Belfast, Cork. In 1910 it was granted to Norwich.
Lord mayors are entitled to be addressed as “right honourable.”
MAYOR OF THE PALACE.—The office of mayor of the
palace was an institution peculiar to the Franks of the Merovingian period.
A landowner who did not manage his own estate placed it in the hands of a
steward (major), who superintended the working of the estate and collected
its revenues. If he had several estates, he appointed a chief steward, who
managed the whole of the estates and was called the major domus. Each
great personage had a major domus—the queen had hers, the king his; and
since the royal house was called the palace, this officer took the name of
“mayor of the palace.” The mayor of the palace, however, did not remain
restricted to domestic functions; he had the discipline of the palace and tried
persons who resided there. Soon his functions expanded. If the king were a
minor, the mayor of the palace supervised his education in the capacity of
guardian (nutricius), and often also occupied himself with affairs of state.
When the king came of age, the mayor exerted himself to keep this power,
and succeeded. In the 7th century he became the head of the administration
and a veritable prime minister. He took part in the nomination of the counts
and dukes; in the king’s absence he presided over the royal tribunal; and he
often commanded the armies. When the custom of commendation
developed, the king charged the mayor of the palace to protect those who
acquired by prescription the prefix of “lord.” In the case of London it seems to date from
1540. It has also been conferred during the closing years of the 19th century by letters
patent on other cities—Birmingham, Liverpool, Manchester, Bristol, Sheffield, Leeds,
Cardiff, Bradford, Newcastle-on-Tyne, Belfast, Cork. In 1910 it was granted to Norwich.
Lord mayors are entitled to be addressed as “right honourable.”
MAYOR OF THE PALACE.—The office of mayor of the
palace was an institution peculiar to the Franks of the Merovingian period.
A landowner who did not manage his own estate placed it in the hands of a
steward (major), who superintended the working of the estate and collected
its revenues. If he had several estates, he appointed a chief steward, who
managed the whole of the estates and was called the major domus. Each
great personage had a major domus—the queen had hers, the king his; and
since the royal house was called the palace, this officer took the name of
“mayor of the palace.” The mayor of the palace, however, did not remain
restricted to domestic functions; he had the discipline of the palace and tried
persons who resided there. Soon his functions expanded. If the king were a
minor, the mayor of the palace supervised his education in the capacity of
guardian (nutricius), and often also occupied himself with affairs of state.
When the king came of age, the mayor exerted himself to keep this power,
and succeeded. In the 7th century he became the head of the administration
and a veritable prime minister. He took part in the nomination of the counts
and dukes; in the king’s absence he presided over the royal tribunal; and he
often commanded the armies. When the custom of commendation
developed, the king charged the mayor of the palace to protect those who
Page 252
had commended themselves to him and to intervene at law on their behalf.
The mayor of the palace thus found himself at the head of the commendati,
just as he was at the head of the functionaries.
It is difficult to trace the names of some of the mayors of the palace, the
post being of almost no significance in the time of Gregory of Tours. When
the office increased in importance the mayors of the palace did not, as has
been thought, pursue an identical policy. Some—for instance, Otto, the
mayor of the palace of Austrasia towards 640—were devoted to the Crown.
On the other hand, mayors like Flaochat (in Burgundy) and Erkinoald (in
Neustria) stirred up the great nobles, who claimed the right to take part in
their nomination, against the king. Others again, sought to exercise the
power in their own name both against the king and against the great nobles
—such as Ebroïn (in Neustria), and, later, the Carolingians Pippin II.,
Charles Martel, and Pippin III., who, after making use of the great nobles,
kept the authority for themselves. In 751 Pippin III., fortified by his
consultation with Pope Zacharias, could quite naturally exchange the title of
mayor for that of king; and when he became king, he suppressed the title of
mayor of the palace. It must be observed that from 639 there were generally
separate mayors of Neustria, Austrasia and Burgundy, even when Austrasia
and Burgundy formed a single kingdom; the mayor was a sign of the
independence of the region. Each mayor, however, sought to supplant the
others; the Pippins and Charles Martel succeeded, and their victory was at
the same time the victory of Austrasia over Neustria and Burgundy.
See G. H. Pertz, Geschichte der merowingischen Hausmeier
(Hanover, 1819); H. Bonnell, De dignitate majoris domus (Berlin,
1858); E. Hermann, Das Hausmeieramt, ein echt germanisches Amt,
vol. ix. of Untersuchungen zur deutschen Staats- und Rechtsgeschichte,
ed. by O. Gierke (Breslau, 1878, seq.); G. Waitz, Deutsche
Verfassungsgeschichte, 3rd ed., revised by K. Zeumer; and Fustel de
The mayor of the palace thus found himself at the head of the commendati,
just as he was at the head of the functionaries.
It is difficult to trace the names of some of the mayors of the palace, the
post being of almost no significance in the time of Gregory of Tours. When
the office increased in importance the mayors of the palace did not, as has
been thought, pursue an identical policy. Some—for instance, Otto, the
mayor of the palace of Austrasia towards 640—were devoted to the Crown.
On the other hand, mayors like Flaochat (in Burgundy) and Erkinoald (in
Neustria) stirred up the great nobles, who claimed the right to take part in
their nomination, against the king. Others again, sought to exercise the
power in their own name both against the king and against the great nobles
—such as Ebroïn (in Neustria), and, later, the Carolingians Pippin II.,
Charles Martel, and Pippin III., who, after making use of the great nobles,
kept the authority for themselves. In 751 Pippin III., fortified by his
consultation with Pope Zacharias, could quite naturally exchange the title of
mayor for that of king; and when he became king, he suppressed the title of
mayor of the palace. It must be observed that from 639 there were generally
separate mayors of Neustria, Austrasia and Burgundy, even when Austrasia
and Burgundy formed a single kingdom; the mayor was a sign of the
independence of the region. Each mayor, however, sought to supplant the
others; the Pippins and Charles Martel succeeded, and their victory was at
the same time the victory of Austrasia over Neustria and Burgundy.
See G. H. Pertz, Geschichte der merowingischen Hausmeier
(Hanover, 1819); H. Bonnell, De dignitate majoris domus (Berlin,
1858); E. Hermann, Das Hausmeieramt, ein echt germanisches Amt,
vol. ix. of Untersuchungen zur deutschen Staats- und Rechtsgeschichte,
ed. by O. Gierke (Breslau, 1878, seq.); G. Waitz, Deutsche
Verfassungsgeschichte, 3rd ed., revised by K. Zeumer; and Fustel de
Page 253
Coulanges, Histoire des institutions politiques de l’ancienne France:
La monarchie franque (Paris, 1888). (C. Pf.)
MAYORUNA, a tribe of South American Indians of Panoan stock.
Their country is between the Ucayali and Javari rivers, north-eastern Peru.
They are a fine race, roaming the forests and living by hunting. They cut
their hair in a line across the forehead and let it hang down their backs.
Many have fair skins and beards, a peculiarity sometimes explained by their
alleged descent from Ursua’s soldiers, but this theory is improbable. They
are famous for the potency of their blow-gun poison.
MAYO-SMITH, RICHMOND (1854-1901), American
economist, was born in Troy, Ohio, on the 9th of February 1854. Educated
at Amherst, and at Berlin and Heidelberg, he became assistant professor of
economics at Columbia University in 1877. He was an adjunct professor
from 1878 to 1883, when he was appointed professor of political economy
and social science, a post which he held until his death on the 11th of
November 1901. He devoted himself especially to the study of statistics,
La monarchie franque (Paris, 1888). (C. Pf.)
MAYORUNA, a tribe of South American Indians of Panoan stock.
Their country is between the Ucayali and Javari rivers, north-eastern Peru.
They are a fine race, roaming the forests and living by hunting. They cut
their hair in a line across the forehead and let it hang down their backs.
Many have fair skins and beards, a peculiarity sometimes explained by their
alleged descent from Ursua’s soldiers, but this theory is improbable. They
are famous for the potency of their blow-gun poison.
MAYO-SMITH, RICHMOND (1854-1901), American
economist, was born in Troy, Ohio, on the 9th of February 1854. Educated
at Amherst, and at Berlin and Heidelberg, he became assistant professor of
economics at Columbia University in 1877. He was an adjunct professor
from 1878 to 1883, when he was appointed professor of political economy
and social science, a post which he held until his death on the 11th of
November 1901. He devoted himself especially to the study of statistics,
Page 254
and was recognized as one of the foremost authorities on the subject. His
works include Emigration and Immigration (1890); Sociology and Statistics
(1895), and Statistics and Economics (1899).
MAYOTTE, one of the Comoro Islands, in the Mozambique
Channel between Madagascar and the African mainland. It has belonged to
France since 1843 (see Comoro Islands).
MAYOW, JOHN (1643-1679), English chemist and physiologist,
was born in London in May 1643. At the age of fifteen he went up to
Wadham College, Oxford, of which he became a scholar a year later, and in
1660 he was elected to a fellowship at All Souls. He graduated in law
(bachelor, 1665, doctor, 1670), but made medicine his profession, and
“became noted for his practice therein, especially in the summer time, in the
city of Bath.” In 1678, on the proposal of R. Hooke, he was chosen a fellow
of the Royal Society. The following year, after a marriage which was “not
altogether to his content,” he died in London in September 1679. He
works include Emigration and Immigration (1890); Sociology and Statistics
(1895), and Statistics and Economics (1899).
MAYOTTE, one of the Comoro Islands, in the Mozambique
Channel between Madagascar and the African mainland. It has belonged to
France since 1843 (see Comoro Islands).
MAYOW, JOHN (1643-1679), English chemist and physiologist,
was born in London in May 1643. At the age of fifteen he went up to
Wadham College, Oxford, of which he became a scholar a year later, and in
1660 he was elected to a fellowship at All Souls. He graduated in law
(bachelor, 1665, doctor, 1670), but made medicine his profession, and
“became noted for his practice therein, especially in the summer time, in the
city of Bath.” In 1678, on the proposal of R. Hooke, he was chosen a fellow
of the Royal Society. The following year, after a marriage which was “not
altogether to his content,” he died in London in September 1679. He
Page 255
published at Oxford in 1668 two tracts, on respiration and rickets, and in
1674 these were reprinted, the former in an enlarged and corrected form,
with three others “De sal-nitro et spiritu nitro-aereo,” “De respiratione
foetus in utero et ovo,” and “De motu musculari et spiritibus animalibus” as
Tractatus quinque medico-physici. The contents of this work, which was
several times republished and translated into Dutch, German and French,
show him to have been an investigator much in advance of his time.
Accepting as proved by Boyle’s experiments that air is necessary for
combustion, he showed that fire is supported not by the air as a whole
but by a “more active and subtle part of it.” This part he called spiritus
igneo-aereus, or sometimes nitro-aereus; for he identified it with one
of the constituents of the acid portion of nitre which he regarded as
formed by the union of fixed alkali with a spiritus acidus. In
combustion the particulae nitro-aereae—either pre-existent in the
thing consumed or supplied by the air—combined with the material
burnt; as he inferred from his observation that antimony, strongly
heated with a burning glass, undergoes an increase of weight which can
be attributed to nothing else but these particles. In respiration he argued
that the same particles are consumed, because he found that when a
small animal and a lighted candle were placed in a closed vessel full of
air the candle first went out and soon afterwards the animal died, but if
there was no candle present it lived twice as long. He concluded that
this constituent of the air is absolutely necessary for life, and supposed
that the lungs separate it from the atmosphere and pass it into the
blood. It is also necessary, he inferred, for all muscular movements,
and he thought there was reason to believe that the sudden contraction
of muscle is produced by its combination with other combustible
(salino-sulphureous) particles in the body; hence the heart, being a
muscle, ceases to beat when respiration is stopped. Animal heat also is
due to the union of nitro-aerial particles, breathed in from the air, with
1674 these were reprinted, the former in an enlarged and corrected form,
with three others “De sal-nitro et spiritu nitro-aereo,” “De respiratione
foetus in utero et ovo,” and “De motu musculari et spiritibus animalibus” as
Tractatus quinque medico-physici. The contents of this work, which was
several times republished and translated into Dutch, German and French,
show him to have been an investigator much in advance of his time.
Accepting as proved by Boyle’s experiments that air is necessary for
combustion, he showed that fire is supported not by the air as a whole
but by a “more active and subtle part of it.” This part he called spiritus
igneo-aereus, or sometimes nitro-aereus; for he identified it with one
of the constituents of the acid portion of nitre which he regarded as
formed by the union of fixed alkali with a spiritus acidus. In
combustion the particulae nitro-aereae—either pre-existent in the
thing consumed or supplied by the air—combined with the material
burnt; as he inferred from his observation that antimony, strongly
heated with a burning glass, undergoes an increase of weight which can
be attributed to nothing else but these particles. In respiration he argued
that the same particles are consumed, because he found that when a
small animal and a lighted candle were placed in a closed vessel full of
air the candle first went out and soon afterwards the animal died, but if
there was no candle present it lived twice as long. He concluded that
this constituent of the air is absolutely necessary for life, and supposed
that the lungs separate it from the atmosphere and pass it into the
blood. It is also necessary, he inferred, for all muscular movements,
and he thought there was reason to believe that the sudden contraction
of muscle is produced by its combination with other combustible
(salino-sulphureous) particles in the body; hence the heart, being a
muscle, ceases to beat when respiration is stopped. Animal heat also is
due to the union of nitro-aerial particles, breathed in from the air, with
Page 256
the combustible particles in the blood, and is further formed by the
combination of these two sets of particles in muscle during violent
exertion. In effect, therefore, Mayow—who also gives a remarkably
correct anatomical description of the mechanism of respiration—
preceded Priestley and Lavoisier by a century in recognizing the
existence of oxygen, under the guise of his spiritus nitro-aereus, as a
separate entity distinct from the general mass of the air; he perceived
the part it plays in combustion and in increasing the weight of the
calces of metals as compared with metals themselves; and, rejecting the
common notions of his time that the use of breathing is to cool the
heart, or assist the passage of the blood from the right to the left side of
the heart, or merely to agitate it, he saw in inspiration a mechanism for
introducing oxygen into the body, where it is consumed for the
production of heat and muscular activity, and even vaguely conceived
of expiration as an excretory process.
MAYSVILLE, a city and the county-seat of Mason county,
Kentucky, U.S.A., on the Ohio river, 60 m. by rail S.E. of Cincinnati. Pop.
(1890) 5358; (1900) 6423 (1155 negroes); (1910) 6141. It is served by the
Louisville & Nashville, and the Chesapeake & Ohio railways, and by
steamboats on the Ohio river. Among its principal buildings are the Mason
county public library (1878), the Federal building and Masonic and Odd
Fellows’ temples. The city lies between the river and a range of hills; at the
back of the hills is a fine farming country, of which tobacco of excellent
combination of these two sets of particles in muscle during violent
exertion. In effect, therefore, Mayow—who also gives a remarkably
correct anatomical description of the mechanism of respiration—
preceded Priestley and Lavoisier by a century in recognizing the
existence of oxygen, under the guise of his spiritus nitro-aereus, as a
separate entity distinct from the general mass of the air; he perceived
the part it plays in combustion and in increasing the weight of the
calces of metals as compared with metals themselves; and, rejecting the
common notions of his time that the use of breathing is to cool the
heart, or assist the passage of the blood from the right to the left side of
the heart, or merely to agitate it, he saw in inspiration a mechanism for
introducing oxygen into the body, where it is consumed for the
production of heat and muscular activity, and even vaguely conceived
of expiration as an excretory process.
MAYSVILLE, a city and the county-seat of Mason county,
Kentucky, U.S.A., on the Ohio river, 60 m. by rail S.E. of Cincinnati. Pop.
(1890) 5358; (1900) 6423 (1155 negroes); (1910) 6141. It is served by the
Louisville & Nashville, and the Chesapeake & Ohio railways, and by
steamboats on the Ohio river. Among its principal buildings are the Mason
county public library (1878), the Federal building and Masonic and Odd
Fellows’ temples. The city lies between the river and a range of hills; at the
back of the hills is a fine farming country, of which tobacco of excellent
Page 257
quality is a leading product. There is a large plant of the American Tobacco
Company at Maysville, and among the city’s manufactures are pulleys,
ploughs, whisky, flour, lumber, furniture, carriages, cigars, foundry and
machine-shop products, bricks and cotton goods. The city is a distributing
point for coal and other products brought to it by Ohio river boats. Formerly
it was one of the principal hemp markets of the country. The place early
became a landing point for immigrants to Kentucky, and in 1784 a double
log cabin and a blockhouse were erected here. It was then called Limestone,
from the creek which flows into the Ohio here, but several years later the
present name was adopted in honour of John May, who with Simon Kenton
laid out the town in 1787, and who in 1790 was killed by the Indians.
Maysville was incorporated as a town in 1787, was chartered as a city in
1833, and became the county-seat in 1848.
In 1830, when the question of “internal improvements” by the
National government was an important political issue, Congress passed
a bill directing the government to aid in building a turnpike road from
Maysville to Lexington. President Andrew Jackson vetoed the bill on
the ground that the proposed improvement was a local rather than a
national one; but one-half the capital was then furnished privately, the
other half was furnished through several state appropriations, and the
road was completed in 1835 and marked the beginning of a system of
turnpike roads built with state aid.
Company at Maysville, and among the city’s manufactures are pulleys,
ploughs, whisky, flour, lumber, furniture, carriages, cigars, foundry and
machine-shop products, bricks and cotton goods. The city is a distributing
point for coal and other products brought to it by Ohio river boats. Formerly
it was one of the principal hemp markets of the country. The place early
became a landing point for immigrants to Kentucky, and in 1784 a double
log cabin and a blockhouse were erected here. It was then called Limestone,
from the creek which flows into the Ohio here, but several years later the
present name was adopted in honour of John May, who with Simon Kenton
laid out the town in 1787, and who in 1790 was killed by the Indians.
Maysville was incorporated as a town in 1787, was chartered as a city in
1833, and became the county-seat in 1848.
In 1830, when the question of “internal improvements” by the
National government was an important political issue, Congress passed
a bill directing the government to aid in building a turnpike road from
Maysville to Lexington. President Andrew Jackson vetoed the bill on
the ground that the proposed improvement was a local rather than a
national one; but one-half the capital was then furnished privately, the
other half was furnished through several state appropriations, and the
road was completed in 1835 and marked the beginning of a system of
turnpike roads built with state aid.
Page 258
MAZAGAN (El Jadīda), a port on the Atlantic coast of Morocco in
33° 16′ N. 8° 26′ W. Pop. (1908), about 12,000, of whom a fourth are Jews
and some 400 Europeans. It is the port for Marrákesh, from which it is 110
m. nearly due north, and also for the fertile province of Dukálla. Mazagan
presents from the sea a very un-Moorish appearance; it has massive
Portuguese walls of hewn stone. The exports, which include beans,
almonds, maize, chick-peas, wool, hides, wax, eggs, &c., were valued at
£360,000 in 1900, £364,000 in 1904, and £248,000 in 1906. The imports
(cotton goods, sugar, tea, rice, &c.) were valued at £280,000 in 1900,
£286,000 in 1904, and £320,000 in 1906. About 46% of the trade is with
Great Britain and 34% with France. Mazagan was built in 1506 by the
Portuguese, who abandoned it to the Moors in 1769 and established a
colony, New Mazagan, on the shores of Para in Brazil.
See A. H. Dyé, “Les ports du Maroc” in Bull. Soc. Geog. Comm.
Paris, xxx. 325-332 (1908), and British consular reports.
MAZAMET, an industrial town of south-western France in the
department of Tarn, 41 m. S.S.E. of Albi by rail. Pop. (1906), town, 11,370;
commune, 14,386. Mazamet is situated on the northern slope of the
Montagnes Noires and on the Arnette, a small sub-tributary of the Agout.
Numerous establishments are employed in wool-spinning and in the
manufacture of “swan-skins” and flannels, and clothing for troops, and
hosiery, and there are important tanneries and leather-dressing, glove and
33° 16′ N. 8° 26′ W. Pop. (1908), about 12,000, of whom a fourth are Jews
and some 400 Europeans. It is the port for Marrákesh, from which it is 110
m. nearly due north, and also for the fertile province of Dukálla. Mazagan
presents from the sea a very un-Moorish appearance; it has massive
Portuguese walls of hewn stone. The exports, which include beans,
almonds, maize, chick-peas, wool, hides, wax, eggs, &c., were valued at
£360,000 in 1900, £364,000 in 1904, and £248,000 in 1906. The imports
(cotton goods, sugar, tea, rice, &c.) were valued at £280,000 in 1900,
£286,000 in 1904, and £320,000 in 1906. About 46% of the trade is with
Great Britain and 34% with France. Mazagan was built in 1506 by the
Portuguese, who abandoned it to the Moors in 1769 and established a
colony, New Mazagan, on the shores of Para in Brazil.
See A. H. Dyé, “Les ports du Maroc” in Bull. Soc. Geog. Comm.
Paris, xxx. 325-332 (1908), and British consular reports.
MAZAMET, an industrial town of south-western France in the
department of Tarn, 41 m. S.S.E. of Albi by rail. Pop. (1906), town, 11,370;
commune, 14,386. Mazamet is situated on the northern slope of the
Montagnes Noires and on the Arnette, a small sub-tributary of the Agout.
Numerous establishments are employed in wool-spinning and in the
manufacture of “swan-skins” and flannels, and clothing for troops, and
hosiery, and there are important tanneries and leather-dressing, glove and
Page 259
dye works. Extensive commerce is carried on in wool and raw hides from
Argentina, Australia and Cape Colony.
MAZANDARAN, a province of northern Persia, lying between
the Caspian Sea and the Elburz range, and bounded E. and W. by the
provinces of Astarabad and Gilan respectively, 220 m. in length and 60 m.
in (mean) breadth, with an area of about 10,000 sq. m. and a population
estimated at from 150,000 to 200,000. Mazandaran comprises two distinct
natural regions presenting the sharpest contrasts in their relief, climate and
products. In the north the Caspian is encircled by the level and swampy
lowlands, varying in breadth from 10 to 30 m., partly under impenetrable
jungle, partly under rice, cotton, sugar and other crops. This section is
fringed northwards by the sandy beach of the Caspian, here almost destitute
of natural harbours, and rises somewhat abruptly inland to the second
section, comprising the northern slopes and spurs of the Elburz, which
approach at some points within 1 or 2 m. of the sea, and are almost
everywhere covered with dense forest. The lowlands, rising but a few feet
above the Caspian, and subject to frequent floodings, are extremely
malarious, while the highlands, culminating with the magnificent
Demavend (19,400 ft.), enjoy a tolerably healthy climate. But the climate,
generally hot and moist in summer, is everywhere capricious and liable to
sudden changes of temperature, whence the prevalence of rheumatism,
dropsy and especially ophthalmia, noticed by all travellers. Snow falls
heavily in the uplands, where it often lies for weeks on the ground. The
Argentina, Australia and Cape Colony.
MAZANDARAN, a province of northern Persia, lying between
the Caspian Sea and the Elburz range, and bounded E. and W. by the
provinces of Astarabad and Gilan respectively, 220 m. in length and 60 m.
in (mean) breadth, with an area of about 10,000 sq. m. and a population
estimated at from 150,000 to 200,000. Mazandaran comprises two distinct
natural regions presenting the sharpest contrasts in their relief, climate and
products. In the north the Caspian is encircled by the level and swampy
lowlands, varying in breadth from 10 to 30 m., partly under impenetrable
jungle, partly under rice, cotton, sugar and other crops. This section is
fringed northwards by the sandy beach of the Caspian, here almost destitute
of natural harbours, and rises somewhat abruptly inland to the second
section, comprising the northern slopes and spurs of the Elburz, which
approach at some points within 1 or 2 m. of the sea, and are almost
everywhere covered with dense forest. The lowlands, rising but a few feet
above the Caspian, and subject to frequent floodings, are extremely
malarious, while the highlands, culminating with the magnificent
Demavend (19,400 ft.), enjoy a tolerably healthy climate. But the climate,
generally hot and moist in summer, is everywhere capricious and liable to
sudden changes of temperature, whence the prevalence of rheumatism,
dropsy and especially ophthalmia, noticed by all travellers. Snow falls
heavily in the uplands, where it often lies for weeks on the ground. The
Page 260
direction of the long sandbanks at the river mouths, which project with
remarkable uniformity from west to east, shows that the prevailing winds
blow from the west and north-west. The rivers themselves, of which there
are as many as fifty, are little more than mountain torrents, all rising on the
northern slopes of Elburz, flowing mostly in independent channels to the
Caspian, and subject to sudden freshets and inundations along their lower
course. The chief are the Sardab-rud, Chalus, Herhaz (Lar in its upper
course), Babul, Tejen and Nika, and all are well stocked with trout, salmon
(azad-mahi), perch (safid-mahi), carp (kupur), bream (subulu), sturgeon
(sag-mahi) and other fish, which with rice form the staple food of the
inhabitants; the sturgeon supplies the caviare for the Russian market. Near
their mouths the rivers, running counter to the prevailing winds and waves
of the Caspian, form long sand-hills 20 to 30 ft. high and about 200 yds.
broad, behind which are developed the so-called múrd-áb, or “dead waters,”
stagnant pools and swamps characteristic of this coast, and a main cause of
its unhealthiness.
The chief products are rice, cotton, sugar, a little silk, and fruits in great
variety, including several kinds of the orange, lemon and citron. Some of
the slopes are covered with extensive thickets of the pomegranate, and the
wild vine climbs to a great height round the trunks of the forest trees. These
woodlands are haunted by the tiger, panther, bear, wolf and wild boar in
considerable numbers. Of the domestic animals, all remarkable for their
small size, the chief are the black, humped cattle somewhat resembling the
Indian variety, and sheep and goats.
Kinneir, Fraser and other observers speak unfavourably of the
Mazandarani people, whom they describe as very ignorant and bigoted,
arrogant, rudely inquisitive and almost insolent towards strangers. The
peasantry, however, are far from dull, and betray much shrewdness
where their interests are concerned. In the healthy districts they are
remarkable uniformity from west to east, shows that the prevailing winds
blow from the west and north-west. The rivers themselves, of which there
are as many as fifty, are little more than mountain torrents, all rising on the
northern slopes of Elburz, flowing mostly in independent channels to the
Caspian, and subject to sudden freshets and inundations along their lower
course. The chief are the Sardab-rud, Chalus, Herhaz (Lar in its upper
course), Babul, Tejen and Nika, and all are well stocked with trout, salmon
(azad-mahi), perch (safid-mahi), carp (kupur), bream (subulu), sturgeon
(sag-mahi) and other fish, which with rice form the staple food of the
inhabitants; the sturgeon supplies the caviare for the Russian market. Near
their mouths the rivers, running counter to the prevailing winds and waves
of the Caspian, form long sand-hills 20 to 30 ft. high and about 200 yds.
broad, behind which are developed the so-called múrd-áb, or “dead waters,”
stagnant pools and swamps characteristic of this coast, and a main cause of
its unhealthiness.
The chief products are rice, cotton, sugar, a little silk, and fruits in great
variety, including several kinds of the orange, lemon and citron. Some of
the slopes are covered with extensive thickets of the pomegranate, and the
wild vine climbs to a great height round the trunks of the forest trees. These
woodlands are haunted by the tiger, panther, bear, wolf and wild boar in
considerable numbers. Of the domestic animals, all remarkable for their
small size, the chief are the black, humped cattle somewhat resembling the
Indian variety, and sheep and goats.
Kinneir, Fraser and other observers speak unfavourably of the
Mazandarani people, whom they describe as very ignorant and bigoted,
arrogant, rudely inquisitive and almost insolent towards strangers. The
peasantry, however, are far from dull, and betray much shrewdness
where their interests are concerned. In the healthy districts they are
Page 261
stout and well made, and are considered a warlike race, furnishing
some cavalry (800 men) and eight battalions of infantry (5600 men) to
government. They speak a marked Persian dialect, but a Tūrki idiom
closely akin to the Turkoman is still current amongst the tribes,
although they have mostly already passed from the nomad to the settled
state. Of these tribes the most numerous are the Modaunlū,
Khojehvand and Abdul Maleki, originally of Lek or Kurd stock,
besides branches of the royal Afshār and Kājār tribes of Tūrki descent.
All these are exempt from taxes in consideration of their military
service.
The export trade is chiefly with Russia from Meshed-i-Sar, the
principal port of the province, to Baku, where European goods are
taken in exchange for the white and coloured calicoes, caviare, rice,
fruits and raw cotton of Mazandarān. Great quantities of rice are also
exported to the interior of Persia, principally to Teheran and Kazvin.
Owing to the almost impenetrable character of the country there are
scarcely any roads accessible to wheeled carriages, and the great
causeway of Shah Abbas along the coast has in many places even
disappeared under the jungle. Two routes, however, lead to Teheran,
one by Firuz Kuh, 180 m. long, the other by Larijan, 144 m. long, both
in tolerably good repair. Except where crossed by these routes the
Elburz forms an almost impassable barrier to the south.
The administration is in the hands of a governor, who appoints the
sub-governors of the nine districts of Amol, Barfarush, Meshed-i-Sar,
Sari, Ashref, Farah-abad, Tunakabun, Kelarrustak and Kujur into which
the province is divided. There is fair security for life and property; and,
although otherwise indifferently administered, the country is quite free
from marauders; but local disturbances have latterly been frequent in
the two last-named districts. The revenue is about £30,000, of which
some cavalry (800 men) and eight battalions of infantry (5600 men) to
government. They speak a marked Persian dialect, but a Tūrki idiom
closely akin to the Turkoman is still current amongst the tribes,
although they have mostly already passed from the nomad to the settled
state. Of these tribes the most numerous are the Modaunlū,
Khojehvand and Abdul Maleki, originally of Lek or Kurd stock,
besides branches of the royal Afshār and Kājār tribes of Tūrki descent.
All these are exempt from taxes in consideration of their military
service.
The export trade is chiefly with Russia from Meshed-i-Sar, the
principal port of the province, to Baku, where European goods are
taken in exchange for the white and coloured calicoes, caviare, rice,
fruits and raw cotton of Mazandarān. Great quantities of rice are also
exported to the interior of Persia, principally to Teheran and Kazvin.
Owing to the almost impenetrable character of the country there are
scarcely any roads accessible to wheeled carriages, and the great
causeway of Shah Abbas along the coast has in many places even
disappeared under the jungle. Two routes, however, lead to Teheran,
one by Firuz Kuh, 180 m. long, the other by Larijan, 144 m. long, both
in tolerably good repair. Except where crossed by these routes the
Elburz forms an almost impassable barrier to the south.
The administration is in the hands of a governor, who appoints the
sub-governors of the nine districts of Amol, Barfarush, Meshed-i-Sar,
Sari, Ashref, Farah-abad, Tunakabun, Kelarrustak and Kujur into which
the province is divided. There is fair security for life and property; and,
although otherwise indifferently administered, the country is quite free
from marauders; but local disturbances have latterly been frequent in
the two last-named districts. The revenue is about £30,000, of which
Page 262
little goes to the state treasury, most being required for the governors,
troops and pensions. The capital is Sari, the other chief towns being
Barfarush, Meshed-i-Sar, Ashref and Farah-abad. (A. H.-S.)
MAZARIN, JULES (1602-1661), French cardinal and
statesman, elder son of a Sicilian, Pietro Mazarini, the intendant of the
household of Philip Colonna, and of his wife Ortensia Buffalini, a
connexion of the Colonnas, was born at Piscina in the Abruzzi on the 14th
of July 1602. He was educated by the Jesuits at Rome till his seventeenth
year, when he accompanied Jerome Colonna as chamberlain to the
university of Alcala in Spain. There he distinguished himself more by his
love of gambling and his gallant adventures than by study, but made himself
a thorough master, not only of the Spanish language and character, but also
of that romantic fashion of Spanish love-making which was to help him
greatly in after life, when he became the servant of a Spanish queen. On his
return to Rome, about 1622, he took his degree as Doctor utriusque juris,
and then became captain of infantry in the regiment of Colonna, which took
part in the war in the Valtelline. During this war he gave proofs of much
diplomatic ability, and Pope Urban VIII. entrusted him, in 1629, with the
difficult task of putting an end to the war of the Mantuan succession. His
success marked him out for further distinction. He was presented to two
canonries in the churches of St John Lateran and Sta Maria Maggiore,
although he had only taken the minor orders, and had never been
consecrated priest; he negotiated the treaty of Turin between France and
troops and pensions. The capital is Sari, the other chief towns being
Barfarush, Meshed-i-Sar, Ashref and Farah-abad. (A. H.-S.)
MAZARIN, JULES (1602-1661), French cardinal and
statesman, elder son of a Sicilian, Pietro Mazarini, the intendant of the
household of Philip Colonna, and of his wife Ortensia Buffalini, a
connexion of the Colonnas, was born at Piscina in the Abruzzi on the 14th
of July 1602. He was educated by the Jesuits at Rome till his seventeenth
year, when he accompanied Jerome Colonna as chamberlain to the
university of Alcala in Spain. There he distinguished himself more by his
love of gambling and his gallant adventures than by study, but made himself
a thorough master, not only of the Spanish language and character, but also
of that romantic fashion of Spanish love-making which was to help him
greatly in after life, when he became the servant of a Spanish queen. On his
return to Rome, about 1622, he took his degree as Doctor utriusque juris,
and then became captain of infantry in the regiment of Colonna, which took
part in the war in the Valtelline. During this war he gave proofs of much
diplomatic ability, and Pope Urban VIII. entrusted him, in 1629, with the
difficult task of putting an end to the war of the Mantuan succession. His
success marked him out for further distinction. He was presented to two
canonries in the churches of St John Lateran and Sta Maria Maggiore,
although he had only taken the minor orders, and had never been
consecrated priest; he negotiated the treaty of Turin between France and
Page 263
Savoy in 1632, became vice-legate at Avignon in 1634, and nuncio at the
court of France from 1634 to 1636. But he began to wish for a wider sphere
than papal negotiations, and, seeing that he had no chance of becoming a
cardinal except by the aid of some great power, he accepted Richelieu’s
offer of entering the service of the king of France, and in 1639 became a
naturalized Frenchman.
In 1640 Richelieu sent him to Savoy, where the regency of Christine, the
duchess of Savoy, and sister of Louis XIII., was disputed by her brothers-in-
law, the princes Maurice and Thomas of Savoy, and he succeeded not only
in firmly establishing Christine but in winning over the princes to France.
This great service was rewarded by his promotion to the rank of cardinal on
the presentation of the king of France in December 1641. On the 4th of
December 1642 Cardinal Richelieu died, and on the very next day the king
sent a circular letter to all officials ordering them to send in their reports to
Cardinal Mazarin, as they had formerly done to Cardinal Richelieu.
Mazarin was thus acknowledged supreme minister, but he still had a
difficult part to play. The king evidently could not live long, and to preserve
power he must make himself necessary to the queen, who would then be
regent, and do this without arousing the suspicions of the king or the
distrust of the queen. His measures were ably taken, and when the king
died, on the 14th of May 1643, to everyone’s surprise her husband’s
minister remained the queen’s. The king had by a royal edict cumbered the
queen-regent with a council and other restrictions, and it was necessary to
get the parlement of Paris to overrule the edict and make the queen absolute
regent, which was done with the greatest complaisance. Now that the queen
was all-powerful, it was expected she would at once dismiss Mazarin and
summon her own friends to power. One of them, Potier, bishop of Beauvais,
already gave himself airs as prime minister, but Mazarin had had the
address to touch both the queen’s heart by his Spanish gallantry and her
desire for her son’s glory by his skilful policy abroad, and he found himself
court of France from 1634 to 1636. But he began to wish for a wider sphere
than papal negotiations, and, seeing that he had no chance of becoming a
cardinal except by the aid of some great power, he accepted Richelieu’s
offer of entering the service of the king of France, and in 1639 became a
naturalized Frenchman.
In 1640 Richelieu sent him to Savoy, where the regency of Christine, the
duchess of Savoy, and sister of Louis XIII., was disputed by her brothers-in-
law, the princes Maurice and Thomas of Savoy, and he succeeded not only
in firmly establishing Christine but in winning over the princes to France.
This great service was rewarded by his promotion to the rank of cardinal on
the presentation of the king of France in December 1641. On the 4th of
December 1642 Cardinal Richelieu died, and on the very next day the king
sent a circular letter to all officials ordering them to send in their reports to
Cardinal Mazarin, as they had formerly done to Cardinal Richelieu.
Mazarin was thus acknowledged supreme minister, but he still had a
difficult part to play. The king evidently could not live long, and to preserve
power he must make himself necessary to the queen, who would then be
regent, and do this without arousing the suspicions of the king or the
distrust of the queen. His measures were ably taken, and when the king
died, on the 14th of May 1643, to everyone’s surprise her husband’s
minister remained the queen’s. The king had by a royal edict cumbered the
queen-regent with a council and other restrictions, and it was necessary to
get the parlement of Paris to overrule the edict and make the queen absolute
regent, which was done with the greatest complaisance. Now that the queen
was all-powerful, it was expected she would at once dismiss Mazarin and
summon her own friends to power. One of them, Potier, bishop of Beauvais,
already gave himself airs as prime minister, but Mazarin had had the
address to touch both the queen’s heart by his Spanish gallantry and her
desire for her son’s glory by his skilful policy abroad, and he found himself
Page 264
able easily to overthrow the clique of Importants, as they were called. That
skilful policy was shown in every arena on which the great Thirty Years’
War was being fought out. Mazarin had inherited the policy of France
during the Thirty Years’ War from Richelieu. He had inherited his desire for
the humiliation of the house of Austria in both its branches, his desire to
push the French frontier to the Rhine and maintain a counterpoise of
German states against Austria, his alliances with the Netherlands and with
Sweden, and his four theatres of war—on the Rhine, in Flanders, in Italy
and in Catalonia.
During the last five years of the great war it was Mazarin alone who
directed the French diplomacy of the period. He it was who made the peace
of Brömsebro between the Danes and the Swedes, and turned the latter once
again against the empire; he it was who sent Lionne to make the peace of
Castro, and combine the princes of North Italy against the Spaniards, and
who made the peace of Ulm between France and Bavaria, thus detaching
the emperor’s best ally. He made one fatal mistake—he dreamt of the
French frontier being the Rhine and the Scheldt, and that a Spanish princess
might bring the Spanish Netherlands as dowry to Louis XIV. This roused
the jealousy of the United Provinces, and they made a separate peace with
Spain in January 1648; but the valour of the French generals made the skill
of the Spanish diplomatists of no avail, for Turenne’s victory at
Zusmarshausen, and Condé’s at Lens, caused the peace of Westphalia to be
definitely signed in October 1648. This celebrated treaty belongs rather to
the history of Germany than to a life of Mazarin; but two questions have
been often asked, whether Mazarin did not delay the peace as long as
possible in order to more completely ruin Germany, and whether Richelieu
would have made a similar peace. To the first question Mazarin’s letters,
published by M. Chéruel, prove a complete negative, for in them appears
the zeal of Mazarin for the peace. On the second point, Richelieu’s letters in
many places indicate that his treatment of the great question of frontier
skilful policy was shown in every arena on which the great Thirty Years’
War was being fought out. Mazarin had inherited the policy of France
during the Thirty Years’ War from Richelieu. He had inherited his desire for
the humiliation of the house of Austria in both its branches, his desire to
push the French frontier to the Rhine and maintain a counterpoise of
German states against Austria, his alliances with the Netherlands and with
Sweden, and his four theatres of war—on the Rhine, in Flanders, in Italy
and in Catalonia.
During the last five years of the great war it was Mazarin alone who
directed the French diplomacy of the period. He it was who made the peace
of Brömsebro between the Danes and the Swedes, and turned the latter once
again against the empire; he it was who sent Lionne to make the peace of
Castro, and combine the princes of North Italy against the Spaniards, and
who made the peace of Ulm between France and Bavaria, thus detaching
the emperor’s best ally. He made one fatal mistake—he dreamt of the
French frontier being the Rhine and the Scheldt, and that a Spanish princess
might bring the Spanish Netherlands as dowry to Louis XIV. This roused
the jealousy of the United Provinces, and they made a separate peace with
Spain in January 1648; but the valour of the French generals made the skill
of the Spanish diplomatists of no avail, for Turenne’s victory at
Zusmarshausen, and Condé’s at Lens, caused the peace of Westphalia to be
definitely signed in October 1648. This celebrated treaty belongs rather to
the history of Germany than to a life of Mazarin; but two questions have
been often asked, whether Mazarin did not delay the peace as long as
possible in order to more completely ruin Germany, and whether Richelieu
would have made a similar peace. To the first question Mazarin’s letters,
published by M. Chéruel, prove a complete negative, for in them appears
the zeal of Mazarin for the peace. On the second point, Richelieu’s letters in
many places indicate that his treatment of the great question of frontier
Page 265
would have been more thorough, but then he would not have been
hampered in France itself.
At home Mazarin’s policy lacked the strength of Richelieu’s. The
Frondes were largely due to his own fault. The arrest of Broussel threw the
people on the side of the parlement. His avarice and unscrupulous
plundering of the revenues of the realm, the enormous fortune which he
thus amassed, his supple ways, his nepotism, and the general lack of public
interest in the great foreign policy of Richelieu, made Mazarin the especial
object of hatred both by bourgeois and nobles. The irritation of the latter
was greatly Mazarin’s own fault; he had tried consistently to play off the
king’s brother Gaston of Orleans against Condé, and their respective
followers against each other, and had also, as his carnets prove, jealously
kept any courtier from getting into the good graces of the queen-regent
except by his means, so that it was not unnatural that the nobility should
hate him, while the queen found herself surrounded by his creatures alone.
Events followed each other quickly; the day of the barricades was followed
by the peace of Ruel, the peace of Ruel by the arrest of the princes, by the
battle of Rethel, and Mazarin’s exile to Brühl before the union of the two
Frondes. It was while in exile at Brühl that Mazarin saw the mistake he had
made in isolating himself and the queen, and that his policy of balancing
every party in the state against each other had made every party distrust
him. So by his counsel the queen, while nominally in league with De Retz
and the parliamentary Fronde, laboured to form a purely royal party,
wearied by civil dissensions, who should act for her and her son’s interest
alone, under the leadership of Mathieu Molé, the famous premier president
of the parlement of Paris. The new party grew in strength, and in January
1652, after exactly a year’s absence, Mazarin returned to the court. Turenne
had now become the royal general, and out-manœuvred Condé, while the
royal party at last grew to such strength in Paris that Condé had to leave the
capital and France. In order to promote a reconciliation with the parlement
hampered in France itself.
At home Mazarin’s policy lacked the strength of Richelieu’s. The
Frondes were largely due to his own fault. The arrest of Broussel threw the
people on the side of the parlement. His avarice and unscrupulous
plundering of the revenues of the realm, the enormous fortune which he
thus amassed, his supple ways, his nepotism, and the general lack of public
interest in the great foreign policy of Richelieu, made Mazarin the especial
object of hatred both by bourgeois and nobles. The irritation of the latter
was greatly Mazarin’s own fault; he had tried consistently to play off the
king’s brother Gaston of Orleans against Condé, and their respective
followers against each other, and had also, as his carnets prove, jealously
kept any courtier from getting into the good graces of the queen-regent
except by his means, so that it was not unnatural that the nobility should
hate him, while the queen found herself surrounded by his creatures alone.
Events followed each other quickly; the day of the barricades was followed
by the peace of Ruel, the peace of Ruel by the arrest of the princes, by the
battle of Rethel, and Mazarin’s exile to Brühl before the union of the two
Frondes. It was while in exile at Brühl that Mazarin saw the mistake he had
made in isolating himself and the queen, and that his policy of balancing
every party in the state against each other had made every party distrust
him. So by his counsel the queen, while nominally in league with De Retz
and the parliamentary Fronde, laboured to form a purely royal party,
wearied by civil dissensions, who should act for her and her son’s interest
alone, under the leadership of Mathieu Molé, the famous premier president
of the parlement of Paris. The new party grew in strength, and in January
1652, after exactly a year’s absence, Mazarin returned to the court. Turenne
had now become the royal general, and out-manœuvred Condé, while the
royal party at last grew to such strength in Paris that Condé had to leave the
capital and France. In order to promote a reconciliation with the parlement
Page 266
of Paris Mazarin had again retired from court, this time to Sedan, in August
1652, but he returned finally in February 1653. Long had been the trial, and
greatly had Mazarin been to blame in allowing the Frondes to come into
existence, but he had retrieved his position by founding that great royal
party which steadily grew until Louis XIV. could fairly have said “L’État,
c’est moi.” As the war had progressed, Mazarin had steadily followed
Richelieu’s policy of weakening the nobles on their country estates.
Whenever he had an opportunity he destroyed a feudal castle, and by
destroying the towers which commanded nearly every town in France, he
freed such towns as Bourges, for instance, from their long practical
subjection to the neighbouring great lord.
The Fronde over, Mazarin had to build up afresh the power of France at
home and abroad. It is to his shame that he did so little at home. Beyond
destroying the brick-and-mortar remains of feudalism, he did nothing for
the people. But abroad his policy was everywhere successful, and opened
the way for the policy of Louis XIV. He at first, by means of an alliance
with Cromwell, recovered the north-western cities of France, though at the
price of yielding Dunkirk to the Protector. On the Baltic, France guaranteed
the Treaty of Oliva between her old allies Sweden, Poland and
Brandenburg, which preserved her influence in that quarter. In Germany he,
through Hugues de Lionne, formed the league of the Rhine, by which the
states along the Rhine bound themselves under the headship of France to be
on their guard against the house of Austria. By such measures Spain was
induced to sue for peace, which was finally signed in the Isle of Pheasants
on the Bidassoa, and is known as the Treaty of the Pyrenees. By it Spain
recovered Franche Comté, but ceded to France Roussillon, and much of
French Flanders; and, what was of greater ultimate importance to Europe,
Louis XIV. was to marry a Spanish princess, who was to renounce her
claims to the Spanish succession if her dowry was paid, which Mazarin
knew could not happen at present from the emptiness of the Spanish
1652, but he returned finally in February 1653. Long had been the trial, and
greatly had Mazarin been to blame in allowing the Frondes to come into
existence, but he had retrieved his position by founding that great royal
party which steadily grew until Louis XIV. could fairly have said “L’État,
c’est moi.” As the war had progressed, Mazarin had steadily followed
Richelieu’s policy of weakening the nobles on their country estates.
Whenever he had an opportunity he destroyed a feudal castle, and by
destroying the towers which commanded nearly every town in France, he
freed such towns as Bourges, for instance, from their long practical
subjection to the neighbouring great lord.
The Fronde over, Mazarin had to build up afresh the power of France at
home and abroad. It is to his shame that he did so little at home. Beyond
destroying the brick-and-mortar remains of feudalism, he did nothing for
the people. But abroad his policy was everywhere successful, and opened
the way for the policy of Louis XIV. He at first, by means of an alliance
with Cromwell, recovered the north-western cities of France, though at the
price of yielding Dunkirk to the Protector. On the Baltic, France guaranteed
the Treaty of Oliva between her old allies Sweden, Poland and
Brandenburg, which preserved her influence in that quarter. In Germany he,
through Hugues de Lionne, formed the league of the Rhine, by which the
states along the Rhine bound themselves under the headship of France to be
on their guard against the house of Austria. By such measures Spain was
induced to sue for peace, which was finally signed in the Isle of Pheasants
on the Bidassoa, and is known as the Treaty of the Pyrenees. By it Spain
recovered Franche Comté, but ceded to France Roussillon, and much of
French Flanders; and, what was of greater ultimate importance to Europe,
Louis XIV. was to marry a Spanish princess, who was to renounce her
claims to the Spanish succession if her dowry was paid, which Mazarin
knew could not happen at present from the emptiness of the Spanish
Page 267
exchequer. He returned to Paris in declining health, and did not long survive
the unhealthy sojourn on the Bidassoa; after some political instruction to his
young master he passed away at Vincennes on the 9th of March 1661,
leaving a fortune estimated at from 18 to 40 million livres behind him, and
his nieces married into the greatest families of France and Italy.
The man who could have had such success, who could have made
the Treaties of Westphalia and the Pyrenees, who could have weathered
the storm of the Fronde, and left France at peace with itself and with
Europe to Louis XIV., must have been a great man; and historians,
relying too much on the brilliant memoirs of his adversaries, like De
Retz, are apt to rank him too low. That he had many a petty fault there
can be no doubt; that he was avaricious and double-dealing was also
undoubted; and his carnets show to what unworthy means he had
recourse to maintain his influence over the queen. What that influence
was will be always debated, but both his carnets and the Brühl letters
show that a real personal affection, amounting to passion on the
queen’s part, existed. Whether they were ever married may be doubted;
but that hypothesis is made more possible by M. Chéruel’s having been
able to prove from Mazarin’s letters that the cardinal himself had never
taken more than the minor orders, which could always be thrown off.
With regard to France he played a more patriotic part than Condé or
Turenne, for he never treated with the Spaniards, and his letters show
that in the midst of his difficulties he followed with intense eagerness
every movement on the frontiers. It is that immense mass of letters that
prove the real greatness of the statesman, and disprove De Retz’s
portrait, which is carefully arranged to show off his enemy against the
might of Richelieu. To concede that the master was the greater man and
the greater statesman does not imply that Mazarin was but a foil to his
predecessor. It is true that we find none of those deep plans for the
internal prosperity of France which shine through Richelieu’s policy.
the unhealthy sojourn on the Bidassoa; after some political instruction to his
young master he passed away at Vincennes on the 9th of March 1661,
leaving a fortune estimated at from 18 to 40 million livres behind him, and
his nieces married into the greatest families of France and Italy.
The man who could have had such success, who could have made
the Treaties of Westphalia and the Pyrenees, who could have weathered
the storm of the Fronde, and left France at peace with itself and with
Europe to Louis XIV., must have been a great man; and historians,
relying too much on the brilliant memoirs of his adversaries, like De
Retz, are apt to rank him too low. That he had many a petty fault there
can be no doubt; that he was avaricious and double-dealing was also
undoubted; and his carnets show to what unworthy means he had
recourse to maintain his influence over the queen. What that influence
was will be always debated, but both his carnets and the Brühl letters
show that a real personal affection, amounting to passion on the
queen’s part, existed. Whether they were ever married may be doubted;
but that hypothesis is made more possible by M. Chéruel’s having been
able to prove from Mazarin’s letters that the cardinal himself had never
taken more than the minor orders, which could always be thrown off.
With regard to France he played a more patriotic part than Condé or
Turenne, for he never treated with the Spaniards, and his letters show
that in the midst of his difficulties he followed with intense eagerness
every movement on the frontiers. It is that immense mass of letters that
prove the real greatness of the statesman, and disprove De Retz’s
portrait, which is carefully arranged to show off his enemy against the
might of Richelieu. To concede that the master was the greater man and
the greater statesman does not imply that Mazarin was but a foil to his
predecessor. It is true that we find none of those deep plans for the
internal prosperity of France which shine through Richelieu’s policy.
Page 268
Mazarin was not a Frenchman, but a citizen of the world, and always
paid most attention to foreign affairs; in his letters all that could teach a
diplomatist is to be found, broad general views of policy, minute details
carefully elaborated, keen insight into men’s characters, cunning
directions when to dissimulate or when to be frank. Italian though he
was by birth, education and nature, France owed him a great debt for
his skilful management during the early years of Louis XIV., and the
king owed him yet more, for he had not only transmitted to him a
nation at peace, but had educated for him his great servants Le Tellier,
Lionne and Colbert. Literary men owed him also much; not only did he
throw his famous library open to them, but he pensioned all their
leaders, including Descartes, Vincent Voiture (1598-1648), Jean Louis
Guez de Balzac (1597-1654) and Pierre Corneille. The last-named
applied, with an adroit allusion to his birthplace, in the dedication of
his Pompée, the line of Virgil:—
“Tu regere imperio populos, Romane, memento.”
(H. M. S.)
Authorities.—All the earlier works on Mazarin, and early accounts
of his administration, of which the best were Bazin’s Histoire de
France sous Louis XIII. et sous le Cardinal Mazarin, 4 vols. (1846),
and Saint-Aulaire’s Histoire de la Fronde, have been superseded by P.
A. Chéruel’s admirable Histoire de France pendant la minorité de
Louis XIV., 4 vols. (1879-1880), which covers from 1643-1651, and its
sequel Histoire de France sous le ministère de Cardinal Mazarin, 2
vols. (1881-1882), which is the first account of the period written by
one able to sift the statements of De Retz and the memoir writers, and
rest upon such documents as Mazarin’s letters and carnets. Mazarin’s
Lettres, which must be carefully studied by any student of the history
of France, have appeared in the Collection des documents inédits, 9
vols. For his carnets reference must be made to V. Cousin’s articles in
paid most attention to foreign affairs; in his letters all that could teach a
diplomatist is to be found, broad general views of policy, minute details
carefully elaborated, keen insight into men’s characters, cunning
directions when to dissimulate or when to be frank. Italian though he
was by birth, education and nature, France owed him a great debt for
his skilful management during the early years of Louis XIV., and the
king owed him yet more, for he had not only transmitted to him a
nation at peace, but had educated for him his great servants Le Tellier,
Lionne and Colbert. Literary men owed him also much; not only did he
throw his famous library open to them, but he pensioned all their
leaders, including Descartes, Vincent Voiture (1598-1648), Jean Louis
Guez de Balzac (1597-1654) and Pierre Corneille. The last-named
applied, with an adroit allusion to his birthplace, in the dedication of
his Pompée, the line of Virgil:—
“Tu regere imperio populos, Romane, memento.”
(H. M. S.)
Authorities.—All the earlier works on Mazarin, and early accounts
of his administration, of which the best were Bazin’s Histoire de
France sous Louis XIII. et sous le Cardinal Mazarin, 4 vols. (1846),
and Saint-Aulaire’s Histoire de la Fronde, have been superseded by P.
A. Chéruel’s admirable Histoire de France pendant la minorité de
Louis XIV., 4 vols. (1879-1880), which covers from 1643-1651, and its
sequel Histoire de France sous le ministère de Cardinal Mazarin, 2
vols. (1881-1882), which is the first account of the period written by
one able to sift the statements of De Retz and the memoir writers, and
rest upon such documents as Mazarin’s letters and carnets. Mazarin’s
Lettres, which must be carefully studied by any student of the history
of France, have appeared in the Collection des documents inédits, 9
vols. For his carnets reference must be made to V. Cousin’s articles in
Page 269
the Journal des Savants, and Chéruel in Revue historique (1877), see
also Chéruel’s Histoire de France pendant la minorité, &c., app. to vol.
iii.; for his early life to Cousin’s Jeunesse de Mazarin (1865) and for
the careers of his nieces to Renée’s Les Nièces de Mazarin (1856). For
the Mazarinades or squibs written against him in Paris during the
Fronde, see C. Moreau’s Bibliographie des mazarinades (1850),
containing an account of 4082 Mazarinades. See also A. Hassall,
Mazarin (1903).
MAZAR-I-SHARIF, a town of Afghanistan, the capital of the
province of Afghan Turkestan. Owing to the importance of the military
cantonment of Takhtapul, and its religious sanctity, it has long ago
supplanted the more ancient capital of Balkh. It is situated in a malarious,
almost desert plain, 9 m. E. of Balkh, and 30 m. S. of the Pata Kesar ferry
on the Oxus river. In this neighbourhood is concentrated most of the Afghan
army north of the Hindu Kush mountains, the fortified cantonment of
Dehdadi having been completed by Sirdar Ghulam Ali Khan and
incorporated with Mazar. Mazar-i-Sharif also contains a celebrated mosque,
from which the town takes its name. It is a huge ornate building with
minarets and a lofty cupola faced with shining blue tiles. It was built by
Sultan Ali Mirza about a.d. 1420, and is held in great veneration by all
Mussulmans, and especially by Shiites, because it is supposed to be the
tomb of Ali, the son-in-law of Mahomet.
also Chéruel’s Histoire de France pendant la minorité, &c., app. to vol.
iii.; for his early life to Cousin’s Jeunesse de Mazarin (1865) and for
the careers of his nieces to Renée’s Les Nièces de Mazarin (1856). For
the Mazarinades or squibs written against him in Paris during the
Fronde, see C. Moreau’s Bibliographie des mazarinades (1850),
containing an account of 4082 Mazarinades. See also A. Hassall,
Mazarin (1903).
MAZAR-I-SHARIF, a town of Afghanistan, the capital of the
province of Afghan Turkestan. Owing to the importance of the military
cantonment of Takhtapul, and its religious sanctity, it has long ago
supplanted the more ancient capital of Balkh. It is situated in a malarious,
almost desert plain, 9 m. E. of Balkh, and 30 m. S. of the Pata Kesar ferry
on the Oxus river. In this neighbourhood is concentrated most of the Afghan
army north of the Hindu Kush mountains, the fortified cantonment of
Dehdadi having been completed by Sirdar Ghulam Ali Khan and
incorporated with Mazar. Mazar-i-Sharif also contains a celebrated mosque,
from which the town takes its name. It is a huge ornate building with
minarets and a lofty cupola faced with shining blue tiles. It was built by
Sultan Ali Mirza about a.d. 1420, and is held in great veneration by all
Mussulmans, and especially by Shiites, because it is supposed to be the
tomb of Ali, the son-in-law of Mahomet.
Page 270
MAZARRÓN, a town of eastern Spain, in the province of Murcia,
19 m. W. of Cartagena. Pop. (1900), 23,284. There are soap and flour mills
and metallurgic factories in the town, and iron, copper and lead mines in the
neighbouring Sierra de Almenara. A railway 5 m. long unites Mazarron to
its port on the Mediterranean, where there is a suburb with 2500 inhabitants
(mostly engaged in fisheries and coasting trade), containing barracks, a
custom-house, and important leadworks. Outside of the suburb there are
saltpans, most of the proceeds of which are exported to Galicia.
MAZATLÁN, a city and port of the state of Sinaloa, Mexico, 120
m. (direct) W.S.W. of the city of Durango, in lat. 23° 12′ N., long 106° 24′
W. Pop. (1895), 15,852; (1900), 17,852. It is the Pacific coast terminus of
the International railway which crosses northern Mexico from Ciudad
Porfirio Diaz, and a port of call for the principal steamship lines on this
coast. The harbour is spacious, but the entrance is obstructed by a bar. The
city is built on a small peninsula. Its public buildings include a fine town-
hall, chamber of commerce, a custom-house and two hospitals, besides
19 m. W. of Cartagena. Pop. (1900), 23,284. There are soap and flour mills
and metallurgic factories in the town, and iron, copper and lead mines in the
neighbouring Sierra de Almenara. A railway 5 m. long unites Mazarron to
its port on the Mediterranean, where there is a suburb with 2500 inhabitants
(mostly engaged in fisheries and coasting trade), containing barracks, a
custom-house, and important leadworks. Outside of the suburb there are
saltpans, most of the proceeds of which are exported to Galicia.
MAZATLÁN, a city and port of the state of Sinaloa, Mexico, 120
m. (direct) W.S.W. of the city of Durango, in lat. 23° 12′ N., long 106° 24′
W. Pop. (1895), 15,852; (1900), 17,852. It is the Pacific coast terminus of
the International railway which crosses northern Mexico from Ciudad
Porfirio Diaz, and a port of call for the principal steamship lines on this
coast. The harbour is spacious, but the entrance is obstructed by a bar. The
city is built on a small peninsula. Its public buildings include a fine town-
hall, chamber of commerce, a custom-house and two hospitals, besides
Page 271
which there is a nautical school and a meteorological station, one of the first
established in Mexico. The harbour is provided with a sea-wall at Olas
Altas. A government wireless telegraph service is maintained between
Mazatlán and La Paz, Lower California. Among the manufactures are saw-
mills, foundries, cotton factories and ropeworks, and the exports are chiefly
hides, ixtle, dried and salted fish, gold, silver and copper (bars and ores),
fruit, rubber, tortoise-shell, and gums and resins.
MAZE, a network of winding paths, a labyrinth (q.v.). The word
means properly a state of confusion or wonder, and is probably of
Scandinavian origin; cf. Norw. mas, exhausting labour, also chatter, masa,
to be busy, also to worry, annoy; Swed. masa, to lounge, move slowly and
lazily, to dream, muse. Skeat (Etym. Dict.) takes the original sense to be
probably “to be lost in thought,” “to dream,” and connects with the root ma-
man-, to think, cf. “mind,” “man,” &c. The word “maze” represents the
addition of an intensive suffix.
established in Mexico. The harbour is provided with a sea-wall at Olas
Altas. A government wireless telegraph service is maintained between
Mazatlán and La Paz, Lower California. Among the manufactures are saw-
mills, foundries, cotton factories and ropeworks, and the exports are chiefly
hides, ixtle, dried and salted fish, gold, silver and copper (bars and ores),
fruit, rubber, tortoise-shell, and gums and resins.
MAZE, a network of winding paths, a labyrinth (q.v.). The word
means properly a state of confusion or wonder, and is probably of
Scandinavian origin; cf. Norw. mas, exhausting labour, also chatter, masa,
to be busy, also to worry, annoy; Swed. masa, to lounge, move slowly and
lazily, to dream, muse. Skeat (Etym. Dict.) takes the original sense to be
probably “to be lost in thought,” “to dream,” and connects with the root ma-
man-, to think, cf. “mind,” “man,” &c. The word “maze” represents the
addition of an intensive suffix.
Page 272
MAZEPA-KOLEDINSKY, IVAN STEPANOVICH
(1644?-1709), hetman of the Cossacks, belonging to a noble Orthodox
family, was born possibly at Mazeptsina, either in 1629 or 1644, the latter
being the more probable date. He was educated at the court of the Polish
king, John Casimir, and completed his studies abroad. An intrigue with a
Polish married lady forced him to fly into the Ukraine. There is a
trustworthy tradition that the infuriated husband tied the naked youth to the
back of a wild horse and sent him forth into the steppe. He was rescued and
cared for by the Dnieperian Cossacks, and speedily became one of their
ablest leaders. In 1687, during a visit to Moscow, he won the favour of the
then all-powerful Vasily Golitsuin, from whom he virtually purchased the
hetmanship of the Cossacks (July 25). He took a very active part in the
Azov campaigns of Peter the Great and won the entire confidence of the
young tsar by his zeal and energy. He was also very serviceable to Peter at
the beginning of the Great Northern War, especially in 1705 and 1706,
when he took part in the Volhynian campaign and helped to construct the
fortress of Pechersk. The power and influence of Mazepa were fully
recognized by Peter the Great. No other Cossack hetman had ever been
treated with such deference at Moscow. He ranked with the highest
dignitaries in the state; he sat at the tsar’s own table. He had been made one
of the first cavaliers of the newly established order of St Andrew, and
Augustus of Poland had bestowed upon him, at Peter’s earnest solicitation,
the universally coveted order of the White Eagle. Mazepa had no
temptations to be anything but loyal, and loyal he would doubtless have
remained had not Charles XII. crossed the Russian frontier. Then it was that
Mazepa, who had had doubts of the issue of the struggle all along, made up
his mind that Charles, not Peter, was going to win, and that it was high time
he looked after his own interests. Besides, he had his personal grievances
against the tsar. He did not like the new ways because they interfered with
his old ones. He was very jealous of the favourite (Menshikov), whom he
(1644?-1709), hetman of the Cossacks, belonging to a noble Orthodox
family, was born possibly at Mazeptsina, either in 1629 or 1644, the latter
being the more probable date. He was educated at the court of the Polish
king, John Casimir, and completed his studies abroad. An intrigue with a
Polish married lady forced him to fly into the Ukraine. There is a
trustworthy tradition that the infuriated husband tied the naked youth to the
back of a wild horse and sent him forth into the steppe. He was rescued and
cared for by the Dnieperian Cossacks, and speedily became one of their
ablest leaders. In 1687, during a visit to Moscow, he won the favour of the
then all-powerful Vasily Golitsuin, from whom he virtually purchased the
hetmanship of the Cossacks (July 25). He took a very active part in the
Azov campaigns of Peter the Great and won the entire confidence of the
young tsar by his zeal and energy. He was also very serviceable to Peter at
the beginning of the Great Northern War, especially in 1705 and 1706,
when he took part in the Volhynian campaign and helped to construct the
fortress of Pechersk. The power and influence of Mazepa were fully
recognized by Peter the Great. No other Cossack hetman had ever been
treated with such deference at Moscow. He ranked with the highest
dignitaries in the state; he sat at the tsar’s own table. He had been made one
of the first cavaliers of the newly established order of St Andrew, and
Augustus of Poland had bestowed upon him, at Peter’s earnest solicitation,
the universally coveted order of the White Eagle. Mazepa had no
temptations to be anything but loyal, and loyal he would doubtless have
remained had not Charles XII. crossed the Russian frontier. Then it was that
Mazepa, who had had doubts of the issue of the struggle all along, made up
his mind that Charles, not Peter, was going to win, and that it was high time
he looked after his own interests. Besides, he had his personal grievances
against the tsar. He did not like the new ways because they interfered with
his old ones. He was very jealous of the favourite (Menshikov), whom he
Page 273
suspected of a design to supplant him. But he proceeded very cautiously.
Indeed, he would have preferred to remain neutral, but he was not strong
enough to stand alone. The crisis came when Peter ordered him to co-
operate actively with the Russian forces in the Ukraine. At this very time he
was in communication with Charles’s first minister, Count Piper, and had
agreed to harbour the Swedes in the Ukraine and close it against the
Russians (Oct. 1708). The last doubt disappeared when Menshikov was sent
to supervise Mazepa. At the approach of his rival the old hetman hastened
to the Swedish outposts at Horki, in Severia. Mazepa’s treason took Peter
completely by surprise. He instantly commanded Menshikov to get a new
hetman elected and raze Baturin, Mazepa’s chief stronghold in the Ukraine,
to the ground. When Charles, a week later, passed Baturin by, all that
remained of the Cossack capital was a heap of smouldering mills and ruined
houses. The total destruction of Baturin, almost in sight of the Swedes,
overawed the bulk of the Cossacks into obedience, and Mazepa’s ancient
prestige was ruined in a day when the metropolitan of Kiev solemnly
excommunicated him from the high altar, and his effigy, after being dragged
with contumely through the mud at Kiev, was publicly burnt by the
common hangman. Henceforth Mazepa, perforce, attached himself to
Charles. What part he took at the battle of Poltava is not quite clear. After
the catastrophe he accompanied Charles to Turkey with some 1500
horsemen (the miserable remnant of his 80,000 warriors). The sultan
refused to surrender him to the tsar, though Peter offered 300,000 ducats for
his head. He died at Bender on the 22nd of August 1709.
See N. I. Kostomarov, Mazepa and the Mazepanites (Russ.) (St
Petersburg), 1885; R. Nisbet Bain, The First Romanovs (London,
1905); S. M. Solovev, History of Russia (Russ.), vol. xv. (St
Petersburg, 1895).
(R. N. B.)
Indeed, he would have preferred to remain neutral, but he was not strong
enough to stand alone. The crisis came when Peter ordered him to co-
operate actively with the Russian forces in the Ukraine. At this very time he
was in communication with Charles’s first minister, Count Piper, and had
agreed to harbour the Swedes in the Ukraine and close it against the
Russians (Oct. 1708). The last doubt disappeared when Menshikov was sent
to supervise Mazepa. At the approach of his rival the old hetman hastened
to the Swedish outposts at Horki, in Severia. Mazepa’s treason took Peter
completely by surprise. He instantly commanded Menshikov to get a new
hetman elected and raze Baturin, Mazepa’s chief stronghold in the Ukraine,
to the ground. When Charles, a week later, passed Baturin by, all that
remained of the Cossack capital was a heap of smouldering mills and ruined
houses. The total destruction of Baturin, almost in sight of the Swedes,
overawed the bulk of the Cossacks into obedience, and Mazepa’s ancient
prestige was ruined in a day when the metropolitan of Kiev solemnly
excommunicated him from the high altar, and his effigy, after being dragged
with contumely through the mud at Kiev, was publicly burnt by the
common hangman. Henceforth Mazepa, perforce, attached himself to
Charles. What part he took at the battle of Poltava is not quite clear. After
the catastrophe he accompanied Charles to Turkey with some 1500
horsemen (the miserable remnant of his 80,000 warriors). The sultan
refused to surrender him to the tsar, though Peter offered 300,000 ducats for
his head. He died at Bender on the 22nd of August 1709.
See N. I. Kostomarov, Mazepa and the Mazepanites (Russ.) (St
Petersburg), 1885; R. Nisbet Bain, The First Romanovs (London,
1905); S. M. Solovev, History of Russia (Russ.), vol. xv. (St
Petersburg, 1895).
(R. N. B.)
Page 274
MAZER, the name of a special type of drinking vessel, properly
made of maple-wood, and so-called from the spotted or “birds-eye”
marking on the wood (Ger. Maser, spot, marking, especially on wood; cf.
“measles”). These drinking vessels are shallow bowls without handles, with
a broad flat foot and a knob or boss in the centre of the inside, known
technically as the “print.” They were made from the 13th to the 16th
centuries, and were the most prized of the various wooden cups in use, and
so were ornamented with a rim of precious metal, generally of silver or
silver gilt; the foot and the “print” being also of metal. The depth of the
mazers seems to have decreased in course of time, those of the 16th century
that survive being much shallower than the earlier examples. There are
examples with wooden covers with a metal handle, such as the Flemish and
German mazers in the Franks Bequest in the British Museum. On the metal
rim is usually an inscription, religious or bacchanalian, and the “print” was
also often decorated. The later mazers sometimes had metal straps between
the rim and the foot.
A very fine mazer with silver gilt ornamentation 3 in. deep and 91⁄2
in. in diameter was sold in the Braikenridge collection in 1908 for
£2300. It bears the London hall-mark of 1534. This example is
illustrated in the article Plate: see also Drinking Vessels.
made of maple-wood, and so-called from the spotted or “birds-eye”
marking on the wood (Ger. Maser, spot, marking, especially on wood; cf.
“measles”). These drinking vessels are shallow bowls without handles, with
a broad flat foot and a knob or boss in the centre of the inside, known
technically as the “print.” They were made from the 13th to the 16th
centuries, and were the most prized of the various wooden cups in use, and
so were ornamented with a rim of precious metal, generally of silver or
silver gilt; the foot and the “print” being also of metal. The depth of the
mazers seems to have decreased in course of time, those of the 16th century
that survive being much shallower than the earlier examples. There are
examples with wooden covers with a metal handle, such as the Flemish and
German mazers in the Franks Bequest in the British Museum. On the metal
rim is usually an inscription, religious or bacchanalian, and the “print” was
also often decorated. The later mazers sometimes had metal straps between
the rim and the foot.
A very fine mazer with silver gilt ornamentation 3 in. deep and 91⁄2
in. in diameter was sold in the Braikenridge collection in 1908 for
£2300. It bears the London hall-mark of 1534. This example is
illustrated in the article Plate: see also Drinking Vessels.
Page 275
MAZURKA (Polish for a woman of the province of Mazovia), a
lively dance, originating in Poland, somewhat resembling the polka.It is
danced in couples, the music being in 3⁄8 or 3⁄4 time.
MAZZARA DEL VALLO, a town of Sicily, in the province of
Trapani, on the south-west coast of the island, 32 m. by rail S. of Trapani.
Pop. (1901), 20,130. It is the seat of a bishop; the cathedral, founded in
1093, was rebuilt in the 17th century. The castle, at the south-eastern angle
of the town walls, was erected in 1073. The mouth of the river, which bears
the same name, serves as a port for small ships only. Mazzara was in origin
a colony of Selinus: it was destroyed in 409, but it is mentioned again as a
Carthaginian fortress in the First Punic War and as a post station on the
Roman coast road, though whether it had municipal rights is doubtful.1 A
few inscriptions of the imperial period exist, but no other remains of
importance. On the west bank of the river are grottoes cut in the rock, of
uncertain date: and there are quarries in the neighbourhood resembling
those of Syracuse, but on a smaller scale.
lively dance, originating in Poland, somewhat resembling the polka.It is
danced in couples, the music being in 3⁄8 or 3⁄4 time.
MAZZARA DEL VALLO, a town of Sicily, in the province of
Trapani, on the south-west coast of the island, 32 m. by rail S. of Trapani.
Pop. (1901), 20,130. It is the seat of a bishop; the cathedral, founded in
1093, was rebuilt in the 17th century. The castle, at the south-eastern angle
of the town walls, was erected in 1073. The mouth of the river, which bears
the same name, serves as a port for small ships only. Mazzara was in origin
a colony of Selinus: it was destroyed in 409, but it is mentioned again as a
Carthaginian fortress in the First Punic War and as a post station on the
Roman coast road, though whether it had municipal rights is doubtful.1 A
few inscriptions of the imperial period exist, but no other remains of
importance. On the west bank of the river are grottoes cut in the rock, of
uncertain date: and there are quarries in the neighbourhood resembling
those of Syracuse, but on a smaller scale.
Page 276
See A. Castiglione, Sulle cose antiche della città di Mazzara
(Alcamo, 1878).
1 Th. Mommsen in Corpus inscr. lat. (Berlin, 1883), x. 739.
MAZZINI, GIUSEPPE (1805-1872), Italian patriot, was born
on the 22nd of June 1805 at Genoa, where his father, Giacomo Mazzini,
was a physician in good practice, and a professor in the university. His
mother is described as having been a woman of great personal beauty, as
well as of active intellect and strong affections. During infancy and
childhood his health was extremely delicate, and it appears that he was
nearly six years of age before he was quite able to walk; but he had already
begun to devour books of all kinds and to show other signs of great
intellectual precocity. He studied Latin with his first tutor, an old priest, but
no one directed his extensive course of reading. He became a student at the
university of Genoa at an unusually early age, and intended to follow his
father’s profession, but being unable to conquer his horror of practical
anatomy, he decided to graduate in law (1826). His exceptional abilities,
together with his remarkable generosity, kindness and loftiness of character,
endeared him to his fellow students. As to his inner life during this period,
we have only one brief but significant sentence; “for a short time,” he says,
“my mind was somewhat tainted by the doctrines of the foreign
materialistic school; but the study of history and the intuitions of conscience
(Alcamo, 1878).
1 Th. Mommsen in Corpus inscr. lat. (Berlin, 1883), x. 739.
MAZZINI, GIUSEPPE (1805-1872), Italian patriot, was born
on the 22nd of June 1805 at Genoa, where his father, Giacomo Mazzini,
was a physician in good practice, and a professor in the university. His
mother is described as having been a woman of great personal beauty, as
well as of active intellect and strong affections. During infancy and
childhood his health was extremely delicate, and it appears that he was
nearly six years of age before he was quite able to walk; but he had already
begun to devour books of all kinds and to show other signs of great
intellectual precocity. He studied Latin with his first tutor, an old priest, but
no one directed his extensive course of reading. He became a student at the
university of Genoa at an unusually early age, and intended to follow his
father’s profession, but being unable to conquer his horror of practical
anatomy, he decided to graduate in law (1826). His exceptional abilities,
together with his remarkable generosity, kindness and loftiness of character,
endeared him to his fellow students. As to his inner life during this period,
we have only one brief but significant sentence; “for a short time,” he says,
“my mind was somewhat tainted by the doctrines of the foreign
materialistic school; but the study of history and the intuitions of conscience
Page 277
—the only tests of truth—soon led me back to the spiritualism of our Italian
fathers.”
The natural bent of his genius was towards literature, and, in the course
of the four years of his nominal connexion with the legal profession, he
wrote a considerable number of essays and reviews, some of which have
been wholly or partially reproduced in the critical and literary volumes of
his Life and Writings. His first essay, characteristically enough on “Dante’s
Love of Country,” was sent to the editor of the Antologia fiorentina in 1826,
but did not appear until some years afterwards in the Subalpino. He was an
ardent supporter of romanticism as against what he called “literary
servitude under the name of classicism”; and in this interest all his critiques
(as, for example, that of Giannoni’s “Exile” in the Indicatore Livornese,
1829) were penned. But in the meantime the “republican instincts” which
he tells us he had inherited from his mother had been developing, and his
sense of the evils under which Italy was groaning had been intensified; and
at the same time he became possessed with the idea that Italians, and he
himself in particular, “could and therefore ought to struggle for liberty of
country.” Therefore, he at once put aside his dearest ambition, that of
producing a complete history of religion, developing his scheme of a new
theology uniting the spiritual with the practical life, and devoted himself to
political thought. His literary articles accordingly became more and more
suggestive of advanced liberalism in politics, and led to the suppression by
government of the Indicatore Genovese and the Indicatore Livornese
successively. Having joined the Carbonari, he soon rose to one of the higher
grades in their hierarchy, and was entrusted with a special secret mission
into Tuscany; but, as his acquaintance grew, his dissatisfaction with the
organization of the society increased, and he was already meditating the
formation of a new association stripped of foolish mysterious and theatrical
formulae, which instead of merely combating existing authorities should
have a definite and purely patriotic aim, when shortly after the French
fathers.”
The natural bent of his genius was towards literature, and, in the course
of the four years of his nominal connexion with the legal profession, he
wrote a considerable number of essays and reviews, some of which have
been wholly or partially reproduced in the critical and literary volumes of
his Life and Writings. His first essay, characteristically enough on “Dante’s
Love of Country,” was sent to the editor of the Antologia fiorentina in 1826,
but did not appear until some years afterwards in the Subalpino. He was an
ardent supporter of romanticism as against what he called “literary
servitude under the name of classicism”; and in this interest all his critiques
(as, for example, that of Giannoni’s “Exile” in the Indicatore Livornese,
1829) were penned. But in the meantime the “republican instincts” which
he tells us he had inherited from his mother had been developing, and his
sense of the evils under which Italy was groaning had been intensified; and
at the same time he became possessed with the idea that Italians, and he
himself in particular, “could and therefore ought to struggle for liberty of
country.” Therefore, he at once put aside his dearest ambition, that of
producing a complete history of religion, developing his scheme of a new
theology uniting the spiritual with the practical life, and devoted himself to
political thought. His literary articles accordingly became more and more
suggestive of advanced liberalism in politics, and led to the suppression by
government of the Indicatore Genovese and the Indicatore Livornese
successively. Having joined the Carbonari, he soon rose to one of the higher
grades in their hierarchy, and was entrusted with a special secret mission
into Tuscany; but, as his acquaintance grew, his dissatisfaction with the
organization of the society increased, and he was already meditating the
formation of a new association stripped of foolish mysterious and theatrical
formulae, which instead of merely combating existing authorities should
have a definite and purely patriotic aim, when shortly after the French
Page 278
revolution of 1830 he was betrayed, while initiating a new member, to the
Piedmontese authorities. He was imprisoned in the fortress of Savona on
the western Riviera for about six months, when, a conviction having been
found impracticable through deficiency of evidence, he was released, but
upon conditions involving so many restrictions of his liberty that he
preferred the alternative of leaving the country. He withdrew accordingly
into France, living chiefly in Marseilles.
While in his lonely cell at Savona, in presence of “those symbols of the
infinite, the sky and the sea,” with a greenfinch for his sole companion, and
having access to no books but “a Tacitus, a Byron, and a Bible,” he had
finally become aware of the great mission or “apostolate” (as he himself
called it) of his life; and soon after his release his prison meditations took
shape in the programme of the organization which was destined soon to
become so famous throughout Europe, that of La Giovine Italia, or Young
Italy. Its publicly avowed aims were to be the liberation of Italy both from
foreign and domestic tyranny, and its unification under a republican form of
government; the means to be used were education, and, where advisable,
insurrection by guerrilla bands; the motto was to be “God and the people,”
and the banner was to bear on one side the words “Unity” and
“Independence” and on the other “Liberty,” “Equality,” and “Humanity,” to
describe respectively the national and the international aims. In April 1831
Charles Albert, “the ex-Carbonaro conspirator of 1821,” succeeded Charles
Felix on the Sardinian throne, and towards the close of that year Mazzini,
making himself, as he afterwards confessed, “the interpreter of a hope
which he did not share,” wrote the new king a letter, published at
Marseilles, urging him to take the lead in the impending struggle for Italian
independence. Clandestinely reprinted, and rapidly circulated all over Italy,
its bold and outspoken words produced a great sensation, but so deep was
the offence it gave to the Sardinian government that orders were issued for
the immediate arrest and imprisonment of the author should he attempt to
Piedmontese authorities. He was imprisoned in the fortress of Savona on
the western Riviera for about six months, when, a conviction having been
found impracticable through deficiency of evidence, he was released, but
upon conditions involving so many restrictions of his liberty that he
preferred the alternative of leaving the country. He withdrew accordingly
into France, living chiefly in Marseilles.
While in his lonely cell at Savona, in presence of “those symbols of the
infinite, the sky and the sea,” with a greenfinch for his sole companion, and
having access to no books but “a Tacitus, a Byron, and a Bible,” he had
finally become aware of the great mission or “apostolate” (as he himself
called it) of his life; and soon after his release his prison meditations took
shape in the programme of the organization which was destined soon to
become so famous throughout Europe, that of La Giovine Italia, or Young
Italy. Its publicly avowed aims were to be the liberation of Italy both from
foreign and domestic tyranny, and its unification under a republican form of
government; the means to be used were education, and, where advisable,
insurrection by guerrilla bands; the motto was to be “God and the people,”
and the banner was to bear on one side the words “Unity” and
“Independence” and on the other “Liberty,” “Equality,” and “Humanity,” to
describe respectively the national and the international aims. In April 1831
Charles Albert, “the ex-Carbonaro conspirator of 1821,” succeeded Charles
Felix on the Sardinian throne, and towards the close of that year Mazzini,
making himself, as he afterwards confessed, “the interpreter of a hope
which he did not share,” wrote the new king a letter, published at
Marseilles, urging him to take the lead in the impending struggle for Italian
independence. Clandestinely reprinted, and rapidly circulated all over Italy,
its bold and outspoken words produced a great sensation, but so deep was
the offence it gave to the Sardinian government that orders were issued for
the immediate arrest and imprisonment of the author should he attempt to
Page 279
cross the frontier. Towards the end of the same year appeared the important
Young Italy “Manifesto,” the substance of which is given in the first volume
of the Life and Writings of Mazzini; and this was followed soon afterwards
by the society’s Journal, which, smuggled across the Italian frontier, had
great success in the objects for which it was written, numerous
“congregations” being formed at Genoa, Leghorn, and elsewhere.
Representations were consequently made by the Sardinian to the French
government, which issued in an order for Mazzini’s withdrawal from
Marseilles (Aug. 1832); he lingered for a few months in concealment, but
ultimately found it necessary to retire into Switzerland.
From this point it is somewhat difficult to follow the career of the
mysterious and terrible conspirator who for twenty years out of the next
thirty led a life of voluntary imprisonment (as he himself tells us) “within
the four walls of a room,” and “kept no record of dates, made no
biographical notes, and preserved no copies of letters.” In 1833, however,
he is known to have been concerned in an abortive revolutionary movement
which took place in the Sardinian army; several executions took place, and
he himself was laid under sentence of death. Before the close of the same
year a similar movement in Genoa had been planned, but failed through the
youth and inexperience of the leaders. At Geneva, also in 1833, Mazzini set
on foot L’Europe Centrale, a journal of which one of the main objects was
the emancipation of Savoy; but he did not confine himself to a merely
literary agitation for this end. Chiefly through his agency a considerable
body of German, Polish and Italian exiles was organized, and an armed
invasion of the duchy planned. The frontier was actually crossed on the 1st
of February 1834, but the attack ignominiously broke down without a shot
having been fired. Mazzini, who personally accompanied the expedition, is
no doubt correct in attributing the failure to dissensions with the Carbonari
leaders in Paris, and to want of a cordial understanding between himself and
the Savoyard Ramorino, who had been chosen as military leader.
Young Italy “Manifesto,” the substance of which is given in the first volume
of the Life and Writings of Mazzini; and this was followed soon afterwards
by the society’s Journal, which, smuggled across the Italian frontier, had
great success in the objects for which it was written, numerous
“congregations” being formed at Genoa, Leghorn, and elsewhere.
Representations were consequently made by the Sardinian to the French
government, which issued in an order for Mazzini’s withdrawal from
Marseilles (Aug. 1832); he lingered for a few months in concealment, but
ultimately found it necessary to retire into Switzerland.
From this point it is somewhat difficult to follow the career of the
mysterious and terrible conspirator who for twenty years out of the next
thirty led a life of voluntary imprisonment (as he himself tells us) “within
the four walls of a room,” and “kept no record of dates, made no
biographical notes, and preserved no copies of letters.” In 1833, however,
he is known to have been concerned in an abortive revolutionary movement
which took place in the Sardinian army; several executions took place, and
he himself was laid under sentence of death. Before the close of the same
year a similar movement in Genoa had been planned, but failed through the
youth and inexperience of the leaders. At Geneva, also in 1833, Mazzini set
on foot L’Europe Centrale, a journal of which one of the main objects was
the emancipation of Savoy; but he did not confine himself to a merely
literary agitation for this end. Chiefly through his agency a considerable
body of German, Polish and Italian exiles was organized, and an armed
invasion of the duchy planned. The frontier was actually crossed on the 1st
of February 1834, but the attack ignominiously broke down without a shot
having been fired. Mazzini, who personally accompanied the expedition, is
no doubt correct in attributing the failure to dissensions with the Carbonari
leaders in Paris, and to want of a cordial understanding between himself and
the Savoyard Ramorino, who had been chosen as military leader.
Page 280
In April 1834 the “Young Europe” association “of men believing in a
future of liberty, equality and fraternity for all mankind, and desirous of
consecrating their thoughts and actions to the realization of that future” was
formed also under the influence of Mazzini’s enthusiasm; it was followed
soon afterwards by a “Young Switzerland” society, having for its leading
idea the formation of an Alpine confederation, to include Switzerland,
Tyrol, Savoy and the rest of the Alpine chain as well. But La Jeune Suisse
newspaper was compelled to stop within a year, and in other respects the
affairs of the struggling patriot became embarrassed. He was permitted to
remain at Grenchen in Solothurn for a while, but at last the Swiss diet,
yielding to strong and persistent pressure from abroad, exiled him about the
end of 1836. In January 1837 he arrived in London, where for many months
he had to carry on a hard fight with poverty and the sense of spiritual
loneliness, so touchingly described by himself in the first volume of the Life
and Writings. Ultimately, as he gained command of the English language,
he began to earn a livelihood by writing review articles, some of which
have since been reprinted, and are of a high order of literary merit; they
include papers on “Italian Literature since 1830” and “Paolo Sarpi” in the
Westminster Review, articles on “Lamennais,” “George Sand,” “Byron and
Goethe” in the Monthly Chronicle, and on “Lamartine,” “Carlyle,” and
“The Minor Works of Dante” in the British and Foreign Review. In 1839 he
entered into relations with the revolutionary committees sitting in Malta and
Paris, and in 1840 he originated a working men’s association, and the
weekly journal entitled Apostolato Popolare, in which the admirable
popular treatise “On the Duties of Man” was commenced. Among the
patriotic and philanthropic labours undertaken by Mazzini during this
period of retirement in London may be mentioned a free evening school
conducted by himself and a few others for some years, at which several
hundreds of Italian children received at least the rudiments of secular and
religious education. He also exposed and combated the infamous traffic
future of liberty, equality and fraternity for all mankind, and desirous of
consecrating their thoughts and actions to the realization of that future” was
formed also under the influence of Mazzini’s enthusiasm; it was followed
soon afterwards by a “Young Switzerland” society, having for its leading
idea the formation of an Alpine confederation, to include Switzerland,
Tyrol, Savoy and the rest of the Alpine chain as well. But La Jeune Suisse
newspaper was compelled to stop within a year, and in other respects the
affairs of the struggling patriot became embarrassed. He was permitted to
remain at Grenchen in Solothurn for a while, but at last the Swiss diet,
yielding to strong and persistent pressure from abroad, exiled him about the
end of 1836. In January 1837 he arrived in London, where for many months
he had to carry on a hard fight with poverty and the sense of spiritual
loneliness, so touchingly described by himself in the first volume of the Life
and Writings. Ultimately, as he gained command of the English language,
he began to earn a livelihood by writing review articles, some of which
have since been reprinted, and are of a high order of literary merit; they
include papers on “Italian Literature since 1830” and “Paolo Sarpi” in the
Westminster Review, articles on “Lamennais,” “George Sand,” “Byron and
Goethe” in the Monthly Chronicle, and on “Lamartine,” “Carlyle,” and
“The Minor Works of Dante” in the British and Foreign Review. In 1839 he
entered into relations with the revolutionary committees sitting in Malta and
Paris, and in 1840 he originated a working men’s association, and the
weekly journal entitled Apostolato Popolare, in which the admirable
popular treatise “On the Duties of Man” was commenced. Among the
patriotic and philanthropic labours undertaken by Mazzini during this
period of retirement in London may be mentioned a free evening school
conducted by himself and a few others for some years, at which several
hundreds of Italian children received at least the rudiments of secular and
religious education. He also exposed and combated the infamous traffic
Page 281
carried on in southern Italy, where scoundrels bought small boys from
poverty-stricken parents and carried them off to England and elsewhere to
grind organs and suffer martyrdom at the hands of cruel taskmasters.
The most memorable episode in his life during the same period was
perhaps that which arose out of the conduct of Sir James Graham, the home
secretary, in systematically, for some months, opening Mazzini’s letters as
they passed through the British post office, and communicating their
contents to the Neapolitan government—a proceeding which was believed
at the time to have led to the arrest and execution of the brothers Bandiera,
Austrian subjects, who had been planning an expedition against Naples,
although the recent publication of Sir James Graham’s life seems to
exonerate him from the charge. The prolonged discussions in parliament,
and the report of the committee appointed to inquire into the matter, did not,
however, lead to any practical result, unless indeed the incidental
vindication of Mazzini’s character, which had been recklessly assailed in
the course of debate. In this connexion Thomas Carlyle wrote to The Times:
“I have had the honour to know Mr Mazzini for a series of years, and,
whatever I may think of his practical insight and skill in worldly affairs, I
can with great freedom testify that he, if I have ever seen one such, is a man
of genius and virtue, one of those rare men, numerable unfortunately but as
units in this world, who are worthy to be called martyr souls; who in
silence, piously in their daily life, practise what is meant by that.”
Mazzini did not share the enthusiastic hopes everywhere raised in the
ranks of the Liberal party throughout Europe by the first acts of Pius IX., in
1846, but at the same time he availed himself, towards the end of 1847, of
the opportunity to publish a letter addressed to the new pope, indicating the
nature of the religious and national mission which the Liberals expected
him to undertake. The leaders of the revolutionary outbreaks in Milan and
Messina in the beginning of 1848 had long been in secret correspondence
poverty-stricken parents and carried them off to England and elsewhere to
grind organs and suffer martyrdom at the hands of cruel taskmasters.
The most memorable episode in his life during the same period was
perhaps that which arose out of the conduct of Sir James Graham, the home
secretary, in systematically, for some months, opening Mazzini’s letters as
they passed through the British post office, and communicating their
contents to the Neapolitan government—a proceeding which was believed
at the time to have led to the arrest and execution of the brothers Bandiera,
Austrian subjects, who had been planning an expedition against Naples,
although the recent publication of Sir James Graham’s life seems to
exonerate him from the charge. The prolonged discussions in parliament,
and the report of the committee appointed to inquire into the matter, did not,
however, lead to any practical result, unless indeed the incidental
vindication of Mazzini’s character, which had been recklessly assailed in
the course of debate. In this connexion Thomas Carlyle wrote to The Times:
“I have had the honour to know Mr Mazzini for a series of years, and,
whatever I may think of his practical insight and skill in worldly affairs, I
can with great freedom testify that he, if I have ever seen one such, is a man
of genius and virtue, one of those rare men, numerable unfortunately but as
units in this world, who are worthy to be called martyr souls; who in
silence, piously in their daily life, practise what is meant by that.”
Mazzini did not share the enthusiastic hopes everywhere raised in the
ranks of the Liberal party throughout Europe by the first acts of Pius IX., in
1846, but at the same time he availed himself, towards the end of 1847, of
the opportunity to publish a letter addressed to the new pope, indicating the
nature of the religious and national mission which the Liberals expected
him to undertake. The leaders of the revolutionary outbreaks in Milan and
Messina in the beginning of 1848 had long been in secret correspondence
Page 282
with Mazzini; and their action, along with the revolution in Paris, brought
him early in the same year to Italy, where he took a great and active interest
in the events which dragged Charles Albert into an unprofitable war with
Austria; he actually for a short time bore arms under Garibaldi immediately
before the reoccupation of Milan, but ultimately, after vain attempts to
maintain the insurrection in the mountain districts, found it necessary to
retire to Lugano. In the beginning of the following year he was nominated a
member of the short-lived provisional government of Tuscany formed after
the flight of the grand-duke, and almost simultaneously, when Rome had, in
consequence of the withdrawal of Pius IX., been proclaimed a republic, he
was declared a member of the constituent assembly there. A month
afterwards, the battle of Novara having again decided against Charles
Albert in the brief struggle with Austria, into which he had once more been
drawn, Mazzini was appointed a member of the Roman triumvirate, with
supreme executive power (March 23, 1849). The opportunity he now had
for showing the administrative and political ability which he was believed
to possess was more apparent than real, for the approach of the professedly
friendly French troops soon led to hostilities, and resulted in a siege which
terminated, towards the end of June, with the assembly’s resolution to
discontinue the defence, and Mazzini’s indignant resignation. That he
succeeded, however, for so long a time, and in circumstances so adverse, in
maintaining a high degree of order within the turbulent city is a fact that
speaks for itself. His diplomacy, backed as it was by no adequate physical
force, naturally showed at the time to very great disadvantage, but his
official correspondence and proclamations can still be read with admiration
and intellectual pleasure, as well as his eloquent vindication of the
revolution in his published “Letter to MM. de Tocqueville and de Falloux.”
The surrender of the city on the 30th of June was followed by Mazzini’s not
too precipitate flight by way of Marseilles into Switzerland, whence he
once more found his way to London. Here in 1850 he became president of
him early in the same year to Italy, where he took a great and active interest
in the events which dragged Charles Albert into an unprofitable war with
Austria; he actually for a short time bore arms under Garibaldi immediately
before the reoccupation of Milan, but ultimately, after vain attempts to
maintain the insurrection in the mountain districts, found it necessary to
retire to Lugano. In the beginning of the following year he was nominated a
member of the short-lived provisional government of Tuscany formed after
the flight of the grand-duke, and almost simultaneously, when Rome had, in
consequence of the withdrawal of Pius IX., been proclaimed a republic, he
was declared a member of the constituent assembly there. A month
afterwards, the battle of Novara having again decided against Charles
Albert in the brief struggle with Austria, into which he had once more been
drawn, Mazzini was appointed a member of the Roman triumvirate, with
supreme executive power (March 23, 1849). The opportunity he now had
for showing the administrative and political ability which he was believed
to possess was more apparent than real, for the approach of the professedly
friendly French troops soon led to hostilities, and resulted in a siege which
terminated, towards the end of June, with the assembly’s resolution to
discontinue the defence, and Mazzini’s indignant resignation. That he
succeeded, however, for so long a time, and in circumstances so adverse, in
maintaining a high degree of order within the turbulent city is a fact that
speaks for itself. His diplomacy, backed as it was by no adequate physical
force, naturally showed at the time to very great disadvantage, but his
official correspondence and proclamations can still be read with admiration
and intellectual pleasure, as well as his eloquent vindication of the
revolution in his published “Letter to MM. de Tocqueville and de Falloux.”
The surrender of the city on the 30th of June was followed by Mazzini’s not
too precipitate flight by way of Marseilles into Switzerland, whence he
once more found his way to London. Here in 1850 he became president of
Page 283
the National Italian Committee, and at the same time entered into close
relations with Ledru-Rollin and Kossuth. He had a firm belief in the value
of revolutionary attempts, however hopeless they might seem; he had a
hand in the abortive rising at Mantua in 1852, and again, in February 1853,
a considerable share in the ill-planned insurrection at Milan on the 6th of
February 1853, the failure of which greatly weakened his influence; once
more, in 1854, he had gone far with preparations for renewed action when
his plans were completely disconcerted by the withdrawal of professed
supporters, and by the action of the French and English governments in
sending ships of war to Naples.
The year 1857 found him yet once more in Italy, where, for complicity in
short-lived émeutes which took place at Genoa, Leghorn and Naples, he
was again laid under sentence of death. Undiscouraged in the pursuit of the
one great aim of his life by any such incidents as these, he returned to
London, where he edited his new journal Pensiero ed Azione, in which the
constant burden of his message to the overcautious practical politicians of
Italy was: “I am but a voice crying Action; but the state of Italy cries for it
also. So do the best men and people of her cities. Do you wish to destroy
my influence? Act.” The same tone was at a somewhat later date assumed in
the letter he wrote to Victor Emmanuel, urging him to put himself at the
head of the movement for Italian unity, and promising republican support.
As regards the events of 1859-1860, however, it may be questioned
whether, through his characteristic inability to distinguish between the
ideally perfect and the practically possible, he did not actually hinder more
than he helped the course of events by which the realization of so much of
the great dream of his life was at last brought about. If Mazzini was the
prophet of Italian unity, and Garibaldi its knight errant, to Cavour alone
belongs the honour of having been the statesman by whom it was finally
accomplished. After the irresistible pressure of the popular movement had
led to the establishment not of an Italian republic but of an Italian kingdom,
relations with Ledru-Rollin and Kossuth. He had a firm belief in the value
of revolutionary attempts, however hopeless they might seem; he had a
hand in the abortive rising at Mantua in 1852, and again, in February 1853,
a considerable share in the ill-planned insurrection at Milan on the 6th of
February 1853, the failure of which greatly weakened his influence; once
more, in 1854, he had gone far with preparations for renewed action when
his plans were completely disconcerted by the withdrawal of professed
supporters, and by the action of the French and English governments in
sending ships of war to Naples.
The year 1857 found him yet once more in Italy, where, for complicity in
short-lived émeutes which took place at Genoa, Leghorn and Naples, he
was again laid under sentence of death. Undiscouraged in the pursuit of the
one great aim of his life by any such incidents as these, he returned to
London, where he edited his new journal Pensiero ed Azione, in which the
constant burden of his message to the overcautious practical politicians of
Italy was: “I am but a voice crying Action; but the state of Italy cries for it
also. So do the best men and people of her cities. Do you wish to destroy
my influence? Act.” The same tone was at a somewhat later date assumed in
the letter he wrote to Victor Emmanuel, urging him to put himself at the
head of the movement for Italian unity, and promising republican support.
As regards the events of 1859-1860, however, it may be questioned
whether, through his characteristic inability to distinguish between the
ideally perfect and the practically possible, he did not actually hinder more
than he helped the course of events by which the realization of so much of
the great dream of his life was at last brought about. If Mazzini was the
prophet of Italian unity, and Garibaldi its knight errant, to Cavour alone
belongs the honour of having been the statesman by whom it was finally
accomplished. After the irresistible pressure of the popular movement had
led to the establishment not of an Italian republic but of an Italian kingdom,
Page 284
Mazzini could honestly enough write, “I too have striven to realize unity
under a monarchical flag,” but candour compelled him to add, “The Italian
people are led astray by a delusion at the present day, a delusion which has
induced them to substitute material for moral unity and their own
reorganization. Not so I. I bow my head sorrowfully to the sovereignty of
the national will; but monarchy will never number me amongst its servants
or followers.” In 1865, by way of protest against the still uncancelled
sentence of death under which he lay, Mazzini was elected by Messina as
delegate to the Italian parliament, but, feeling himself unable to take the
oath of allegiance to the monarchy, he never took his seat. In the following
year, when a general amnesty was granted after the cession of Venice to
Italy, the sentence of death was at last removed, but he declined to accept
such an “offer of oblivion and pardon for having loved Italy above all
earthly things.” In May 1869 he was again expelled from Switzerland at the
instance of the Italian government for having conspired with Garibaldi;
after a few months spent in England he set out (1870) for Sicily, but was
promptly arrested at sea and carried to Gaeta, where he was imprisoned for
two months. Events soon made it evident that there was little danger to fear
from the contemplated rising, and the occasion of the birth of a prince was
seized for restoring him to liberty. The remainder of his life, spent partly in
London and partly at Lugano, presents no noteworthy incidents. For some
time his health had been far from satisfactory, but the immediate cause of
his death was an attack of pleurisy with which he was seized at Pisa, and
which terminated fatally on the 10th of March 1872. The Italian parliament
by a unanimous vote expressed the national sorrow with which the tidings
of his death had been received, the president pronouncing an eloquent
eulogy on the departed patriot as a model of disinterestedness and self-
denial, and one who had dedicated his whole life ungrudgingly to the cause
of his country’s freedom. A public funeral took place at Pisa on the 14th of
March, and the remains were afterwards conveyed to Genoa.
under a monarchical flag,” but candour compelled him to add, “The Italian
people are led astray by a delusion at the present day, a delusion which has
induced them to substitute material for moral unity and their own
reorganization. Not so I. I bow my head sorrowfully to the sovereignty of
the national will; but monarchy will never number me amongst its servants
or followers.” In 1865, by way of protest against the still uncancelled
sentence of death under which he lay, Mazzini was elected by Messina as
delegate to the Italian parliament, but, feeling himself unable to take the
oath of allegiance to the monarchy, he never took his seat. In the following
year, when a general amnesty was granted after the cession of Venice to
Italy, the sentence of death was at last removed, but he declined to accept
such an “offer of oblivion and pardon for having loved Italy above all
earthly things.” In May 1869 he was again expelled from Switzerland at the
instance of the Italian government for having conspired with Garibaldi;
after a few months spent in England he set out (1870) for Sicily, but was
promptly arrested at sea and carried to Gaeta, where he was imprisoned for
two months. Events soon made it evident that there was little danger to fear
from the contemplated rising, and the occasion of the birth of a prince was
seized for restoring him to liberty. The remainder of his life, spent partly in
London and partly at Lugano, presents no noteworthy incidents. For some
time his health had been far from satisfactory, but the immediate cause of
his death was an attack of pleurisy with which he was seized at Pisa, and
which terminated fatally on the 10th of March 1872. The Italian parliament
by a unanimous vote expressed the national sorrow with which the tidings
of his death had been received, the president pronouncing an eloquent
eulogy on the departed patriot as a model of disinterestedness and self-
denial, and one who had dedicated his whole life ungrudgingly to the cause
of his country’s freedom. A public funeral took place at Pisa on the 14th of
March, and the remains were afterwards conveyed to Genoa.
Page 285
( )
The published writings of Mazzini, mostly occasional, are very
voluminous. An edition was begun by himself and continued by A.
Saffi, Scritti editi e inediti di Giuseppe Mazzini, in 18 vols. (Milan and
Rome, 1861-1891); many of the most important are found in the
partially autobiographical Life and Writings of Joseph Mazzini (1864-
1870) and the two most systematic—Thoughts upon Democracy in
Europe, a remarkable series of criticisms on Benthamism, St
Simonianism, Fourierism, and other economic and socialistic schools
of the day, and the treatise On the Duties of Man, an admirable primer
of ethics, dedicated to the Italian working class—will be found in
Joseph Mazzini: a Memoir, by Mrs E. A. Venturi (London, 1875).
Mazzini’s “first great sacrifice,” he tells us, was “the renunciation of
the career of literature for the more direct path of political action,” and
as late as 1861 we find him still recurring to the long-cherished hope of
being able to leave the stormy arena of politics and consecrate the last
years of his life to the dream of his youth. He had specially
contemplated three considerable literary undertakings—a volume of
Thoughts on Religion, a popular History of Italy, to enable the working
classes to apprehend what he conceived to be the “mission” of Italy in
God’s providential ordering of the world, and a comprehensive
collection of translations of ancient and modern classics into Italian.
None of these was actually achieved. No one, however, can read even
the briefest and most occasional writing of Mazzini without gaining
some impression of the simple grandeur of the man, the lofty elevation
of his moral tone, his unwavering faith in the living God, who is ever
revealing Himself in the progressive development of humanity. His last
public utterance is to be found in a highly characteristic article on
Renan’s Réforme Morale et Intellectuelle, finished on the 3rd of March
1872, and published in the Fortnightly Review for February 1874. Of
the 40,000 letters of Mazzini only a small part have been published. In
The published writings of Mazzini, mostly occasional, are very
voluminous. An edition was begun by himself and continued by A.
Saffi, Scritti editi e inediti di Giuseppe Mazzini, in 18 vols. (Milan and
Rome, 1861-1891); many of the most important are found in the
partially autobiographical Life and Writings of Joseph Mazzini (1864-
1870) and the two most systematic—Thoughts upon Democracy in
Europe, a remarkable series of criticisms on Benthamism, St
Simonianism, Fourierism, and other economic and socialistic schools
of the day, and the treatise On the Duties of Man, an admirable primer
of ethics, dedicated to the Italian working class—will be found in
Joseph Mazzini: a Memoir, by Mrs E. A. Venturi (London, 1875).
Mazzini’s “first great sacrifice,” he tells us, was “the renunciation of
the career of literature for the more direct path of political action,” and
as late as 1861 we find him still recurring to the long-cherished hope of
being able to leave the stormy arena of politics and consecrate the last
years of his life to the dream of his youth. He had specially
contemplated three considerable literary undertakings—a volume of
Thoughts on Religion, a popular History of Italy, to enable the working
classes to apprehend what he conceived to be the “mission” of Italy in
God’s providential ordering of the world, and a comprehensive
collection of translations of ancient and modern classics into Italian.
None of these was actually achieved. No one, however, can read even
the briefest and most occasional writing of Mazzini without gaining
some impression of the simple grandeur of the man, the lofty elevation
of his moral tone, his unwavering faith in the living God, who is ever
revealing Himself in the progressive development of humanity. His last
public utterance is to be found in a highly characteristic article on
Renan’s Réforme Morale et Intellectuelle, finished on the 3rd of March
1872, and published in the Fortnightly Review for February 1874. Of
the 40,000 letters of Mazzini only a small part have been published. In
Page 286
1887 two hundred unpublished letters were printed at Turin (Duecento
lettere inedite di Giuseppe Mazzini), in 1895 the Lettres intimes were
published in Paris, and in 1905 Francesco Rosso published Lettre
inedite di Giuseppe Mazzini (Turin, 1905). A popular edition of
Mazzini’s writings has been undertaken by order of the Italian
government.
For Mazzini’s biography see Jessie White Mario, Della vita di
Giuseppe Mazzini (Milan, 1886), a useful if somewhat too enthusiastic
work; Bolton King, Mazzini (London, 1903); Count von Schack,
Joseph Mazzini und die italienische Einheit (Stuttgart, 1891). A.
Luzio’s Giuseppe Mazzini (Milan, 1905) contains a great deal of
valuable information, bibliographical and other, and Dora Melegari in
La giovine Italia e Giuseppe Mazzini (Milan, 1906) publishes the
correspondence between Mazzini and Luigi A. Melegari during the
early days of “Young Italy.” For the literary side of Mazzini’s life see
Peretti, Gli scritti letterarii di Giuseppe Mazzini (Turin, 1904). (L. V.*)
MAZZONI, GIACOMO (1548-1598), Italian philosopher, was
born at Cesena and died at Ferrara. A member of a noble family and highly
educated, he was one of the most eminent savants of the period. He
occupied chairs in the universities of Pisa and Rome, was one of the
founders of the Della Crusca Academy, and had the distinction, it is said, of
thrice vanquishing the Admirable Crichton in dialectic. His chief work in
philosophy was an attempt to reconcile Plato and Aristotle, and in this spirit
lettere inedite di Giuseppe Mazzini), in 1895 the Lettres intimes were
published in Paris, and in 1905 Francesco Rosso published Lettre
inedite di Giuseppe Mazzini (Turin, 1905). A popular edition of
Mazzini’s writings has been undertaken by order of the Italian
government.
For Mazzini’s biography see Jessie White Mario, Della vita di
Giuseppe Mazzini (Milan, 1886), a useful if somewhat too enthusiastic
work; Bolton King, Mazzini (London, 1903); Count von Schack,
Joseph Mazzini und die italienische Einheit (Stuttgart, 1891). A.
Luzio’s Giuseppe Mazzini (Milan, 1905) contains a great deal of
valuable information, bibliographical and other, and Dora Melegari in
La giovine Italia e Giuseppe Mazzini (Milan, 1906) publishes the
correspondence between Mazzini and Luigi A. Melegari during the
early days of “Young Italy.” For the literary side of Mazzini’s life see
Peretti, Gli scritti letterarii di Giuseppe Mazzini (Turin, 1904). (L. V.*)
MAZZONI, GIACOMO (1548-1598), Italian philosopher, was
born at Cesena and died at Ferrara. A member of a noble family and highly
educated, he was one of the most eminent savants of the period. He
occupied chairs in the universities of Pisa and Rome, was one of the
founders of the Della Crusca Academy, and had the distinction, it is said, of
thrice vanquishing the Admirable Crichton in dialectic. His chief work in
philosophy was an attempt to reconcile Plato and Aristotle, and in this spirit
Page 287
he published in 1597 a treatise In universam Platonis et Aristotelis
philosophiam praecludia. He wrote also De triplici hominum vita, wherein
he outlined a theory of the infinite perfection and development of nature.
Apart from philosophy, he was prominent in literature as the champion of
Dante, and produced two works in the poet’s defence: Discorso composto in
difesa della comedia di Dante (1572), and Della difesa della comedia di
Dante (1587, reprinted 1688). He was an authority on ancient languages
and philology, and gave a great impetus to the scientific study of the Italian
language.
MAZZONI, GUIDO (1859- ), Italian poet, was born at
Florence, and educated at Pisa and Bologna. In 1887 he became professor
of Italian at Padua, and in 1894 at Florence. He was much influenced by
Carducci, and became prominent both as a prolific and well-read critic and
as a poet of individual distinction. His chief volumes of verse are Versi
(1880), Nuove poesie (1886), Poesie (1891), Voci della vita (1893).
philosophiam praecludia. He wrote also De triplici hominum vita, wherein
he outlined a theory of the infinite perfection and development of nature.
Apart from philosophy, he was prominent in literature as the champion of
Dante, and produced two works in the poet’s defence: Discorso composto in
difesa della comedia di Dante (1572), and Della difesa della comedia di
Dante (1587, reprinted 1688). He was an authority on ancient languages
and philology, and gave a great impetus to the scientific study of the Italian
language.
MAZZONI, GUIDO (1859- ), Italian poet, was born at
Florence, and educated at Pisa and Bologna. In 1887 he became professor
of Italian at Padua, and in 1894 at Florence. He was much influenced by
Carducci, and became prominent both as a prolific and well-read critic and
as a poet of individual distinction. His chief volumes of verse are Versi
(1880), Nuove poesie (1886), Poesie (1891), Voci della vita (1893).
Page 288
MEAD, LARKIN GOLDSMITH (1835- ), American
sculptor, was born at Chesterfield, New Hampshire, on the 3rd of January
1835. He was a pupil (1853-1855) of Henry Kirke Brown. During the early
part of the Civil War he was at the front for six months, with the army of the
Potomac, as an artist for Harper’s Weekly; and in 1862-1865 he was in Italy,
being for part of the time attached to the United States consulate at Venice,
while William D. Howells, his brother-in-law, was consul. He returned to
America in 1865, but subsequently went back to Italy and lived at Florence.
His first important work was a statue of Ethan Allen, now at the State
House, Montpelier, Vermont. His principal works are: the monument to
President Lincoln, Springfield, Illinois; “Ethan Allen” (1876), National Hall
of Statuary, Capitol, Washington; an heroic marble statue, “The Father of
Waters,” New Orleans; and “Triumph of Ceres,” made for the Columbian
Exposition, Chicago.
His brother, William Rutherford Mead (1846- ), graduated at
Amherst College in 1867, and studied architecture in New York under
Russell Sturgis, and also abroad. In 1879 he and J. F. McKim, with whom
he had been in partnership for two years as architects, were joined by
Stanford White, and formed the well-known firm of McKim, Mead &
White.
MEAD, RICHARD (1673-1754), English physician, eleventh
child of Matthew Mead (1630-1699), Independent divine, was born on the
sculptor, was born at Chesterfield, New Hampshire, on the 3rd of January
1835. He was a pupil (1853-1855) of Henry Kirke Brown. During the early
part of the Civil War he was at the front for six months, with the army of the
Potomac, as an artist for Harper’s Weekly; and in 1862-1865 he was in Italy,
being for part of the time attached to the United States consulate at Venice,
while William D. Howells, his brother-in-law, was consul. He returned to
America in 1865, but subsequently went back to Italy and lived at Florence.
His first important work was a statue of Ethan Allen, now at the State
House, Montpelier, Vermont. His principal works are: the monument to
President Lincoln, Springfield, Illinois; “Ethan Allen” (1876), National Hall
of Statuary, Capitol, Washington; an heroic marble statue, “The Father of
Waters,” New Orleans; and “Triumph of Ceres,” made for the Columbian
Exposition, Chicago.
His brother, William Rutherford Mead (1846- ), graduated at
Amherst College in 1867, and studied architecture in New York under
Russell Sturgis, and also abroad. In 1879 he and J. F. McKim, with whom
he had been in partnership for two years as architects, were joined by
Stanford White, and formed the well-known firm of McKim, Mead &
White.
MEAD, RICHARD (1673-1754), English physician, eleventh
child of Matthew Mead (1630-1699), Independent divine, was born on the
Page 289
11th of August 1673 at Stepney, London. He studied at Utrecht for three
years under J. G. Graevius; having decided to follow the medical
profession, he then went to Leiden and attended the lectures of Paul
Hermann and Archibald Pitcairne. In 1695 he graduated in philosophy and
physic at Padua, and in 1696 he returned to London, entering at once on a
successful practice. His Mechanical Account of Poisons appeared in 1702,
and in 1703 he was admitted to the Royal Society, to whose Transactions he
contributed in that year a paper on the parasitic nature of scabies. In the
same year he was elected physician to St Thomas’s Hospital, and appointed
to read anatomical lectures at the Surgeons’ Hall. On the death of John
Radcliffe in 1714 Mead became the recognized head of his profession; he
attended Queen Anne on her deathbed, and in 1727 was appointed
physician to George II., having previously served him in that capacity when
he was prince of Wales. He died in London on the 16th of February 1754.
Besides the Mechanical Account of Poisons (2nd ed., 1708), Mead
published a treatise De imperio solis et lunae in corpora humana et
morbis inde oriundis (1704), A Short Discourse concerning Pestilential
Contagion, and the Method to be used to prevent it (1720), De variolis
et morbillis dissertatio (1747), Medica sacra, sive de morbis
insignioribus qui in bibliis memorantur commentarius (1748), On the
Scurvy (1749), and Monita et praecepta medica (1751). A Life of Mead
by Dr Matthew Maty appeared in 1755.
years under J. G. Graevius; having decided to follow the medical
profession, he then went to Leiden and attended the lectures of Paul
Hermann and Archibald Pitcairne. In 1695 he graduated in philosophy and
physic at Padua, and in 1696 he returned to London, entering at once on a
successful practice. His Mechanical Account of Poisons appeared in 1702,
and in 1703 he was admitted to the Royal Society, to whose Transactions he
contributed in that year a paper on the parasitic nature of scabies. In the
same year he was elected physician to St Thomas’s Hospital, and appointed
to read anatomical lectures at the Surgeons’ Hall. On the death of John
Radcliffe in 1714 Mead became the recognized head of his profession; he
attended Queen Anne on her deathbed, and in 1727 was appointed
physician to George II., having previously served him in that capacity when
he was prince of Wales. He died in London on the 16th of February 1754.
Besides the Mechanical Account of Poisons (2nd ed., 1708), Mead
published a treatise De imperio solis et lunae in corpora humana et
morbis inde oriundis (1704), A Short Discourse concerning Pestilential
Contagion, and the Method to be used to prevent it (1720), De variolis
et morbillis dissertatio (1747), Medica sacra, sive de morbis
insignioribus qui in bibliis memorantur commentarius (1748), On the
Scurvy (1749), and Monita et praecepta medica (1751). A Life of Mead
by Dr Matthew Maty appeared in 1755.
Page 290
MEAD. (1) A word now only used more or less poetically for the
commoner form “meadow,” properly land laid down for grass and cut for
hay, but often extended in meaning to include pasture-land. “Meadow”
represents the oblique case, maédwe, of O. Eng. maéd, which comes from
the root seen in “mow”; the word, therefore, means “mowed land.” Cognate
words appear in other Teutonic languages, a familiar instance being Ger.
matt, seen in place-names such as Zermatt, Andermatt, &c. (See Grass.) (2)
The name of a drink made by the fermentation of honey mixed with water.
Alcoholic drinks made from honey were common in ancient times, and
during the middle ages throughout Europe. The Greeks and Romans knew
of such under the names of ὁδρόμελι and hydromel; mulsum was a form of
mead with the addition of wine. The word is common to Teutonic languages
(cf. Du. mede, Ger. Met or Meth), and is cognate with Gr. μέθυ, wine, and
Sansk. mádhu, sweet drink. “Metheglin,” another word for mead, properly a
medicated or spiced form of the drink, is an adaptation of the Welsh
meddyglyn, which is derived from meddyg, healing (Lat. medicus) and llyn,
liquor. It therefore means “spiced or medicated drink,” and is not
etymologically connected with “mead.”
MEADE, GEORGE GORDON (1815-1872), American
soldier, was born of American parentage at Cadiz, Spain, on the 31st of
December 1815. On graduation at the United States Military Academy in
1835, he served in Florida with the 3rd Artillery against the Seminoles.
Resigning from the army in 1836, he became a civil engineer and
commoner form “meadow,” properly land laid down for grass and cut for
hay, but often extended in meaning to include pasture-land. “Meadow”
represents the oblique case, maédwe, of O. Eng. maéd, which comes from
the root seen in “mow”; the word, therefore, means “mowed land.” Cognate
words appear in other Teutonic languages, a familiar instance being Ger.
matt, seen in place-names such as Zermatt, Andermatt, &c. (See Grass.) (2)
The name of a drink made by the fermentation of honey mixed with water.
Alcoholic drinks made from honey were common in ancient times, and
during the middle ages throughout Europe. The Greeks and Romans knew
of such under the names of ὁδρόμελι and hydromel; mulsum was a form of
mead with the addition of wine. The word is common to Teutonic languages
(cf. Du. mede, Ger. Met or Meth), and is cognate with Gr. μέθυ, wine, and
Sansk. mádhu, sweet drink. “Metheglin,” another word for mead, properly a
medicated or spiced form of the drink, is an adaptation of the Welsh
meddyglyn, which is derived from meddyg, healing (Lat. medicus) and llyn,
liquor. It therefore means “spiced or medicated drink,” and is not
etymologically connected with “mead.”
MEADE, GEORGE GORDON (1815-1872), American
soldier, was born of American parentage at Cadiz, Spain, on the 31st of
December 1815. On graduation at the United States Military Academy in
1835, he served in Florida with the 3rd Artillery against the Seminoles.
Resigning from the army in 1836, he became a civil engineer and
Page 291
constructor of railways, and was engaged under the war department in
survey work. In 1842 he was appointed a second lieutenant in the corps of
the topographical engineers. In the war with Mexico he was on the staffs
successively of Generals Taylor, J. Worth and Robert Patterson, and was
brevetted for gallant conduct at Monterey. Until the Civil War he was
engaged in various engineering works, mainly in connexion with
lighthouses, and later as a captain of topographical engineers in the survey
of the northern lakes. In 1861 he was appointed brigadier-general of
volunteers, and had command of the 2nd brigade of the Pennsylvania
Reserves in the Army of the Potomac under General M’Call. He served in
the Seven Days, receiving a severe wound at the action of Frazier’s Farm.
He was absent from his command until the second battle of Bull Run, after
which he obtained the command of his division. He distinguished himself
greatly at the battles of South Mountain and Antietam. At Fredericksburg he
and his division won great distinction by their attack on the position held by
Jackson’s corps, and Meade was promoted major-general of volunteers, to
date from the 29th of November. Soon afterwards he was placed in
command of the V. corps. At Chancellorsville he displayed great intrepidity
and energy, and on the eve of the battle of Gettysburg was appointed to
succeed Hooker. The choice was unexpected, but Meade justified it by his
conduct of the operations, and in the famous three days’ battle he inflicted a
complete defeat on General Lee’s army. His reward was the commission of
brigadier-general in the regular army. In the autumn of 1863 a war of
manœuvre was fought between the two commanders, on the whole
favourably to the Union arms. Grant, commanding all the armies of the
United States, joined the Army of the Potomac in the spring of 1864, and
remained with it until the end of the war; but he continued Meade in his
command, and successfully urged his appointment as major-general in the
regular army (Aug. 18, 1864), eulogizing him as the commander who had
successfully met and defeated the best general and the strongest army on
survey work. In 1842 he was appointed a second lieutenant in the corps of
the topographical engineers. In the war with Mexico he was on the staffs
successively of Generals Taylor, J. Worth and Robert Patterson, and was
brevetted for gallant conduct at Monterey. Until the Civil War he was
engaged in various engineering works, mainly in connexion with
lighthouses, and later as a captain of topographical engineers in the survey
of the northern lakes. In 1861 he was appointed brigadier-general of
volunteers, and had command of the 2nd brigade of the Pennsylvania
Reserves in the Army of the Potomac under General M’Call. He served in
the Seven Days, receiving a severe wound at the action of Frazier’s Farm.
He was absent from his command until the second battle of Bull Run, after
which he obtained the command of his division. He distinguished himself
greatly at the battles of South Mountain and Antietam. At Fredericksburg he
and his division won great distinction by their attack on the position held by
Jackson’s corps, and Meade was promoted major-general of volunteers, to
date from the 29th of November. Soon afterwards he was placed in
command of the V. corps. At Chancellorsville he displayed great intrepidity
and energy, and on the eve of the battle of Gettysburg was appointed to
succeed Hooker. The choice was unexpected, but Meade justified it by his
conduct of the operations, and in the famous three days’ battle he inflicted a
complete defeat on General Lee’s army. His reward was the commission of
brigadier-general in the regular army. In the autumn of 1863 a war of
manœuvre was fought between the two commanders, on the whole
favourably to the Union arms. Grant, commanding all the armies of the
United States, joined the Army of the Potomac in the spring of 1864, and
remained with it until the end of the war; but he continued Meade in his
command, and successfully urged his appointment as major-general in the
regular army (Aug. 18, 1864), eulogizing him as the commander who had
successfully met and defeated the best general and the strongest army on
Page 292
the Confederate side. After the war Meade commanded successively the
military division of the Atlantic, the department of the east, the third
military district (Georgia and Alabama) and the department of the south. He
died at Philadelphia on the 6th of November, 1872. The degree of LL.D.
was conferred upon him by Harvard University, and his scientific
attainments were recognized by the American Philosophical Society and the
Philadelphia Academy of Natural Sciences. There are statues of General
Meade in Philadelphia and at Gettysburg.
See I. R. Pennypacker, General Meade (“Great Commanders” series,
New York, 1901).
MEADE, WILLIAM (1789-1862), American Protestant
Episcopal bishop, the son of Richard Kidder Meade (1746-1805), one of
General Washington’s aides during the War of Independence, was born on
the 11th of November 1789, near Millwood, in that part of Frederick county
which is now Clarke county, Virginia. He graduated as valedictorian in
1808 at the college of New Jersey (Princeton); studied theology under the
Rev. Walter Addison of Maryland, and in Princeton; was ordained deacon in
1811 and priest in 1814; and preached both in the Stone Chapel, Millwood,
and in Christ Church, Alexandria, for some time. He became assistant
bishop of Virginia in 1829; was pastor of Christ Church, Norfolk, in 1834-
1836; in 1841 became bishop of Virginia; and in 1842-1862 was president
of the Protestant Episcopal Theological Seminary in Virginia, near
military division of the Atlantic, the department of the east, the third
military district (Georgia and Alabama) and the department of the south. He
died at Philadelphia on the 6th of November, 1872. The degree of LL.D.
was conferred upon him by Harvard University, and his scientific
attainments were recognized by the American Philosophical Society and the
Philadelphia Academy of Natural Sciences. There are statues of General
Meade in Philadelphia and at Gettysburg.
See I. R. Pennypacker, General Meade (“Great Commanders” series,
New York, 1901).
MEADE, WILLIAM (1789-1862), American Protestant
Episcopal bishop, the son of Richard Kidder Meade (1746-1805), one of
General Washington’s aides during the War of Independence, was born on
the 11th of November 1789, near Millwood, in that part of Frederick county
which is now Clarke county, Virginia. He graduated as valedictorian in
1808 at the college of New Jersey (Princeton); studied theology under the
Rev. Walter Addison of Maryland, and in Princeton; was ordained deacon in
1811 and priest in 1814; and preached both in the Stone Chapel, Millwood,
and in Christ Church, Alexandria, for some time. He became assistant
bishop of Virginia in 1829; was pastor of Christ Church, Norfolk, in 1834-
1836; in 1841 became bishop of Virginia; and in 1842-1862 was president
of the Protestant Episcopal Theological Seminary in Virginia, near
Page 293
Alexandria, delivering an annual course of lectures on pastoral theology. In
1819 he had acted as the agent of the American Colonization Society to
purchase slaves, illegally brought into Georgia, which had become the
property of that state and were sold publicly at Milledgeville. He had been
prominent in the work of the Education Society, which was organized in
1818 to advance funds to needy students for the ministry of the American
Episcopal Church, and in the establishment of the Theological Seminary
near Alexandria, as he was afterwards in the work of the American Tract
Society, and the Bible Society. He was a founder and president of the
Evangelical Knowledge Society (1847), which, opposing what it considered
the heterodoxy of many of the books published by the Sunday School
Union, attempted to displace them by issuing works of a more evangelical
type. A low Churchman, he strongly opposed Tractarianism. He was active
in the case against Bishop Henry Ustick Onderdonk (1789-1858) of
Pennsylvania, who because of intemperance was forced to resign and was
suspended from the ministry in 1844; in that against Bishop Benjamin
Tredwell Onderdonk (1791-1861) of New York, who in 1845 was
suspended from the ministry on the charge of intoxication and improper
conduct; and in that against Bishop G. W. Doane of New Jersey. He fought
against the threatening secession of Virginia, but acquiesced in the decision
of the state and became presiding bishop of the Southern Church. He died
in Richmond, Virginia, on the 14th of March 1862.
Among his publications, besides many sermons, were A Brief
Review of the Episcopal Church in Virginia (1845); Wilberforce,
Cranmer, Jewett and the Prayer Book on the Incarnation (1850);
Reasons for Loving the Episcopal Church (1852); and Old Churches,
Ministers and Families of Virginia (1857); a storehouse of material on
the ecclesiastical history of the state.
See the Life by John Johns (Baltimore, 1867).
1819 he had acted as the agent of the American Colonization Society to
purchase slaves, illegally brought into Georgia, which had become the
property of that state and were sold publicly at Milledgeville. He had been
prominent in the work of the Education Society, which was organized in
1818 to advance funds to needy students for the ministry of the American
Episcopal Church, and in the establishment of the Theological Seminary
near Alexandria, as he was afterwards in the work of the American Tract
Society, and the Bible Society. He was a founder and president of the
Evangelical Knowledge Society (1847), which, opposing what it considered
the heterodoxy of many of the books published by the Sunday School
Union, attempted to displace them by issuing works of a more evangelical
type. A low Churchman, he strongly opposed Tractarianism. He was active
in the case against Bishop Henry Ustick Onderdonk (1789-1858) of
Pennsylvania, who because of intemperance was forced to resign and was
suspended from the ministry in 1844; in that against Bishop Benjamin
Tredwell Onderdonk (1791-1861) of New York, who in 1845 was
suspended from the ministry on the charge of intoxication and improper
conduct; and in that against Bishop G. W. Doane of New Jersey. He fought
against the threatening secession of Virginia, but acquiesced in the decision
of the state and became presiding bishop of the Southern Church. He died
in Richmond, Virginia, on the 14th of March 1862.
Among his publications, besides many sermons, were A Brief
Review of the Episcopal Church in Virginia (1845); Wilberforce,
Cranmer, Jewett and the Prayer Book on the Incarnation (1850);
Reasons for Loving the Episcopal Church (1852); and Old Churches,
Ministers and Families of Virginia (1857); a storehouse of material on
the ecclesiastical history of the state.
See the Life by John Johns (Baltimore, 1867).
Page 294
MEADVILLE, a city and the county-seat of Crawford county,
Pennsylvania, U.S.A., on French Creek, 36 m. S. of Erie. Pop. (1900),
10,291, of whom 912 were foreign-born and 173 were negroes; (1910
census) 12,780. It is served by the Erie, and the Bessemer & Lake Erie
railways. Meadville has three public parks, two general hospitals and a
public library, and is the seat of the Pennsylvania College of Music, of a
commercial college, of the Meadville Theological School (1844, Unitarian),
and of Allegheny College (co-educational), which was opened in 1815,
came under the general patronage of the Methodist Episcopal Church in
1833, and in 1909 had 322 students (200 men and 122 women). Meadville
is the commercial centre of a good agricultural region, which also abounds
in oil and natural gas. The Erie Railroad has extensive shops here, which in
1905 employed 46.7% of the total number of wage-earners, and there are
various manufactures. The factory product in 1905 was valued at
$2,074,600, being 24.4% more than that of 1900. Meadville, the oldest
settlement in N.W. Pennsylvania, was founded as a fortified post by David
Mead in 1793, laid out as a town in 1795, incorporated as a borough in
1823 and chartered as a city in 1866.
Pennsylvania, U.S.A., on French Creek, 36 m. S. of Erie. Pop. (1900),
10,291, of whom 912 were foreign-born and 173 were negroes; (1910
census) 12,780. It is served by the Erie, and the Bessemer & Lake Erie
railways. Meadville has three public parks, two general hospitals and a
public library, and is the seat of the Pennsylvania College of Music, of a
commercial college, of the Meadville Theological School (1844, Unitarian),
and of Allegheny College (co-educational), which was opened in 1815,
came under the general patronage of the Methodist Episcopal Church in
1833, and in 1909 had 322 students (200 men and 122 women). Meadville
is the commercial centre of a good agricultural region, which also abounds
in oil and natural gas. The Erie Railroad has extensive shops here, which in
1905 employed 46.7% of the total number of wage-earners, and there are
various manufactures. The factory product in 1905 was valued at
$2,074,600, being 24.4% more than that of 1900. Meadville, the oldest
settlement in N.W. Pennsylvania, was founded as a fortified post by David
Mead in 1793, laid out as a town in 1795, incorporated as a borough in
1823 and chartered as a city in 1866.
Page 295
MEAGHER, THOMAS FRANCIS (1823-1867), Irish
nationalist and American soldier, was born in Waterford, Ireland, on the 3rd
of August 1823. He graduated at Stonyhurst College, Lancashire, in 1843,
and in 1844 began the study of law at Dublin. He became a member of the
Young Ireland Party in 1845, and in 1847 was one of the founders of the
Irish Confederation. In March 1848 he made a speech before the
Confederation which led to his arrest for sedition, but at his trial the jury
failed to agree and he was discharged. In the following July the
Confederation created a “war directory” of five, of which Meagher was a
member, and he and William Smith O’Brien travelled through Ireland for
the purpose of starting a revolution. The attempt proved abortive; Meagher
was arrested in August, and in October was tried for high treason before a
special commission at Clonmel. He was found guilty and was condemned
to death, but his sentence was commuted to life imprisonment in Van
Diemen’s Land, whither he was transported in the summer of 1849. Early in
1852 he escaped, and in May reached New York City. He made a tour of the
cities of the United States as a popular lecturer, and then studied law and
was admitted to the New York bar in 1855. He made two unsuccessful
ventures in journalism, and in 1857 went to Central America, where he
acquired material for another series of lectures. In 1861 he was captain of a
company (which he had raised) in the 69th regiment of New York
volunteers and fought at the first battle of Bull Run; he then organized an
Irish brigade, of whose first regiment he was colonel until the 3rd of
nationalist and American soldier, was born in Waterford, Ireland, on the 3rd
of August 1823. He graduated at Stonyhurst College, Lancashire, in 1843,
and in 1844 began the study of law at Dublin. He became a member of the
Young Ireland Party in 1845, and in 1847 was one of the founders of the
Irish Confederation. In March 1848 he made a speech before the
Confederation which led to his arrest for sedition, but at his trial the jury
failed to agree and he was discharged. In the following July the
Confederation created a “war directory” of five, of which Meagher was a
member, and he and William Smith O’Brien travelled through Ireland for
the purpose of starting a revolution. The attempt proved abortive; Meagher
was arrested in August, and in October was tried for high treason before a
special commission at Clonmel. He was found guilty and was condemned
to death, but his sentence was commuted to life imprisonment in Van
Diemen’s Land, whither he was transported in the summer of 1849. Early in
1852 he escaped, and in May reached New York City. He made a tour of the
cities of the United States as a popular lecturer, and then studied law and
was admitted to the New York bar in 1855. He made two unsuccessful
ventures in journalism, and in 1857 went to Central America, where he
acquired material for another series of lectures. In 1861 he was captain of a
company (which he had raised) in the 69th regiment of New York
volunteers and fought at the first battle of Bull Run; he then organized an
Irish brigade, of whose first regiment he was colonel until the 3rd of
Page 296
February 1862, when he was appointed to the command of this organization
with the rank of brigadier-general. He took part in the siege of Yorktown,
the battle of Fair Oaks, the seven days’ battle before Richmond, and the
battles of Antietam, Fredericksburg, where he was wounded, and
Chancellorsville, where his brigade was reduced in numbers to less than a
regiment, and General Meagher resigned his commission. On the 23rd of
December 1863 his resignation was cancelled, and he was assigned to the
command of the military district of Etowah, with headquarters at
Chattanooga. At the close of the war he was appointed by President Johnson
secretary of Montana Territory, and there, in the absence of the territorial
governor, he acted as governor from September 1866 until his death from
accidental drowning in the Missouri River near Fort Benton, Montana, on
the 1st of July 1867. He published Speeches on the Legislative
Independence of Ireland (1852).
W. F. Lyons, in Brigadier-General Thomas Francis Meagher (New
York, 1870), gives a eulogistic account of his career.
MEAL. (1) (A word common to Teutonic languages, cf. Ger. Mehl,
Du. meel; the ultimate source is the root seen in various Teutonic words
meaning “to grind,” and in Eng. “mill,” Lat. mola, molěre, Gr. μύλη), a
powder made from the edible part of any grain or pulse, with the exception
of wheat, which is known as “flour.” In America the word is specifically
applied to the meal produced from Indian corn or maize, as in Scotland and
with the rank of brigadier-general. He took part in the siege of Yorktown,
the battle of Fair Oaks, the seven days’ battle before Richmond, and the
battles of Antietam, Fredericksburg, where he was wounded, and
Chancellorsville, where his brigade was reduced in numbers to less than a
regiment, and General Meagher resigned his commission. On the 23rd of
December 1863 his resignation was cancelled, and he was assigned to the
command of the military district of Etowah, with headquarters at
Chattanooga. At the close of the war he was appointed by President Johnson
secretary of Montana Territory, and there, in the absence of the territorial
governor, he acted as governor from September 1866 until his death from
accidental drowning in the Missouri River near Fort Benton, Montana, on
the 1st of July 1867. He published Speeches on the Legislative
Independence of Ireland (1852).
W. F. Lyons, in Brigadier-General Thomas Francis Meagher (New
York, 1870), gives a eulogistic account of his career.
MEAL. (1) (A word common to Teutonic languages, cf. Ger. Mehl,
Du. meel; the ultimate source is the root seen in various Teutonic words
meaning “to grind,” and in Eng. “mill,” Lat. mola, molěre, Gr. μύλη), a
powder made from the edible part of any grain or pulse, with the exception
of wheat, which is known as “flour.” In America the word is specifically
applied to the meal produced from Indian corn or maize, as in Scotland and
Page 297
Ireland to that produced from oats, while in South Africa the ears of the
Indian corn itself are called “mealies.” (2) Properly, eating and drinking at
regular stated times of the day, as breakfast, dinner, &c., hence taking of
food at any time and also the food provided. The word was in O.E. mael,
which also had the meanings (now lost) of time, mark, measure, &c., which
still appear in many forms of the word in Teutonic languages; thus Ger. mal,
time, mark, cf. Denkmal, monument, Mahl, meal, repast, or Du. maal,
Swed. mal, also with both meanings. The ultimate source is the pre-
Teutonic root me- ma-, to measure, and the word thus stood for a marked-
out point of time.
MEALIE, the South African name for Indian corn or maize. The
word as spelled represents the pronunciation of the Cape Dutch milje, an
adaptation of milho (da India), the millet of India, the Portuguese name for
millet, used in South Africa for maize.
Indian corn itself are called “mealies.” (2) Properly, eating and drinking at
regular stated times of the day, as breakfast, dinner, &c., hence taking of
food at any time and also the food provided. The word was in O.E. mael,
which also had the meanings (now lost) of time, mark, measure, &c., which
still appear in many forms of the word in Teutonic languages; thus Ger. mal,
time, mark, cf. Denkmal, monument, Mahl, meal, repast, or Du. maal,
Swed. mal, also with both meanings. The ultimate source is the pre-
Teutonic root me- ma-, to measure, and the word thus stood for a marked-
out point of time.
MEALIE, the South African name for Indian corn or maize. The
word as spelled represents the pronunciation of the Cape Dutch milje, an
adaptation of milho (da India), the millet of India, the Portuguese name for
millet, used in South Africa for maize.
Page 298
MEAN, an homonymous word, the chief uses of which may be
divided thus. (1) A verb with two principal applications, to intend, purpose
or design, and to signify. This word is in O.E. maenan, and cognate forms
appear in other Teutonic languages, cf. Du. meenen, Ger. meinen. The
ultimate origin is usually taken to be the root men-, to think, the root of
“mind.” (2) An adjective and substantive meaning “that which is in the
middle.” This is derived through the O. Fr. men, meien or moien, modern
moyen, from the late Lat. adjective medianus, from medius, middle. The law
French form mesne is still preserved in certain legal phrases (see Mesne).
The adjective “mean” is chiefly used in the sense of “average,” as in mean
temperature, mean birth or death rate, &c.
“Mean” as a substantive has the following principal applications; it is
used of that quality, course of action, condition, state, &c., which is equally
distant from two extremes, as in such phrases as the “golden (or happy)
mean.” For the philosophic application see Aristotle and Ethics.
In mathematics, the term “mean,” in its most general sense, is given to
some function of two or more quantities which (1) becomes equal to each of
the quantities when they themselves are made equal, and (2) is unaffected in
value when the quantities suffer any transpositions. The three commonest
means are the arithmetical, geometrical, and harmonic; of less importance
are the contraharmonical, arithmetico-geometrical, and quadratic.
From the sense of that which stands between two things, “mean,” or the
plural “means,” often with a singular construction, takes the further
significance of agency, instrument, &c., of which that produces some result,
hence resources capable of producing a result, particularly the pecuniary or
other resources by which a person is enabled to live, and so used either of
employment or of property, wealth, &c. There are many adverbial phrases,
such as “by all means,” “by no means,” &c., which are extensions of
“means” in the sense of agency.
divided thus. (1) A verb with two principal applications, to intend, purpose
or design, and to signify. This word is in O.E. maenan, and cognate forms
appear in other Teutonic languages, cf. Du. meenen, Ger. meinen. The
ultimate origin is usually taken to be the root men-, to think, the root of
“mind.” (2) An adjective and substantive meaning “that which is in the
middle.” This is derived through the O. Fr. men, meien or moien, modern
moyen, from the late Lat. adjective medianus, from medius, middle. The law
French form mesne is still preserved in certain legal phrases (see Mesne).
The adjective “mean” is chiefly used in the sense of “average,” as in mean
temperature, mean birth or death rate, &c.
“Mean” as a substantive has the following principal applications; it is
used of that quality, course of action, condition, state, &c., which is equally
distant from two extremes, as in such phrases as the “golden (or happy)
mean.” For the philosophic application see Aristotle and Ethics.
In mathematics, the term “mean,” in its most general sense, is given to
some function of two or more quantities which (1) becomes equal to each of
the quantities when they themselves are made equal, and (2) is unaffected in
value when the quantities suffer any transpositions. The three commonest
means are the arithmetical, geometrical, and harmonic; of less importance
are the contraharmonical, arithmetico-geometrical, and quadratic.
From the sense of that which stands between two things, “mean,” or the
plural “means,” often with a singular construction, takes the further
significance of agency, instrument, &c., of which that produces some result,
hence resources capable of producing a result, particularly the pecuniary or
other resources by which a person is enabled to live, and so used either of
employment or of property, wealth, &c. There are many adverbial phrases,
such as “by all means,” “by no means,” &c., which are extensions of
“means” in the sense of agency.
Page 299
The word “mean” (like the French moyen) had also the sense of
middling, moderate, and this considerably influenced the uses of “mean”
(3). This, which is now chiefly used in the sense of inferior, low, ignoble, or
of avaricious, penurious, “stingy,” meant originally that which is common
to more persons or things than one. The word in O. E. is gemaéne, and is
represented in the modern Ger. gemein, common. It is cognate with Lat.
communis, from which “common” is derived. The descent in meaning from
that which is shared alike by several to that which is inferior, vulgar or low,
is paralleled by the uses of “common.”
In astronomy the “mean sun” is a fictitious sun which moves uniformly
in the celestial equator and has its right ascension always equal to the sun’s
mean longitude. The time recorded by the mean sun is termed mean-solar or
clock time; it is regular as distinct from the non-uniform solar or sun-dial
time. The “mean moon” is a fictitious moon which moves around the earth
with a uniform velocity and in the same time as the real moon. The “mean
longitude” of a planet is the longitude of the “mean” planet, i.e. a fictitious
planet performing uniform revolutions in the same time as the real planet.
The arithmetical mean of n quantities is the sum of the quantities
divided by their number n. The geometrical mean of n quantities is the
nth root of their product. The harmonic mean of n quantities is the
arithmetical mean of their reciprocals. The significance of the word
“mean,” i.e., middle, is seen by considering 3 instead of n quantities;
these will be denoted by a, b, c. The arithmetic mean b, is seen to be
such that the terms a, b, c are in arithmetical progression, i.e. b = 1⁄2(a +
c); the geometrical mean b places a, b, c in geometrical progression, i.e.
in the proportion a : b :: b : c or b2 = ac; and the harmonic mean places
the quantities in harmonic proportion, i.e. a : c :: a − b : b − c, or b =
2ac/(a + c). The contraharmonical mean is the quantity b given by the
proportion a : c :: b − c : a − b, i.e. b = (a2 + c2)/(a + c). The
middling, moderate, and this considerably influenced the uses of “mean”
(3). This, which is now chiefly used in the sense of inferior, low, ignoble, or
of avaricious, penurious, “stingy,” meant originally that which is common
to more persons or things than one. The word in O. E. is gemaéne, and is
represented in the modern Ger. gemein, common. It is cognate with Lat.
communis, from which “common” is derived. The descent in meaning from
that which is shared alike by several to that which is inferior, vulgar or low,
is paralleled by the uses of “common.”
In astronomy the “mean sun” is a fictitious sun which moves uniformly
in the celestial equator and has its right ascension always equal to the sun’s
mean longitude. The time recorded by the mean sun is termed mean-solar or
clock time; it is regular as distinct from the non-uniform solar or sun-dial
time. The “mean moon” is a fictitious moon which moves around the earth
with a uniform velocity and in the same time as the real moon. The “mean
longitude” of a planet is the longitude of the “mean” planet, i.e. a fictitious
planet performing uniform revolutions in the same time as the real planet.
The arithmetical mean of n quantities is the sum of the quantities
divided by their number n. The geometrical mean of n quantities is the
nth root of their product. The harmonic mean of n quantities is the
arithmetical mean of their reciprocals. The significance of the word
“mean,” i.e., middle, is seen by considering 3 instead of n quantities;
these will be denoted by a, b, c. The arithmetic mean b, is seen to be
such that the terms a, b, c are in arithmetical progression, i.e. b = 1⁄2(a +
c); the geometrical mean b places a, b, c in geometrical progression, i.e.
in the proportion a : b :: b : c or b2 = ac; and the harmonic mean places
the quantities in harmonic proportion, i.e. a : c :: a − b : b − c, or b =
2ac/(a + c). The contraharmonical mean is the quantity b given by the
proportion a : c :: b − c : a − b, i.e. b = (a2 + c2)/(a + c). The
Page 300
arithmetico-geometrical mean of two quantities is obtained by first
forming the geometrical and arithmetical means, then forming the
means of these means, and repeating the process until the numbers
become equal. They were invented by Gauss to facilitate the
computation of elliptic integrals. The quadratic mean of n quantities is
the square root of the arithmetical mean of their squares.
MEASLES, (Morbilli, Rubeola; the M. E. word is maseles, properly
a diminutive of a word meaning “spot,” O.H.G. māsa, cf. “mazer”; the
equivalent is Ger. Masern; Fr. Rougeole), an acute infectious disease
occurring mostly in children. It is mentioned in the writings of Rhazes and
others of the Arabian physicians in the 10th century. For long, however, it
was held to be a variety of small-pox. After the non-identity of these two
diseases had been established, measles and scarlet-fever continued to be
confounded with each other; and in the account given by Thomas
Sydenham of epidemics of measles in London in 1670 and 1674 it is
evident that even that accurate observer had not as yet clearly perceived
their pathological distinction, although it would seem to have been made a
century earlier by Giovanni Filippo Ingrassias (1510-1580), a physician of
Palermo. The specific micro-organism responsible for measles has not been
definitely isolated.
Its progress is marked by several stages more or less sharply defined.
After the reception of the contagion into the system, there follows a period
of incubation or latency during which scarcely any disturbance of the health
forming the geometrical and arithmetical means, then forming the
means of these means, and repeating the process until the numbers
become equal. They were invented by Gauss to facilitate the
computation of elliptic integrals. The quadratic mean of n quantities is
the square root of the arithmetical mean of their squares.
MEASLES, (Morbilli, Rubeola; the M. E. word is maseles, properly
a diminutive of a word meaning “spot,” O.H.G. māsa, cf. “mazer”; the
equivalent is Ger. Masern; Fr. Rougeole), an acute infectious disease
occurring mostly in children. It is mentioned in the writings of Rhazes and
others of the Arabian physicians in the 10th century. For long, however, it
was held to be a variety of small-pox. After the non-identity of these two
diseases had been established, measles and scarlet-fever continued to be
confounded with each other; and in the account given by Thomas
Sydenham of epidemics of measles in London in 1670 and 1674 it is
evident that even that accurate observer had not as yet clearly perceived
their pathological distinction, although it would seem to have been made a
century earlier by Giovanni Filippo Ingrassias (1510-1580), a physician of
Palermo. The specific micro-organism responsible for measles has not been
definitely isolated.
Its progress is marked by several stages more or less sharply defined.
After the reception of the contagion into the system, there follows a period
of incubation or latency during which scarcely any disturbance of the health
Page 301
is perceptible. This period generally lasts for from ten to fourteen days,
when it is followed by the invasion of the symptoms specially characteristic
of measles. These consist in the somewhat sudden onset of acute catarrh of
the mucous membranes. At this stage minute white spots in the buccal
mucous membrane frequently occur; when they do, they are diagnostic of
the disease. Sneezing, accompanied with a watery discharge, sometimes
bleeding, from the nose, redness and watering of the eyes, cough of a short,
frequent, and noisy character, with little or no expectoration, hoarseness of
the voice, and occasionally sickness and diarrhoea, are the chief local
phenomena of this stage. With these there is well-marked febrile
disturbance, the temperature being elevated (102°-104° F.), and the pulse
rapid, while headache, thirst, and restlessness are usually present. In some
instances, these initial symptoms are slight, and the child is allowed to
associate with others at a time when, as will be afterwards seen, the
contagion of the disease is most active. In rare cases, especially in young
children, convulsions usher in, or occur in the course of, this stage of
invasion, which lasts as a rule for four or five days, the febrile symptoms,
however, showing some tendency to undergo abatement after the second
day. On the fourth or fifth day after the invasion, sometimes later, rarely
earlier, the characteristic eruption appears on the skin, being first noticed on
the brow, cheeks, chin, also behind the ears, and on the neck. It consists of
small spots of a dusky red or crimson colour, just like flea-bites, slightly
elevated above the surface, at first isolated, but tending to become grouped
into patches of irregular, occasionally crescentic, outline, with portions of
skin free from the eruption intervening. The face acquires a swollen and
bloated appearance, which, taken with the catarrh of the nostrils and eyes, is
almost characteristic, and renders the diagnosis at this stage a matter of no
difficulty. The eruption spreads downwards over the body and limbs, which
are soon thickly studded with the red spots or patches. Sometimes these
become confluent over a considerable surface. The rash continues to come
when it is followed by the invasion of the symptoms specially characteristic
of measles. These consist in the somewhat sudden onset of acute catarrh of
the mucous membranes. At this stage minute white spots in the buccal
mucous membrane frequently occur; when they do, they are diagnostic of
the disease. Sneezing, accompanied with a watery discharge, sometimes
bleeding, from the nose, redness and watering of the eyes, cough of a short,
frequent, and noisy character, with little or no expectoration, hoarseness of
the voice, and occasionally sickness and diarrhoea, are the chief local
phenomena of this stage. With these there is well-marked febrile
disturbance, the temperature being elevated (102°-104° F.), and the pulse
rapid, while headache, thirst, and restlessness are usually present. In some
instances, these initial symptoms are slight, and the child is allowed to
associate with others at a time when, as will be afterwards seen, the
contagion of the disease is most active. In rare cases, especially in young
children, convulsions usher in, or occur in the course of, this stage of
invasion, which lasts as a rule for four or five days, the febrile symptoms,
however, showing some tendency to undergo abatement after the second
day. On the fourth or fifth day after the invasion, sometimes later, rarely
earlier, the characteristic eruption appears on the skin, being first noticed on
the brow, cheeks, chin, also behind the ears, and on the neck. It consists of
small spots of a dusky red or crimson colour, just like flea-bites, slightly
elevated above the surface, at first isolated, but tending to become grouped
into patches of irregular, occasionally crescentic, outline, with portions of
skin free from the eruption intervening. The face acquires a swollen and
bloated appearance, which, taken with the catarrh of the nostrils and eyes, is
almost characteristic, and renders the diagnosis at this stage a matter of no
difficulty. The eruption spreads downwards over the body and limbs, which
are soon thickly studded with the red spots or patches. Sometimes these
become confluent over a considerable surface. The rash continues to come
Page 302
out for two or three days, and then begins to fade in the order in which it
first showed itself, namely from above downwards. By the end of about a
week after its first appearance scarcely any trace of the eruption remains
beyond a faint staining of the skin. Usually during convalescence slight
peeling of the epidermis takes place, but much less distinctly than is the
case in scarlet fever. At the commencement of the eruptive stage the fever,
catarrh, and other constitutional disturbance, which were present from the
beginning, become aggravated, the temperature often rising to 105° or
more, and there is headache, thirst, furred tongue, and soreness of the
throat, upon which red patches similar to those on the surface of the body
may be observed. These symptoms usually decline as soon as the rash has
attained its maximum, and often there occurs a sudden and extensive fall of
temperature, indicating that the crisis of the disease has been reached. In
favourable cases convalescence proceeds rapidly, the patient feeling
perfectly well even before the rash has faded from the skin.
Measles may, however, occur in a very malignant form, in which the
symptoms throughout are of urgent character, the rash but feebly developed,
and of dark purple hue, while there is great prostration, accompanied with
intense catarrh of the respiratory or gastro-intestinal mucous membrane.
Such cases are rare, occurring mostly in circumstances of bad hygiene, both
as regards the individual and his surroundings. On the other hand, cases of
measles are often of so mild a form throughout that the patient can scarcely
be persuaded to submit to treatment.
Measles as a disease derives its chief importance from the risk, by no
means slight, of certain complications which are apt to arise during its
course, more especially inflammatory affections of the respiratory organs.
These are most liable to occur in the colder seasons of the year and in very
young and delicate children. It has been already stated that irritation of the
respiratory passages is one of the symptoms characteristic of measles, but
first showed itself, namely from above downwards. By the end of about a
week after its first appearance scarcely any trace of the eruption remains
beyond a faint staining of the skin. Usually during convalescence slight
peeling of the epidermis takes place, but much less distinctly than is the
case in scarlet fever. At the commencement of the eruptive stage the fever,
catarrh, and other constitutional disturbance, which were present from the
beginning, become aggravated, the temperature often rising to 105° or
more, and there is headache, thirst, furred tongue, and soreness of the
throat, upon which red patches similar to those on the surface of the body
may be observed. These symptoms usually decline as soon as the rash has
attained its maximum, and often there occurs a sudden and extensive fall of
temperature, indicating that the crisis of the disease has been reached. In
favourable cases convalescence proceeds rapidly, the patient feeling
perfectly well even before the rash has faded from the skin.
Measles may, however, occur in a very malignant form, in which the
symptoms throughout are of urgent character, the rash but feebly developed,
and of dark purple hue, while there is great prostration, accompanied with
intense catarrh of the respiratory or gastro-intestinal mucous membrane.
Such cases are rare, occurring mostly in circumstances of bad hygiene, both
as regards the individual and his surroundings. On the other hand, cases of
measles are often of so mild a form throughout that the patient can scarcely
be persuaded to submit to treatment.
Measles as a disease derives its chief importance from the risk, by no
means slight, of certain complications which are apt to arise during its
course, more especially inflammatory affections of the respiratory organs.
These are most liable to occur in the colder seasons of the year and in very
young and delicate children. It has been already stated that irritation of the
respiratory passages is one of the symptoms characteristic of measles, but
Page 303
that this subsides with the decline of the eruption. Not unfrequently,
however, these symptoms, instead of abating, become aggravated, and
bronchitis of the capillary form (see Bronchitis), or pneumonia, generally
of the diffuse or lobular variety (see Pneumonia), supervene. By far the
greater proportion of the mortality in measles is due to its complications, of
which those just mentioned are the most common, but which also include
inflammatory affections of the larynx, with attacks resembling croup, and
also diarrhoea assuming a dysenteric character. Or there may remain as
direct results of the disease chronic ophthalmia, or discharge from the ears
with deafness, and occasionally a form of gangrene affecting the tissues of
the mouth or cheeks and other parts of the body, leading to disfigurement
and gravely endangering life.
Apart from those immediate risks there appears to be a tendency in many
cases for the disease to leave behind a weakened and vulnerable condition
of the general health, which may render children, previously robust, delicate
and liable to chest complaints, and is in not a few instances the precursor of
some of those tubercular affections to which the period of childhood and
youth is liable. These various effects or sequelae of measles indicate that
although in itself a comparatively mild ailment, it should not be regarded
with indifference. Indeed it is doubtful whether any other disease of early
life demands more careful watching as to its influence on the health.
Happily many of those attending evils may by proper management be
averted.
Measles is a disease of the earlier years of childhood. Like other
infectious maladies, it is admittedly rare, though not unknown, in nurslings
or infants under six months old. It is comparatively seldom met with in
adults, but this is due to the fact that most persons have undergone an attack
in early life. Where this has not been the case, the old suffer equally with
the young. All races of men appear liable to this disease, provided that
however, these symptoms, instead of abating, become aggravated, and
bronchitis of the capillary form (see Bronchitis), or pneumonia, generally
of the diffuse or lobular variety (see Pneumonia), supervene. By far the
greater proportion of the mortality in measles is due to its complications, of
which those just mentioned are the most common, but which also include
inflammatory affections of the larynx, with attacks resembling croup, and
also diarrhoea assuming a dysenteric character. Or there may remain as
direct results of the disease chronic ophthalmia, or discharge from the ears
with deafness, and occasionally a form of gangrene affecting the tissues of
the mouth or cheeks and other parts of the body, leading to disfigurement
and gravely endangering life.
Apart from those immediate risks there appears to be a tendency in many
cases for the disease to leave behind a weakened and vulnerable condition
of the general health, which may render children, previously robust, delicate
and liable to chest complaints, and is in not a few instances the precursor of
some of those tubercular affections to which the period of childhood and
youth is liable. These various effects or sequelae of measles indicate that
although in itself a comparatively mild ailment, it should not be regarded
with indifference. Indeed it is doubtful whether any other disease of early
life demands more careful watching as to its influence on the health.
Happily many of those attending evils may by proper management be
averted.
Measles is a disease of the earlier years of childhood. Like other
infectious maladies, it is admittedly rare, though not unknown, in nurslings
or infants under six months old. It is comparatively seldom met with in
adults, but this is due to the fact that most persons have undergone an attack
in early life. Where this has not been the case, the old suffer equally with
the young. All races of men appear liable to this disease, provided that
Page 304
which constitutes the essential factor in its origin and spread exists, namely,
contagion. Some countries enjoy long immunity from outbreaks of measles,
but it has frequently been found in such cases that when the contagion has
once been introduced the disease extends with great rapidity and virulence.
This was shown by the epidemic in the Faroe Islands in 1846, where, within
six months after the arrival of a single case of measles, more than three-
fourths of the entire population were attacked and many perished; and the
similarly produced and still more destructive outbreak in Fiji in 1875, in
which it was estimated that about one-fourth of the inhabitants died from
the disease in about three months. In both these cases the great mortality
was due to the complications of the malady, specially induced by
overcrowding, insanitary surroundings, the absence of proper nourishment
and nursing for the sick, and the utter prostration and terror of the people,
and to the disease being specially malignant, occurring on what might be
termed virgin soil.1 It may be regarded as an invariable rule that the first
epidemic of any disease in a community is specially virulent, each
successive attack conferring a certain immunity.
In many lands, such as the United Kingdom, measles is rarely absent,
especially from large centres of population, where sporadic cases are found
at all seasons. Every now and then epidemics arise from the extension of
the disease among those members of a community who have not been in
some measure protected by a previous attack. There are few diseases so
contagious as measles, and its rapid spread in epidemic outbreaks is no
doubt due to the well-ascertained fact that contagion is most potent in the
earlier stages, even before its real nature has been evinced by the
characteristic appearances on the skin. Hence the difficulty of timely
isolation, and the readiness with which the disease is spread in schools and
families. The contagion is present in the skin and the various secretions.
While the contagion is generally direct, it can also be conveyed by the
particles from the nose and mouth which, after being expelled, become dry
contagion. Some countries enjoy long immunity from outbreaks of measles,
but it has frequently been found in such cases that when the contagion has
once been introduced the disease extends with great rapidity and virulence.
This was shown by the epidemic in the Faroe Islands in 1846, where, within
six months after the arrival of a single case of measles, more than three-
fourths of the entire population were attacked and many perished; and the
similarly produced and still more destructive outbreak in Fiji in 1875, in
which it was estimated that about one-fourth of the inhabitants died from
the disease in about three months. In both these cases the great mortality
was due to the complications of the malady, specially induced by
overcrowding, insanitary surroundings, the absence of proper nourishment
and nursing for the sick, and the utter prostration and terror of the people,
and to the disease being specially malignant, occurring on what might be
termed virgin soil.1 It may be regarded as an invariable rule that the first
epidemic of any disease in a community is specially virulent, each
successive attack conferring a certain immunity.
In many lands, such as the United Kingdom, measles is rarely absent,
especially from large centres of population, where sporadic cases are found
at all seasons. Every now and then epidemics arise from the extension of
the disease among those members of a community who have not been in
some measure protected by a previous attack. There are few diseases so
contagious as measles, and its rapid spread in epidemic outbreaks is no
doubt due to the well-ascertained fact that contagion is most potent in the
earlier stages, even before its real nature has been evinced by the
characteristic appearances on the skin. Hence the difficulty of timely
isolation, and the readiness with which the disease is spread in schools and
families. The contagion is present in the skin and the various secretions.
While the contagion is generally direct, it can also be conveyed by the
particles from the nose and mouth which, after being expelled, become dry
Page 305
and are conveyed as dust on clothes, toys, &c. Fortunately the germs of
measles do not retain their virulence long under such conditions, comparing
favourably with those of some other diseases.
Treatment.—The treatment embraces the preventive measures to be
adopted by the isolation of the sick at as early a period as possible.
Epidemics have often, especially in limited localities, been curtailed by
such a precaution. In families with little house accommodation this measure
is frequently, for the reason given regarding the communicable period of the
disease, ineffectual; nevertheless where practicable it ought to be tried. The
unaffected children should be kept from school for a time (probably about
three weeks from the outbreak in the family would suffice if no other case
occur in the interval), and all clothing in contact with the patient or nurses
should be disinfected. In extensive epidemics it is often desirable to close
the schools for a time. As regards special treatment, in an ordinary case of
measles little is required beyond what is necessary in febrile conditions
generally. Confinement to bed in a somewhat darkened room, into which,
however, air is freely admitted; light, nourishing, liquid diet (soups, milk,
&c.), water almost ad lib. to drink, and mild diaphoretic remedies such as
the acetate of ammonia or ipecacuanha, are all that is necessary in the
febrile stage. When the fever is very severe, sponging the body generally or
the chest and arms affords relief. The serious chest complications of
measles are to be dealt with by those measures applicable for the relief of
the particular symptoms (see Bronchitis; Pneumonia). The preparations of
ammonia are of special efficacy. During convalescence the patient must be
guarded from exposure to cold, and for a time after recovery the state of the
health ought to be watched with a view of averting the evils, both local and
constitutional, which too often follow this disease.
“German measles” (Rötheln, or Epidemic Roseola) is a term applied
to a contagious eruptive disorder having certain points of resemblance
measles do not retain their virulence long under such conditions, comparing
favourably with those of some other diseases.
Treatment.—The treatment embraces the preventive measures to be
adopted by the isolation of the sick at as early a period as possible.
Epidemics have often, especially in limited localities, been curtailed by
such a precaution. In families with little house accommodation this measure
is frequently, for the reason given regarding the communicable period of the
disease, ineffectual; nevertheless where practicable it ought to be tried. The
unaffected children should be kept from school for a time (probably about
three weeks from the outbreak in the family would suffice if no other case
occur in the interval), and all clothing in contact with the patient or nurses
should be disinfected. In extensive epidemics it is often desirable to close
the schools for a time. As regards special treatment, in an ordinary case of
measles little is required beyond what is necessary in febrile conditions
generally. Confinement to bed in a somewhat darkened room, into which,
however, air is freely admitted; light, nourishing, liquid diet (soups, milk,
&c.), water almost ad lib. to drink, and mild diaphoretic remedies such as
the acetate of ammonia or ipecacuanha, are all that is necessary in the
febrile stage. When the fever is very severe, sponging the body generally or
the chest and arms affords relief. The serious chest complications of
measles are to be dealt with by those measures applicable for the relief of
the particular symptoms (see Bronchitis; Pneumonia). The preparations of
ammonia are of special efficacy. During convalescence the patient must be
guarded from exposure to cold, and for a time after recovery the state of the
health ought to be watched with a view of averting the evils, both local and
constitutional, which too often follow this disease.
“German measles” (Rötheln, or Epidemic Roseola) is a term applied
to a contagious eruptive disorder having certain points of resemblance
Page 306
to measles, and also to scarlet fever, but exhibiting its distinct
individuality in the fact that it protects from neither of these diseases. It
occurs most commonly in children, but frequently in adults also, and is
occasionally seen in extensive epidemics. Beyond confinement to the
house in the eruptive stage, which, from the slight symptoms
experienced, is often difficult of accomplishment, no special treatment
is called for. There is little doubt that the disease is often mistaken for
true measles, and many of the alleged second attacks of the latter
malady are probably cases of rötheln. The chief points of difference are
the following: (1) The absence of distinct premonitory symptoms, the
stage of invasion, which in measles is usually of four days’ duration,
and accompanied with well-marked fever and catarrh, being in rötheln
either wholly absent or exceedingly slight, enduring only for one day.
(2) The eruption of rötheln, which, although as regards its locality and
manner of progress similar to measles, differs somewhat in its
appearance, the spots being of smaller size, paler colour, and with less
tendency to grouping in crescentic patches. The rash attains its
maximum in about one day, and quickly disappears. There is not the
same increase of temperature in this stage as in measles. (3) The
presence of white spots on the buccal mucous membrane, in the case of
measles. (4) The milder character of the symptoms of rötheln
throughout its whole course, and the absence of complications and of
liability to subsequent impairment of health such as have been seen to
appertain to measles.
1 Transactions of the Epidemiological Society (London, 1877).
individuality in the fact that it protects from neither of these diseases. It
occurs most commonly in children, but frequently in adults also, and is
occasionally seen in extensive epidemics. Beyond confinement to the
house in the eruptive stage, which, from the slight symptoms
experienced, is often difficult of accomplishment, no special treatment
is called for. There is little doubt that the disease is often mistaken for
true measles, and many of the alleged second attacks of the latter
malady are probably cases of rötheln. The chief points of difference are
the following: (1) The absence of distinct premonitory symptoms, the
stage of invasion, which in measles is usually of four days’ duration,
and accompanied with well-marked fever and catarrh, being in rötheln
either wholly absent or exceedingly slight, enduring only for one day.
(2) The eruption of rötheln, which, although as regards its locality and
manner of progress similar to measles, differs somewhat in its
appearance, the spots being of smaller size, paler colour, and with less
tendency to grouping in crescentic patches. The rash attains its
maximum in about one day, and quickly disappears. There is not the
same increase of temperature in this stage as in measles. (3) The
presence of white spots on the buccal mucous membrane, in the case of
measles. (4) The milder character of the symptoms of rötheln
throughout its whole course, and the absence of complications and of
liability to subsequent impairment of health such as have been seen to
appertain to measles.
1 Transactions of the Epidemiological Society (London, 1877).
Page 307
MEAT, a word originally applied to food in general, and so still used
in such phrases as “meat and drink”; but now, except as an archaism,
generally used of the flesh of certain domestic animals, slaughtered for
human food by butchers, “butcher’s meat,” as opposed to “game,” that of
wild animals, “fish” or “poultry.” Cognate forms of the O. Eng. mete are
found in certain Teutonic languages, e.g. Swed. mat, Dan. mad and O. H.
Ger. Maz. The ultimate origin has been disputed; the New English
Dictionary considers probable a connexion with the root med-, “to be fat,”
seen in Sansk. mēda, Lat. madere, “to be wet,” and Eng. “mast,” the fruit of
the beech as food for pigs.
See Dietetics; Food Preservation; Public Health; Agriculture; and
the sections dealing with agricultural statistics under the names of the
various countries.
MEATH (pronounced with th soft, as in the), a county of Ireland in
the province of Leinster, bounded E. by the Irish Sea, S.E. by Dublin, S. by
in such phrases as “meat and drink”; but now, except as an archaism,
generally used of the flesh of certain domestic animals, slaughtered for
human food by butchers, “butcher’s meat,” as opposed to “game,” that of
wild animals, “fish” or “poultry.” Cognate forms of the O. Eng. mete are
found in certain Teutonic languages, e.g. Swed. mat, Dan. mad and O. H.
Ger. Maz. The ultimate origin has been disputed; the New English
Dictionary considers probable a connexion with the root med-, “to be fat,”
seen in Sansk. mēda, Lat. madere, “to be wet,” and Eng. “mast,” the fruit of
the beech as food for pigs.
See Dietetics; Food Preservation; Public Health; Agriculture; and
the sections dealing with agricultural statistics under the names of the
various countries.
MEATH (pronounced with th soft, as in the), a county of Ireland in
the province of Leinster, bounded E. by the Irish Sea, S.E. by Dublin, S. by
Page 308
Kildare and King’s County, W. by Westmeath, N.W. by Cavan and
Monaghan, and N.E. by Louth. Area 579,320 acres, or about 905 sq. m. In
some districts the surface is varied by hills and swells, which to the west
reach a considerable elevation, although the general features of a fine
champain country are never lost. The coast, low and shelving, extends about
10 m., but there is no harbour of importance. Laytown is a small seaside
resort, 5 m. S.E. of Drogheda. The Boyne enters the county at its south-
western extremity, and flowing north-east to Drogheda divides it into two
almost equal parts. At Navan it receives the Blackwater, which flows south-
west from Cavan. Both these rivers are noted for their trout, and salmon are
taken in the Boyne. The Boyne is navigable for barges as far as Navan
whence a canal is carried to Trim. The Royal Canal passes along the
southern boundary of the county from Dublin.
In the north is a broken country of Silurian rocks with much igneous
material, partly contemporaneous, partly intrusive, near Slane.
Carboniferous Limestone stretches from the Boyne valley to the Dublin
border, giving rise to a flat plain especially suitable for grazing.
Outliers of higher Carboniferous strata occur on the surface; but the
Coal Measures have all been removed by denudation.
The climate is genial and favourable for all kinds of crops, there
being less rain than even in the neighbouring counties. Except a small
portion occupied by the Bog of Allen, the county is verdant and fertile.
The soil is principally a rich deep loam resting on limestone gravel, but
varies from a strong clayey loam to a light sandy gravel. The
proportion of tillage to pasturage is roughly as 1 to 31⁄2. Oats, potatoes
and turnips are the principal crops, but all decrease. The numbers of
cattle, sheep and poultry, however, are increasing or well maintained.
Agriculture is almost the sole industry, but coarse linen is woven by
hand-looms, and there are a few woollen manufactories. The main line
Monaghan, and N.E. by Louth. Area 579,320 acres, or about 905 sq. m. In
some districts the surface is varied by hills and swells, which to the west
reach a considerable elevation, although the general features of a fine
champain country are never lost. The coast, low and shelving, extends about
10 m., but there is no harbour of importance. Laytown is a small seaside
resort, 5 m. S.E. of Drogheda. The Boyne enters the county at its south-
western extremity, and flowing north-east to Drogheda divides it into two
almost equal parts. At Navan it receives the Blackwater, which flows south-
west from Cavan. Both these rivers are noted for their trout, and salmon are
taken in the Boyne. The Boyne is navigable for barges as far as Navan
whence a canal is carried to Trim. The Royal Canal passes along the
southern boundary of the county from Dublin.
In the north is a broken country of Silurian rocks with much igneous
material, partly contemporaneous, partly intrusive, near Slane.
Carboniferous Limestone stretches from the Boyne valley to the Dublin
border, giving rise to a flat plain especially suitable for grazing.
Outliers of higher Carboniferous strata occur on the surface; but the
Coal Measures have all been removed by denudation.
The climate is genial and favourable for all kinds of crops, there
being less rain than even in the neighbouring counties. Except a small
portion occupied by the Bog of Allen, the county is verdant and fertile.
The soil is principally a rich deep loam resting on limestone gravel, but
varies from a strong clayey loam to a light sandy gravel. The
proportion of tillage to pasturage is roughly as 1 to 31⁄2. Oats, potatoes
and turnips are the principal crops, but all decrease. The numbers of
cattle, sheep and poultry, however, are increasing or well maintained.
Agriculture is almost the sole industry, but coarse linen is woven by
hand-looms, and there are a few woollen manufactories. The main line
Page 309
of the Midland Great Western railway skirts the southern boundary,
with a branch line north from Clonsilla to Navan and Kingscourt
(county Cavan). From Kilmessan on this line a branch serves Trim and
Athboy. From Drogheda (county Louth) a branch of the Great Northern
railway crosses the county from east to West by Navan and Kells to
Oldcastle.
The population (76,111 in 1891; 67,497 in 1901) suffers a large
decrease, considerably above the average of Irish counties, and
emigration is heavy. Nearly 93% are Roman Catholics. The chief towns
are Navan (pop. 3839), Kells (2428) and Trim (1513), the county town.
Lesser market towns are Oldcastle and Athboy, an ancient town which
received a charter from Henry IV. The county includes eighteen
baronies. Assizes are held at Trim, and quarter sessions at Kells, Navan
and Trim. The county is in the Protestant dioceses of Armagh, Kilmore
and Meath, and in the Roman Catholic dioceses of Armagh and Meath.
Before the Union in 1800 it sent fourteen members to parliament, but
now only two members are returned, for the north and south divisions
of the county respectively.
History and Antiquities.—A district known as Meath (Midhe), and
including the present county of Meath as well as Westmeath and Longford,
with parts of Cavan, Kildare and King’s County, was formed by Tuathal (c.
130) into a kingdom to serve as mensal land or personal estate of the Ard Ri
or over-king of Ireland. Kings of Meath reigned until 1173, and the title was
claimed as late as the 15th century by their descendants, but at the date
mentioned Hugh de Lacy obtained the lordship of the country and was
confirmed in it by Henry II. Meath thus came into the English “Pale.” But
though it was declared a county in the reign of Edward I. (1296), and
though it came by descent into the possession of the Crown in the person of
Edward IV., it was long before it was fully subdued and its boundaries
with a branch line north from Clonsilla to Navan and Kingscourt
(county Cavan). From Kilmessan on this line a branch serves Trim and
Athboy. From Drogheda (county Louth) a branch of the Great Northern
railway crosses the county from east to West by Navan and Kells to
Oldcastle.
The population (76,111 in 1891; 67,497 in 1901) suffers a large
decrease, considerably above the average of Irish counties, and
emigration is heavy. Nearly 93% are Roman Catholics. The chief towns
are Navan (pop. 3839), Kells (2428) and Trim (1513), the county town.
Lesser market towns are Oldcastle and Athboy, an ancient town which
received a charter from Henry IV. The county includes eighteen
baronies. Assizes are held at Trim, and quarter sessions at Kells, Navan
and Trim. The county is in the Protestant dioceses of Armagh, Kilmore
and Meath, and in the Roman Catholic dioceses of Armagh and Meath.
Before the Union in 1800 it sent fourteen members to parliament, but
now only two members are returned, for the north and south divisions
of the county respectively.
History and Antiquities.—A district known as Meath (Midhe), and
including the present county of Meath as well as Westmeath and Longford,
with parts of Cavan, Kildare and King’s County, was formed by Tuathal (c.
130) into a kingdom to serve as mensal land or personal estate of the Ard Ri
or over-king of Ireland. Kings of Meath reigned until 1173, and the title was
claimed as late as the 15th century by their descendants, but at the date
mentioned Hugh de Lacy obtained the lordship of the country and was
confirmed in it by Henry II. Meath thus came into the English “Pale.” But
though it was declared a county in the reign of Edward I. (1296), and
though it came by descent into the possession of the Crown in the person of
Edward IV., it was long before it was fully subdued and its boundaries
Page 310
clearly defined. In 1543 Westmeath was created a county apart from that of
Meath, but as late as 1598 Meath was still regarded as a province by some,
who included in it the counties Westmeath, East Meath, Longford and
Cavan. In the early part of the 17th century it was at last established as a
county, and no longer considered as a fifth province of Ireland.
There are two ancient round towers, the one at Kells and the other in the
churchyard of Donaghmore, near Navan. By the river Boyne near Slane
there is an extensive ancient burial-place called Brugh. Here are some
twenty burial mounds, the largest of which is that of New Grange, a domed
tumulus erected above a circular chamber, which is entered by a narrow
passage enclosed by great upright blocks of stone, covered with carvings.
The mound is surrounded by remains of a stone circle, and the whole forms
one of the most remarkable extant erections of its kind. Tara (q.v.) is famous
in history, especially as the seat of a royal palace referred to in the well-
known lines of Thomas Moore. Monastic buildings were very numerous in
Meath, among the more important ruins being those of Duleek, which is
said to have been the first ecclesiastical building in Ireland of stone and
mortar; the extensive remains of Bective Abbey; and those of Clonard,
where also were a cathedral and a famous college. Of the old fortresses, the
castle of Trim still presents an imposing appearance. There are many fine
old mansions.
Meath, but as late as 1598 Meath was still regarded as a province by some,
who included in it the counties Westmeath, East Meath, Longford and
Cavan. In the early part of the 17th century it was at last established as a
county, and no longer considered as a fifth province of Ireland.
There are two ancient round towers, the one at Kells and the other in the
churchyard of Donaghmore, near Navan. By the river Boyne near Slane
there is an extensive ancient burial-place called Brugh. Here are some
twenty burial mounds, the largest of which is that of New Grange, a domed
tumulus erected above a circular chamber, which is entered by a narrow
passage enclosed by great upright blocks of stone, covered with carvings.
The mound is surrounded by remains of a stone circle, and the whole forms
one of the most remarkable extant erections of its kind. Tara (q.v.) is famous
in history, especially as the seat of a royal palace referred to in the well-
known lines of Thomas Moore. Monastic buildings were very numerous in
Meath, among the more important ruins being those of Duleek, which is
said to have been the first ecclesiastical building in Ireland of stone and
mortar; the extensive remains of Bective Abbey; and those of Clonard,
where also were a cathedral and a famous college. Of the old fortresses, the
castle of Trim still presents an imposing appearance. There are many fine
old mansions.
Page 311
MEAUX, a town of northern France, capital of an arrondissement in
the department of Seine-et-Marne, and chief town of the agricultural region
of Brie, 28 m. E.N.E. of Paris by rail. Pop. (1906), 11,089. The town proper
stands on an eminence on the right bank of the Marne; on the left bank lies
the old suburb of Le Marché, with which it is united by a bridge of the 16th
century. Two rows of picturesque mills of the same period are built across
the river. The cathedral of St Stephen dates from the 12th to the 16th
centuries, and was restored in the 19th century. Of the two western towers,
the completed one is that to the north of the façade, the other being
disfigured by an unsightly slate roof. The building, which is 275 ft. long and
105 ft. high, consists of a short nave, with aisles, a fine transept, a choir and
a sanctuary. The choir contains the statue and the tomb of Bossuet, bishop
from 1681 to 1704, and the pulpit of the cathedral has been reconstructed
with the panels of that from which the “eagle of Meaux” used to preach.
The transept terminates at each end in a fine portal surmounted by a rose-
window. The episcopal palace (17th century) has several curious old rooms;
the buildings of the choir school are likewise of some archaeological
interest. A statue of General Raoult (1870) stands in one of the squares.
Meaux is the centre of a considerable trade in cereals, wool, Brie cheeses,
and other farm-produce, while its mills provide much of the flour with
which Paris is supplied. Other industries are saw-milling, metal-founding,
distilling, the preparation of vermicelli and preserved vegetables, and the
manufacture of mustard, hosiery, plaster and machinery. There are nursery-
gardens in the vicinity. The Canal de l’Ourcq, which surrounds the town,
and the Marne furnish the means of transport. Meaux is the seat of a
bishopric dating from the 4th century, and has among its public institutions
a sub-prefecture, and tribunals of first instance and of commerce.
In the Roman period Meaux was the capital of the Meldi, a small Gallic
tribe, and in the middle ages of the Brie. It formed part of the kingdom of
the department of Seine-et-Marne, and chief town of the agricultural region
of Brie, 28 m. E.N.E. of Paris by rail. Pop. (1906), 11,089. The town proper
stands on an eminence on the right bank of the Marne; on the left bank lies
the old suburb of Le Marché, with which it is united by a bridge of the 16th
century. Two rows of picturesque mills of the same period are built across
the river. The cathedral of St Stephen dates from the 12th to the 16th
centuries, and was restored in the 19th century. Of the two western towers,
the completed one is that to the north of the façade, the other being
disfigured by an unsightly slate roof. The building, which is 275 ft. long and
105 ft. high, consists of a short nave, with aisles, a fine transept, a choir and
a sanctuary. The choir contains the statue and the tomb of Bossuet, bishop
from 1681 to 1704, and the pulpit of the cathedral has been reconstructed
with the panels of that from which the “eagle of Meaux” used to preach.
The transept terminates at each end in a fine portal surmounted by a rose-
window. The episcopal palace (17th century) has several curious old rooms;
the buildings of the choir school are likewise of some archaeological
interest. A statue of General Raoult (1870) stands in one of the squares.
Meaux is the centre of a considerable trade in cereals, wool, Brie cheeses,
and other farm-produce, while its mills provide much of the flour with
which Paris is supplied. Other industries are saw-milling, metal-founding,
distilling, the preparation of vermicelli and preserved vegetables, and the
manufacture of mustard, hosiery, plaster and machinery. There are nursery-
gardens in the vicinity. The Canal de l’Ourcq, which surrounds the town,
and the Marne furnish the means of transport. Meaux is the seat of a
bishopric dating from the 4th century, and has among its public institutions
a sub-prefecture, and tribunals of first instance and of commerce.
In the Roman period Meaux was the capital of the Meldi, a small Gallic
tribe, and in the middle ages of the Brie. It formed part of the kingdom of
Page 312
Austrasia, and afterwards belonged to the counts of Vermandois and
Champagne, the latter of whom established important markets on the left
bank of the Marne. Its communal charter, received from them, is dated
1179. A treaty signed at Meaux in 1229 after the Albigensian War sealed the
submission of Raymond VII., count of Toulouse. The town suffered much
during the Jacquerie, the peasants receiving a severe check there in 1358;
during the Hundred Years’ War; and also during the Religious Wars, in
which it was an important Protestant centre. It was the first town which
opened its gates to Henry IV. in 1594. On the high-road for invaders
marching on Paris from the east of France, Meaux saw its environs ravaged
by the army of Lorraine in 1652, and was laid under heavy requisitions in
1814, 1815 and 1870. In September 1567 Meaux was the scene of an
attempt made by the Protestants to seize the French king Charles IX., and
his mother Catherine de’ Medici. The plot, which is sometimes called the
“enterprise of Meaux,” failed, the king and queen with their courtiers
escaping to Paris. This conduct, however, on the part of the Huguenots had
doubtless some share in influencing Charles to assent to the massacre of St
Bartholomew.
MECCA (Arab. Makkah),1 the chief town of the Hejaz in Arabia, and
the great holy city of Islām. It is situated two camel marches (the resting-
place being Bahra or Hadda), or about 45 m. almost due E., from Jidda on
the Red Sea. Thus on a rough estimate Mecca lies in 21° 25′ N., 39° 50′ E.
It is said in the Koran (Sur. xiv. 40) that Mecca lies in a sterile valley, and
Champagne, the latter of whom established important markets on the left
bank of the Marne. Its communal charter, received from them, is dated
1179. A treaty signed at Meaux in 1229 after the Albigensian War sealed the
submission of Raymond VII., count of Toulouse. The town suffered much
during the Jacquerie, the peasants receiving a severe check there in 1358;
during the Hundred Years’ War; and also during the Religious Wars, in
which it was an important Protestant centre. It was the first town which
opened its gates to Henry IV. in 1594. On the high-road for invaders
marching on Paris from the east of France, Meaux saw its environs ravaged
by the army of Lorraine in 1652, and was laid under heavy requisitions in
1814, 1815 and 1870. In September 1567 Meaux was the scene of an
attempt made by the Protestants to seize the French king Charles IX., and
his mother Catherine de’ Medici. The plot, which is sometimes called the
“enterprise of Meaux,” failed, the king and queen with their courtiers
escaping to Paris. This conduct, however, on the part of the Huguenots had
doubtless some share in influencing Charles to assent to the massacre of St
Bartholomew.
MECCA (Arab. Makkah),1 the chief town of the Hejaz in Arabia, and
the great holy city of Islām. It is situated two camel marches (the resting-
place being Bahra or Hadda), or about 45 m. almost due E., from Jidda on
the Red Sea. Thus on a rough estimate Mecca lies in 21° 25′ N., 39° 50′ E.
It is said in the Koran (Sur. xiv. 40) that Mecca lies in a sterile valley, and
Page 313
the old geographers observe that the whole Haram or sacred territory round
the city is almost without cultivation or date palms, while fruit trees,
springs, wells, gardens and green valleys are found immediately beyond.
Mecca in fact lies in the heart of a mass of rough hills, intersected by a
labyrinth of narrow valleys and passes, and projecting into the Tehāma or
low country on the Red Sea, in front of the great mountain wall that divides
the coast-lands from the central plateau, though in turn they are themselves
separated from the sea by a second curtain of hills forming the western wall
of the great Wādi Marr. The inner mountain wall is pierced by only two
great passes, and the valleys descending from these embrace on both sides
the Mecca hills.
Holding this position commanding two great routes between the lowlands
and inner Arabia, and situated in a narrow and barren valley incapable of
supporting an urban population, Mecca must have been from the first a
commercial centre.2 In the palmy days of South Arabia it was probably a
station on the great incense route, and thus Ptolemy may have learned the
name, which he writes Makoraba. At all events, long before Mahomet we
find Mecca established in the twofold quality of a commercial centre and a
privileged holy place, surrounded by an inviolable territory (the Haram),
which was not the sanctuary of a single tribe but a place of pilgrimage,
where religious observances were associated with a series of annual fairs at
different points in the vicinity. Indeed in the unsettled state of the country
commerce was possible only under the sanctions of religion, and through
the provisions of the sacred truce which prohibited war for four months of
the year, three of these being the month of pilgrimage, with those
immediately preceding and following. The first of the series of fairs in
which the Meccans had an interest was at Okaz on the easier road between
Mecca and Taif, where there was also a sanctuary, and from it the visitors
moved on to points still nearer Mecca (Majanna, and finally Dhul-Majāz,
on the flank of Jebel Kabkab behind Arafa) where further fairs were held,3
the city is almost without cultivation or date palms, while fruit trees,
springs, wells, gardens and green valleys are found immediately beyond.
Mecca in fact lies in the heart of a mass of rough hills, intersected by a
labyrinth of narrow valleys and passes, and projecting into the Tehāma or
low country on the Red Sea, in front of the great mountain wall that divides
the coast-lands from the central plateau, though in turn they are themselves
separated from the sea by a second curtain of hills forming the western wall
of the great Wādi Marr. The inner mountain wall is pierced by only two
great passes, and the valleys descending from these embrace on both sides
the Mecca hills.
Holding this position commanding two great routes between the lowlands
and inner Arabia, and situated in a narrow and barren valley incapable of
supporting an urban population, Mecca must have been from the first a
commercial centre.2 In the palmy days of South Arabia it was probably a
station on the great incense route, and thus Ptolemy may have learned the
name, which he writes Makoraba. At all events, long before Mahomet we
find Mecca established in the twofold quality of a commercial centre and a
privileged holy place, surrounded by an inviolable territory (the Haram),
which was not the sanctuary of a single tribe but a place of pilgrimage,
where religious observances were associated with a series of annual fairs at
different points in the vicinity. Indeed in the unsettled state of the country
commerce was possible only under the sanctions of religion, and through
the provisions of the sacred truce which prohibited war for four months of
the year, three of these being the month of pilgrimage, with those
immediately preceding and following. The first of the series of fairs in
which the Meccans had an interest was at Okaz on the easier road between
Mecca and Taif, where there was also a sanctuary, and from it the visitors
moved on to points still nearer Mecca (Majanna, and finally Dhul-Majāz,
on the flank of Jebel Kabkab behind Arafa) where further fairs were held,3
Page 314
culminating in the special religious ceremonies of the great feast at ‘Arafa,
Quzaḥ (Mozdalifa), and Mecca itself. The system of intercalation in the
lunar calendar of the heathen Arabs was designed to secure that the feast
should always fall at the time when the hides, fruits and other merchandise
were ready for market,4 and the Meccans, who knew how to attract the
Bedouins by hospitality, bought up these wares in exchange for imported
goods, and so became the leaders of the international trade of Arabia. Their
caravans traversed the length and breadth of the peninsula. Syria, and
especially Gaza, was their chief goal. The Syrian caravan intercepted, on its
return, at Badr (see Mahomet) represented capital to the value of £20,000,
an enormous sum for those days.5
The victory of Mahommedanism made a vast change in the position of
Mecca. The merchant aristocracy became satraps or pensioners of a great
empire; but the seat of dominion was removed beyond the desert, and
though Mecca and the Hejāz strove for a time to maintain political as well
as religious predominance, the struggle was vain, and terminated on the
death of Ibn Zubair, the Meccan pretendant to the caliphate, when the city
was taken by Hajjāj (a.d. 692). The sanctuary and feast of Mecca received,
however, a new prestige from the victory of Islām. Purged of elements
obviously heathen, the Ka‘ba became the holiest site, and the pilgrimage the
most sacred ritual observance of Mahommedanism, drawing worshippers
from so wide a circle that the confluence of the petty traders of the desert
was no longer the main feature of the holy season. The pilgrimage retained
its importance for the commercial well-being of Mecca; to this day the
Meccans live by the Hajj—letting rooms, acting as guides and directors in
the sacred ceremonies, as contractors and touts for land and sea transport, as
well as exploiting the many benefactions that flow to the holy city; while
the surrounding Bedouins derive support from the camel-transport it
demands and from the subsidies by which they are engaged to protect or
abstain from molesting the pilgrim caravans. But the ancient “fairs of
Quzaḥ (Mozdalifa), and Mecca itself. The system of intercalation in the
lunar calendar of the heathen Arabs was designed to secure that the feast
should always fall at the time when the hides, fruits and other merchandise
were ready for market,4 and the Meccans, who knew how to attract the
Bedouins by hospitality, bought up these wares in exchange for imported
goods, and so became the leaders of the international trade of Arabia. Their
caravans traversed the length and breadth of the peninsula. Syria, and
especially Gaza, was their chief goal. The Syrian caravan intercepted, on its
return, at Badr (see Mahomet) represented capital to the value of £20,000,
an enormous sum for those days.5
The victory of Mahommedanism made a vast change in the position of
Mecca. The merchant aristocracy became satraps or pensioners of a great
empire; but the seat of dominion was removed beyond the desert, and
though Mecca and the Hejāz strove for a time to maintain political as well
as religious predominance, the struggle was vain, and terminated on the
death of Ibn Zubair, the Meccan pretendant to the caliphate, when the city
was taken by Hajjāj (a.d. 692). The sanctuary and feast of Mecca received,
however, a new prestige from the victory of Islām. Purged of elements
obviously heathen, the Ka‘ba became the holiest site, and the pilgrimage the
most sacred ritual observance of Mahommedanism, drawing worshippers
from so wide a circle that the confluence of the petty traders of the desert
was no longer the main feature of the holy season. The pilgrimage retained
its importance for the commercial well-being of Mecca; to this day the
Meccans live by the Hajj—letting rooms, acting as guides and directors in
the sacred ceremonies, as contractors and touts for land and sea transport, as
well as exploiting the many benefactions that flow to the holy city; while
the surrounding Bedouins derive support from the camel-transport it
demands and from the subsidies by which they are engaged to protect or
abstain from molesting the pilgrim caravans. But the ancient “fairs of
Page 315
heathenism” were given up, and the traffic of the pilgrim season, sanctioned
by the Prophet in Sur. ii. 194, was concentrated at Minā and Mecca, where
most of the pilgrims still have something to buy or sell, so that Minā, after
the sacrifice of the feast day, presents the aspect of a huge international
fancy fair.6 In the middle ages this trade was much more important than it is
now. Ibn Jubair (ed. Wright, p. 118 seq.) in the 12th century describes the
mart of Mecca in the eight days following the feast as full of gems,
unguents, precious drugs, and all rare merchandise from India, Irāk,
Khorāsān, and every part of the Moslem world.
The hills east and west of Mecca, which are partly built over and rise
several hundred feet above the valley, so enclose the city that the ancient
walls only barred the valley at three points, where three gates led into the
town. In the time of Ibn Jubair the gates still stood though the walls were
ruined, but now the gates have only left their names to quarters of the town.
At the northern or upper end was the Bāb el Mā‘lā, or gate of the upper
quarter, whence the road continues up the valley towards Minā and Arafa as
well as towards Zeima and the Nejd. Beyond the gate, in a place called the
Hajūn, is the chief cemetery, commonly called el Mā‘lā, and said to be the
resting-place of many of the companions of Mahomet. Here a cross-road,
running over the hill to join the main Medina road from the western gate,
turns off to the west by the pass of Kadā, the point from which the troops of
the Prophet stormed the city (a.h. 8).7 Here too the body of Ibn Zubair was
hung on a cross by Ḥajjāj. The lower or southern gate, at the Masfala
quarter, opened on the Yemen road, where the rain-water from Mecca flows
off into an open valley. Beyond, there are mountains on both sides; on that
to the east, commanding the town, is the great castle, a fortress of
considerable strength. The third or western gate, Bāb el-Omra (formerly
also Bāb el-Zāhir, from a village of that name), lay almost opposite the
great mosque, and opened on a road leading westwards round the southern
spurs of the Red Mountain. This is the way to Wādi Fātima and Medīna, the
by the Prophet in Sur. ii. 194, was concentrated at Minā and Mecca, where
most of the pilgrims still have something to buy or sell, so that Minā, after
the sacrifice of the feast day, presents the aspect of a huge international
fancy fair.6 In the middle ages this trade was much more important than it is
now. Ibn Jubair (ed. Wright, p. 118 seq.) in the 12th century describes the
mart of Mecca in the eight days following the feast as full of gems,
unguents, precious drugs, and all rare merchandise from India, Irāk,
Khorāsān, and every part of the Moslem world.
The hills east and west of Mecca, which are partly built over and rise
several hundred feet above the valley, so enclose the city that the ancient
walls only barred the valley at three points, where three gates led into the
town. In the time of Ibn Jubair the gates still stood though the walls were
ruined, but now the gates have only left their names to quarters of the town.
At the northern or upper end was the Bāb el Mā‘lā, or gate of the upper
quarter, whence the road continues up the valley towards Minā and Arafa as
well as towards Zeima and the Nejd. Beyond the gate, in a place called the
Hajūn, is the chief cemetery, commonly called el Mā‘lā, and said to be the
resting-place of many of the companions of Mahomet. Here a cross-road,
running over the hill to join the main Medina road from the western gate,
turns off to the west by the pass of Kadā, the point from which the troops of
the Prophet stormed the city (a.h. 8).7 Here too the body of Ibn Zubair was
hung on a cross by Ḥajjāj. The lower or southern gate, at the Masfala
quarter, opened on the Yemen road, where the rain-water from Mecca flows
off into an open valley. Beyond, there are mountains on both sides; on that
to the east, commanding the town, is the great castle, a fortress of
considerable strength. The third or western gate, Bāb el-Omra (formerly
also Bāb el-Zāhir, from a village of that name), lay almost opposite the
great mosque, and opened on a road leading westwards round the southern
spurs of the Red Mountain. This is the way to Wādi Fātima and Medīna, the
Page 316
Jidda road branching off from it to the left. Considerable suburbs now lie
outside the quarter named after this gate; in the middle ages a pleasant
country road led for some miles through partly cultivated land with good
wells, as far as the boundary of the sacred territory and gathering place of
the pilgrims at Tanīm, near the mosque of Ayesha. This is the spot on the
Medīna road now called the Omra, from a ceremonial connected with it
which will be mentioned below.
The length of the sinuous main axis of the city from the farthest suburbs
on the Medina road to the suburbs in the extreme north, now frequented by
Bedouins, is, according to Burckhardt, 3500 paces.8 About the middle of
this line the longitudinal thoroughfares are pushed aside by the vast
courtyard and colonnades composing the great mosque, which, with its
spacious arcades surrounding the Ka‘ba and other holy places, and its seven
minarets, forms the only prominent architectural feature of the city. The
mosque is enclosed by houses with windows opening on the arcades and
commanding a view of the Ka‘ba. Immediately beyond these, on the side
facing Jebel Abu Kobais, a broad street runs south-east and north-west
across the valley. This is the Mas‘ā (sacred course) between the eminences
of Safā and Merwa, and has been from very early times one of the most
lively bazaars and the centre of Meccan life. The other chief bazaars are
also near the mosque in smaller streets. The general aspect of the town is
picturesque; the streets are fairly spacious, though ill-kept and filthy; the
houses are all of stone, many of them well-built and four or five storeys
high, with terraced roofs and large projecting windows as in Jidda—a style
of building which has not varied materially since the 10th century
(Mukaddasī, p. 71), and gains in effect from the way in which the dwellings
run up the sides and spurs of the mountains. Of public institutions there are
baths, ribāṭs, or hospices, for poor pilgrims from India, Java, &c., a hospital
and a public kitchen for the poor.
outside the quarter named after this gate; in the middle ages a pleasant
country road led for some miles through partly cultivated land with good
wells, as far as the boundary of the sacred territory and gathering place of
the pilgrims at Tanīm, near the mosque of Ayesha. This is the spot on the
Medīna road now called the Omra, from a ceremonial connected with it
which will be mentioned below.
The length of the sinuous main axis of the city from the farthest suburbs
on the Medina road to the suburbs in the extreme north, now frequented by
Bedouins, is, according to Burckhardt, 3500 paces.8 About the middle of
this line the longitudinal thoroughfares are pushed aside by the vast
courtyard and colonnades composing the great mosque, which, with its
spacious arcades surrounding the Ka‘ba and other holy places, and its seven
minarets, forms the only prominent architectural feature of the city. The
mosque is enclosed by houses with windows opening on the arcades and
commanding a view of the Ka‘ba. Immediately beyond these, on the side
facing Jebel Abu Kobais, a broad street runs south-east and north-west
across the valley. This is the Mas‘ā (sacred course) between the eminences
of Safā and Merwa, and has been from very early times one of the most
lively bazaars and the centre of Meccan life. The other chief bazaars are
also near the mosque in smaller streets. The general aspect of the town is
picturesque; the streets are fairly spacious, though ill-kept and filthy; the
houses are all of stone, many of them well-built and four or five storeys
high, with terraced roofs and large projecting windows as in Jidda—a style
of building which has not varied materially since the 10th century
(Mukaddasī, p. 71), and gains in effect from the way in which the dwellings
run up the sides and spurs of the mountains. Of public institutions there are
baths, ribāṭs, or hospices, for poor pilgrims from India, Java, &c., a hospital
and a public kitchen for the poor.
Page 317
The mosque is at the same time the university hall, where between two
pilgrim seasons lectures are delivered on Mahommedan law, doctrine and
connected branches of science. A poorly provided public library is open to
the use of students. The madrassehs or buildings around the mosque,
originally intended as lodgings for students and professors, have long been
let out to rich pilgrims. The minor places of visitation for pilgrims, such as
the birthplaces of the prophet and his chief followers, are not notable.9 Both
these and the court of the great mosque lie beneath the general level of the
city, the site having been gradually raised by accumulated rubbish. The
town in fact has little air of antiquity; genuine Arab buildings do not last
long, especially in a valley periodically ravaged by tremendous floods when
the tropical rains burst on the surrounding hills. The history of Mecca is full
of the record of these inundations, unsuccessfully combated by the great
dam drawn across the valley by the caliph Omar (Kutbeddin, p. 76), and
later works of Mahdī.10
The fixed population of Mecca in 1878 was estimated by Assistant-
Surgeon ‘Abd el-Razzāq at 50,000 to 60,000; there is a large floating
population—and that not merely at the proper season of pilgrimage, the
pilgrims of one season often beginning to arrive before those of the former
season have all dispersed. At the height of the season the town is much
overcrowded, and the entire want of a drainage system is severely felt.
Fortunately good water is tolerably plentiful; for, though the wells are
mostly undrinkable, and even the famous Zamzam water only available for
medicinal or religious purposes, the underground conduit from beyond
Arafa, completed by Sultan Selim II. in 1571, supplies to the public
fountains a sweet and light water, containing, according to ‘Abd el-Razzāq,
a large amount of chlorides. The water is said to be free to townsmen, but is
sold to the pilgrims at a rather high rate.11
pilgrim seasons lectures are delivered on Mahommedan law, doctrine and
connected branches of science. A poorly provided public library is open to
the use of students. The madrassehs or buildings around the mosque,
originally intended as lodgings for students and professors, have long been
let out to rich pilgrims. The minor places of visitation for pilgrims, such as
the birthplaces of the prophet and his chief followers, are not notable.9 Both
these and the court of the great mosque lie beneath the general level of the
city, the site having been gradually raised by accumulated rubbish. The
town in fact has little air of antiquity; genuine Arab buildings do not last
long, especially in a valley periodically ravaged by tremendous floods when
the tropical rains burst on the surrounding hills. The history of Mecca is full
of the record of these inundations, unsuccessfully combated by the great
dam drawn across the valley by the caliph Omar (Kutbeddin, p. 76), and
later works of Mahdī.10
The fixed population of Mecca in 1878 was estimated by Assistant-
Surgeon ‘Abd el-Razzāq at 50,000 to 60,000; there is a large floating
population—and that not merely at the proper season of pilgrimage, the
pilgrims of one season often beginning to arrive before those of the former
season have all dispersed. At the height of the season the town is much
overcrowded, and the entire want of a drainage system is severely felt.
Fortunately good water is tolerably plentiful; for, though the wells are
mostly undrinkable, and even the famous Zamzam water only available for
medicinal or religious purposes, the underground conduit from beyond
Arafa, completed by Sultan Selim II. in 1571, supplies to the public
fountains a sweet and light water, containing, according to ‘Abd el-Razzāq,
a large amount of chlorides. The water is said to be free to townsmen, but is
sold to the pilgrims at a rather high rate.11
Page 318
Medieval writers celebrate the copious supplies, especially of fine fruits,
brought to the city from Tāif and other fertile parts of Arabia. These fruits
are still famous; rice and other foreign products are brought by sea to Jidda;
mutton, milk and butter are plentifully supplied from the desert.12 The
industries all centre in the pilgrimage; the chief object of every Meccan—
from the notables and sheikhs, who use their influence to gain custom for
the Jidda speculators in the pilgrim traffic, down to the cicerones, pilgrim
brokers, lodging-house keepers, and mendicants at the holy places—being
to pillage the visitor in every possible way. The fanaticism of the Meccan is
an affair of the purse; the mongrel population (for the town is by no means
purely Arab) has exchanged the virtues of the Bedouin for the worst
corruptions of Eastern town life, without casting off the ferocity of the
desert, and it is hardly possible to find a worse certificate of character than
the three parallel gashes on each cheek, called Tashrīṭ, which are the
customary mark of birth in the holy city. The unspeakable vices of Mecca
are a scandal to all Islām, and a constant source of wonder to pious
pilgrims.13 The slave trade has connexions with the pilgrimage which are
not thoroughly clear; but under cover of the pilgrimage a great deal of
importation and exportation of slaves goes on.
Since the fall of Ibn Zubair the political position of Mecca has always
been dependent on the movements of the greater Mahommedan world. In
the splendid times of the caliphs immense sums were lavished upon the
pilgrimage and the holy city; and conversely the decay of the central
authority of Islām brought with it a long period of faction, wars and misery,
in which the most notable episode was the sack of Mecca by the
Carmathians at the pilgrimage season of a.d. 930. The victors carried off the
“black stone,” which was not restored for twenty-two years, and then only
for a great ransom, when it was plain that even the loss of its palladium
could not destroy the sacred character of the city. Under the Fatimites
Egyptian influence began to be strong in Mecca; it was opposed by the
brought to the city from Tāif and other fertile parts of Arabia. These fruits
are still famous; rice and other foreign products are brought by sea to Jidda;
mutton, milk and butter are plentifully supplied from the desert.12 The
industries all centre in the pilgrimage; the chief object of every Meccan—
from the notables and sheikhs, who use their influence to gain custom for
the Jidda speculators in the pilgrim traffic, down to the cicerones, pilgrim
brokers, lodging-house keepers, and mendicants at the holy places—being
to pillage the visitor in every possible way. The fanaticism of the Meccan is
an affair of the purse; the mongrel population (for the town is by no means
purely Arab) has exchanged the virtues of the Bedouin for the worst
corruptions of Eastern town life, without casting off the ferocity of the
desert, and it is hardly possible to find a worse certificate of character than
the three parallel gashes on each cheek, called Tashrīṭ, which are the
customary mark of birth in the holy city. The unspeakable vices of Mecca
are a scandal to all Islām, and a constant source of wonder to pious
pilgrims.13 The slave trade has connexions with the pilgrimage which are
not thoroughly clear; but under cover of the pilgrimage a great deal of
importation and exportation of slaves goes on.
Since the fall of Ibn Zubair the political position of Mecca has always
been dependent on the movements of the greater Mahommedan world. In
the splendid times of the caliphs immense sums were lavished upon the
pilgrimage and the holy city; and conversely the decay of the central
authority of Islām brought with it a long period of faction, wars and misery,
in which the most notable episode was the sack of Mecca by the
Carmathians at the pilgrimage season of a.d. 930. The victors carried off the
“black stone,” which was not restored for twenty-two years, and then only
for a great ransom, when it was plain that even the loss of its palladium
could not destroy the sacred character of the city. Under the Fatimites
Egyptian influence began to be strong in Mecca; it was opposed by the
Page 319
sultans of Yemen, while native princes claiming descent from the Prophet—
the Hāshimite amīrs of Mecca, and after them the amīrs of the house of
Qatāda (since 1202)—attained to great authority and aimed at
independence; but soon after the final fall of the Abbasids the Egyptian
overlordship was definitely established by sultan Bībars (a.d. 1269). The
Turkish conquest of Egypt transferred the supremacy to the Ottoman sultans
(1517), who treated Mecca with much favour, and during the 16th century
executed great works in the sanctuary and temple. The Ottoman power,
however, became gradually almost nominal, and that of the amīrs or sherīfs
increased in proportion, culminating under Ghālib, whose accession dates
from 1786. Then followed the wars of the Wahhābīs (see Arabia and
Wahhābīs) and the restoration of Turkish rule by the troops of Mehemet
‘Ali. By him the dignity of sherīf was deprived of much of its weight, and
in 1827 a change of dynasty was effected by the appointment of Ibn ‘Aun.
Afterwards Turkish authority again decayed. Mecca is, however, officially
the capital of a Turkish province, and has a governor-general and a Turkish
garrison, while Mahommedan law is administered by a judge sent from
Constantinople. But the real sovereign of Mecca and the Hejāz is the sherīf,
who, as head of a princely family claiming descent from the Prophet, holds
a sort of feudal position. The dignity of sherīf (or grand sherīf, as Europeans
usually say for the sake of distinction, since all the kin of the princely
houses reckoning descent from the Prophet are also named sherīfs),
although by no means a religious pontificate, is highly respected owing to
its traditional descent in the line of Hasan, son of the fourth caliph ‘Ali.
From a political point of view the sherīf is the modern counterpart of the
ancient amīrs of Mecca, who were named in the public prayers immediately
after the reigning caliph. When the great Mahommedan sultanates had
become too much occupied in internecine wars to maintain order in the
distant Hejāz, those branches of the Hassanids which from the beginning of
Islam had retained rural property in Arabia usurped power in the holy cities
the Hāshimite amīrs of Mecca, and after them the amīrs of the house of
Qatāda (since 1202)—attained to great authority and aimed at
independence; but soon after the final fall of the Abbasids the Egyptian
overlordship was definitely established by sultan Bībars (a.d. 1269). The
Turkish conquest of Egypt transferred the supremacy to the Ottoman sultans
(1517), who treated Mecca with much favour, and during the 16th century
executed great works in the sanctuary and temple. The Ottoman power,
however, became gradually almost nominal, and that of the amīrs or sherīfs
increased in proportion, culminating under Ghālib, whose accession dates
from 1786. Then followed the wars of the Wahhābīs (see Arabia and
Wahhābīs) and the restoration of Turkish rule by the troops of Mehemet
‘Ali. By him the dignity of sherīf was deprived of much of its weight, and
in 1827 a change of dynasty was effected by the appointment of Ibn ‘Aun.
Afterwards Turkish authority again decayed. Mecca is, however, officially
the capital of a Turkish province, and has a governor-general and a Turkish
garrison, while Mahommedan law is administered by a judge sent from
Constantinople. But the real sovereign of Mecca and the Hejāz is the sherīf,
who, as head of a princely family claiming descent from the Prophet, holds
a sort of feudal position. The dignity of sherīf (or grand sherīf, as Europeans
usually say for the sake of distinction, since all the kin of the princely
houses reckoning descent from the Prophet are also named sherīfs),
although by no means a religious pontificate, is highly respected owing to
its traditional descent in the line of Hasan, son of the fourth caliph ‘Ali.
From a political point of view the sherīf is the modern counterpart of the
ancient amīrs of Mecca, who were named in the public prayers immediately
after the reigning caliph. When the great Mahommedan sultanates had
become too much occupied in internecine wars to maintain order in the
distant Hejāz, those branches of the Hassanids which from the beginning of
Islam had retained rural property in Arabia usurped power in the holy cities
Page 320
and the adjacent Bedouin territories. About a.d. 960 they established a sort
of kingdom with Mecca as capital. The influence of the princes of Mecca
has varied from time to time, according to the strength of the foreign
protectorate in the Hejāz or in consequence of feuds among the branches of
the house; until about 1882 it was for most purposes much greater than that
of the Turks. The latter were strong enough to hold the garrisoned towns,
and thus the sultan was able within certain limits—playing off one against
the other the two rival branches of the aristocracy, viz. the kin of Ghālib and
the house of Ibn‘Aun—to assert the right of designating or removing the
sherīf, to whom in turn he owed the possibility of maintaining, with the aid
of considerable pensions, the semblance of his much-prized lordship over
the holy cities. The grand sherīf can muster a considerable force of
freedmen and clients, and his kin, holding wells and lands in various places
through the Hejāz, act as his deputies and administer the old Arabic
customary law to the Bedouin. To this influence the Hejāz owes what little
of law and order it enjoys. During the last quarter of the 19th century
Turkish influence became preponderant in western Arabia, and the railway
from Syria to the Hejāz tended to consolidate the sultan’s supremacy. After
the sherīfs, the principal family of Mecca is the house of Shaibah, which
holds the hereditary custodianship of the Ka‘ba.
The Great Mosque and the Ka‘ba.—Long before Mahomet the chief
sanctuary of Mecca was the Ka‘ba, a rude stone building without windows,
and having a door 7 ft. from the ground; and so named from its resemblance
to a monstrous astragalus (die) of about 40 ft. cube, though the shapeless
structure is not really an exact cube nor even exactly rectangular.14 The
Ka‘ba has been rebuilt more than once since Mahomet purged it of idols
and adopted it as the chief sanctuary of Islām, but the old form has been
preserved, except in secondary details;15 so that the “Ancient House,” as it
is titled, is still essentially a heathen temple, adapted to the worship of Islām
by the clumsy fiction that it was built by Abraham and Ishmael by divine
of kingdom with Mecca as capital. The influence of the princes of Mecca
has varied from time to time, according to the strength of the foreign
protectorate in the Hejāz or in consequence of feuds among the branches of
the house; until about 1882 it was for most purposes much greater than that
of the Turks. The latter were strong enough to hold the garrisoned towns,
and thus the sultan was able within certain limits—playing off one against
the other the two rival branches of the aristocracy, viz. the kin of Ghālib and
the house of Ibn‘Aun—to assert the right of designating or removing the
sherīf, to whom in turn he owed the possibility of maintaining, with the aid
of considerable pensions, the semblance of his much-prized lordship over
the holy cities. The grand sherīf can muster a considerable force of
freedmen and clients, and his kin, holding wells and lands in various places
through the Hejāz, act as his deputies and administer the old Arabic
customary law to the Bedouin. To this influence the Hejāz owes what little
of law and order it enjoys. During the last quarter of the 19th century
Turkish influence became preponderant in western Arabia, and the railway
from Syria to the Hejāz tended to consolidate the sultan’s supremacy. After
the sherīfs, the principal family of Mecca is the house of Shaibah, which
holds the hereditary custodianship of the Ka‘ba.
The Great Mosque and the Ka‘ba.—Long before Mahomet the chief
sanctuary of Mecca was the Ka‘ba, a rude stone building without windows,
and having a door 7 ft. from the ground; and so named from its resemblance
to a monstrous astragalus (die) of about 40 ft. cube, though the shapeless
structure is not really an exact cube nor even exactly rectangular.14 The
Ka‘ba has been rebuilt more than once since Mahomet purged it of idols
and adopted it as the chief sanctuary of Islām, but the old form has been
preserved, except in secondary details;15 so that the “Ancient House,” as it
is titled, is still essentially a heathen temple, adapted to the worship of Islām
by the clumsy fiction that it was built by Abraham and Ishmael by divine
Page 321
revelation as a temple of pure monotheism, and that it was only temporarily
perverted to idol worship from the time when ‘Amr ibn Lohai introduced
the statue of Hobal from Syria16 till the victory of Islam. This fiction has
involved the superinduction of a new mythology over the old heathen ritual,
which remains practically unchanged. Thus the chief object of veneration is
the black stone, which is fixed in the external angle facing Safā. The
building is not exactly oriented, but it may be called the south-east corner.
Its technical name is the black corner, the others being named the Yemen
(south-west), Syrian (north-west), and Irāk (north-east) corners, from the
lands to which they approximately point. The black stone is a small dark
mass a span long, with an aspect suggesting volcanic or meteoric origin,
fixed at such a height that it can be conveniently kissed by a person of
middle size. It was broken by fire in the siege of a.d. 683 (not, as many
authors relate, by the Carmathians), and the pieces are kept together by a
silver setting. The history of this heavenly stone, given by Gabriel to
Abraham, does not conceal the fact that it was originally a fetish, the most
venerated of a multitude of idols and sacred stones which stood all round
the sanctuary in the time of Mahomet. The Prophet destroyed the idols, but
he left the characteristic form of worship—the ṭawāf, or sevenfold circuit of
the sanctuary, the worshipper kissing or touching the objects of his
veneration—and besides the black stone he recognized the so-called
“southern” stone, the same presumably as that which is still touched in the
ṭawāf at the Yemen corner (Muh. in Med. pp. 336, 425). The ceremony of
the ṭawāf and the worship of stone fetishes was common to Mecca with
other ancient Arabian sanctuaries.17 It was, as it still is, a frequent religious
exercise of the Meccans, and the first duty of one who returned to the city
or arrived there under a vow of pilgrimage; and thus the outside of the
Ka‘ba was and is more important than the inside. Islām did away with the
worship of idols; what was lost in interest by their suppression has been
supplied by the invention of spots consecrated by recollections of Abraham,
perverted to idol worship from the time when ‘Amr ibn Lohai introduced
the statue of Hobal from Syria16 till the victory of Islam. This fiction has
involved the superinduction of a new mythology over the old heathen ritual,
which remains practically unchanged. Thus the chief object of veneration is
the black stone, which is fixed in the external angle facing Safā. The
building is not exactly oriented, but it may be called the south-east corner.
Its technical name is the black corner, the others being named the Yemen
(south-west), Syrian (north-west), and Irāk (north-east) corners, from the
lands to which they approximately point. The black stone is a small dark
mass a span long, with an aspect suggesting volcanic or meteoric origin,
fixed at such a height that it can be conveniently kissed by a person of
middle size. It was broken by fire in the siege of a.d. 683 (not, as many
authors relate, by the Carmathians), and the pieces are kept together by a
silver setting. The history of this heavenly stone, given by Gabriel to
Abraham, does not conceal the fact that it was originally a fetish, the most
venerated of a multitude of idols and sacred stones which stood all round
the sanctuary in the time of Mahomet. The Prophet destroyed the idols, but
he left the characteristic form of worship—the ṭawāf, or sevenfold circuit of
the sanctuary, the worshipper kissing or touching the objects of his
veneration—and besides the black stone he recognized the so-called
“southern” stone, the same presumably as that which is still touched in the
ṭawāf at the Yemen corner (Muh. in Med. pp. 336, 425). The ceremony of
the ṭawāf and the worship of stone fetishes was common to Mecca with
other ancient Arabian sanctuaries.17 It was, as it still is, a frequent religious
exercise of the Meccans, and the first duty of one who returned to the city
or arrived there under a vow of pilgrimage; and thus the outside of the
Ka‘ba was and is more important than the inside. Islām did away with the
worship of idols; what was lost in interest by their suppression has been
supplied by the invention of spots consecrated by recollections of Abraham,
Page 322
Ishmael and Hagar, or held to be acceptable places of prayer. Thus the space
of ten spans between the black stone and the door, which is on the east side,
between the black and Irāk corners, and a man’s height from the ground, is
called the Multazam, and here prayer should be offered after the ṭawāf with
outstretched arms and breast pressed against the house. On the other side of
the door, against the same wall, is a shallow trough, which is said to mark
the original site of the stone on which Abraham stood to build the Ka‘ba.
Here the growth of the legend can be traced, for the place is now called the
“kneading-place” (Ma‘jan), where the cement for the Ka‘ba was prepared.
This name and story do not appear in the older accounts. Once more, on the
north side of the Ka‘ba, there projects a low semicircular wall of marble,
with an opening at each end between it and the walls of the house. The
space within is paved with mosaic, and is called the Ḥijr. It is included in
the ṭawāf, and two slabs of verde antico within it are called the graves of
Ishmael and Hagar, and are places of acceptable prayer. Even the golden or
gilded mīzāb (water-spout) that projects into the Ḥijr marks a place where
prayer is heard, and another such place is the part of the west wall close to
the Yemen corner.
The feeling of religious conservatism which has preserved the structural
rudeness of the Ka‘ba did not prohibit costly surface decoration. In
Mahomet’s time the outer walls were covered by a veil (or kiswa) of striped
Yemen cloth. The caliphs substituted a covering of figured brocade, and the
Egyptian government still sends with each pilgrim caravan from Cairo a
new kiswa of black brocade, adorned with a broad band embroidered with
golden inscriptions from the Korān, as well as a richer curtain for the
door.18 The door of two leaves, with its posts and lintel, is of silver gilt.
The interior of the Ka‘ba is now opened but a few times every year for
the general public, which ascends by the portable staircase brought forward
for the purpose. Foreigners can obtain admission at any time for a special
of ten spans between the black stone and the door, which is on the east side,
between the black and Irāk corners, and a man’s height from the ground, is
called the Multazam, and here prayer should be offered after the ṭawāf with
outstretched arms and breast pressed against the house. On the other side of
the door, against the same wall, is a shallow trough, which is said to mark
the original site of the stone on which Abraham stood to build the Ka‘ba.
Here the growth of the legend can be traced, for the place is now called the
“kneading-place” (Ma‘jan), where the cement for the Ka‘ba was prepared.
This name and story do not appear in the older accounts. Once more, on the
north side of the Ka‘ba, there projects a low semicircular wall of marble,
with an opening at each end between it and the walls of the house. The
space within is paved with mosaic, and is called the Ḥijr. It is included in
the ṭawāf, and two slabs of verde antico within it are called the graves of
Ishmael and Hagar, and are places of acceptable prayer. Even the golden or
gilded mīzāb (water-spout) that projects into the Ḥijr marks a place where
prayer is heard, and another such place is the part of the west wall close to
the Yemen corner.
The feeling of religious conservatism which has preserved the structural
rudeness of the Ka‘ba did not prohibit costly surface decoration. In
Mahomet’s time the outer walls were covered by a veil (or kiswa) of striped
Yemen cloth. The caliphs substituted a covering of figured brocade, and the
Egyptian government still sends with each pilgrim caravan from Cairo a
new kiswa of black brocade, adorned with a broad band embroidered with
golden inscriptions from the Korān, as well as a richer curtain for the
door.18 The door of two leaves, with its posts and lintel, is of silver gilt.
The interior of the Ka‘ba is now opened but a few times every year for
the general public, which ascends by the portable staircase brought forward
for the purpose. Foreigners can obtain admission at any time for a special
Page 323
fee. The modern descriptions, from observations made under difficulties,
are not very complete. Little change, however, seems to have been made
since the time of Ibn Jubair, who describes the floor and walls as overlaid
with richly variegated marbles, and the upper half of the walls as plated
with silver thickly gilt, while the roof was veiled with coloured silk.
Modern writers describe the place as windowless, but Ibn Jubair mentions
five windows of rich stained glass from Irāk. Between the three pillars of
teak hung thirteen silver lamps. A chest in the corner to the left of one
entering contained Korans, and at the Irāk corner a space was cut off
enclosing the stair that leads to the roof. The door to this stair (called the
door of mercy—Bāb el-Raḥma) was plated with silver by the caliph
Motawakkil. Here, in the time of Ibn Jubair, the Maqām or standing stone
of Abraham was usually placed for better security, but brought out on great
occasions.19
The houses of ancient Mecca pressed close upon the Ka‘ba, the noblest
families, who traced their descent from Ḳoṣai, the reputed founder of the
city, having their dwellings immediately round the sanctuary. To the north
of the Ka‘ba was the Dār el-Nadwa, or place of assembly of the Koreish.
The multiplication of pilgrims after Islām soon made it necessary to clear
away the nearest dwellings and enlarge the place of prayer around the
Ancient House. Omar, Othmān and Ibn Jubair had all a share in this work,
but the great founder of the mosque in its present form, with its spacious
area and deep colonnades, was the caliph Mahdī, who spent enormous sums
in bringing costly pillars from Egypt and Syria. The work was still
incomplete at his death in a.d. 785, and was finished in less sumptuous style
by his successor. Subsequent repairs and additions, extending down to
Turkish times, have left little of Mahdī’s work untouched, though a few of
the pillars probably date from his days. There are more than five hundred
pillars in all, of very various style and workmanship, and the enclosure—
are not very complete. Little change, however, seems to have been made
since the time of Ibn Jubair, who describes the floor and walls as overlaid
with richly variegated marbles, and the upper half of the walls as plated
with silver thickly gilt, while the roof was veiled with coloured silk.
Modern writers describe the place as windowless, but Ibn Jubair mentions
five windows of rich stained glass from Irāk. Between the three pillars of
teak hung thirteen silver lamps. A chest in the corner to the left of one
entering contained Korans, and at the Irāk corner a space was cut off
enclosing the stair that leads to the roof. The door to this stair (called the
door of mercy—Bāb el-Raḥma) was plated with silver by the caliph
Motawakkil. Here, in the time of Ibn Jubair, the Maqām or standing stone
of Abraham was usually placed for better security, but brought out on great
occasions.19
The houses of ancient Mecca pressed close upon the Ka‘ba, the noblest
families, who traced their descent from Ḳoṣai, the reputed founder of the
city, having their dwellings immediately round the sanctuary. To the north
of the Ka‘ba was the Dār el-Nadwa, or place of assembly of the Koreish.
The multiplication of pilgrims after Islām soon made it necessary to clear
away the nearest dwellings and enlarge the place of prayer around the
Ancient House. Omar, Othmān and Ibn Jubair had all a share in this work,
but the great founder of the mosque in its present form, with its spacious
area and deep colonnades, was the caliph Mahdī, who spent enormous sums
in bringing costly pillars from Egypt and Syria. The work was still
incomplete at his death in a.d. 785, and was finished in less sumptuous style
by his successor. Subsequent repairs and additions, extending down to
Turkish times, have left little of Mahdī’s work untouched, though a few of
the pillars probably date from his days. There are more than five hundred
pillars in all, of very various style and workmanship, and the enclosure—
Page 324
250 paces in length and 200 in breadth, according to Burckhardt’s
measurement—is entered by nineteen archways irregularly disposed.
After the Ka‘ba the principal points of interest in the mosque are the well
Zamzam and the Maqām Ibrāhīm. The former is a deep shaft enclosed in a
massive vaulted building paved with marble, and, according to
Mahommedan tradition, is the source (corresponding to the Beer-lahai-roi
of Gen. xvi. 14) from which Hagar drew water for her son Ishmael. The
legend tells that the well was long covered up and rediscovered by ‘Abd al-
Moṭṭalib, the grandfather of the Prophet. Sacred wells are familiar features
of Semitic sanctuaries, and Islām, retaining the well, made a quasi-biblical
story for it, and endowed its tepid waters with miraculous curative virtues.
They are eagerly drunk by the pilgrims, or when poured over the body are
held to give a miraculous refreshment after the fatigues of religious
exercise; and the manufacture of bottles or jars for carrying the water to
distant countries is quite a trade. Ibn Jubair mentions a curious superstition
of the Meccans, who believed that the water rose in the shaft at the full
moon of the month Shaban. On this occasion a great crowd, especially of
young people, thronged round the well with shouts of religious enthusiasm,
while the servants of the well dashed buckets of water over their heads. The
Maqām of Abraham is also connected with a relic of heathenism, the
ancient holy stone which once stood on the Ma‘jan, and is said to bear the
prints of the patriarch’s feet. The whole legend of this stone, which is full of
miraculous incidents, seems to have arisen from a misconception, the
Maqām Ibrāhīm in the Korān meaning the sanctuary itself; but the stone,
which is a block about 3 spans in height and 2 in breadth, and in shape “like
a potter’s furnace” (Ibn Jubair), is certainly very ancient. No one is now
allowed to see it, though the box in which it lies can be seen or touched
through a grating in the little chapel that surrounds it. In the middle ages it
was sometimes shown, and Ibn Jubair describes the pious enthusiasm with
which he drank Zamzam water poured on the footprints. It was covered
measurement—is entered by nineteen archways irregularly disposed.
After the Ka‘ba the principal points of interest in the mosque are the well
Zamzam and the Maqām Ibrāhīm. The former is a deep shaft enclosed in a
massive vaulted building paved with marble, and, according to
Mahommedan tradition, is the source (corresponding to the Beer-lahai-roi
of Gen. xvi. 14) from which Hagar drew water for her son Ishmael. The
legend tells that the well was long covered up and rediscovered by ‘Abd al-
Moṭṭalib, the grandfather of the Prophet. Sacred wells are familiar features
of Semitic sanctuaries, and Islām, retaining the well, made a quasi-biblical
story for it, and endowed its tepid waters with miraculous curative virtues.
They are eagerly drunk by the pilgrims, or when poured over the body are
held to give a miraculous refreshment after the fatigues of religious
exercise; and the manufacture of bottles or jars for carrying the water to
distant countries is quite a trade. Ibn Jubair mentions a curious superstition
of the Meccans, who believed that the water rose in the shaft at the full
moon of the month Shaban. On this occasion a great crowd, especially of
young people, thronged round the well with shouts of religious enthusiasm,
while the servants of the well dashed buckets of water over their heads. The
Maqām of Abraham is also connected with a relic of heathenism, the
ancient holy stone which once stood on the Ma‘jan, and is said to bear the
prints of the patriarch’s feet. The whole legend of this stone, which is full of
miraculous incidents, seems to have arisen from a misconception, the
Maqām Ibrāhīm in the Korān meaning the sanctuary itself; but the stone,
which is a block about 3 spans in height and 2 in breadth, and in shape “like
a potter’s furnace” (Ibn Jubair), is certainly very ancient. No one is now
allowed to see it, though the box in which it lies can be seen or touched
through a grating in the little chapel that surrounds it. In the middle ages it
was sometimes shown, and Ibn Jubair describes the pious enthusiasm with
which he drank Zamzam water poured on the footprints. It was covered
Page 325
with inscriptions in an unknown character, one of which was copied by
Fākihī in his history of Mecca. To judge by the facsimile in Dozy’s
Israeliten te Mekka, the character is probably essentially one with that of
the Syrian Safā inscriptions, which extended through the Nejd and into the
Ḥejāz.20
Safā and Merwa.—In religious importance these two points or
“hills,” connected by the Mas‘ā, stand second only to the Ka‘ba. Safā is
an elevated platform surmounted by a triple arch, and approached by a
flight of steps.21 It lies south-east of the Ka‘ba, facing the black corner,
and 76 paces from the “Gate of Safā,” which is architecturally the chief
gate of the mosque. Merwa is a similar platform, formerly covered with
a single arch, on the opposite side of the valley. It stands on a spur of
the Red Mountain called Jebel Kuayḳian. The course between these
two sacred points is 493 paces long, and the religious ceremony called
the “sa‘y” consists in traversing it seven times, beginning and ending at
Safā. The lowest part of the course, between the so-called green
milestones, is done at a run. This ceremony, which, as we shall
presently see, is part of the omra, is generally said to be performed in
memory of Hagar, who ran to and fro between the two eminences
vainly seeking water for her son. The observance, however, is certainly
of pagan origin; and at one time there were idols on both the so-called
hills (see especially Azraqī, pp. 74, 78).
The Ceremonies and the Pilgrimage.—Before Islām the Ka‘ba was
the local sanctuary of the Meccans, where they prayed and did
sacrifice, where oaths were administered and hard cases submitted to
divine sentence according to the immemorial custom of Semitic
shrines. But, besides this, Mecca was already a place of pilgrimage.
Pilgrimage with the ancient Arabs was the fulfilment of a vow, which
appears to have generally terminated—at least on the part of the well-
Fākihī in his history of Mecca. To judge by the facsimile in Dozy’s
Israeliten te Mekka, the character is probably essentially one with that of
the Syrian Safā inscriptions, which extended through the Nejd and into the
Ḥejāz.20
Safā and Merwa.—In religious importance these two points or
“hills,” connected by the Mas‘ā, stand second only to the Ka‘ba. Safā is
an elevated platform surmounted by a triple arch, and approached by a
flight of steps.21 It lies south-east of the Ka‘ba, facing the black corner,
and 76 paces from the “Gate of Safā,” which is architecturally the chief
gate of the mosque. Merwa is a similar platform, formerly covered with
a single arch, on the opposite side of the valley. It stands on a spur of
the Red Mountain called Jebel Kuayḳian. The course between these
two sacred points is 493 paces long, and the religious ceremony called
the “sa‘y” consists in traversing it seven times, beginning and ending at
Safā. The lowest part of the course, between the so-called green
milestones, is done at a run. This ceremony, which, as we shall
presently see, is part of the omra, is generally said to be performed in
memory of Hagar, who ran to and fro between the two eminences
vainly seeking water for her son. The observance, however, is certainly
of pagan origin; and at one time there were idols on both the so-called
hills (see especially Azraqī, pp. 74, 78).
The Ceremonies and the Pilgrimage.—Before Islām the Ka‘ba was
the local sanctuary of the Meccans, where they prayed and did
sacrifice, where oaths were administered and hard cases submitted to
divine sentence according to the immemorial custom of Semitic
shrines. But, besides this, Mecca was already a place of pilgrimage.
Pilgrimage with the ancient Arabs was the fulfilment of a vow, which
appears to have generally terminated—at least on the part of the well-
Page 326
to-do—in a sacrificial feast. A vow of pilgrimage might be directed to
other sanctuaries than Mecca—the technical word for it (ihlāl) is
applied, for example, to the pilgrimage to Manāt (Bakri, p. 519). He
who was under such a vow was bound by ceremonial observances of
abstinence from certain acts (e.g. hunting) and sensual pleasures, and in
particular was forbidden to shear or comb his hair till the fulfilment of
the vow. This old Semitic usage has its close parallel in the vow of the
Nazarite. It was not peculiarly connected with Mecca; at Tāif, for
example, it was customary on return to the city after an absence to
present oneself at the sanctuary, and there shear the hair (Muh. in Med.,
p. 381). Pilgrimages to Mecca were not tied to a single time, but they
were naturally associated with festive occasions, and especially with
the great annual feast and market. The pilgrimage was so intimately
connected with the well-being of Mecca, and had already such a hold
on the Arabs round about, that Mahomet could not afford to sacrifice it
to an abstract purity of religion, and thus the old usages were
transplanted into Islām in the double form of the omra or vow of
pilgrimage to Mecca, which can be discharged at any time, and the ḥajj
or pilgrimage at the great annual feast. The latter closes with a visit to
the Ka‘ba, but its essential ceremonies lie outside Mecca, at the
neighbouring shrines where the old Arabs gathered before the Meccan
fair.
The omra begins at some point outside the Ḥaram (or holy territory),
generally at Tanim, both for convenience sake and because Ayesha
began the omra there in the year 10 of the Hegira. The pilgrim enters
the Ḥaram in the antique and scanty pilgrimage dress (iḥrām),
consisting of two cloths wound round his person in a way prescribed
by ritual. His devotion is expressed in shouts of “Labbeyka” (a word of
obscure origin and meaning); he enters the great mosque, performs the
ṭawāf and the sa‘y22 and then has his head shaved and resumes his
other sanctuaries than Mecca—the technical word for it (ihlāl) is
applied, for example, to the pilgrimage to Manāt (Bakri, p. 519). He
who was under such a vow was bound by ceremonial observances of
abstinence from certain acts (e.g. hunting) and sensual pleasures, and in
particular was forbidden to shear or comb his hair till the fulfilment of
the vow. This old Semitic usage has its close parallel in the vow of the
Nazarite. It was not peculiarly connected with Mecca; at Tāif, for
example, it was customary on return to the city after an absence to
present oneself at the sanctuary, and there shear the hair (Muh. in Med.,
p. 381). Pilgrimages to Mecca were not tied to a single time, but they
were naturally associated with festive occasions, and especially with
the great annual feast and market. The pilgrimage was so intimately
connected with the well-being of Mecca, and had already such a hold
on the Arabs round about, that Mahomet could not afford to sacrifice it
to an abstract purity of religion, and thus the old usages were
transplanted into Islām in the double form of the omra or vow of
pilgrimage to Mecca, which can be discharged at any time, and the ḥajj
or pilgrimage at the great annual feast. The latter closes with a visit to
the Ka‘ba, but its essential ceremonies lie outside Mecca, at the
neighbouring shrines where the old Arabs gathered before the Meccan
fair.
The omra begins at some point outside the Ḥaram (or holy territory),
generally at Tanim, both for convenience sake and because Ayesha
began the omra there in the year 10 of the Hegira. The pilgrim enters
the Ḥaram in the antique and scanty pilgrimage dress (iḥrām),
consisting of two cloths wound round his person in a way prescribed
by ritual. His devotion is expressed in shouts of “Labbeyka” (a word of
obscure origin and meaning); he enters the great mosque, performs the
ṭawāf and the sa‘y22 and then has his head shaved and resumes his
Page 327
common dress. This ceremony is now generally combined with the
ḥajj, or is performed by every stranger or traveller when he enters
Mecca, and the iḥrām (which involves the acts of abstinence already
referred to) is assumed at a considerable distance from the city. But it is
also proper during one’s residence in the holy city to perform at least
one omra from Tanim in connexion with a visit to the mosque of
Ayesha there. The triviality of these rites is ill concealed by the legends
of the sa‘y of Hagar and of the ṭawāf being first performed by Adam in
imitation of the circuit of the angels about the throne of God; the
meaning of their ceremonies seems to have been almost a blank to the
Arabs before Islām, whose religion had become a mere formal
tradition. We do not even know to what deity the worship expressed in
the ṭawāf was properly addressed. There is a tradition that the Ka‘ba
was a temple of Saturn (Shahrastānī, p. 431); perhaps the most
distinctive feature of the shrine may be sought in the sacred doves
which still enjoy the protection of the sanctuary. These recall the sacred
doves of Ascalon (Philo vi. 200 of Richter’s ed.), and suggests Venus-
worship as at least one element (cf. Herod i. 131, iii. 8; Ephr. Syr., Op.
Syr. ii. 457).
To the ordinary pilgrim the omra has become so much an episode of
the ḥajj that it is described by some European pilgrims as a mere visit
to the mosque of Ayesha; a better conception of its original significance
is got from the Meccan feast of the seventh month (Rajab), graphically
described by Ibn Jubair from his observations in a.d. 1184. Rajab was
one of the ancient sacred months, and the feast, which extended
through the whole month and was a joyful season of hospitality and
thanksgiving, no doubt represents the ancient feasts of Mecca more
exactly than the ceremonies of the ḥajj, in which old usage has been
overlaid by traditions and glosses of Islām. The omra was performed
by crowds from day to day, especially at new and full moon.23 The new
ḥajj, or is performed by every stranger or traveller when he enters
Mecca, and the iḥrām (which involves the acts of abstinence already
referred to) is assumed at a considerable distance from the city. But it is
also proper during one’s residence in the holy city to perform at least
one omra from Tanim in connexion with a visit to the mosque of
Ayesha there. The triviality of these rites is ill concealed by the legends
of the sa‘y of Hagar and of the ṭawāf being first performed by Adam in
imitation of the circuit of the angels about the throne of God; the
meaning of their ceremonies seems to have been almost a blank to the
Arabs before Islām, whose religion had become a mere formal
tradition. We do not even know to what deity the worship expressed in
the ṭawāf was properly addressed. There is a tradition that the Ka‘ba
was a temple of Saturn (Shahrastānī, p. 431); perhaps the most
distinctive feature of the shrine may be sought in the sacred doves
which still enjoy the protection of the sanctuary. These recall the sacred
doves of Ascalon (Philo vi. 200 of Richter’s ed.), and suggests Venus-
worship as at least one element (cf. Herod i. 131, iii. 8; Ephr. Syr., Op.
Syr. ii. 457).
To the ordinary pilgrim the omra has become so much an episode of
the ḥajj that it is described by some European pilgrims as a mere visit
to the mosque of Ayesha; a better conception of its original significance
is got from the Meccan feast of the seventh month (Rajab), graphically
described by Ibn Jubair from his observations in a.d. 1184. Rajab was
one of the ancient sacred months, and the feast, which extended
through the whole month and was a joyful season of hospitality and
thanksgiving, no doubt represents the ancient feasts of Mecca more
exactly than the ceremonies of the ḥajj, in which old usage has been
overlaid by traditions and glosses of Islām. The omra was performed
by crowds from day to day, especially at new and full moon.23 The new
Page 328
moon celebration was nocturnal; the road to Tanim, the Mas‘ā, and the
mosque were brilliantly illuminated; and the appearing of the moon
was greeted with noisy music. A genuine old Arab market was held, for
the wild Bedouins of the Yemen mountains came in thousands to barter
their cattle and fruits for clothing, and deemed that to absent
themselves would bring drought and cattle plague in their homes.
Though ignorant of the legal ritual and prayers, they performed the
ṭawāf with enthusiasm, throwing themselves against the Ka‘ba and
clinging to its curtains as a child clings to its mother. They also made a
point of entering the Ka‘ba. The 29th of the month was the feast day of
the Meccan women, when they and their little ones had the Ka‘ba to
themselves without the presence even of the Sheybās.
The central and essential ceremonies of the ḥajj or greater
pilgrimage are those of the day of Arafa, the 9th of the “pilgrimage
month” (Dhu‘l Ḥijja), the last of the Arab year; and every Moslem who
is his own master, and can command the necessary means, is bound to
join in these once in his life, or to have them fulfilled by a substitute on
his behalf and at his expense. By them the pilgrim becomes as pure
from sin as when he was born, and gains for the rest of his life the
honourable title of ḥajj. Neglect of many other parts of the pilgrim
ceremonial may be compensated by offerings, but to miss the “stand”
(woqūf) at Arafa is to miss the pilgrimage. Arafa or Arafat is a space,
artificially limited, round a small isolated hill called the Hill of Mercy,
a little way outside the holy territory, on the road from Mecca to Taif.
One leaving Mecca after midday can easily reach the place on foot the
same evening. The road is first northwards along the Mecca valley and
then turns eastward. It leads through the straggling village of Mina,
occupying a long narrow valley (Wādi Mina), two to three hours from
Mecca, and thence by the mosque of Mozdalifa over a narrow pass
opening out into the plain of Arafa, which is an expansion of the great
mosque were brilliantly illuminated; and the appearing of the moon
was greeted with noisy music. A genuine old Arab market was held, for
the wild Bedouins of the Yemen mountains came in thousands to barter
their cattle and fruits for clothing, and deemed that to absent
themselves would bring drought and cattle plague in their homes.
Though ignorant of the legal ritual and prayers, they performed the
ṭawāf with enthusiasm, throwing themselves against the Ka‘ba and
clinging to its curtains as a child clings to its mother. They also made a
point of entering the Ka‘ba. The 29th of the month was the feast day of
the Meccan women, when they and their little ones had the Ka‘ba to
themselves without the presence even of the Sheybās.
The central and essential ceremonies of the ḥajj or greater
pilgrimage are those of the day of Arafa, the 9th of the “pilgrimage
month” (Dhu‘l Ḥijja), the last of the Arab year; and every Moslem who
is his own master, and can command the necessary means, is bound to
join in these once in his life, or to have them fulfilled by a substitute on
his behalf and at his expense. By them the pilgrim becomes as pure
from sin as when he was born, and gains for the rest of his life the
honourable title of ḥajj. Neglect of many other parts of the pilgrim
ceremonial may be compensated by offerings, but to miss the “stand”
(woqūf) at Arafa is to miss the pilgrimage. Arafa or Arafat is a space,
artificially limited, round a small isolated hill called the Hill of Mercy,
a little way outside the holy territory, on the road from Mecca to Taif.
One leaving Mecca after midday can easily reach the place on foot the
same evening. The road is first northwards along the Mecca valley and
then turns eastward. It leads through the straggling village of Mina,
occupying a long narrow valley (Wādi Mina), two to three hours from
Mecca, and thence by the mosque of Mozdalifa over a narrow pass
opening out into the plain of Arafa, which is an expansion of the great
Page 329
Wādi Naman, through which the Taif road descends from Mount Kara.
The lofty and rugged mountains of the Hodheyl tower over the plain on
the north side and overshadow the little Hill of Mercy, which is one of
those bosses of weathered granite so common in the Hejāz. Arafa lay
quite near Dhul-Majaz, where, according to Arabian tradition, a great
fair was held from the 1st to the 8th of the pilgrimage month; and the
ceremonies from which the ḥajj was derived were originally an
appendix to this fair. Now, on the contrary, the pilgrim is expected to
follow as closely as may be the movements of the prophet at his
“farewell pilgrimage” in the year 10 of the Hegira (a.d. 632). He
therefore leaves Mecca in pilgrim garb on the 8th of Dhu‘l Ḥijja, called
the day of tarwīya (an obscure and pre-Islamic name), and, strictly
speaking, should spend the night at Mina. It is now, however,
customary to go right on and encamp at once at Arafa. The night should
be spent in devotion, but the coffee booths do a lively trade, and songs
are as common as prayers. Next forenoon the pilgrim is free to move
about, and towards midday he may if he please hear a sermon. In the
afternoon the essential ceremony begins; it consists simply in
“standing” on Arafa shouting “Labbeyka” and reciting prayers and
texts till sunset. After the sun is down the vast assemblage breaks up,
and a rush (technically ifāḍa, daf‘, nafr) is made in the utmost
confusion to Mozdalifa, where the night prayer is said and the night
spent. Before sunrise next morning (the 10th) a second “stand” like that
on Arafa is made for a short time by torchlight round the mosque of
Mozdalifa, but before the sun is fairly up all must be in motion in the
second ifāḍa towards Mina. The day thus begun is the “day of
sacrifice,” and has four ceremonies—(1) to pelt with seven stones a
cairn (jamrat al ‘aqaba) at the eastern end of W. Mina, (2) to slay a
victim at Mina and hold a sacrificial meal, part of the flesh being also
dried and so preserved, or given to the poor,24 (3) to be shaved and so
The lofty and rugged mountains of the Hodheyl tower over the plain on
the north side and overshadow the little Hill of Mercy, which is one of
those bosses of weathered granite so common in the Hejāz. Arafa lay
quite near Dhul-Majaz, where, according to Arabian tradition, a great
fair was held from the 1st to the 8th of the pilgrimage month; and the
ceremonies from which the ḥajj was derived were originally an
appendix to this fair. Now, on the contrary, the pilgrim is expected to
follow as closely as may be the movements of the prophet at his
“farewell pilgrimage” in the year 10 of the Hegira (a.d. 632). He
therefore leaves Mecca in pilgrim garb on the 8th of Dhu‘l Ḥijja, called
the day of tarwīya (an obscure and pre-Islamic name), and, strictly
speaking, should spend the night at Mina. It is now, however,
customary to go right on and encamp at once at Arafa. The night should
be spent in devotion, but the coffee booths do a lively trade, and songs
are as common as prayers. Next forenoon the pilgrim is free to move
about, and towards midday he may if he please hear a sermon. In the
afternoon the essential ceremony begins; it consists simply in
“standing” on Arafa shouting “Labbeyka” and reciting prayers and
texts till sunset. After the sun is down the vast assemblage breaks up,
and a rush (technically ifāḍa, daf‘, nafr) is made in the utmost
confusion to Mozdalifa, where the night prayer is said and the night
spent. Before sunrise next morning (the 10th) a second “stand” like that
on Arafa is made for a short time by torchlight round the mosque of
Mozdalifa, but before the sun is fairly up all must be in motion in the
second ifāḍa towards Mina. The day thus begun is the “day of
sacrifice,” and has four ceremonies—(1) to pelt with seven stones a
cairn (jamrat al ‘aqaba) at the eastern end of W. Mina, (2) to slay a
victim at Mina and hold a sacrificial meal, part of the flesh being also
dried and so preserved, or given to the poor,24 (3) to be shaved and so
Page 330
terminate the iḥrām, (4) to make the third ifāḍa, i.e. go to Mecca and
perform the ṭawāf and sa‘y (‘omrat al-ifāḍa), returning thereafter to
Mina. The sacrifice and visit to Mecca may, however, be delayed till
the 11th, 12th or 13th. These are the days of Mina, a fair and joyous
feast, with no special ceremony except that each day the pilgrim is
expected to throw seven stones at the jamrat al ‘aqaba, and also at
each of two similar cairns in the valley. The stones are thrown in the
name of Allah, and are generally thought to be directed at the devil.
This is, however, a custom older than Islām, and a tradition in Azraqī,
p. 412, represents it as an act of worship to idols at Mina. As the stones
are thrown on the days of the fair, it is not unlikely that they have
something to do with the old Arab mode of closing a sale by the
purchaser throwing a stone (Bīrūnī, p. 328).25 The pilgrims leave Mina
on the 12th or 13th, and the ḥajj is then over. (See further
Mahommedan Religion.)
The colourless character of these ceremonies is plainly due to the
fact that they are nothing more than expurgated heathen rites. In Islām
proper they have no raison d’être; the legends about Adam and Eve on
Arafa, about Abraham’s sacrifice of the ram at Thabii by Mina,
imitated in the sacrifices of the pilgrimage, are clumsy afterthoughts, as
appears from their variations and only partial acceptance. It is not so
easy to get at the nature of the original rites, which Islām was careful to
suppress. But we find mention of practices condemned by the
orthodox, or forming no part of the Moslem ritual, which may be
regarded as traces of an older ceremonial. Such are nocturnal
illuminations at Mina (Ibn Baṭūta i. 396), Arafa and Mozdalifa (Ibn
Jubair, 179), and ṭawāfs performed by the ignorant at holy spots at
Arafa not recognized by law (Snouck-Hurgronje p. 149 sqq.). We know
that the rites at Mozdalifa were originally connected with a holy hill
bearing the name of the god Quzah (the Edomite Kozē) whose bow is
perform the ṭawāf and sa‘y (‘omrat al-ifāḍa), returning thereafter to
Mina. The sacrifice and visit to Mecca may, however, be delayed till
the 11th, 12th or 13th. These are the days of Mina, a fair and joyous
feast, with no special ceremony except that each day the pilgrim is
expected to throw seven stones at the jamrat al ‘aqaba, and also at
each of two similar cairns in the valley. The stones are thrown in the
name of Allah, and are generally thought to be directed at the devil.
This is, however, a custom older than Islām, and a tradition in Azraqī,
p. 412, represents it as an act of worship to idols at Mina. As the stones
are thrown on the days of the fair, it is not unlikely that they have
something to do with the old Arab mode of closing a sale by the
purchaser throwing a stone (Bīrūnī, p. 328).25 The pilgrims leave Mina
on the 12th or 13th, and the ḥajj is then over. (See further
Mahommedan Religion.)
The colourless character of these ceremonies is plainly due to the
fact that they are nothing more than expurgated heathen rites. In Islām
proper they have no raison d’être; the legends about Adam and Eve on
Arafa, about Abraham’s sacrifice of the ram at Thabii by Mina,
imitated in the sacrifices of the pilgrimage, are clumsy afterthoughts, as
appears from their variations and only partial acceptance. It is not so
easy to get at the nature of the original rites, which Islām was careful to
suppress. But we find mention of practices condemned by the
orthodox, or forming no part of the Moslem ritual, which may be
regarded as traces of an older ceremonial. Such are nocturnal
illuminations at Mina (Ibn Baṭūta i. 396), Arafa and Mozdalifa (Ibn
Jubair, 179), and ṭawāfs performed by the ignorant at holy spots at
Arafa not recognized by law (Snouck-Hurgronje p. 149 sqq.). We know
that the rites at Mozdalifa were originally connected with a holy hill
bearing the name of the god Quzah (the Edomite Kozē) whose bow is
Page 331
the rainbow, and there is reason to think that the ifāḍas from Arafa and
Quzah, which were not made as now after sunset and before sunrise,
but when the sun rested on the tops of the mountains, were ceremonies
of farewell and salutation to the sun-god.
The statistics of the pilgrimage cannot be given with certainty and
vary much from year to year. The quarantine office keeps a record of
arrivals by sea at Jidda (66,000 for 1904); but to these must be added
those travelling by land from Cairo, Damascus and Irāk, the pilgrims
who reach Medina from Yanbu and go on to Mecca, and those from all
parts of the peninsula. Burckhardt in 1814 estimated the crowd at Arafa
at 70,000, Burton in 1853 at 50,000, ‘Abd el-Razzāk in 1858 at 60,000.
This great assemblage is always a dangerous centre of infection, and
the days of Mina especially, spent under circumstances originally
adapted only for a Bedouin fair, with no provisions for proper
cleanliness, and with the air full of the smell of putrefying offal and
flesh drying in the sun, produce much sickness.
Literature.—Besides the Arabic geographers and cosmographers,
we have Ibn ‘Abd Rabbih’s description of the mosque, early in the 10th
century (‘Iḳd Farīd, Cairo ed., iii. 362 sqq.), but above all the
admirable record of Ibn Jubair (a.d. 1184), by far the best account
extant of Mecca and the pilgrimage. It has been much pillaged by Ibn
Baṭūta. The Arabic historians are largely occupied with fabulous matter
as to Mecca before Islām; for these legends the reader may refer to C.
de Perceval’s Essai. How little confidence can be placed in the pre-
Islamic history appears very clearly from the distorted accounts of
Abraha’s excursion against the Hejāz, which fell but a few years before
the birth of the Prophet, and is the first event in Meccan history which
has confirmation from other sources. See Nöldeke’s version of Ţabarī,
p. 204 sqq. For the period of the Prophet, Ibn Hishām and Wāḳidī are
Quzah, which were not made as now after sunset and before sunrise,
but when the sun rested on the tops of the mountains, were ceremonies
of farewell and salutation to the sun-god.
The statistics of the pilgrimage cannot be given with certainty and
vary much from year to year. The quarantine office keeps a record of
arrivals by sea at Jidda (66,000 for 1904); but to these must be added
those travelling by land from Cairo, Damascus and Irāk, the pilgrims
who reach Medina from Yanbu and go on to Mecca, and those from all
parts of the peninsula. Burckhardt in 1814 estimated the crowd at Arafa
at 70,000, Burton in 1853 at 50,000, ‘Abd el-Razzāk in 1858 at 60,000.
This great assemblage is always a dangerous centre of infection, and
the days of Mina especially, spent under circumstances originally
adapted only for a Bedouin fair, with no provisions for proper
cleanliness, and with the air full of the smell of putrefying offal and
flesh drying in the sun, produce much sickness.
Literature.—Besides the Arabic geographers and cosmographers,
we have Ibn ‘Abd Rabbih’s description of the mosque, early in the 10th
century (‘Iḳd Farīd, Cairo ed., iii. 362 sqq.), but above all the
admirable record of Ibn Jubair (a.d. 1184), by far the best account
extant of Mecca and the pilgrimage. It has been much pillaged by Ibn
Baṭūta. The Arabic historians are largely occupied with fabulous matter
as to Mecca before Islām; for these legends the reader may refer to C.
de Perceval’s Essai. How little confidence can be placed in the pre-
Islamic history appears very clearly from the distorted accounts of
Abraha’s excursion against the Hejāz, which fell but a few years before
the birth of the Prophet, and is the first event in Meccan history which
has confirmation from other sources. See Nöldeke’s version of Ţabarī,
p. 204 sqq. For the period of the Prophet, Ibn Hishām and Wāḳidī are
Page 332
valuable sources in topography as well as history. Of the special
histories and descriptions of Mecca published by Wüstenfeld
(Chroniken der Stadt Mekka, 3 vols., 1857-1859, with an abstract in
German, 1861), the most valuable is that of Azraqī. It has passed
through the hands of several editors, but the oldest part goes back to
the beginning of the 9th Christian century. Kutbeddin’s history (vol. iii.
of the Chroniken) goes down with the additions of his nephew to a.d.
1592.
Of European descriptions of Mecca from personal observation the
best is Burckhardt’s Travels in Arabia (cited above from the 8vo ed.,
1829). The Travels of Aly Bey (Badia, London, 1816) describe a visit in
1807; Burton’s Pilgrimage (3rd ed., 1879) often supplements
Burckhardt; Von Maltzan’s Wallfahrt nach Mekka (1865) is lively but
very slight. ‘Abd el-Razzāq’s report to the government of India on the
pilgrimage of 1858 is specially directed to sanitary questions; C.
Snouck-Hurgronje, Mekka (2 vols., and a collection of photographs,
The Hague, 1888-1889), gives a description of the Meccan sanctuary
and of the public and private life of the Meccans as observed by the
author during a sojourn in the holy city in 1884-1885 and a political
history of Mecca from native sources from the Hegira till 1884. For the
pilgrimage see particularly Snouck-Hurgronje, Het Mekkaansche Feest
(Leiden, 1880). (W. R. S.)
1 A variant of the name Makkah is Bakkah (Sur. iii. 90; Bakrī, 155 seq.). For other
names and honorific epithets of the city see Bakrī, ut supra, Azraqī, p. 197, Yāqūt iv. 617
seq. The lists are in part corrupt, and some of the names (Kūthā and ‘Arsh or ‘Ursh, “the
huts”) are not properly names of the town as a whole.
2 Mecca, says one of its citizens, in Wāqidī (Kremer’s ed., p. 196, or Muh. in Med. p.
100), is a settlement formed for trade with Syria in summer and Abyssinia in winter, and
cannot continue to exist if the trade is interrupted.
histories and descriptions of Mecca published by Wüstenfeld
(Chroniken der Stadt Mekka, 3 vols., 1857-1859, with an abstract in
German, 1861), the most valuable is that of Azraqī. It has passed
through the hands of several editors, but the oldest part goes back to
the beginning of the 9th Christian century. Kutbeddin’s history (vol. iii.
of the Chroniken) goes down with the additions of his nephew to a.d.
1592.
Of European descriptions of Mecca from personal observation the
best is Burckhardt’s Travels in Arabia (cited above from the 8vo ed.,
1829). The Travels of Aly Bey (Badia, London, 1816) describe a visit in
1807; Burton’s Pilgrimage (3rd ed., 1879) often supplements
Burckhardt; Von Maltzan’s Wallfahrt nach Mekka (1865) is lively but
very slight. ‘Abd el-Razzāq’s report to the government of India on the
pilgrimage of 1858 is specially directed to sanitary questions; C.
Snouck-Hurgronje, Mekka (2 vols., and a collection of photographs,
The Hague, 1888-1889), gives a description of the Meccan sanctuary
and of the public and private life of the Meccans as observed by the
author during a sojourn in the holy city in 1884-1885 and a political
history of Mecca from native sources from the Hegira till 1884. For the
pilgrimage see particularly Snouck-Hurgronje, Het Mekkaansche Feest
(Leiden, 1880). (W. R. S.)
1 A variant of the name Makkah is Bakkah (Sur. iii. 90; Bakrī, 155 seq.). For other
names and honorific epithets of the city see Bakrī, ut supra, Azraqī, p. 197, Yāqūt iv. 617
seq. The lists are in part corrupt, and some of the names (Kūthā and ‘Arsh or ‘Ursh, “the
huts”) are not properly names of the town as a whole.
2 Mecca, says one of its citizens, in Wāqidī (Kremer’s ed., p. 196, or Muh. in Med. p.
100), is a settlement formed for trade with Syria in summer and Abyssinia in winter, and
cannot continue to exist if the trade is interrupted.
Page 333
3 The details are variously related. See Bīrūnī, p. 328 (E. T., p. 324); Asma‘i in Yāqūt,
iii. 705, iv. 416, 421; Azraqī, p. 129 seq.; Bakrī, p. 661. Jebel Kabkab is a great mountain
occupying the angle between W. Namān and the plain of Arafa. The peak is due north of
Sheddād, the hamlet which Burckhardt (i. 115) calls Shedad. According to Azraqī, p. 80,
the last shrine visited was that of the three trees of Uzzā in W. Nakhla.
4 So we are told by Bīrūnī, p. 62 (E. T., 73).
5 Wāqidī, ed. Kremer, pp. 20, 21; Muh. in Med. p. 39.
6 The older fairs were not entirely deserted till the troubles of the last days of the
Omayyads (Azraqī, p. 131).
7 This is the cross-road traversed by Burckhardt (i. 109), and described by him as cut
through the rocks with much labour.
8 Iṣṭakhrī gives the length of the city proper from north to south as 2 m., and the
greatest breadth from the Jiyād quarter east of the great mosque across the valley and up
the western slopes as two-thirds of the length.
9 For details as to the ancient quarters of Mecca, where the several families or septs
lived apart, see Azraqī, 455 pp. seq., and compare Ya‘qūbī, ed. Juynboll, p. 100. The minor
sacred places are described at length by Azraqī and Ibn Jubair. They are either connected
with genuine memories of the Prophet and his times, or have spurious legends to conceal
the fact that they were originally holy stones, wells, or the like, of heathen sanctity.
10 Balādhurī, in his chapter on the floods of Mecca (pp. 53 seq.), says that ‘Omar built
two dams.
11 The aqueduct is the successor of an older one associated with the names of Zobaida,
wife of Harūn al-Rashīd, and other benefactors. But the old aqueduct was frequently out of
repair, and seems to have played but a secondary part in the medieval water supply. Even
the new aqueduct gave no adequate supply in Burckhardt’s time.
12 In Ibn Jubair’s time large supplies were brought from the Yemen mountains.
13 The corruption of manners in Mecca is no new thing. See the letter of the caliph
Mahdi on the subject; Wüstenfeld, Chron. Mek., iv. 168.
14 The exact measurements (which, however, vary according to different authorities)
are stated to be: sides 37 ft. 2 in. and 38 ft. 4 in.; ends 31 ft. 7 in. and 29 ft.; height 35 ft.
15 The Ka‘ba of Mahomet’s time was the successor of an older building, said to have
been destroyed by fire. It was constructed in the still usual rude style of Arabic masonry,
with string courses of timber between the stones (like Solomon’s Temple). The roof rested
on six pillars; the door was raised above the ground and approached by a stair (probably on
iii. 705, iv. 416, 421; Azraqī, p. 129 seq.; Bakrī, p. 661. Jebel Kabkab is a great mountain
occupying the angle between W. Namān and the plain of Arafa. The peak is due north of
Sheddād, the hamlet which Burckhardt (i. 115) calls Shedad. According to Azraqī, p. 80,
the last shrine visited was that of the three trees of Uzzā in W. Nakhla.
4 So we are told by Bīrūnī, p. 62 (E. T., 73).
5 Wāqidī, ed. Kremer, pp. 20, 21; Muh. in Med. p. 39.
6 The older fairs were not entirely deserted till the troubles of the last days of the
Omayyads (Azraqī, p. 131).
7 This is the cross-road traversed by Burckhardt (i. 109), and described by him as cut
through the rocks with much labour.
8 Iṣṭakhrī gives the length of the city proper from north to south as 2 m., and the
greatest breadth from the Jiyād quarter east of the great mosque across the valley and up
the western slopes as two-thirds of the length.
9 For details as to the ancient quarters of Mecca, where the several families or septs
lived apart, see Azraqī, 455 pp. seq., and compare Ya‘qūbī, ed. Juynboll, p. 100. The minor
sacred places are described at length by Azraqī and Ibn Jubair. They are either connected
with genuine memories of the Prophet and his times, or have spurious legends to conceal
the fact that they were originally holy stones, wells, or the like, of heathen sanctity.
10 Balādhurī, in his chapter on the floods of Mecca (pp. 53 seq.), says that ‘Omar built
two dams.
11 The aqueduct is the successor of an older one associated with the names of Zobaida,
wife of Harūn al-Rashīd, and other benefactors. But the old aqueduct was frequently out of
repair, and seems to have played but a secondary part in the medieval water supply. Even
the new aqueduct gave no adequate supply in Burckhardt’s time.
12 In Ibn Jubair’s time large supplies were brought from the Yemen mountains.
13 The corruption of manners in Mecca is no new thing. See the letter of the caliph
Mahdi on the subject; Wüstenfeld, Chron. Mek., iv. 168.
14 The exact measurements (which, however, vary according to different authorities)
are stated to be: sides 37 ft. 2 in. and 38 ft. 4 in.; ends 31 ft. 7 in. and 29 ft.; height 35 ft.
15 The Ka‘ba of Mahomet’s time was the successor of an older building, said to have
been destroyed by fire. It was constructed in the still usual rude style of Arabic masonry,
with string courses of timber between the stones (like Solomon’s Temple). The roof rested
on six pillars; the door was raised above the ground and approached by a stair (probably on
Page 334
account of the floods which often swept the valley); and worshippers left their shoes under
the stair before entering. During the first siege of Mecca (a.d. 683), the building was
burned down, the Ibn Zubair reconstructed it on an enlarged scale and in better style of
solid ashlar-work. After his death his most glaring innovations (the introduction of two
doors on a level with the ground, and the extension of the building lengthwise to include
the Ḥijr) were corrected by Ḥajjāj, under orders from the caliph, but the building retained
its more solid structure. The roof now rested on three pillars, and the height was raised one-
half. The Ka‘ba was again entirely rebuilt after the flood of a.d. 1626, but since Ḥajjāj
there seem to have been no structural changes.
16 Hobal was set up within the Temple over the pit that contained the sacred treasures.
His chief function was connected with the sacred lot to which the Meccans were
accustomed to betake themselves in all matters of difficulty.
17 See Ibn Hishām i. 54, Azraḳī p. 80 (‘Uzzā in Baṭn Marr); Yāḳūt iii. 705 (Otheydā);
Bar Hebraeus on Psalm xii. 9. Stones worshipped by circling round them bore the name
dawār or duwār (Krehl, Rel. d. Araber, p. 69). The later Arabs not unnaturally viewed such
cultus as imitated from that of Mecca (Yāqūt iv. 622, cf. Dozy, Israeliten te Mekka, p. 125,
who draws very perverse inferences).
18 The old kiswa is removed on the 25th day of the month before the pilgrimage, and
fragments of it are bought by the pilgrims as charms. Till the 10th day of the pilgrimage
month the Ka‘ba is bare.
19 Before Islām the Ka‘ba was opened every Monday and Thursday; in the time of Ibn
Jubair it was opened with considerable ceremony every Monday and Friday, and daily in
the month Rajab. But, though prayer within the building is favoured by the example of the
Prophet, it is not compulsory on the Moslem, and even in the time of Ibn Baṭūṭa the
opportunities of entrance were reduced to Friday and the birthday of the Prophet.
20 See De Vogué, Syrie centrale: inscr. sem.; Lady Anne Blunt Pilgrimage of Nejd, ii.,
and W. R. Smith, in the Athenaeum, March 20, 1880.
21 Ibn Jubair speaks of fourteen steps, Ali Bey of four, Burckhardt of three. The
surrounding ground no doubt has risen so that the old name “hill of Safā” is now
inapplicable.
22 The latter perhaps was no part of the ancient omra; see Snouck-Hurgronje, Het
Mekkaansche Feest (1880) p. 115 sqq.
23 The 27th was also a great day, but this day was in commemoration of the rebuilding
of the Ka‘ba by Ibn Jubair.
24 The sacrifice is not indispensable except for those who can afford it and are
combining the hajj with the omra.
the stair before entering. During the first siege of Mecca (a.d. 683), the building was
burned down, the Ibn Zubair reconstructed it on an enlarged scale and in better style of
solid ashlar-work. After his death his most glaring innovations (the introduction of two
doors on a level with the ground, and the extension of the building lengthwise to include
the Ḥijr) were corrected by Ḥajjāj, under orders from the caliph, but the building retained
its more solid structure. The roof now rested on three pillars, and the height was raised one-
half. The Ka‘ba was again entirely rebuilt after the flood of a.d. 1626, but since Ḥajjāj
there seem to have been no structural changes.
16 Hobal was set up within the Temple over the pit that contained the sacred treasures.
His chief function was connected with the sacred lot to which the Meccans were
accustomed to betake themselves in all matters of difficulty.
17 See Ibn Hishām i. 54, Azraḳī p. 80 (‘Uzzā in Baṭn Marr); Yāḳūt iii. 705 (Otheydā);
Bar Hebraeus on Psalm xii. 9. Stones worshipped by circling round them bore the name
dawār or duwār (Krehl, Rel. d. Araber, p. 69). The later Arabs not unnaturally viewed such
cultus as imitated from that of Mecca (Yāqūt iv. 622, cf. Dozy, Israeliten te Mekka, p. 125,
who draws very perverse inferences).
18 The old kiswa is removed on the 25th day of the month before the pilgrimage, and
fragments of it are bought by the pilgrims as charms. Till the 10th day of the pilgrimage
month the Ka‘ba is bare.
19 Before Islām the Ka‘ba was opened every Monday and Thursday; in the time of Ibn
Jubair it was opened with considerable ceremony every Monday and Friday, and daily in
the month Rajab. But, though prayer within the building is favoured by the example of the
Prophet, it is not compulsory on the Moslem, and even in the time of Ibn Baṭūṭa the
opportunities of entrance were reduced to Friday and the birthday of the Prophet.
20 See De Vogué, Syrie centrale: inscr. sem.; Lady Anne Blunt Pilgrimage of Nejd, ii.,
and W. R. Smith, in the Athenaeum, March 20, 1880.
21 Ibn Jubair speaks of fourteen steps, Ali Bey of four, Burckhardt of three. The
surrounding ground no doubt has risen so that the old name “hill of Safā” is now
inapplicable.
22 The latter perhaps was no part of the ancient omra; see Snouck-Hurgronje, Het
Mekkaansche Feest (1880) p. 115 sqq.
23 The 27th was also a great day, but this day was in commemoration of the rebuilding
of the Ka‘ba by Ibn Jubair.
24 The sacrifice is not indispensable except for those who can afford it and are
combining the hajj with the omra.
Page 335
25 On the similar pelting of the supposed graves of Abū Lahab and his wife (Ibn Jubair,
p. 110) and of Abū Righāl at Mughammas, see Nöldeke’s translation of Tabarī, 208.
MECHANICS. The subject of mechanics may be divided into two
parts: (1) theoretical or abstract mechanics, and (2) applied mechanics.
1. Theoretical Mechanics
Historically theoretical mechanics began with the study of practical
contrivances such as the lever, and the name mechanics (Gr. τὰ μηχανικά),
which might more properly be restricted to the theory of mechanisms, and
which was indeed used in this narrower sense by Newton, has clung to it,
although the subject has long attained a far wider scope. In recent times it
has been proposed to adopt the term dynamics (from Gr. δύναμις force,) as
including the whole science of the action of force on bodies, whether at rest
or in motion. The subject is usually expounded under the two divisions of
statics and kinetics, the former dealing with the conditions of rest or
equilibrium and the latter with the phenomena of motion as affected by
force. To this latter division the old name of dynamics (in a restricted sense)
is still often applied. The mere geometrical description and analysis of
various types of motion, apart from the consideration of the forces
concerned, belongs to kinematics. This is sometimes discussed as a separate
theory, but for our present purposes it is more convenient to introduce
kinematical motions as they are required. We follow also the traditional
p. 110) and of Abū Righāl at Mughammas, see Nöldeke’s translation of Tabarī, 208.
MECHANICS. The subject of mechanics may be divided into two
parts: (1) theoretical or abstract mechanics, and (2) applied mechanics.
1. Theoretical Mechanics
Historically theoretical mechanics began with the study of practical
contrivances such as the lever, and the name mechanics (Gr. τὰ μηχανικά),
which might more properly be restricted to the theory of mechanisms, and
which was indeed used in this narrower sense by Newton, has clung to it,
although the subject has long attained a far wider scope. In recent times it
has been proposed to adopt the term dynamics (from Gr. δύναμις force,) as
including the whole science of the action of force on bodies, whether at rest
or in motion. The subject is usually expounded under the two divisions of
statics and kinetics, the former dealing with the conditions of rest or
equilibrium and the latter with the phenomena of motion as affected by
force. To this latter division the old name of dynamics (in a restricted sense)
is still often applied. The mere geometrical description and analysis of
various types of motion, apart from the consideration of the forces
concerned, belongs to kinematics. This is sometimes discussed as a separate
theory, but for our present purposes it is more convenient to introduce
kinematical motions as they are required. We follow also the traditional
Page 336
practice of dealing first with statics and then with kinetics. This is, in the
main, the historical order of development, and for purposes of exposition it
has many advantages. The laws of equilibrium are, it is true, necessarily
included as a particular case under those of motion; but there is no real
inconvenience in formulating as the basis of statics a few provisional
postulates which are afterwards seen to be comprehended in a more general
scheme.
The whole subject rests ultimately on the Newtonian laws of motion and
on some natural extensions of them. As these laws are discussed under a
separate heading (Motion, Laws of), it is here only necessary to indicate the
standpoint from which the present article is written. It is a purely empirical
one. Guided by experience, we are able to frame rules which enable us to
say with more or less accuracy what will be the consequences, or what were
the antecedents, of a given state of things. These rules are sometimes
dignified by the name of “laws of nature,” but they have relation to our
present state of knowledge and to the degree of skill with which we have
succeeded in giving more or less compact expression to it. They are
therefore liable to be modified from time to time, or to be superseded by
more convenient or more comprehensive modes of statement. Again, we do
not aim at anything so hopeless, or indeed so useless, as a complete
description of any phenomenon. Some features are naturally more
important or more interesting to us than others; by their relative simplicity
and evident constancy they have the first hold on our attention, whilst those
which are apparently accidental and vary from one occasion to another arc
ignored, or postponed for later examination. It follows that for the purposes
of such description as is possible some process of abstraction is inevitable if
our statements are to be simple and definite. Thus in studying the flight of a
stone through the air we replace the body in imagination by a mathematical
point endowed with a mass-coefficient. The size and shape, the complicated
spinning motion which it is seen to execute, the internal strains and
main, the historical order of development, and for purposes of exposition it
has many advantages. The laws of equilibrium are, it is true, necessarily
included as a particular case under those of motion; but there is no real
inconvenience in formulating as the basis of statics a few provisional
postulates which are afterwards seen to be comprehended in a more general
scheme.
The whole subject rests ultimately on the Newtonian laws of motion and
on some natural extensions of them. As these laws are discussed under a
separate heading (Motion, Laws of), it is here only necessary to indicate the
standpoint from which the present article is written. It is a purely empirical
one. Guided by experience, we are able to frame rules which enable us to
say with more or less accuracy what will be the consequences, or what were
the antecedents, of a given state of things. These rules are sometimes
dignified by the name of “laws of nature,” but they have relation to our
present state of knowledge and to the degree of skill with which we have
succeeded in giving more or less compact expression to it. They are
therefore liable to be modified from time to time, or to be superseded by
more convenient or more comprehensive modes of statement. Again, we do
not aim at anything so hopeless, or indeed so useless, as a complete
description of any phenomenon. Some features are naturally more
important or more interesting to us than others; by their relative simplicity
and evident constancy they have the first hold on our attention, whilst those
which are apparently accidental and vary from one occasion to another arc
ignored, or postponed for later examination. It follows that for the purposes
of such description as is possible some process of abstraction is inevitable if
our statements are to be simple and definite. Thus in studying the flight of a
stone through the air we replace the body in imagination by a mathematical
point endowed with a mass-coefficient. The size and shape, the complicated
spinning motion which it is seen to execute, the internal strains and
Page 337
vibrations which doubtless take place, are all sacrificed in the mental
picture in order that attention may be concentrated on those features of the
phenomenon which are in the first place most interesting to us. At a later
stage in our subject the conception of the ideal rigid body is introduced; this
enables us to fill in some details which were previously wanting, but others
are still omitted. Again, the conception of a force as concentrated in a
mathematical line is as unreal as that of a mass concentrated in a point, but
it is a convenient fiction for our purpose, owing to the simplicity which it
lends to our statements.
The laws which are to be imposed on these ideal representations are in
the first instance largely at our choice. Any scheme of abstract dynamics
constructed in this way, provided it be self-consistent, is mathematically
legitimate; but from the physical point of view we require that it should
help us to picture the sequence of phenomena as they actually occur. Its
success or failure in this respect can only be judged a posteriori by
comparison of the results to which it leads with the facts. It is to be noticed,
moreover, that all available tests apply only to the scheme as a whole;
owing to the complexity of phenomena we cannot submit any one of its
postulates to verification apart from the rest.
It is from this point of view that the question of relativity of motion,
which is often felt to be a stumbling-block on the very threshold of the
subject, is to be judged. By “motion” we mean of necessity motion relative
to some frame of reference which is conventionally spoken of as “fixed.” In
the earlier stages of our subject this may be any rigid, or apparently rigid,
structure fixed relatively to the earth. If we meet with phenomena which do
not fit easily into this view, we have the alternatives either to modify our
assumed laws of motion, or to call to our aid adventitious forces, or to
examine whether the discrepancy can be reconciled by the simpler
expedient of a new basis of reference. It is hardly necessary to say that the
picture in order that attention may be concentrated on those features of the
phenomenon which are in the first place most interesting to us. At a later
stage in our subject the conception of the ideal rigid body is introduced; this
enables us to fill in some details which were previously wanting, but others
are still omitted. Again, the conception of a force as concentrated in a
mathematical line is as unreal as that of a mass concentrated in a point, but
it is a convenient fiction for our purpose, owing to the simplicity which it
lends to our statements.
The laws which are to be imposed on these ideal representations are in
the first instance largely at our choice. Any scheme of abstract dynamics
constructed in this way, provided it be self-consistent, is mathematically
legitimate; but from the physical point of view we require that it should
help us to picture the sequence of phenomena as they actually occur. Its
success or failure in this respect can only be judged a posteriori by
comparison of the results to which it leads with the facts. It is to be noticed,
moreover, that all available tests apply only to the scheme as a whole;
owing to the complexity of phenomena we cannot submit any one of its
postulates to verification apart from the rest.
It is from this point of view that the question of relativity of motion,
which is often felt to be a stumbling-block on the very threshold of the
subject, is to be judged. By “motion” we mean of necessity motion relative
to some frame of reference which is conventionally spoken of as “fixed.” In
the earlier stages of our subject this may be any rigid, or apparently rigid,
structure fixed relatively to the earth. If we meet with phenomena which do
not fit easily into this view, we have the alternatives either to modify our
assumed laws of motion, or to call to our aid adventitious forces, or to
examine whether the discrepancy can be reconciled by the simpler
expedient of a new basis of reference. It is hardly necessary to say that the
Page 338
latter procedure has hitherto been found to be adequate. As a first step we
adopt a system of rectangular axes whose origin is fixed in the earth, but
whose directions are fixed by relation to the stars; in the planetary theory
the origin is transferred to the sun, and afterwards to the mass-centre of the
solar system; and so on. At each step there is a gain in accuracy and
comprehensiveness; and the conviction is cherished that some system of
rectangular axes exists with respect to which the Newtonian scheme holds
with all imaginable accuracy.
A similar account might be given of the conception of time as a
measurable quantity, but the remarks which it is necessary to make under
this head will find a place later.
The following synopsis shows the scheme on which the treatment is
based:—
Part 1.—Statics.
1. Statics of a particle.
2. Statics of a system of particles.
3. Plane kinematics of a rigid body.
4. Plane statics.
5. Graphical statics.
6. Theory of frames.
7. Three-dimensional kinematics of a rigid body.
8. Three-dimensional statics.
9. Work.
10. Statics of inextensible chains.
11. Theory of mass-systems.
Part 2.—Kinetics.
12. Rectilinear motion.
adopt a system of rectangular axes whose origin is fixed in the earth, but
whose directions are fixed by relation to the stars; in the planetary theory
the origin is transferred to the sun, and afterwards to the mass-centre of the
solar system; and so on. At each step there is a gain in accuracy and
comprehensiveness; and the conviction is cherished that some system of
rectangular axes exists with respect to which the Newtonian scheme holds
with all imaginable accuracy.
A similar account might be given of the conception of time as a
measurable quantity, but the remarks which it is necessary to make under
this head will find a place later.
The following synopsis shows the scheme on which the treatment is
based:—
Part 1.—Statics.
1. Statics of a particle.
2. Statics of a system of particles.
3. Plane kinematics of a rigid body.
4. Plane statics.
5. Graphical statics.
6. Theory of frames.
7. Three-dimensional kinematics of a rigid body.
8. Three-dimensional statics.
9. Work.
10. Statics of inextensible chains.
11. Theory of mass-systems.
Part 2.—Kinetics.
12. Rectilinear motion.
Page 339
13. General motion of a particle.
14. Central forces. Hodograph.
15. Kinetics of a system of discrete particles.
16. Kinetics of a rigid body. Fundamental principles.
17. Two-dimensional problems.
18. Equations of motion in three dimensions.
19. Free motion of a solid.
20. Motion of a solid of revolution.
21. Moving axes of reference.
22. Equations of motion in generalized co-ordinates.
23. Stability of equilibrium. Theory of vibrations.
Part I.—Statics
§ 1. Statics of a Particle.—By a particle is meant a body whose position
can for the purpose in hand be sufficiently specified by a mathematical
point. It need not be “infinitely small,” or even small compared with
ordinary standards; thus in astronomy such vast bodies as the sun, the earth,
and the other planets can for many purposes be treated merely as points
endowed with mass.
A force is conceived as an effort having a certain direction and a certain
magnitude. It is therefore adequately represented, for mathematical
purposes, by a straight line AB drawn in the direction in question, of length
proportional (on any convenient scale) to the magnitude of the force. In
other words, a force is mathematically of the nature of a “vector” (see
Vector Analysis, Quaternions). In most questions of pure statics we are
concerned only with the ratios of the various forces which enter into the
problem, so that it is indifferent what unit of force is adopted. For many
purposes a gravitational system of measurement is most natural; thus we
14. Central forces. Hodograph.
15. Kinetics of a system of discrete particles.
16. Kinetics of a rigid body. Fundamental principles.
17. Two-dimensional problems.
18. Equations of motion in three dimensions.
19. Free motion of a solid.
20. Motion of a solid of revolution.
21. Moving axes of reference.
22. Equations of motion in generalized co-ordinates.
23. Stability of equilibrium. Theory of vibrations.
Part I.—Statics
§ 1. Statics of a Particle.—By a particle is meant a body whose position
can for the purpose in hand be sufficiently specified by a mathematical
point. It need not be “infinitely small,” or even small compared with
ordinary standards; thus in astronomy such vast bodies as the sun, the earth,
and the other planets can for many purposes be treated merely as points
endowed with mass.
A force is conceived as an effort having a certain direction and a certain
magnitude. It is therefore adequately represented, for mathematical
purposes, by a straight line AB drawn in the direction in question, of length
proportional (on any convenient scale) to the magnitude of the force. In
other words, a force is mathematically of the nature of a “vector” (see
Vector Analysis, Quaternions). In most questions of pure statics we are
concerned only with the ratios of the various forces which enter into the
problem, so that it is indifferent what unit of force is adopted. For many
purposes a gravitational system of measurement is most natural; thus we
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speak of a force of so many pounds or so many kilogrammes. The
“absolute” system of measurement will be referred to below in Part II.,
Kinetics. It is to be remembered that all “force” is of the nature of a push or
a pull, and that according to the accepted terminology of modern mechanics
such phrases as “force of inertia,” “accelerating force,” “moving force,”
once classical, are proscribed. This rigorous limitation of the meaning of the
word is of comparatively recent origin, and it is perhaps to be regretted that
some more technical term has not been devised, but the convention must
now be regarded as established.
Fig. 1.
The fundamental postulate of this part of our subject is that the two
forces acting on a particle may be compounded by the “parallelogram rule.”
Thus, if the two forces P,Q be represented by the lines OA, OB, they can be
replaced by a single force R represented by the diagonal OC of the
parallelogram determined by OA, OB. This is of course a physical
assumption whose propriety is justified solely by experience. We shall see
later that it is implied in Newton’s statement of his Second Law of motion.
In modern language, forces are compounded by “vector-addition”; thus, if
we draw in succession vectors HK>, KL> to represent P, Q, the force R is
represented by the vector HL> which is the “geometric sum” of HK>, KL>.
“absolute” system of measurement will be referred to below in Part II.,
Kinetics. It is to be remembered that all “force” is of the nature of a push or
a pull, and that according to the accepted terminology of modern mechanics
such phrases as “force of inertia,” “accelerating force,” “moving force,”
once classical, are proscribed. This rigorous limitation of the meaning of the
word is of comparatively recent origin, and it is perhaps to be regretted that
some more technical term has not been devised, but the convention must
now be regarded as established.
Fig. 1.
The fundamental postulate of this part of our subject is that the two
forces acting on a particle may be compounded by the “parallelogram rule.”
Thus, if the two forces P,Q be represented by the lines OA, OB, they can be
replaced by a single force R represented by the diagonal OC of the
parallelogram determined by OA, OB. This is of course a physical
assumption whose propriety is justified solely by experience. We shall see
later that it is implied in Newton’s statement of his Second Law of motion.
In modern language, forces are compounded by “vector-addition”; thus, if
we draw in succession vectors HK>, KL> to represent P, Q, the force R is
represented by the vector HL> which is the “geometric sum” of HK>, KL>.
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By successive applications of the above rule any number of forces acting
on a particle may be replaced by a single force which is the vector-sum of
the given forces: this single force is called the resultant. Thus if AB>, BC>,
CD> ..., HK> be vectors representing the given forces, the resultant will be
given by AK>. It will be understood that the figure ABCD ... K need not be
confined to one plane.
Fig. 2.
If, in particular, the point K coincides with A, so that the resultant
vanishes, the given system of forces is said to be in equilibrium—i.e. the
particle could remain permanently at rest under its action. This is the
proposition known as the polygon of forces. In the particular case of three
forces it reduces to the triangle of forces, viz. “If three forces acting on a
particle are represented as to magnitude and direction by the sides of a
triangle taken in order, they are in equilibrium.”
A sort of converse proposition is frequently useful, viz. if three forces
acting on a particle be in equilibrium, and any triangle be constructed
whose sides are respectively parallel to the forces, the magnitudes of the
forces will be to one another as the corresponding sides of the triangle. This
follows from the fact that all such triangles are necessarily similar.
on a particle may be replaced by a single force which is the vector-sum of
the given forces: this single force is called the resultant. Thus if AB>, BC>,
CD> ..., HK> be vectors representing the given forces, the resultant will be
given by AK>. It will be understood that the figure ABCD ... K need not be
confined to one plane.
Fig. 2.
If, in particular, the point K coincides with A, so that the resultant
vanishes, the given system of forces is said to be in equilibrium—i.e. the
particle could remain permanently at rest under its action. This is the
proposition known as the polygon of forces. In the particular case of three
forces it reduces to the triangle of forces, viz. “If three forces acting on a
particle are represented as to magnitude and direction by the sides of a
triangle taken in order, they are in equilibrium.”
A sort of converse proposition is frequently useful, viz. if three forces
acting on a particle be in equilibrium, and any triangle be constructed
whose sides are respectively parallel to the forces, the magnitudes of the
forces will be to one another as the corresponding sides of the triangle. This
follows from the fact that all such triangles are necessarily similar.
Page 342
Fig. 3.
As a simple example of the geometrical method of treating statical
problems we may consider the equilibrium of a particle on a “rough”
inclined plane. The usual empirical law of sliding friction is that the
mutual action between two plane surfaces in contact, or between a
particle and a curve or surface, cannot make with the normal an angle
exceeding a certain limit λ called the angle of friction. If the conditions
of equilibrium require an obliquity greater than this, sliding will take
place. The precise value of λ will vary with the nature and condition of
the surfaces in contact. In the case of a body simply resting on an
inclined plane, the reaction must of course be vertical, for equilibrium,
and the slope α of the plane must therefore not exceed λ. For this
reason λ is also known as the angle of repose. If α > λ, a force P must
be applied in order to maintain equilibrium; let θ be the inclination of P
to the plane, as shown in the left-hand diagram. The relations between
this force P, the gravity W of the body, and the reaction S of the plane
are then determined by a triangle of forces HKL. Since the inclination
of S to the normal cannot exceed λ on either side, the value of P must
lie between two limits which are represented by L1H, L2H, in the right-
hand diagram. Denoting these limits by P1, P2, we have
As a simple example of the geometrical method of treating statical
problems we may consider the equilibrium of a particle on a “rough”
inclined plane. The usual empirical law of sliding friction is that the
mutual action between two plane surfaces in contact, or between a
particle and a curve or surface, cannot make with the normal an angle
exceeding a certain limit λ called the angle of friction. If the conditions
of equilibrium require an obliquity greater than this, sliding will take
place. The precise value of λ will vary with the nature and condition of
the surfaces in contact. In the case of a body simply resting on an
inclined plane, the reaction must of course be vertical, for equilibrium,
and the slope α of the plane must therefore not exceed λ. For this
reason λ is also known as the angle of repose. If α > λ, a force P must
be applied in order to maintain equilibrium; let θ be the inclination of P
to the plane, as shown in the left-hand diagram. The relations between
this force P, the gravity W of the body, and the reaction S of the plane
are then determined by a triangle of forces HKL. Since the inclination
of S to the normal cannot exceed λ on either side, the value of P must
lie between two limits which are represented by L1H, L2H, in the right-
hand diagram. Denoting these limits by P1, P2, we have
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P1/W = L1H/HK = sin (α − λ)/cos (θ + λ),
P2/W = L2H/HK = sin (α + λ)/cos (θ − λ).
It appears, moreover, that if θ be varied P will be least when L1H is at
right angles to KL1, in which case P1 = W sin (α − λ), corresponding to
θ = −λ.
Just as two or more forces can be
combined into a single resultant, so a
single force may be resolved into
components acting in assigned
directions. Thus a force can be
uniquely resolved into two
components acting in two assigned
directions in the same plane with it by an inversion of the parallelogram
construction of fig. 1. If, as is usually most convenient, the two assigned
directions are at right angles, the two components of a force P will be P cos
θ, P sin θ, where θ is the inclination of P to the direction of the former
component. This leads to formulae for the analytical reduction of a system
of coplanar forces acting on a particle. Adopting rectangular axes Ox, Oy, in
the plane of the forces, and distinguishing the various forces of the system
by suffixes, we can replace the system by two forces X, Y, in the direction
of co-ordinate axes; viz.—
X = P1 cos θ1 + P2 cos θ2 + ... = Σ (P cos θ),
Y = P1 sin θ1 + P2 sin θ2 + ... = Σ (P sin θ).
(1)
These two forces X, Y, may be combined into a single resultant R making
an angle φ with Ox, provided
X = R cos φ, Y = R sin φ,
(2)
P2/W = L2H/HK = sin (α + λ)/cos (θ − λ).
It appears, moreover, that if θ be varied P will be least when L1H is at
right angles to KL1, in which case P1 = W sin (α − λ), corresponding to
θ = −λ.
Just as two or more forces can be
combined into a single resultant, so a
single force may be resolved into
components acting in assigned
directions. Thus a force can be
uniquely resolved into two
components acting in two assigned
directions in the same plane with it by an inversion of the parallelogram
construction of fig. 1. If, as is usually most convenient, the two assigned
directions are at right angles, the two components of a force P will be P cos
θ, P sin θ, where θ is the inclination of P to the direction of the former
component. This leads to formulae for the analytical reduction of a system
of coplanar forces acting on a particle. Adopting rectangular axes Ox, Oy, in
the plane of the forces, and distinguishing the various forces of the system
by suffixes, we can replace the system by two forces X, Y, in the direction
of co-ordinate axes; viz.—
X = P1 cos θ1 + P2 cos θ2 + ... = Σ (P cos θ),
Y = P1 sin θ1 + P2 sin θ2 + ... = Σ (P sin θ).
(1)
These two forces X, Y, may be combined into a single resultant R making
an angle φ with Ox, provided
X = R cos φ, Y = R sin φ,
(2)
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whence
R2 = X2 + Y2, tan φ = Y/X.
(3)
For equilibrium we must have R = 0, which requires X = 0, Y = 0; in words,
the sum of the components of the system must be zero for each of two
perpendicular directions in the plane.
A similar procedure applies to a
three-dimensional system. Thus if, O
being the origin, OH> represent any
force P of the system, the planes
drawn through H parallel to the co-
ordinate planes will enclose with the
latter a parallelepiped, and it is evident
Fig. 5.
that OH> is the geometric sum of OA>,
AN>, NH>, or OA>, OB>, OC>, in the figure. Hence P is equivalent to three
forces Pl, Pm, Pn acting along Ox, Oy, Oz, respectively, where l, m, n, are
the “direction-ratios” of OH>. The whole system can be reduced in this way
to three forces
X = Σ (Pl), Y = Σ (Pm), Z = Σ (Pn),
(4)
acting along the co-ordinate axes. These can again be combined into a
single resultant R acting in the direction (λ, μ, ν), provided
X = Rλ, Y = Rμ, Z = Rν.
(5)
If the axes are rectangular, the direction-ratios become direction-cosines, so
that λ2 + μ2 + ν2 = 1, whence
R2 = X2 + Y2 + Z2.
R2 = X2 + Y2, tan φ = Y/X.
(3)
For equilibrium we must have R = 0, which requires X = 0, Y = 0; in words,
the sum of the components of the system must be zero for each of two
perpendicular directions in the plane.
A similar procedure applies to a
three-dimensional system. Thus if, O
being the origin, OH> represent any
force P of the system, the planes
drawn through H parallel to the co-
ordinate planes will enclose with the
latter a parallelepiped, and it is evident
Fig. 5.
that OH> is the geometric sum of OA>,
AN>, NH>, or OA>, OB>, OC>, in the figure. Hence P is equivalent to three
forces Pl, Pm, Pn acting along Ox, Oy, Oz, respectively, where l, m, n, are
the “direction-ratios” of OH>. The whole system can be reduced in this way
to three forces
X = Σ (Pl), Y = Σ (Pm), Z = Σ (Pn),
(4)
acting along the co-ordinate axes. These can again be combined into a
single resultant R acting in the direction (λ, μ, ν), provided
X = Rλ, Y = Rμ, Z = Rν.
(5)
If the axes are rectangular, the direction-ratios become direction-cosines, so
that λ2 + μ2 + ν2 = 1, whence
R2 = X2 + Y2 + Z2.
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(6)
The conditions of equilibrium are X = 0, Y = 0, Z = 0.
§ 2. Statics of a System of Particles.—We assume that the mutual forces
between the pairs of particles, whatever their nature, are subject to the “Law
of Action and Reaction” (Newton’s Third Law); i.e. the force exerted by a
particle A on a particle B, and the force exerted by B on A, are equal and
opposite in the line AB. The problem of determining the possible
configurations of equilibrium of a system of particles subject to extraneous
forces which are known functions of the positions of the particles, and to
internal forces which are known functions of the distances of the pairs of
particles between which they act, is in general determinate. For if n be the
number of particles, the 3n conditions of equilibrium (three for each
particle) are equal in number to the 3n Cartesian (or other) co-ordinates of
the particles, which are to be found. If the system be subject to frictionless
constraints, e.g. if some of the particles be constrained to lie on smooth
surfaces, or if pairs of particles be connected by inextensible strings, then
for each geometrical relation thus introduced we have an unknown reaction
(e.g. the pressure of the smooth surface, or the tension of the string), so that
the problem is still determinate.
Fig. 6.
The conditions of equilibrium are X = 0, Y = 0, Z = 0.
§ 2. Statics of a System of Particles.—We assume that the mutual forces
between the pairs of particles, whatever their nature, are subject to the “Law
of Action and Reaction” (Newton’s Third Law); i.e. the force exerted by a
particle A on a particle B, and the force exerted by B on A, are equal and
opposite in the line AB. The problem of determining the possible
configurations of equilibrium of a system of particles subject to extraneous
forces which are known functions of the positions of the particles, and to
internal forces which are known functions of the distances of the pairs of
particles between which they act, is in general determinate. For if n be the
number of particles, the 3n conditions of equilibrium (three for each
particle) are equal in number to the 3n Cartesian (or other) co-ordinates of
the particles, which are to be found. If the system be subject to frictionless
constraints, e.g. if some of the particles be constrained to lie on smooth
surfaces, or if pairs of particles be connected by inextensible strings, then
for each geometrical relation thus introduced we have an unknown reaction
(e.g. the pressure of the smooth surface, or the tension of the string), so that
the problem is still determinate.
Fig. 6.
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The case of the
funicular polygon will be
of use to us later. A
number of particles
attached at various
points of a string are
acted on by given
extraneous forces P1, P2, Fig. 7.
P3 ... respectively. The
relation between the
three forces acting on
any particle, viz. the
extraneous force and the
tensions in the two
adjacent portions of the Fig. 8.
string can be exhibited
by means of a triangle of forces; and if the successive triangles be
drawn to the same scale they can be fitted together so as to constitute a
single force-diagram, as shown in fig. 6. This diagram consists of a
polygon whose successive sides represent the given forces P1, P2, P3 ...,
and of a series of lines connecting the vertices with a point O. These
latter lines measure the tensions in the successive portions of string. As
a special, but very important case, the forces P1, P2, P3 ... may be
parallel, e.g. they may be the weights of the several particles. The
polygon of forces is then made up of segments of a vertical line. We
note that the tensions have now the same horizontal projection
(represented by the dotted line in fig. 7). It is further of interest to note
that if the weights be all equal, and at equal horizontal intervals, the
vertices of the funicular will lie on a parabola whose axis is vertical. To
prove this statement, let A, B, C, D ... be successive vertices, and let H,
funicular polygon will be
of use to us later. A
number of particles
attached at various
points of a string are
acted on by given
extraneous forces P1, P2, Fig. 7.
P3 ... respectively. The
relation between the
three forces acting on
any particle, viz. the
extraneous force and the
tensions in the two
adjacent portions of the Fig. 8.
string can be exhibited
by means of a triangle of forces; and if the successive triangles be
drawn to the same scale they can be fitted together so as to constitute a
single force-diagram, as shown in fig. 6. This diagram consists of a
polygon whose successive sides represent the given forces P1, P2, P3 ...,
and of a series of lines connecting the vertices with a point O. These
latter lines measure the tensions in the successive portions of string. As
a special, but very important case, the forces P1, P2, P3 ... may be
parallel, e.g. they may be the weights of the several particles. The
polygon of forces is then made up of segments of a vertical line. We
note that the tensions have now the same horizontal projection
(represented by the dotted line in fig. 7). It is further of interest to note
that if the weights be all equal, and at equal horizontal intervals, the
vertices of the funicular will lie on a parabola whose axis is vertical. To
prove this statement, let A, B, C, D ... be successive vertices, and let H,
Page 347
K ... be the middle points of AC, BD ...; then BH, CK ... will be vertical
by the hypothesis, and since the geometric sum of BA>, BC> is
represented by 2BH>, the tension in BA: tension in BC: weight at B
as BA : BC : 2BH.
The tensions in the successive portions of the string are therefore
proportional to the respective lengths, and the lines BH, CK ... are all
equal. Hence AD, BC are parallel and are bisected by the same vertical
line; and a parabola with vertical axis can therefore be described
through A, B, C, D. The same holds for the four points B, C, D, E and
so on; but since a parabola is uniquely determined by the direction of
its axis and by three points on the curve, the successive parabolas
ABCD, BCDE, CDEF ... must be coincident.
§ 3. Plane Kinematics of a Rigid Body.—The ideal rigid body is one in
which the distance between any two points is invariable. For the present we
confine ourselves to the consideration of displacements in two dimensions,
so that the body is adequately represented by a thin lamina or plate.
The position of a lamina
movable in its own plane is
determinate when we know the
positions of any two points A, B
of it. Since the four co-ordinates
(Cartesian or other) of these two
points are connected by the Fig. 9.
relation which expresses the
invariability of the length AB, it is plain that virtually three independent
elements are required and suffice to specify the position of the lamina. For
instance, the lamina may in general be fixed by connecting any three points
of it by rigid links to three fixed points in its plane. The three independent
by the hypothesis, and since the geometric sum of BA>, BC> is
represented by 2BH>, the tension in BA: tension in BC: weight at B
as BA : BC : 2BH.
The tensions in the successive portions of the string are therefore
proportional to the respective lengths, and the lines BH, CK ... are all
equal. Hence AD, BC are parallel and are bisected by the same vertical
line; and a parabola with vertical axis can therefore be described
through A, B, C, D. The same holds for the four points B, C, D, E and
so on; but since a parabola is uniquely determined by the direction of
its axis and by three points on the curve, the successive parabolas
ABCD, BCDE, CDEF ... must be coincident.
§ 3. Plane Kinematics of a Rigid Body.—The ideal rigid body is one in
which the distance between any two points is invariable. For the present we
confine ourselves to the consideration of displacements in two dimensions,
so that the body is adequately represented by a thin lamina or plate.
The position of a lamina
movable in its own plane is
determinate when we know the
positions of any two points A, B
of it. Since the four co-ordinates
(Cartesian or other) of these two
points are connected by the Fig. 9.
relation which expresses the
invariability of the length AB, it is plain that virtually three independent
elements are required and suffice to specify the position of the lamina. For
instance, the lamina may in general be fixed by connecting any three points
of it by rigid links to three fixed points in its plane. The three independent
Page 348
elements may be chosen in a variety of ways (e.g. they may be the lengths
of the three links in the above example). They may be called (in a
generalized sense) the co-ordinates of the lamina. The lamina when
perfectly free to move in its own plane is said to have three degrees of
freedom.
By a theorem due to M. Chasles
any displacement whatever of the
lamina in its own plane is equivalent
to a rotation about some finite or
infinitely distant point J. For
suppose that in consequence of the
displacement a point of the lamina is
brought from A to B, whilst the point
Fig. 10.
of the lamina which was originally
at B is brought to C. Since AB, BC,
are two different positions of the
same line in the lamina they are
equal, and it is evident that the
rotation could have been effected by
a rotation about J, the centre of the
circle ABC, through an angle AJB.
Fig. 11.
As a special case the three points A,
B, C may be in a straight line; J is then at infinity and the displacement is
equivalent to a pure translation, since every point of the lamina is now
displaced parallel to AB through a space equal to AB.
Next, consider any continuous motion of the lamina. The latter may be
brought from any one of its positions to a neighbouring one by a rotation
about the proper centre. The limiting position J of this centre, when the two
positions are taken infinitely close to one another, is called the
of the three links in the above example). They may be called (in a
generalized sense) the co-ordinates of the lamina. The lamina when
perfectly free to move in its own plane is said to have three degrees of
freedom.
By a theorem due to M. Chasles
any displacement whatever of the
lamina in its own plane is equivalent
to a rotation about some finite or
infinitely distant point J. For
suppose that in consequence of the
displacement a point of the lamina is
brought from A to B, whilst the point
Fig. 10.
of the lamina which was originally
at B is brought to C. Since AB, BC,
are two different positions of the
same line in the lamina they are
equal, and it is evident that the
rotation could have been effected by
a rotation about J, the centre of the
circle ABC, through an angle AJB.
Fig. 11.
As a special case the three points A,
B, C may be in a straight line; J is then at infinity and the displacement is
equivalent to a pure translation, since every point of the lamina is now
displaced parallel to AB through a space equal to AB.
Next, consider any continuous motion of the lamina. The latter may be
brought from any one of its positions to a neighbouring one by a rotation
about the proper centre. The limiting position J of this centre, when the two
positions are taken infinitely close to one another, is called the
Page 349
instantaneous centre. If P, P′ be consecutive positions of the same point,
and δθ the corresponding angle of rotation, then ultimately PP′ is at right
angles to JP and equal to JP·δθ. The instantaneous centre will have a certain
locus in space, and a certain locus in the lamina. These two loci are called
pole-curves or centrodes, and are sometimes distinguished as the space-
centrode and the body-centrode, respectively. In the continuous motion in
question the latter curve rolls without slipping on the former (M. Chasles).
Consider in fact any series of successive positions 1, 2, 3... of the lamina
(fig. 11); and let J12, J23, J34... be the positions in space of the centres of the
rotations by which the lamina can be brought from the first position to the
second, from the second to the third, and so on. Further, in the position 1,
let J12, J′23, J′34 ... be the points of the lamina which have become the
successive centres of rotation. The given series of positions will be assumed
in succession if we imagine the lamina to rotate first about J12 until J′23
comes into coincidence with J23, then about J23 until J′34 comes into
coincidence with J34, and so on. This is equivalent to imagining the polygon
J12 J′23 J′34 ..., supposed fixed in the lamina, to roll on the polygon J12 J23
J34 ..., which is supposed fixed in space. By imagining the successive
positions to be taken infinitely close to one another we derive the theorem
stated. The particular case where both centrodes are circles is specially
important in mechanism.
The theory may be illustrated
by the case of “three-bar
motion.” Let ABCD be any
quadrilateral formed of jointed
links. If, AB being held fixed,
the quadrilateral be slightly
deformed, it is obvious that the
instantaneous centre J will be at
and δθ the corresponding angle of rotation, then ultimately PP′ is at right
angles to JP and equal to JP·δθ. The instantaneous centre will have a certain
locus in space, and a certain locus in the lamina. These two loci are called
pole-curves or centrodes, and are sometimes distinguished as the space-
centrode and the body-centrode, respectively. In the continuous motion in
question the latter curve rolls without slipping on the former (M. Chasles).
Consider in fact any series of successive positions 1, 2, 3... of the lamina
(fig. 11); and let J12, J23, J34... be the positions in space of the centres of the
rotations by which the lamina can be brought from the first position to the
second, from the second to the third, and so on. Further, in the position 1,
let J12, J′23, J′34 ... be the points of the lamina which have become the
successive centres of rotation. The given series of positions will be assumed
in succession if we imagine the lamina to rotate first about J12 until J′23
comes into coincidence with J23, then about J23 until J′34 comes into
coincidence with J34, and so on. This is equivalent to imagining the polygon
J12 J′23 J′34 ..., supposed fixed in the lamina, to roll on the polygon J12 J23
J34 ..., which is supposed fixed in space. By imagining the successive
positions to be taken infinitely close to one another we derive the theorem
stated. The particular case where both centrodes are circles is specially
important in mechanism.
The theory may be illustrated
by the case of “three-bar
motion.” Let ABCD be any
quadrilateral formed of jointed
links. If, AB being held fixed,
the quadrilateral be slightly
deformed, it is obvious that the
instantaneous centre J will be at
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the intersection of the straight Fig. 12.
lines AD, BC, since the
displacements of the points D,
C are necessarily at right angles
to AD, BC, respectively. Hence
these displacements are
proportional to JD, JC, and
therefore to DD′ CC′, where
C′D′ is any line drawn parallel
to CD, meeting BC, AD in C′,
D′, respectively. The
Fig. 13.
determination of the centrodes
in three-bar motion is in general complicated, but in one case, that of
the “crossed parallelogram” (fig. 13), they assume simple forms. We
then have AB = DC and AD = BC, and from the symmetries of the
figure it is plain that
AJ + JB = CJ + JD = AD.
Hence the locus of J relative to AB, and the locus relative to CD are
equal ellipses of which A, B and C, D are respectively the foci. It may
be noticed that the lamina in fig. 9 is not, strictly speaking, fixed, but
admits of infinitesimal displacement, whenever the directions of the
three links are concurrent (or parallel).
The matter may of course be treated analytically, but we shall only
require the formula for infinitely small displacements. If the origin of
rectangular axes fixed in the lamina be shifted through a space whose
projections on the original directions of the axes are λ, μ, and if the axes are
simultaneously turned through an angle ε, the co-ordinates of a point of the
lamina, relative to the original axes, are changed from x, y to λ + x cos ε − y
lines AD, BC, since the
displacements of the points D,
C are necessarily at right angles
to AD, BC, respectively. Hence
these displacements are
proportional to JD, JC, and
therefore to DD′ CC′, where
C′D′ is any line drawn parallel
to CD, meeting BC, AD in C′,
D′, respectively. The
Fig. 13.
determination of the centrodes
in three-bar motion is in general complicated, but in one case, that of
the “crossed parallelogram” (fig. 13), they assume simple forms. We
then have AB = DC and AD = BC, and from the symmetries of the
figure it is plain that
AJ + JB = CJ + JD = AD.
Hence the locus of J relative to AB, and the locus relative to CD are
equal ellipses of which A, B and C, D are respectively the foci. It may
be noticed that the lamina in fig. 9 is not, strictly speaking, fixed, but
admits of infinitesimal displacement, whenever the directions of the
three links are concurrent (or parallel).
The matter may of course be treated analytically, but we shall only
require the formula for infinitely small displacements. If the origin of
rectangular axes fixed in the lamina be shifted through a space whose
projections on the original directions of the axes are λ, μ, and if the axes are
simultaneously turned through an angle ε, the co-ordinates of a point of the
lamina, relative to the original axes, are changed from x, y to λ + x cos ε − y
Page 351
sin ε, μ + x sin ε + y cos ε, or λ + x − yε, μ + xε + y, ultimately. Hence the
component displacements are ultimately
δx = λ − yε, δy = μ + xε
(1)
If we equate these to zero we get the co-ordinates of the instantaneous
centre.
§ 4. Plane Statics.—The statics of a rigid body rests on the following two
assumptions:—
(i) A force may be supposed to be applied indifferently at any point in its
line of action. In other words, a force is of the nature of a “bound” or
“localized” vector; it is regarded as resident in a certain line, but has no
special reference to any particular point of the line.
(ii) Two forces in intersecting lines may be replaced by a force which is
their geometric sum, acting through the intersection. The theory of parallel
forces is included as a limiting case. For if O, A, B be any three points, and
m, n any scalar quantities, we have in vectors
m · OA> + n · OB> = (m + n) OC>,
component displacements are ultimately
δx = λ − yε, δy = μ + xε
(1)
If we equate these to zero we get the co-ordinates of the instantaneous
centre.
§ 4. Plane Statics.—The statics of a rigid body rests on the following two
assumptions:—
(i) A force may be supposed to be applied indifferently at any point in its
line of action. In other words, a force is of the nature of a “bound” or
“localized” vector; it is regarded as resident in a certain line, but has no
special reference to any particular point of the line.
(ii) Two forces in intersecting lines may be replaced by a force which is
their geometric sum, acting through the intersection. The theory of parallel
forces is included as a limiting case. For if O, A, B be any three points, and
m, n any scalar quantities, we have in vectors
m · OA> + n · OB> = (m + n) OC>,
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( )
provided
m · CA> + n · CB> = 0.
(2)
Hence if forces P, Q act in OA, OB, the resultant R will pass through C, provided
m = P/OA, n = Q/OB;
also
R = P·OC/OA + Q·OC/OB,
(3)
and
P · AC : Q·CB = OA : OB.
(4)
These formulae give a means of constructing the resultant by means of any
transversal AB cutting the lines of action. If we now imagine the point O to
recede to infinity, the forces P, Q and the resultant R are parallel, and we have
R = P + Q, P·AC = Q·CB.
(5)
When P, Q have opposite signs the point C
divides AB externally on the side of the
greater force. The investigation fails when P
+ Q = 0, since it leads to an infinitely small
resultant acting in an infinitely distant line. A
combination of two equal, parallel, but
oppositely directed forces cannot in fact be
Fig. 14.
replaced by anything simpler, and must
therefore be recognized as an independent
entity in statics. It was called by L. Poinsot, who first systematically investigated
its properties, a couple.
provided
m · CA> + n · CB> = 0.
(2)
Hence if forces P, Q act in OA, OB, the resultant R will pass through C, provided
m = P/OA, n = Q/OB;
also
R = P·OC/OA + Q·OC/OB,
(3)
and
P · AC : Q·CB = OA : OB.
(4)
These formulae give a means of constructing the resultant by means of any
transversal AB cutting the lines of action. If we now imagine the point O to
recede to infinity, the forces P, Q and the resultant R are parallel, and we have
R = P + Q, P·AC = Q·CB.
(5)
When P, Q have opposite signs the point C
divides AB externally on the side of the
greater force. The investigation fails when P
+ Q = 0, since it leads to an infinitely small
resultant acting in an infinitely distant line. A
combination of two equal, parallel, but
oppositely directed forces cannot in fact be
Fig. 14.
replaced by anything simpler, and must
therefore be recognized as an independent
entity in statics. It was called by L. Poinsot, who first systematically investigated
its properties, a couple.
Page 353
We now restrict ourselves for the present to the systems of forces in one plane.
By successive applications of (ii) any such coplanar system can in general be
reduced to a single resultant acting in a definite line. As exceptional cases the
system may reduce to a couple, or it may be in equilibrium.
The moment of a force about a point O is
the product of the force into the
perpendicular drawn to its line of action
from O, this perpendicular being reckoned
positive or negative according as O lies on
one side or other of the line of action. If we
Fig. 15. mark off a segment AB along the line of
action so as to represent the force
completely, the moment is represented as to
magnitude by twice the area of the triangle
OAB, and the usual convention as to sign is
that the area is to be reckoned positive or
negative according as the letters O, A, B,
Fig. 16. occur in “counter-clockwise” or
“clockwise” order.
The sum of the moments of two forces about any point O is equal to the
moment of their resultant (P. Varignon, 1687). Let AB, AC (fig. 16) represent the
two forces, AD their resultant; we have to prove that the sum of the triangles
OAB, OAC is equal to the triangle OAD, regard being had to signs. Since the
side OA is common, we have to prove that the sum of the perpendiculars from B
and C on OA is equal to the perpendicular from D on OA, these perpendiculars
being reckoned positive or negative according as they lie to the right or left of
AO. Regarded as a statement concerning the orthogonal projections of the
vectors AB> and AC> (or BD), and of their sum AD>, on a line perpendicular to
AO, this is obvious.
It is now evident that in the process of reduction of a coplanar system no
change is made at any stage either in the sum of the projections of the forces on
By successive applications of (ii) any such coplanar system can in general be
reduced to a single resultant acting in a definite line. As exceptional cases the
system may reduce to a couple, or it may be in equilibrium.
The moment of a force about a point O is
the product of the force into the
perpendicular drawn to its line of action
from O, this perpendicular being reckoned
positive or negative according as O lies on
one side or other of the line of action. If we
Fig. 15. mark off a segment AB along the line of
action so as to represent the force
completely, the moment is represented as to
magnitude by twice the area of the triangle
OAB, and the usual convention as to sign is
that the area is to be reckoned positive or
negative according as the letters O, A, B,
Fig. 16. occur in “counter-clockwise” or
“clockwise” order.
The sum of the moments of two forces about any point O is equal to the
moment of their resultant (P. Varignon, 1687). Let AB, AC (fig. 16) represent the
two forces, AD their resultant; we have to prove that the sum of the triangles
OAB, OAC is equal to the triangle OAD, regard being had to signs. Since the
side OA is common, we have to prove that the sum of the perpendiculars from B
and C on OA is equal to the perpendicular from D on OA, these perpendiculars
being reckoned positive or negative according as they lie to the right or left of
AO. Regarded as a statement concerning the orthogonal projections of the
vectors AB> and AC> (or BD), and of their sum AD>, on a line perpendicular to
AO, this is obvious.
It is now evident that in the process of reduction of a coplanar system no
change is made at any stage either in the sum of the projections of the forces on
Page 354
any line or in the sum of their moments about any point. It follows that the single
resultant to which the system in general reduces is uniquely determinate, i.e. it
acts in a definite line and has a definite magnitude and sense. Again it is
necessary and sufficient for equilibrium that the sum of the projections of the
forces on each of two perpendicular directions should vanish, and (moreover)
that the sum of the moments about some one point should be zero. The fact that
three independent conditions must hold for equilibrium is important. The
conditions may of course be expressed in different (but equivalent) forms; e.g.
the sum of the moments of the forces about each of the three points which are not
collinear must be zero.
Fig. 17.
The particular case of three forces is of interest. If they are not all parallel they
must be concurrent, and their vector-sum must be zero. Thus three forces acting
perpendicular to the sides of a triangle at the middle points will be in equilibrium
provided they are proportional to the respective sides, and act all inwards or all
outwards. This result is easily extended to the case of a polygon of any number
of sides; it has an important application in hydrostatics.
Again, suppose we have a bar AB resting with its ends on two smooth
inclined planes which face each other. Let G be the centre of gravity (§ 11),
and let AG = a, GB = b. Let α, β be the inclinations of the planes, and θ the
angle which the bar makes with the vertical. The position of equilibrium is
determined by the consideration that the reactions at A and B, which are by
hypothesis normal to the planes, must meet at a point J on the vertical
through G. Hence
resultant to which the system in general reduces is uniquely determinate, i.e. it
acts in a definite line and has a definite magnitude and sense. Again it is
necessary and sufficient for equilibrium that the sum of the projections of the
forces on each of two perpendicular directions should vanish, and (moreover)
that the sum of the moments about some one point should be zero. The fact that
three independent conditions must hold for equilibrium is important. The
conditions may of course be expressed in different (but equivalent) forms; e.g.
the sum of the moments of the forces about each of the three points which are not
collinear must be zero.
Fig. 17.
The particular case of three forces is of interest. If they are not all parallel they
must be concurrent, and their vector-sum must be zero. Thus three forces acting
perpendicular to the sides of a triangle at the middle points will be in equilibrium
provided they are proportional to the respective sides, and act all inwards or all
outwards. This result is easily extended to the case of a polygon of any number
of sides; it has an important application in hydrostatics.
Again, suppose we have a bar AB resting with its ends on two smooth
inclined planes which face each other. Let G be the centre of gravity (§ 11),
and let AG = a, GB = b. Let α, β be the inclinations of the planes, and θ the
angle which the bar makes with the vertical. The position of equilibrium is
determined by the consideration that the reactions at A and B, which are by
hypothesis normal to the planes, must meet at a point J on the vertical
through G. Hence
Page 355
JG/a = sin (θ − α) / sin α, JG/b = sin (θ + β) / sin β,
whence
a cot α − b cot β
cot θ = .
a+b (6)
If the bar is uniform we have a = b, and
cot θ = 1⁄2 (cot α − cot β).
(7)
The problem of a rod suspended by strings attached to two points of it is
virtually identical, the tensions of the strings taking the place of the
reactions of the planes.
Fig. 18.
Just as a system of forces is in general equivalent to a single force, so a given
force can conversely be replaced by combinations of other forces, in various
ways. For instance, a given force (and consequently a system of forces) can be
replaced in one and only one way by three forces acting in three assigned straight
lines, provided these lines be not concurrent or parallel. Thus if the three lines
form a triangle ABC, and if the given force F meet BC in H, then F can be
resolved into two components acting in HA, BC, respectively. And the force in
HA can be resolved into two components acting in BC, CA, respectively. A
whence
a cot α − b cot β
cot θ = .
a+b (6)
If the bar is uniform we have a = b, and
cot θ = 1⁄2 (cot α − cot β).
(7)
The problem of a rod suspended by strings attached to two points of it is
virtually identical, the tensions of the strings taking the place of the
reactions of the planes.
Fig. 18.
Just as a system of forces is in general equivalent to a single force, so a given
force can conversely be replaced by combinations of other forces, in various
ways. For instance, a given force (and consequently a system of forces) can be
replaced in one and only one way by three forces acting in three assigned straight
lines, provided these lines be not concurrent or parallel. Thus if the three lines
form a triangle ABC, and if the given force F meet BC in H, then F can be
resolved into two components acting in HA, BC, respectively. And the force in
HA can be resolved into two components acting in BC, CA, respectively. A
Page 356
simple graphical construction is indicated in fig. 19, where the dotted lines are
parallel. As an example, any system of forces acting on the lamina in fig. 9 is
balanced by three determinate tensions (or thrusts) in the three links, provided the
directions of the latter are not concurrent.
Fig. 19.
If P, Q, R, be any three forces acting along BC, CA, AB, respectively, the
line of action of the resultant is determined by the consideration that the sum
of the moments about any point on it must vanish. Hence in “trilinear” co-
ordinates, with ABC as fundamental triangle, its equation is Pα + Qβ + Rγ =
0. If P : Q : R = a : b : c, where a, b, c are the lengths of the sides, this
becomes the “line at infinity,” and the forces reduce to a couple.
The sum of the moments of the two
forces of a couple is the same about any
point in the plane. Thus in the figure the
sum of the moments about O is P·OA −
P·OB or P·AB, which is independent of
the position of O. This sum is called the
moment of the couple; it must of course Fig. 20.
have the proper sign attributed to it. It
easily follows that any two couples of the same moment are equivalent, and that
any number of couples can be replaced by a single couple whose moment is the
sum of their moments. Since a couple is for our purposes sufficiently represented
by its moment, it has been proposed to substitute the name torque (or twisting
effort), as free from the suggestion of any special pair of forces.
parallel. As an example, any system of forces acting on the lamina in fig. 9 is
balanced by three determinate tensions (or thrusts) in the three links, provided the
directions of the latter are not concurrent.
Fig. 19.
If P, Q, R, be any three forces acting along BC, CA, AB, respectively, the
line of action of the resultant is determined by the consideration that the sum
of the moments about any point on it must vanish. Hence in “trilinear” co-
ordinates, with ABC as fundamental triangle, its equation is Pα + Qβ + Rγ =
0. If P : Q : R = a : b : c, where a, b, c are the lengths of the sides, this
becomes the “line at infinity,” and the forces reduce to a couple.
The sum of the moments of the two
forces of a couple is the same about any
point in the plane. Thus in the figure the
sum of the moments about O is P·OA −
P·OB or P·AB, which is independent of
the position of O. This sum is called the
moment of the couple; it must of course Fig. 20.
have the proper sign attributed to it. It
easily follows that any two couples of the same moment are equivalent, and that
any number of couples can be replaced by a single couple whose moment is the
sum of their moments. Since a couple is for our purposes sufficiently represented
by its moment, it has been proposed to substitute the name torque (or twisting
effort), as free from the suggestion of any special pair of forces.
Page 357
A system of forces represented completely by the sides of a plane polygon
taken in order is equivalent to a couple whose moment is represented by twice
the area of the polygon; this is proved by taking moments about any point. If the
polygon intersects itself, care must be taken to attribute to the different parts of
the area their proper signs.
Fig. 21.
Again, any coplanar system of forces can be replaced by a single force R
acting at any assigned point O, together with a couple G. The force R is the
geometric sum of the given forces, and the moment (G) of the couple is equal to
the sum of the moments of the given forces about O. The value of G will in
general vary with the position of O, and will vanish when O lies on the line of
action of the single resultant.
The formal analytical reduction of a
system of coplanar forces is as follows.
Let (x1, y1), (x2, y2), ... be the
rectangular co-ordinates of any points
A1, A2, ... on the lines of action of the
respective forces. The force at A1 may
be replaced by its components X1, Y1,
parallel to the co-ordinate axes; that at
Fig. 22.
A2 by its components X2, Y2, and so
on. Introducing at O two equal and opposite forces ±X1 in Ox, we see that X1 at
A1 may be replaced by an equal and parallel force at O together with a couple
−y1X1. Similarly the force Y1 at A1 may be replaced by a force Y1 at O together
taken in order is equivalent to a couple whose moment is represented by twice
the area of the polygon; this is proved by taking moments about any point. If the
polygon intersects itself, care must be taken to attribute to the different parts of
the area their proper signs.
Fig. 21.
Again, any coplanar system of forces can be replaced by a single force R
acting at any assigned point O, together with a couple G. The force R is the
geometric sum of the given forces, and the moment (G) of the couple is equal to
the sum of the moments of the given forces about O. The value of G will in
general vary with the position of O, and will vanish when O lies on the line of
action of the single resultant.
The formal analytical reduction of a
system of coplanar forces is as follows.
Let (x1, y1), (x2, y2), ... be the
rectangular co-ordinates of any points
A1, A2, ... on the lines of action of the
respective forces. The force at A1 may
be replaced by its components X1, Y1,
parallel to the co-ordinate axes; that at
Fig. 22.
A2 by its components X2, Y2, and so
on. Introducing at O two equal and opposite forces ±X1 in Ox, we see that X1 at
A1 may be replaced by an equal and parallel force at O together with a couple
−y1X1. Similarly the force Y1 at A1 may be replaced by a force Y1 at O together
Page 358
with a couple x1Y1. The forces X1, Y1, at O can thus be transferred to O provided
we introduce a couple x1Y1 − y1X1. Treating the remaining forces in the same
way we get a force X1 + X2 + ... or Σ(X) along Ox, a force Y1 + Y2 + ... or Σ(Y)
along Oy, and a couple (x1Y1 − y1X1) + (x2Y2 − y2X2) + ... or Σ(xY − yX). The
three conditions of equilibrium are therefore
Σ(X) = 0, Σ(Y) = 0, Σ(xY − yX) = 0.
(8)
If O′ be a point whose co-ordinates are (ξ, η), the moment of the couple when
the forces are transferred to O′ as a new origin will be Σ{(x − ξ) Y − (y − η) X}.
This vanishes, i.e. the system reduces to a single resultant through O′, provided
−ξ·Σ(Y) + η·Σ(X) + Σ(xY − yX) = 0.
(9)
If ξ, η be regarded as current co-ordinates, this is the equation of the line of
action of the single resultant to which the system is in general reducible.
If the forces are all parallel, making say an angle θ with Ox, we may write X1
= P1 cos θ, Y1 = P1 sin θ, X2 = P2 cos θ, Y2 = P2 sin θ, .... The equation (9) then
becomes
{Σ(xP) − ξ·Σ(P)} sin θ − {Σ(yP) − η·Σ(P)} cos θ = 0.
(10)
If the forces P1, P2, ... be turned in the same sense through the same angle about
the respective points A1, A2, ... so as to remain parallel, the value of θ is alone
altered, and the resultant Σ(P) passes always through the point
Σ(xP) Σ(yP)
x= , y= ,
Σ(P) Σ(P) (11)
which is determined solely by the configuration of the points A1, A2, ... and by
the ratios P1 : P2 : ... of the forces acting at them respectively. This point is called
the centre of the given system of parallel forces; it is finite and determinate
unless Σ(P) = 0. A geometrical proof of this theorem, which is not restricted to a
we introduce a couple x1Y1 − y1X1. Treating the remaining forces in the same
way we get a force X1 + X2 + ... or Σ(X) along Ox, a force Y1 + Y2 + ... or Σ(Y)
along Oy, and a couple (x1Y1 − y1X1) + (x2Y2 − y2X2) + ... or Σ(xY − yX). The
three conditions of equilibrium are therefore
Σ(X) = 0, Σ(Y) = 0, Σ(xY − yX) = 0.
(8)
If O′ be a point whose co-ordinates are (ξ, η), the moment of the couple when
the forces are transferred to O′ as a new origin will be Σ{(x − ξ) Y − (y − η) X}.
This vanishes, i.e. the system reduces to a single resultant through O′, provided
−ξ·Σ(Y) + η·Σ(X) + Σ(xY − yX) = 0.
(9)
If ξ, η be regarded as current co-ordinates, this is the equation of the line of
action of the single resultant to which the system is in general reducible.
If the forces are all parallel, making say an angle θ with Ox, we may write X1
= P1 cos θ, Y1 = P1 sin θ, X2 = P2 cos θ, Y2 = P2 sin θ, .... The equation (9) then
becomes
{Σ(xP) − ξ·Σ(P)} sin θ − {Σ(yP) − η·Σ(P)} cos θ = 0.
(10)
If the forces P1, P2, ... be turned in the same sense through the same angle about
the respective points A1, A2, ... so as to remain parallel, the value of θ is alone
altered, and the resultant Σ(P) passes always through the point
Σ(xP) Σ(yP)
x= , y= ,
Σ(P) Σ(P) (11)
which is determined solely by the configuration of the points A1, A2, ... and by
the ratios P1 : P2 : ... of the forces acting at them respectively. This point is called
the centre of the given system of parallel forces; it is finite and determinate
unless Σ(P) = 0. A geometrical proof of this theorem, which is not restricted to a
Page 359
two-dimensional system, is given later (§ 11). It contains the theory of the centre
of gravity as ordinarily understood. For if we have an assemblage of particles
whose mutual distances are small compared with the dimensions of the earth, the
forces of gravity on them constitute a system of sensibly parallel forces, sensibly
proportional to the respective masses. If now the assemblage be brought into any
other position relative to the earth, without alteration of the mutual distances, this
is equivalent to a rotation of the directions of the forces relatively to the
assemblage, the ratios of the forces remaining unaltered. Hence there is a certain
point, fixed relatively to the assemblage, through which the resultant of
gravitational action always passes; this resultant is moreover equal to the sum of
the forces on the several particles.
Fig. 23.
The theorem that any coplanar system of forces can be reduced to a force
acting through any assigned point, together with a couple, has an important
illustration in the theory of the distribution of shearing stress and bending
moment in a horizontal beam, or other structure, subject to vertical
extraneous forces. If we consider any vertical section P, the forces exerted
across the section by the portion of the structure on one side on the portion
on the other may be reduced to a vertical force F at P and a couple M. The
force measures the shearing stress, and the couple the bending moment at P;
we will reckon these quantities positive when the senses are as indicated in
the figure.
If the remaining forces acting on the portion of the structure on either side
of P are known, then resolving vertically we find F, and taking moments
of gravity as ordinarily understood. For if we have an assemblage of particles
whose mutual distances are small compared with the dimensions of the earth, the
forces of gravity on them constitute a system of sensibly parallel forces, sensibly
proportional to the respective masses. If now the assemblage be brought into any
other position relative to the earth, without alteration of the mutual distances, this
is equivalent to a rotation of the directions of the forces relatively to the
assemblage, the ratios of the forces remaining unaltered. Hence there is a certain
point, fixed relatively to the assemblage, through which the resultant of
gravitational action always passes; this resultant is moreover equal to the sum of
the forces on the several particles.
Fig. 23.
The theorem that any coplanar system of forces can be reduced to a force
acting through any assigned point, together with a couple, has an important
illustration in the theory of the distribution of shearing stress and bending
moment in a horizontal beam, or other structure, subject to vertical
extraneous forces. If we consider any vertical section P, the forces exerted
across the section by the portion of the structure on one side on the portion
on the other may be reduced to a vertical force F at P and a couple M. The
force measures the shearing stress, and the couple the bending moment at P;
we will reckon these quantities positive when the senses are as indicated in
the figure.
If the remaining forces acting on the portion of the structure on either side
of P are known, then resolving vertically we find F, and taking moments
Page 360
about P we find M. Again if PQ be any segment of the beam which is free
from load, Q lying to the right of P, we find
FP = F Q , MP − MQ = −F·PQ;
(12)
hence F is constant between the loads, whilst M decreases as we travel to
the right, with a constant gradient −F. If PQ be a short segment containing
an isolated load W, we have
FQ − FP = −W, MQ = MP;
(13)
hence F is discontinuous at a
concentrated load, diminishing by an
amount equal to the load as we pass the
loaded point to the right, whilst M is
continuous. Accordingly the graph of F
for any system of isolated loads will
consist of a series of horizontal lines,
whilst that of M will be a continuous
polygon.
To pass to the case of continuous
loads, let x be measured horizontally Fig. 24.
along the beam to the right. The load on
an element δx of the beam may be represented by wδx, where w is in
general a function of x. The equations (12) are now replaced by
δF = −wδx, δM = −Fδx,
whence
FQ − FP = − ∫QP w dx, MQ − MP = − ∫QP F dx.
(14)
from load, Q lying to the right of P, we find
FP = F Q , MP − MQ = −F·PQ;
(12)
hence F is constant between the loads, whilst M decreases as we travel to
the right, with a constant gradient −F. If PQ be a short segment containing
an isolated load W, we have
FQ − FP = −W, MQ = MP;
(13)
hence F is discontinuous at a
concentrated load, diminishing by an
amount equal to the load as we pass the
loaded point to the right, whilst M is
continuous. Accordingly the graph of F
for any system of isolated loads will
consist of a series of horizontal lines,
whilst that of M will be a continuous
polygon.
To pass to the case of continuous
loads, let x be measured horizontally Fig. 24.
along the beam to the right. The load on
an element δx of the beam may be represented by wδx, where w is in
general a function of x. The equations (12) are now replaced by
δF = −wδx, δM = −Fδx,
whence
FQ − FP = − ∫QP w dx, MQ − MP = − ∫QP F dx.
(14)
Page 361
The latter relation shows that the
bending moment varies as the area cut
off by the ordinate in the graph of F. In
the case of uniform load we have
F = −wx + A, M = 1⁄2wx2 − Ax + B,
(15)
where the arbitrary constants A,B are
to be determined by the conditions of
the special problem, e.g. the conditions
Fig. 25.
at the ends of the beam. The graph of F
is a straight line; that of M is a parabola with vertical axis. In all cases the
graphs due to different distributions of load may be superposed. The figure
shows the case of a uniform heavy beam supported at its ends.
Fig. 26.
§ 5. Graphical Statics.—A graphical method of reducing a plane system of
forces was introduced by C. Culmann (1864). It involves the construction of two
figures, a force-diagram and a funicular polygon. The force-diagram is
constructed by placing end to end a series of vectors representing the given
forces in magnitude and direction, and joining the vertices of the polygon thus
formed to an arbitrary pole O. The funicular or link polygon has its vertices on
bending moment varies as the area cut
off by the ordinate in the graph of F. In
the case of uniform load we have
F = −wx + A, M = 1⁄2wx2 − Ax + B,
(15)
where the arbitrary constants A,B are
to be determined by the conditions of
the special problem, e.g. the conditions
Fig. 25.
at the ends of the beam. The graph of F
is a straight line; that of M is a parabola with vertical axis. In all cases the
graphs due to different distributions of load may be superposed. The figure
shows the case of a uniform heavy beam supported at its ends.
Fig. 26.
§ 5. Graphical Statics.—A graphical method of reducing a plane system of
forces was introduced by C. Culmann (1864). It involves the construction of two
figures, a force-diagram and a funicular polygon. The force-diagram is
constructed by placing end to end a series of vectors representing the given
forces in magnitude and direction, and joining the vertices of the polygon thus
formed to an arbitrary pole O. The funicular or link polygon has its vertices on
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the lines of action of the given forces, and its sides respectively parallel to the
lines drawn from O in the force-diagram; in particular, the two sides meeting in
any vertex are respectively parallel to the lines drawn from O to the ends of that
side of the force-polygon which represents the corresponding force. The relations
will be understood from the annexed diagram, where corresponding lines in the
force-diagram (to the right) and the funicular (to the left) are numbered similarly.
The sides of the force-polygon may in the first instance be arranged in any order;
the force-diagram can then be completed in a doubly infinite number of ways,
owing to the arbitrary position of O; and for each force-diagram a simply infinite
number of funiculars can be drawn. The two diagrams being supposed
constructed, it is seen that each of the given systems of forces can be replaced by
two components acting in the sides of the funicular which meet at the
corresponding vertex, and that the magnitudes of these components will be given
by the corresponding triangle of forces in the force-diagram; thus the force 1 in
the figure is equivalent to two forces represented by 01 and 12. When this
process of replacement is complete, each terminated side of the funicular is the
seat of two forces which neutralize one another, and there remain only two
uncompensated forces, viz., those resident in the first and last sides of the
funicular. If these sides intersect, the resultant acts through the intersection, and
its magnitude and direction are given by the line joining the first and last sides of
the force-polygon (see fig. 26, where the resultant of the four given forces is
denoted by R). As a special case it may happen that the force-polygon is closed,
i.e. its first and last points coincide; the first and last sides of the funicular will
then be parallel (unless they coincide), and the two uncompensated forces form a
couple. If, however, the first and last sides of the funicular coincide, the two
outstanding forces neutralize one another, and we have equilibrium. Hence the
necessary and sufficient conditions of equilibrium are that the force-polygon and
the funicular should both be closed. This is illustrated by fig. 26 if we imagine
the force R, reversed, to be included in the system of given forces.
It is evident that a system of jointed bars having the shape of the funicular
polygon would be in equilibrium under the action of the given forces, supposed
applied to the joints; moreover any bar in which the stress is of the nature of a
lines drawn from O in the force-diagram; in particular, the two sides meeting in
any vertex are respectively parallel to the lines drawn from O to the ends of that
side of the force-polygon which represents the corresponding force. The relations
will be understood from the annexed diagram, where corresponding lines in the
force-diagram (to the right) and the funicular (to the left) are numbered similarly.
The sides of the force-polygon may in the first instance be arranged in any order;
the force-diagram can then be completed in a doubly infinite number of ways,
owing to the arbitrary position of O; and for each force-diagram a simply infinite
number of funiculars can be drawn. The two diagrams being supposed
constructed, it is seen that each of the given systems of forces can be replaced by
two components acting in the sides of the funicular which meet at the
corresponding vertex, and that the magnitudes of these components will be given
by the corresponding triangle of forces in the force-diagram; thus the force 1 in
the figure is equivalent to two forces represented by 01 and 12. When this
process of replacement is complete, each terminated side of the funicular is the
seat of two forces which neutralize one another, and there remain only two
uncompensated forces, viz., those resident in the first and last sides of the
funicular. If these sides intersect, the resultant acts through the intersection, and
its magnitude and direction are given by the line joining the first and last sides of
the force-polygon (see fig. 26, where the resultant of the four given forces is
denoted by R). As a special case it may happen that the force-polygon is closed,
i.e. its first and last points coincide; the first and last sides of the funicular will
then be parallel (unless they coincide), and the two uncompensated forces form a
couple. If, however, the first and last sides of the funicular coincide, the two
outstanding forces neutralize one another, and we have equilibrium. Hence the
necessary and sufficient conditions of equilibrium are that the force-polygon and
the funicular should both be closed. This is illustrated by fig. 26 if we imagine
the force R, reversed, to be included in the system of given forces.
It is evident that a system of jointed bars having the shape of the funicular
polygon would be in equilibrium under the action of the given forces, supposed
applied to the joints; moreover any bar in which the stress is of the nature of a
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tension (as distinguished from a thrust) might be replaced by a string. This is the
origin of the names “link-polygon” and “funicular” (cf. § 2).
If funiculars be drawn for two positions O, O′ of the pole in the force-
diagram, their corresponding sides will intersect on a straight line parallel to
OO′. This is essentially a theorem of projective geometry, but the following
statical proof is interesting. Let AB (fig. 27) be any side of the force-
polygon, and construct the corresponding portions of the two diagrams, first
with O and then with O′ as pole. The force corresponding to AB may be
replaced by the two components marked x, y; and a force corresponding to
BA may be represented by the two components marked x′, y′. Hence the
forces x, y, x′, y′ are in equilibrium. Now x, x′ have a resultant through H,
represented in magnitude and direction by OO′, whilst y, y′ have a resultant
through K represented in magnitude and direction by O′O. Hence HK must
be parallel to OO′. This theorem enables us, when one funicular has been
drawn, to construct any other without further reference to the force-diagram.
Fig. 27.
The complete figures obtained by drawing first the force-diagrams of a
system of forces in equilibrium with two distinct poles O, O′, and secondly
the corresponding funiculars, have various interesting relations. In the first
place, each of these figures may be conceived as an orthogonal projection of
origin of the names “link-polygon” and “funicular” (cf. § 2).
If funiculars be drawn for two positions O, O′ of the pole in the force-
diagram, their corresponding sides will intersect on a straight line parallel to
OO′. This is essentially a theorem of projective geometry, but the following
statical proof is interesting. Let AB (fig. 27) be any side of the force-
polygon, and construct the corresponding portions of the two diagrams, first
with O and then with O′ as pole. The force corresponding to AB may be
replaced by the two components marked x, y; and a force corresponding to
BA may be represented by the two components marked x′, y′. Hence the
forces x, y, x′, y′ are in equilibrium. Now x, x′ have a resultant through H,
represented in magnitude and direction by OO′, whilst y, y′ have a resultant
through K represented in magnitude and direction by O′O. Hence HK must
be parallel to OO′. This theorem enables us, when one funicular has been
drawn, to construct any other without further reference to the force-diagram.
Fig. 27.
The complete figures obtained by drawing first the force-diagrams of a
system of forces in equilibrium with two distinct poles O, O′, and secondly
the corresponding funiculars, have various interesting relations. In the first
place, each of these figures may be conceived as an orthogonal projection of
Page 364
a closed plane-faced polyhedron. As regards the former figure this is evident
at once; viz. the polyhedron consists of two pyramids with vertices
represented by O, O′, and a common base whose perimeter is represented by
the force-polygon (only one of these is shown in fig. 28). As regards the
funicular diagram, let LM be the line on which the pairs of corresponding
sides of the two polygons meet, and through it draw any two planes ω, ω′.
Through the vertices A, B, C, ... and A′, B′, C′, ... of the two funiculars draw
normals to the plane of the diagram, to meet ω and ω′ respectively. The
points thus obtained are evidently the vertices of a polyhedron with plane
faces.
Fig. 28.
at once; viz. the polyhedron consists of two pyramids with vertices
represented by O, O′, and a common base whose perimeter is represented by
the force-polygon (only one of these is shown in fig. 28). As regards the
funicular diagram, let LM be the line on which the pairs of corresponding
sides of the two polygons meet, and through it draw any two planes ω, ω′.
Through the vertices A, B, C, ... and A′, B′, C′, ... of the two funiculars draw
normals to the plane of the diagram, to meet ω and ω′ respectively. The
points thus obtained are evidently the vertices of a polyhedron with plane
faces.
Fig. 28.
Page 365
Fig. 29.
To every line in either of the original figures corresponds of course a
parallel line in the other; moreover, it is seen that concurrent lines in either
figure correspond to lines forming a closed polygon in the other. Two plane
figures so related are called reciprocal, since the properties of the first figure
in relation to the second are the same as those of the second with respect to
the first. A still simpler instance of reciprocal figures is supplied by the case
of concurrent forces in equilibrium (fig. 29). The theory of these reciprocal
figures was first studied by J. Clerk Maxwell, who showed amongst other
things that a reciprocal can always be drawn to any figure which is the
orthogonal projection of a plane-faced polyhedron. If in fact we take the
pole of each face of such a polyhedron with respect to a paraboloid of
revolution, these poles will be the vertices of a second polyhedron whose
edges are the “conjugate lines” of those of the former. If we project both
polyhedra orthogonally on a plane perpendicular to the axis of the
paraboloid, we obtain two figures which are reciprocal, except that
corresponding lines are orthogonal instead of parallel. Another proof will be
indicated later (§ 8) in connexion with the properties of the linear complex.
It is convenient to have a notation which shall put in evidence the reciprocal
character. For this purpose we may designate the points in one figure by
letters A, B, C, ... and the corresponding polygons in the other figure by the
same letters; a line joining two points A, B in one figure will then
correspond to the side common to the two polygons A, B in the other. This
notation was employed by R. H. Bow in connexion with the theory of
frames (§ 6, and see also Applied Mechanics below) where reciprocal
diagrams are frequently of use (cf. Diagram).
When the given forces are all parallel, the force-polygon consists of a
series of segments of a straight line. This case has important practical
applications; for instance we may use the method to find the pressures on
the supports of a beam loaded in any given manner. Thus if AB, BC, CD
represent the given loads, in the force-diagram, we construct the sides
To every line in either of the original figures corresponds of course a
parallel line in the other; moreover, it is seen that concurrent lines in either
figure correspond to lines forming a closed polygon in the other. Two plane
figures so related are called reciprocal, since the properties of the first figure
in relation to the second are the same as those of the second with respect to
the first. A still simpler instance of reciprocal figures is supplied by the case
of concurrent forces in equilibrium (fig. 29). The theory of these reciprocal
figures was first studied by J. Clerk Maxwell, who showed amongst other
things that a reciprocal can always be drawn to any figure which is the
orthogonal projection of a plane-faced polyhedron. If in fact we take the
pole of each face of such a polyhedron with respect to a paraboloid of
revolution, these poles will be the vertices of a second polyhedron whose
edges are the “conjugate lines” of those of the former. If we project both
polyhedra orthogonally on a plane perpendicular to the axis of the
paraboloid, we obtain two figures which are reciprocal, except that
corresponding lines are orthogonal instead of parallel. Another proof will be
indicated later (§ 8) in connexion with the properties of the linear complex.
It is convenient to have a notation which shall put in evidence the reciprocal
character. For this purpose we may designate the points in one figure by
letters A, B, C, ... and the corresponding polygons in the other figure by the
same letters; a line joining two points A, B in one figure will then
correspond to the side common to the two polygons A, B in the other. This
notation was employed by R. H. Bow in connexion with the theory of
frames (§ 6, and see also Applied Mechanics below) where reciprocal
diagrams are frequently of use (cf. Diagram).
When the given forces are all parallel, the force-polygon consists of a
series of segments of a straight line. This case has important practical
applications; for instance we may use the method to find the pressures on
the supports of a beam loaded in any given manner. Thus if AB, BC, CD
represent the given loads, in the force-diagram, we construct the sides
Page 366
corresponding to OA, OB, OC, OD in the funicular; we then draw the
closing line of the funicular polygon, and a parallel OE to it in the force
diagram. The segments DE, EA then represent the upward pressures of the
two supports on the beam, which pressures together with the given loads
constitute a system of forces in equilibrium. The pressures of the beam on
the supports are of course represented by ED, AE. The two diagrams are
portions of reciprocal figures, so that Bow’s notation is applicable.
Fig. 30.
closing line of the funicular polygon, and a parallel OE to it in the force
diagram. The segments DE, EA then represent the upward pressures of the
two supports on the beam, which pressures together with the given loads
constitute a system of forces in equilibrium. The pressures of the beam on
the supports are of course represented by ED, AE. The two diagrams are
portions of reciprocal figures, so that Bow’s notation is applicable.
Fig. 30.
Page 367
Fig. 31.
A graphical method can also be applied to find the moment of a force, or
of a system of forces, about any assigned point P. Let F be a force
represented by AB in the force-diagram. Draw a parallel through P to meet
the sides of the funicular which correspond to OA, OB in the points H, K. If
R be the intersection of these sides, the triangles OAB, RHK are similar, and
if the perpendiculars OM, RN be drawn we have
HK·OM = AB·RN = F·RN,
which is the moment of F about P. If the given forces are all parallel (say
vertical) OM is the same for all, and the moments of the several forces about
P are represented on a certain scale by the lengths intercepted by the
successive pairs of sides on the vertical through P. Moreover, the moments
are compounded by adding (geometrically) the corresponding lengths HK.
Hence if a system of vertical forces be in equilibrium, so that the funicular
polygon is closed, the length which this polygon intercepts on the vertical
through any point P gives the sum of the moments about P of all the forces
on one side of this vertical. For instance, in the case of a beam in
equilibrium under any given loads and the reactions at the supports, we get a
graphical representation of the distribution of bending moment over the
beam. The construction in fig. 30 can easily be adjusted so that the closing
line shall be horizontal; and the figure then becomes identical with the
bending-moment diagram of § 4. If we wish to study the effects of a
movable load, or system of loads, in different positions on the beam, it is
only necessary to shift the lines of action of the pressures of the supports
relatively to the funicular, keeping them at the same, distance apart; the only
change is then in the position of the closing line of the funicular. It may be
remarked that since this line joins homologous points of two “similar” rows
it will envelope a parabola.
The “centre” (§ 4) of a system of parallel forces of given magnitudes, acting at
given points, is easily determined graphically. We have only to construct the line
A graphical method can also be applied to find the moment of a force, or
of a system of forces, about any assigned point P. Let F be a force
represented by AB in the force-diagram. Draw a parallel through P to meet
the sides of the funicular which correspond to OA, OB in the points H, K. If
R be the intersection of these sides, the triangles OAB, RHK are similar, and
if the perpendiculars OM, RN be drawn we have
HK·OM = AB·RN = F·RN,
which is the moment of F about P. If the given forces are all parallel (say
vertical) OM is the same for all, and the moments of the several forces about
P are represented on a certain scale by the lengths intercepted by the
successive pairs of sides on the vertical through P. Moreover, the moments
are compounded by adding (geometrically) the corresponding lengths HK.
Hence if a system of vertical forces be in equilibrium, so that the funicular
polygon is closed, the length which this polygon intercepts on the vertical
through any point P gives the sum of the moments about P of all the forces
on one side of this vertical. For instance, in the case of a beam in
equilibrium under any given loads and the reactions at the supports, we get a
graphical representation of the distribution of bending moment over the
beam. The construction in fig. 30 can easily be adjusted so that the closing
line shall be horizontal; and the figure then becomes identical with the
bending-moment diagram of § 4. If we wish to study the effects of a
movable load, or system of loads, in different positions on the beam, it is
only necessary to shift the lines of action of the pressures of the supports
relatively to the funicular, keeping them at the same, distance apart; the only
change is then in the position of the closing line of the funicular. It may be
remarked that since this line joins homologous points of two “similar” rows
it will envelope a parabola.
The “centre” (§ 4) of a system of parallel forces of given magnitudes, acting at
given points, is easily determined graphically. We have only to construct the line
Page 368
of action of the resultant for each of two arbitrary directions of the forces; the
intersection of the two lines gives the point required. The construction is neatest
if the two arbitrary directions are taken at right angles to one another.
§ 6. Theory of Frames.—A frame is a structure made up of pieces, or
members, each of which has two joints connecting it with other members. In a
two-dimensional frame, each joint may be conceived as consisting of a small
cylindrical pin fitting accurately and smoothly into holes drilled through the
members which it connects. This supposition is a somewhat ideal one, and is
often only roughly approximated to in practice. We shall suppose, in the first
instance, that extraneous forces act on the frame at the joints only, i.e. on the
pins.
On this assumption, the reactions on any member at its two joints must be
equal and opposite. This combination of equal and opposite forces is called the
stress in the member; it may be a tension or a thrust. For diagrammatic purposes
each member is sufficiently represented by a straight line terminating at the two
joints; these lines will be referred to as the bars of the frame.
Fig. 32.
intersection of the two lines gives the point required. The construction is neatest
if the two arbitrary directions are taken at right angles to one another.
§ 6. Theory of Frames.—A frame is a structure made up of pieces, or
members, each of which has two joints connecting it with other members. In a
two-dimensional frame, each joint may be conceived as consisting of a small
cylindrical pin fitting accurately and smoothly into holes drilled through the
members which it connects. This supposition is a somewhat ideal one, and is
often only roughly approximated to in practice. We shall suppose, in the first
instance, that extraneous forces act on the frame at the joints only, i.e. on the
pins.
On this assumption, the reactions on any member at its two joints must be
equal and opposite. This combination of equal and opposite forces is called the
stress in the member; it may be a tension or a thrust. For diagrammatic purposes
each member is sufficiently represented by a straight line terminating at the two
joints; these lines will be referred to as the bars of the frame.
Fig. 32.
Page 369
In structural applications a frame must be stiff, or rigid, i.e. it must be
incapable of deformation without alteration of length in at least one of its bars. It
is said to be just rigid if it ceases to be rigid when any one of its bars is removed.
A frame which has more bars than are essential for rigidity may be called over-
rigid; such a frame is in general self-stressed, i.e. it is in a state of stress
independently of the action of extraneous forces. A plane frame of n joints which
is just rigid (as regards deformation in its own plane) has 2n − 3 bars, for if one
bar be held fixed the 2(n − 2) co-ordinates of the remaining n − 2 joints must just
be determined by the lengths of the remaining bars. The total number of bars is
therefore 2(n − 2) + 1. When a plane frame which is just rigid is subject to a
given system of equilibrating extraneous forces (in its own plane) acting on the
joints, the stresses in the bars are in general uniquely determinate. For the
conditions of equilibrium of the forces on each pin furnish 2n equations, viz. two
for each point, which are linear in respect of the stresses and the extraneous
forces. This system of equations must involve the three conditions of equilibrium
of the extraneous forces which are already identically satisfied, by hypothesis;
there remain therefore 2n − 3 independent relations to determine the 2n − 3
unknown stresses. A frame of n joints and 2n − 3 bars may of course fail to be
rigid owing to some parts being over-stiff whilst others are deformable; in such a
case it will be found that the statical equations, apart from the three identical
relations imposed by the equilibrium of the extraneous forces, are not all
independent but are equivalent to less than 2n − 3 relations. Another exceptional
case, known as the critical case, will be noticed later (§ 9).
A plane frame which can be built up from a single bar by successive steps, at
each of which a new joint is introduced by two new bars meeting there, is called
a simple frame; it is obviously just rigid. The stresses produced by extraneous
forces in a simple frame can be found by considering the equilibrium of the
various joints in a proper succession; and if the graphical method be employed
the various polygons of force can be combined into a single force-diagram. This
procedure was introduced by W. J. M. Rankine and J. Clerk Maxwell (1864). It
may be noticed that if we take an arbitrary pole in the force-diagram, and draw a
corresponding funicular in the skeleton diagram which represents the frame
incapable of deformation without alteration of length in at least one of its bars. It
is said to be just rigid if it ceases to be rigid when any one of its bars is removed.
A frame which has more bars than are essential for rigidity may be called over-
rigid; such a frame is in general self-stressed, i.e. it is in a state of stress
independently of the action of extraneous forces. A plane frame of n joints which
is just rigid (as regards deformation in its own plane) has 2n − 3 bars, for if one
bar be held fixed the 2(n − 2) co-ordinates of the remaining n − 2 joints must just
be determined by the lengths of the remaining bars. The total number of bars is
therefore 2(n − 2) + 1. When a plane frame which is just rigid is subject to a
given system of equilibrating extraneous forces (in its own plane) acting on the
joints, the stresses in the bars are in general uniquely determinate. For the
conditions of equilibrium of the forces on each pin furnish 2n equations, viz. two
for each point, which are linear in respect of the stresses and the extraneous
forces. This system of equations must involve the three conditions of equilibrium
of the extraneous forces which are already identically satisfied, by hypothesis;
there remain therefore 2n − 3 independent relations to determine the 2n − 3
unknown stresses. A frame of n joints and 2n − 3 bars may of course fail to be
rigid owing to some parts being over-stiff whilst others are deformable; in such a
case it will be found that the statical equations, apart from the three identical
relations imposed by the equilibrium of the extraneous forces, are not all
independent but are equivalent to less than 2n − 3 relations. Another exceptional
case, known as the critical case, will be noticed later (§ 9).
A plane frame which can be built up from a single bar by successive steps, at
each of which a new joint is introduced by two new bars meeting there, is called
a simple frame; it is obviously just rigid. The stresses produced by extraneous
forces in a simple frame can be found by considering the equilibrium of the
various joints in a proper succession; and if the graphical method be employed
the various polygons of force can be combined into a single force-diagram. This
procedure was introduced by W. J. M. Rankine and J. Clerk Maxwell (1864). It
may be noticed that if we take an arbitrary pole in the force-diagram, and draw a
corresponding funicular in the skeleton diagram which represents the frame
Page 370
together with the lines of action of the extraneous forces, we obtain two complete
reciprocal figures, in Maxwell’s sense. It is accordingly convenient to use Bow’s
notation (§ 5), and to distinguish the several compartments of the frame-diagram
by letters. See fig. 33, where the successive triangles in the diagram of forces
may be constructed in the order XYZ, ZXA, AZB. The class of “simple” frames
includes many of the frameworks used in the construction of roofs, lattice girders
and suspension bridges; a number of examples will be found in the article
Bridges. By examining the senses in which the respective forces act at each joint
we can ascertain which members are in tension and which are in thrust; in fig. 33
this is indicated by the directions of the arrowheads.
Fig. 33.
When a frame, though just rigid, is
not “simple” in the above sense, the
preceding method must be replaced, or
supplemented, by one or other of
various artifices. In some cases the
method of sections is sufficient for the
purpose. If an ideal section be drawn
Fig. 34.
across the frame, the extraneous forces
on either side must be in equilibrium with the forces in the bars cut across; and if
the section can be drawn so as to cut only three bars, the forces in these can be
reciprocal figures, in Maxwell’s sense. It is accordingly convenient to use Bow’s
notation (§ 5), and to distinguish the several compartments of the frame-diagram
by letters. See fig. 33, where the successive triangles in the diagram of forces
may be constructed in the order XYZ, ZXA, AZB. The class of “simple” frames
includes many of the frameworks used in the construction of roofs, lattice girders
and suspension bridges; a number of examples will be found in the article
Bridges. By examining the senses in which the respective forces act at each joint
we can ascertain which members are in tension and which are in thrust; in fig. 33
this is indicated by the directions of the arrowheads.
Fig. 33.
When a frame, though just rigid, is
not “simple” in the above sense, the
preceding method must be replaced, or
supplemented, by one or other of
various artifices. In some cases the
method of sections is sufficient for the
purpose. If an ideal section be drawn
Fig. 34.
across the frame, the extraneous forces
on either side must be in equilibrium with the forces in the bars cut across; and if
the section can be drawn so as to cut only three bars, the forces in these can be
Page 371
found, since the problem reduces to that of resolving a given force into three
components acting in three given lines (§ 4). The “critical case” where the
directions of the three bars are concurrent is of course excluded. Another method,
always available, will be explained under “Work” (§ 9).
When extraneous forces act on the bars themselves the stress in each bar
no longer consists of a simple longitudinal tension or thrust. To find the
reactions at the joints we may proceed as follows. Each extraneous force W
acting on a bar may be replaced (in an infinite number of ways) by two
components P, Q in lines through the centres of the pins at the extremities.
In practice the forces W are usually vertical, and the components P, Q are
then conveniently taken to be vertical also. We first alter the problem by
transferring the forces P, Q to the pins. The stresses in the bars, in the
problem as thus modified, may be supposed found by the preceding
methods; it remains to infer from the results thus obtained the reactions in
the original form of the problem. To find the pressure exerted by a bar AB
on the pin A we compound with the force in AB given by the diagram a
force equal to P. Conversely, to find the pressure of the pin A on the bar AB
we must compound with the force given by the diagram a force equal and
opposite to P. This question arises in practice in the theory of “three-jointed”
structures; for the purpose in hand such a structure is sufficiently
represented by two bars AB, BC. The right-hand figure represents a portion
of the force-diagram; in particular ZX> represents the pressure of AB on B in
the modified problem where the loads W1 and W2 on the two bars are
replaced by loads P1, Q1, and P2, Q2 respectively, acting on the pins.
Compounding with this XV>, which represents Q1, we get the actual pressure
ZV> exerted by AB on B. The directions and magnitudes of the reactions at
A and C are then easily ascertained. On account of its practical importance
several other graphical solutions of this problem have been devised.
components acting in three given lines (§ 4). The “critical case” where the
directions of the three bars are concurrent is of course excluded. Another method,
always available, will be explained under “Work” (§ 9).
When extraneous forces act on the bars themselves the stress in each bar
no longer consists of a simple longitudinal tension or thrust. To find the
reactions at the joints we may proceed as follows. Each extraneous force W
acting on a bar may be replaced (in an infinite number of ways) by two
components P, Q in lines through the centres of the pins at the extremities.
In practice the forces W are usually vertical, and the components P, Q are
then conveniently taken to be vertical also. We first alter the problem by
transferring the forces P, Q to the pins. The stresses in the bars, in the
problem as thus modified, may be supposed found by the preceding
methods; it remains to infer from the results thus obtained the reactions in
the original form of the problem. To find the pressure exerted by a bar AB
on the pin A we compound with the force in AB given by the diagram a
force equal to P. Conversely, to find the pressure of the pin A on the bar AB
we must compound with the force given by the diagram a force equal and
opposite to P. This question arises in practice in the theory of “three-jointed”
structures; for the purpose in hand such a structure is sufficiently
represented by two bars AB, BC. The right-hand figure represents a portion
of the force-diagram; in particular ZX> represents the pressure of AB on B in
the modified problem where the loads W1 and W2 on the two bars are
replaced by loads P1, Q1, and P2, Q2 respectively, acting on the pins.
Compounding with this XV>, which represents Q1, we get the actual pressure
ZV> exerted by AB on B. The directions and magnitudes of the reactions at
A and C are then easily ascertained. On account of its practical importance
several other graphical solutions of this problem have been devised.
Page 372
Fig. 35.
§ 7. Three-dimensional Kinematics of a Rigid Body.—The position of a rigid
body is determined when we know the positions of three points A, B, C of it
which are not collinear, for the position of any other point P is then determined
by the three distances PA, PB, PC. The nine co-ordinates (Cartesian or other) of
A, B, C are subject to the three relations which express the invariability of the
distances BC, CA, AB, and are therefore equivalent to six independent quantities.
Hence a rigid body not constrained in any way is said to have six degrees of
freedom. Conversely, any six geometrical relations restrict the body in general to
one or other of a series of definite positions, none of which can be departed from
without violating the conditions in question. For instance, the position of a
theodolite is fixed by the fact that its rounded feet rest in contact with six given
plane surfaces. Again, a rigid three-dimensional frame can be rigidly fixed
relatively to the earth by means of six links.
§ 7. Three-dimensional Kinematics of a Rigid Body.—The position of a rigid
body is determined when we know the positions of three points A, B, C of it
which are not collinear, for the position of any other point P is then determined
by the three distances PA, PB, PC. The nine co-ordinates (Cartesian or other) of
A, B, C are subject to the three relations which express the invariability of the
distances BC, CA, AB, and are therefore equivalent to six independent quantities.
Hence a rigid body not constrained in any way is said to have six degrees of
freedom. Conversely, any six geometrical relations restrict the body in general to
one or other of a series of definite positions, none of which can be departed from
without violating the conditions in question. For instance, the position of a
theodolite is fixed by the fact that its rounded feet rest in contact with six given
plane surfaces. Again, a rigid three-dimensional frame can be rigidly fixed
relatively to the earth by means of six links.
Page 373
Fig. 36. Fig. 37.
The six independent quantities, or “co-ordinates,” which serve to specify
the position of a rigid body in space may of course be chosen in an endless
variety of ways. We may, for instance, employ the three Cartesian co-
ordinates of a particular point O of the body, and three angular co-ordinates
which express the orientation of the body with respect to O. Thus in fig. 36,
if OA, OB, OC be three mutually perpendicular lines in the solid, we may
denote by θ the angle which OC makes with a fixed direction OZ, by ψ the
azimuth of the plane ZOC measured from some fixed plane through OZ, and
by φ the inclination of the plane COA to the plane ZOC. In fig. 36 these
various lines and planes are represented by their intersections with a unit
sphere having O as centre. This very useful, although unsymmetrical,
system of angular co-ordinates was introduced by L. Euler. It is exemplified
in “Cardan’s suspension,” as used in connexion with a compass-bowl or a
gyroscope. Thus in the gyroscope the “flywheel” (represented by the globe
in fig. 37) can turn about a diameter OC of a ring which is itself free to turn
about a diametral axis OX at right angles to the former; this axis is carried
by a second ring which is free to turn about a fixed diameter OZ, which is at
right angles to OX.
The six independent quantities, or “co-ordinates,” which serve to specify
the position of a rigid body in space may of course be chosen in an endless
variety of ways. We may, for instance, employ the three Cartesian co-
ordinates of a particular point O of the body, and three angular co-ordinates
which express the orientation of the body with respect to O. Thus in fig. 36,
if OA, OB, OC be three mutually perpendicular lines in the solid, we may
denote by θ the angle which OC makes with a fixed direction OZ, by ψ the
azimuth of the plane ZOC measured from some fixed plane through OZ, and
by φ the inclination of the plane COA to the plane ZOC. In fig. 36 these
various lines and planes are represented by their intersections with a unit
sphere having O as centre. This very useful, although unsymmetrical,
system of angular co-ordinates was introduced by L. Euler. It is exemplified
in “Cardan’s suspension,” as used in connexion with a compass-bowl or a
gyroscope. Thus in the gyroscope the “flywheel” (represented by the globe
in fig. 37) can turn about a diameter OC of a ring which is itself free to turn
about a diametral axis OX at right angles to the former; this axis is carried
by a second ring which is free to turn about a fixed diameter OZ, which is at
right angles to OX.
Page 374
We proceed to sketch the theory of
the finite displacements of a rigid body.
It was shown by Euler (1776) that any
displacement in which one point O of
the body is fixed is equivalent to a pure
rotation about some axis through O.
Imagine two spheres of equal radius
with O as their common centre, one
Fig. 10.
fixed in the body and moving with it,
the other fixed in space. In any
displacement about O as a fixed point,
the former sphere slides over the latter,
as in a “ball-and-socket” joint. Suppose
that as the result of the displacement a
point of the moving sphere is brought
Fig. 38.
from A to B, whilst the point which was
at B is brought to C (cf. fig. 10). Let J be
the pole of the circle ABC (usually a
“small circle” of the fixed sphere), and
join JA, JB, JC, AB, BC by great-circle
arcs. The spherical isosceles triangles
AJB, BJC are congruent, and we see Fig. 39.
that AB can be brought into the position
BC by a rotation about the axis OJ through an angle AJB.
It is convenient to distinguish the two senses in which rotation may take place
about an axis OA by opposite signs. We shall reckon a rotation as positive when
it is related to the direction from O to A as the direction of rotation is related to
that of translation in a right-handed screw. Thus a negative rotation about OA
may be regarded as a positive rotation about OA′, the prolongation of AO. Now
suppose that a body receives first a positive rotation α about OA, and secondly a
positive rotation β about OB; and let A, B be the intersections of these axes with
a sphere described about O as centre. If we construct the spherical triangles ABC,
the finite displacements of a rigid body.
It was shown by Euler (1776) that any
displacement in which one point O of
the body is fixed is equivalent to a pure
rotation about some axis through O.
Imagine two spheres of equal radius
with O as their common centre, one
Fig. 10.
fixed in the body and moving with it,
the other fixed in space. In any
displacement about O as a fixed point,
the former sphere slides over the latter,
as in a “ball-and-socket” joint. Suppose
that as the result of the displacement a
point of the moving sphere is brought
Fig. 38.
from A to B, whilst the point which was
at B is brought to C (cf. fig. 10). Let J be
the pole of the circle ABC (usually a
“small circle” of the fixed sphere), and
join JA, JB, JC, AB, BC by great-circle
arcs. The spherical isosceles triangles
AJB, BJC are congruent, and we see Fig. 39.
that AB can be brought into the position
BC by a rotation about the axis OJ through an angle AJB.
It is convenient to distinguish the two senses in which rotation may take place
about an axis OA by opposite signs. We shall reckon a rotation as positive when
it is related to the direction from O to A as the direction of rotation is related to
that of translation in a right-handed screw. Thus a negative rotation about OA
may be regarded as a positive rotation about OA′, the prolongation of AO. Now
suppose that a body receives first a positive rotation α about OA, and secondly a
positive rotation β about OB; and let A, B be the intersections of these axes with
a sphere described about O as centre. If we construct the spherical triangles ABC,
Page 375
ABC′ (fig. 38), having in each case the angles at A and B equal to 1⁄2α and 1⁄2β
respectively, it is evident that the first rotation will bring a point from C to C′ and
that the second will bring it back to C; the result is therefore equivalent to a
rotation about OC. We note also that if the given rotations had been effected in
the inverse order, the axis of the resultant rotation would have been OC′, so that
finite rotations do not obey the “commutative law.” To find the angle of the
equivalent rotation, in the actual case, suppose that the second rotation (about
OB) brings a point from A to A′. The spherical triangles ABC, A′BC (fig. 39) are
“symmetrically equal,” and the angle of the resultant rotation, viz. ACA′, is 2π −
2C. This is equivalent to a negative rotation 2C about OC, whence the theorem
that the effect of three successive positive rotations 2A, 2B, 2C about OA, OB,
OC, respectively, is to leave the body in its original position, provided the circuit
ABC is left-handed as seen from O. This theorem is due to O. Rodrigues (1840).
The composition of finite rotations about parallel axes is a particular case of the
preceding; the radius of the sphere is now infinite, and the triangles are plane.
In any continuous motion of a solid about a fixed point O, the limiting position
of the axis of the rotation by which the body can be brought from any one of its
positions to a consecutive one is called the instantaneous axis. This axis traces
out a certain cone in the body, and a certain cone in space, and the continuous
motion in question may be represented as consisting in a rolling of the former
cone on the latter. The proof is similar to that of the corresponding theorem of
plane kinematics (§ 3).
It follows from Euler’s theorem that the most general displacement of a rigid
body may be effected by a pure translation which brings any one point of it to its
final position O, followed by a pure rotation about some axis through O. Those
planes in the body which are perpendicular to this axis obviously remain parallel
to their original positions. Hence, if σ, σ′ denote the initial and final positions of
any figure in one of these planes, the displacement could evidently have been
effected by (1) a translation perpendicular to the planes in question, bringing σ
into some position σ″ in the plane of σ′, and (2) a rotation about a normal to the
planes, bringing σ″ into coincidence with σ (§ 3). In other words, the most
respectively, it is evident that the first rotation will bring a point from C to C′ and
that the second will bring it back to C; the result is therefore equivalent to a
rotation about OC. We note also that if the given rotations had been effected in
the inverse order, the axis of the resultant rotation would have been OC′, so that
finite rotations do not obey the “commutative law.” To find the angle of the
equivalent rotation, in the actual case, suppose that the second rotation (about
OB) brings a point from A to A′. The spherical triangles ABC, A′BC (fig. 39) are
“symmetrically equal,” and the angle of the resultant rotation, viz. ACA′, is 2π −
2C. This is equivalent to a negative rotation 2C about OC, whence the theorem
that the effect of three successive positive rotations 2A, 2B, 2C about OA, OB,
OC, respectively, is to leave the body in its original position, provided the circuit
ABC is left-handed as seen from O. This theorem is due to O. Rodrigues (1840).
The composition of finite rotations about parallel axes is a particular case of the
preceding; the radius of the sphere is now infinite, and the triangles are plane.
In any continuous motion of a solid about a fixed point O, the limiting position
of the axis of the rotation by which the body can be brought from any one of its
positions to a consecutive one is called the instantaneous axis. This axis traces
out a certain cone in the body, and a certain cone in space, and the continuous
motion in question may be represented as consisting in a rolling of the former
cone on the latter. The proof is similar to that of the corresponding theorem of
plane kinematics (§ 3).
It follows from Euler’s theorem that the most general displacement of a rigid
body may be effected by a pure translation which brings any one point of it to its
final position O, followed by a pure rotation about some axis through O. Those
planes in the body which are perpendicular to this axis obviously remain parallel
to their original positions. Hence, if σ, σ′ denote the initial and final positions of
any figure in one of these planes, the displacement could evidently have been
effected by (1) a translation perpendicular to the planes in question, bringing σ
into some position σ″ in the plane of σ′, and (2) a rotation about a normal to the
planes, bringing σ″ into coincidence with σ (§ 3). In other words, the most
Page 376
general displacement is equivalent to a translation parallel to a certain axis
combined with a rotation about that axis; i.e. it may be described as a twist about
a certain screw. In particular cases, of course, the translation, or the rotation, may
vanish.
The preceding theorem, which is due to Michel Chasles (1830), may be
proved in various other interesting ways. Thus if a point of the body be
displaced from A to B, whilst the point which was at B is displaced to C, and
that which was at C to D, the four points A, B, C, D lie on a helix whose
axis is the common perpendicular to the bisectors of the angles ABC, BCD.
This is the axis of the required screw; the amount of the translation is
measured by the projection of AB or BC or CD on the axis; and the angle of
rotation is given by the inclination of the aforesaid bisectors. This
construction was given by M. W. Crofton. Again, H. Wiener and W.
Burnside have employed the half-turn (i.e. a rotation through two right
angles) as the fundamental operation. This has the advantage that it is
completely specified by the axis of the rotation, the sense being immaterial.
Successive half-turns about parallel axes a, b are equivalent to a translation
measured by double the distance between these axes in the direction from a
to b. Successive half-turns about intersecting axes a, b are equivalent to a
rotation about the common perpendicular to a, b at their intersection, of
amount equal to twice the acute angle between them, in the direction from a
to b. Successive half-turns about two skew axes a, b are equivalent to a twist
about a screw whose axis is the common perpendicular to a, b, the
translation being double the shortest distance, and the angle of rotation
being twice the acute angle between a, b, in the direction from a to b. It is
easily shown that any displacement whatever is equivalent to two half-turns
and therefore to a screw.
In mechanics we are specially concerned with the theory of infinitesimal
displacements. This is included in the preceding, but it is simpler in that the
various operations are commutative. An infinitely small rotation about any axis is
conveniently represented geometrically by a length AB measures along the axis
combined with a rotation about that axis; i.e. it may be described as a twist about
a certain screw. In particular cases, of course, the translation, or the rotation, may
vanish.
The preceding theorem, which is due to Michel Chasles (1830), may be
proved in various other interesting ways. Thus if a point of the body be
displaced from A to B, whilst the point which was at B is displaced to C, and
that which was at C to D, the four points A, B, C, D lie on a helix whose
axis is the common perpendicular to the bisectors of the angles ABC, BCD.
This is the axis of the required screw; the amount of the translation is
measured by the projection of AB or BC or CD on the axis; and the angle of
rotation is given by the inclination of the aforesaid bisectors. This
construction was given by M. W. Crofton. Again, H. Wiener and W.
Burnside have employed the half-turn (i.e. a rotation through two right
angles) as the fundamental operation. This has the advantage that it is
completely specified by the axis of the rotation, the sense being immaterial.
Successive half-turns about parallel axes a, b are equivalent to a translation
measured by double the distance between these axes in the direction from a
to b. Successive half-turns about intersecting axes a, b are equivalent to a
rotation about the common perpendicular to a, b at their intersection, of
amount equal to twice the acute angle between them, in the direction from a
to b. Successive half-turns about two skew axes a, b are equivalent to a twist
about a screw whose axis is the common perpendicular to a, b, the
translation being double the shortest distance, and the angle of rotation
being twice the acute angle between a, b, in the direction from a to b. It is
easily shown that any displacement whatever is equivalent to two half-turns
and therefore to a screw.
In mechanics we are specially concerned with the theory of infinitesimal
displacements. This is included in the preceding, but it is simpler in that the
various operations are commutative. An infinitely small rotation about any axis is
conveniently represented geometrically by a length AB measures along the axis
Page 377
and proportional to the angle of rotation,
with the convention that the direction from A
to B shall be related to the rotation as is the
direction of translation to that of rotation in a
right-handed screw. The consequent
displacement of any point P will then be at
right angles to the plane PAB, its amount Fig. 16.
will be represented by double the area of the
triangle PAB, and its sense will depend on the cyclical order of the letters P, A, B.
If AB, AC represent infinitesimal rotations about intersecting axes, the
consequent displacement of any point O in the plane BAC will be at right angles
to this plane, and will be represented by twice the sum of the areas OAB, OAC,
taken with proper signs. It follows by analogy with the theory of moments (§ 4)
that the resultant rotation will be represented by AD, the vector-sum of AB, AC
(see fig. 16). It is easily inferred as a limiting case, or proved directly, that two
infinitesimal rotations α, β about parallel axes are equivalent to a rotation α + β
about a parallel axis in the same plane with the two former, and dividing a
common perpendicular AB in a point C so that AC/CB = β/α. If the rotations are
equal and opposite, so that α + β = 0, the point C is at infinity, and the effect is a
translation perpendicular to the plane of the two given axes, of amount α·AB. It
thus appears that an infinitesimal rotation is of the nature of a “localized vector,”
and is subject in all respects to the same mathematical laws as a force, conceived
as acting on a rigid body. Moreover, that an infinitesimal translation is analogous
to a couple and follows the same laws. These results are due to Poinsot.
The analytical treatment of small displacements is as follows. We first suppose
that one point O of the body is fixed, and take this as the origin of a “right-
handed” system of rectangular co-ordinates; i.e. the positive directions of the
axes are assumed to be so arranged that a positive rotation of 90° about Ox
would bring Oy into the position of Oz, and so on. The displacement will consist
of an infinitesimal rotation ε about some axis through O, whose direction-cosines
are, say, l, m, n. From the equivalence of a small rotation to a localized vector it
with the convention that the direction from A
to B shall be related to the rotation as is the
direction of translation to that of rotation in a
right-handed screw. The consequent
displacement of any point P will then be at
right angles to the plane PAB, its amount Fig. 16.
will be represented by double the area of the
triangle PAB, and its sense will depend on the cyclical order of the letters P, A, B.
If AB, AC represent infinitesimal rotations about intersecting axes, the
consequent displacement of any point O in the plane BAC will be at right angles
to this plane, and will be represented by twice the sum of the areas OAB, OAC,
taken with proper signs. It follows by analogy with the theory of moments (§ 4)
that the resultant rotation will be represented by AD, the vector-sum of AB, AC
(see fig. 16). It is easily inferred as a limiting case, or proved directly, that two
infinitesimal rotations α, β about parallel axes are equivalent to a rotation α + β
about a parallel axis in the same plane with the two former, and dividing a
common perpendicular AB in a point C so that AC/CB = β/α. If the rotations are
equal and opposite, so that α + β = 0, the point C is at infinity, and the effect is a
translation perpendicular to the plane of the two given axes, of amount α·AB. It
thus appears that an infinitesimal rotation is of the nature of a “localized vector,”
and is subject in all respects to the same mathematical laws as a force, conceived
as acting on a rigid body. Moreover, that an infinitesimal translation is analogous
to a couple and follows the same laws. These results are due to Poinsot.
The analytical treatment of small displacements is as follows. We first suppose
that one point O of the body is fixed, and take this as the origin of a “right-
handed” system of rectangular co-ordinates; i.e. the positive directions of the
axes are assumed to be so arranged that a positive rotation of 90° about Ox
would bring Oy into the position of Oz, and so on. The displacement will consist
of an infinitesimal rotation ε about some axis through O, whose direction-cosines
are, say, l, m, n. From the equivalence of a small rotation to a localized vector it
Page 378
follows that the rotation ε will be equivalent to rotations ξ, η, ζ about Ox, Oy, Oz,
respectively, provided
ξ = lε, η = mε, ζ = nε,
(1)
and we note that
ξ2 + η2 + ζ2 = ε2.
(2)
Thus in the case of fig. 36 it may be required to connect the infinitesimal
rotations ξ, η, ζ about OA, OB, OC with the variations of the angular co-
ordinates θ, ψ, φ. The displacement of the point C of the body is made up of
δθ tangential to the meridian ZC and sin θ δψ perpendicular to the plane of
this meridian. Hence, resolving along the tangents to the arcs BC, CA,
respectively, we have
ξ = δθ sin φ − sin θ δψ cos φ, η = δθ cos φ + sin θ δψ sin φ.
(3)
Again, consider the point of the solid
which was initially at A′ in the figure.
This is displaced relatively to A′ through
a space δψ perpendicular to the plane of
the meridian, whilst A′ itself is displaced
through a space cos θ δψ in the same
direction. Hence
ζ = δφ + cos θ δψ.
(4)
Fig. 40.
To find the component displacements of
a point P of the body, whose co-ordinates are x, y, z, we draw PL normal to the
plane yOz, and LH, LK perpendicular to Oy, Oz, respectively. The displacement
of P parallel to Ox is the same as that of L, which is made up of ηz and −ζy. In
this way we obtain the formulae
respectively, provided
ξ = lε, η = mε, ζ = nε,
(1)
and we note that
ξ2 + η2 + ζ2 = ε2.
(2)
Thus in the case of fig. 36 it may be required to connect the infinitesimal
rotations ξ, η, ζ about OA, OB, OC with the variations of the angular co-
ordinates θ, ψ, φ. The displacement of the point C of the body is made up of
δθ tangential to the meridian ZC and sin θ δψ perpendicular to the plane of
this meridian. Hence, resolving along the tangents to the arcs BC, CA,
respectively, we have
ξ = δθ sin φ − sin θ δψ cos φ, η = δθ cos φ + sin θ δψ sin φ.
(3)
Again, consider the point of the solid
which was initially at A′ in the figure.
This is displaced relatively to A′ through
a space δψ perpendicular to the plane of
the meridian, whilst A′ itself is displaced
through a space cos θ δψ in the same
direction. Hence
ζ = δφ + cos θ δψ.
(4)
Fig. 40.
To find the component displacements of
a point P of the body, whose co-ordinates are x, y, z, we draw PL normal to the
plane yOz, and LH, LK perpendicular to Oy, Oz, respectively. The displacement
of P parallel to Ox is the same as that of L, which is made up of ηz and −ζy. In
this way we obtain the formulae
Page 379
δx = ηz − ζy, δy = ζx − ξz, δz = ξy − ηx.
(5)
The most general case is derived from this by adding the component
displacements λ, μ, ν (say) of the point which was at O; thus
δx = λ + ηz − ζy,
δy = μ + ζx − ξz,
δz = ν + ξy − ηx.
(6)
The displacement is thus expressed in terms of the six independent quantities
ξ, η, ζ, λ, μ, ν. The points whose displacements are in the direction of the
resultant axis of rotation are determined by δx : δy : δz = ξ : η : ζ, or
(λ + ηz − ζy)/ξ = (μ + ζx − ξz)/η = (ν + ξy − ηx)/ζ.
(7)
These are the equations of a straight line, and the displacement is in fact
equivalent to a twist about a screw having this line as axis. The translation
parallel to this axis is
lδx + mδy + nδz = (λξ + μη + νζ)/ε.
(8)
The linear magnitude which measures the ratio of translation to rotation in a
screw is called the pitch. In the present case the pitch is
(λξ + μη + νζ) / (ξ2 + η2 + ζ2).
(9)
Since ξ2 + η2 + ζ2, or ε2, is necessarily an absolute invariant for all
transformations of the (rectangular) co-ordinate axes, we infer that λξ + μη + νζ
is also an absolute invariant. When the latter invariant, but not the former,
vanishes, the displacement is equivalent to a pure rotation.
If the small displacements of a rigid body be subject to one constraint,
e.g. if a point of the body be restricted to lie on a given surface, the
(5)
The most general case is derived from this by adding the component
displacements λ, μ, ν (say) of the point which was at O; thus
δx = λ + ηz − ζy,
δy = μ + ζx − ξz,
δz = ν + ξy − ηx.
(6)
The displacement is thus expressed in terms of the six independent quantities
ξ, η, ζ, λ, μ, ν. The points whose displacements are in the direction of the
resultant axis of rotation are determined by δx : δy : δz = ξ : η : ζ, or
(λ + ηz − ζy)/ξ = (μ + ζx − ξz)/η = (ν + ξy − ηx)/ζ.
(7)
These are the equations of a straight line, and the displacement is in fact
equivalent to a twist about a screw having this line as axis. The translation
parallel to this axis is
lδx + mδy + nδz = (λξ + μη + νζ)/ε.
(8)
The linear magnitude which measures the ratio of translation to rotation in a
screw is called the pitch. In the present case the pitch is
(λξ + μη + νζ) / (ξ2 + η2 + ζ2).
(9)
Since ξ2 + η2 + ζ2, or ε2, is necessarily an absolute invariant for all
transformations of the (rectangular) co-ordinate axes, we infer that λξ + μη + νζ
is also an absolute invariant. When the latter invariant, but not the former,
vanishes, the displacement is equivalent to a pure rotation.
If the small displacements of a rigid body be subject to one constraint,
e.g. if a point of the body be restricted to lie on a given surface, the
Page 380
mathematical expression of this fact leads to a homogeneous linear equation
between the infinitesimals ξ, η, ζ, λ, μ, ν, say
Aξ + Bη + Cζ + Fλ + Gμ + Hν = 0.
(10)
The quantities ξ, η, ζ, λ, μ, ν are no longer independent, and the body has
now only five degrees of freedom. Every additional constraint introduces an
additional equation of the type (10) and reduces the number of degrees of
freedom by one. In Sir R. S. Ball’s Theory of Screws an analysis is made of
the possible displacements of a body which has respectively two, three, four,
five degrees of freedom. We will briefly notice the case of two degrees,
which involves an interesting generalization of the method (already
explained) of compounding rotations about intersecting axes. We assume
that the body receives arbitrary twists about two given screws, and it is
required to determine the character of the resultant displacement. We
examine first the case where the axes of the two screws are at right angles
and intersect. We take these as axes of x and y; then if ξ, η be the component
rotations about them, we have
λ = hξ, μ = kη, ν = 0,
(11)
where h, k, are the pitches of the two given screws. The equations (7) of the
axis of the resultant screw then reduce to
x/ξ = y/η, z(ξ2 + η2) = (k − h) ξη.
(12)
Hence, whatever the ratio ξ : η, the axis of the resultant screw lies on the
conoidal surface
z (x2 + y2) = cxy,
(13)
where c = 1⁄2(k − h). The co-ordinates of any point on (13) may be written
between the infinitesimals ξ, η, ζ, λ, μ, ν, say
Aξ + Bη + Cζ + Fλ + Gμ + Hν = 0.
(10)
The quantities ξ, η, ζ, λ, μ, ν are no longer independent, and the body has
now only five degrees of freedom. Every additional constraint introduces an
additional equation of the type (10) and reduces the number of degrees of
freedom by one. In Sir R. S. Ball’s Theory of Screws an analysis is made of
the possible displacements of a body which has respectively two, three, four,
five degrees of freedom. We will briefly notice the case of two degrees,
which involves an interesting generalization of the method (already
explained) of compounding rotations about intersecting axes. We assume
that the body receives arbitrary twists about two given screws, and it is
required to determine the character of the resultant displacement. We
examine first the case where the axes of the two screws are at right angles
and intersect. We take these as axes of x and y; then if ξ, η be the component
rotations about them, we have
λ = hξ, μ = kη, ν = 0,
(11)
where h, k, are the pitches of the two given screws. The equations (7) of the
axis of the resultant screw then reduce to
x/ξ = y/η, z(ξ2 + η2) = (k − h) ξη.
(12)
Hence, whatever the ratio ξ : η, the axis of the resultant screw lies on the
conoidal surface
z (x2 + y2) = cxy,
(13)
where c = 1⁄2(k − h). The co-ordinates of any point on (13) may be written
Page 381
x = r cos θ, y = r sin θ, z = c sin 2θ;
(14)
hence if we imagine a curve of sines to be traced on a circular cylinder so
that the circumference just includes two complete undulations, a straight
line cutting the axis of the cylinder at right angles and meeting this curve
will generate the surface. This is called a cylindroid. Again, the pitch of the
resultant screw is
p = (λξ + μη) / (ξ2 + η2) = h cos2 θ + k sin2 θ.
(15)
From Sir Robert S. Ball’s Theory of Screws.
Fig. 41.
The distribution of pitch among the various screws has therefore a simple
relation to the pitch-conic
hx2 + ky2 = const;
(16)
(14)
hence if we imagine a curve of sines to be traced on a circular cylinder so
that the circumference just includes two complete undulations, a straight
line cutting the axis of the cylinder at right angles and meeting this curve
will generate the surface. This is called a cylindroid. Again, the pitch of the
resultant screw is
p = (λξ + μη) / (ξ2 + η2) = h cos2 θ + k sin2 θ.
(15)
From Sir Robert S. Ball’s Theory of Screws.
Fig. 41.
The distribution of pitch among the various screws has therefore a simple
relation to the pitch-conic
hx2 + ky2 = const;
(16)
Page 382
viz. the pitch of any screw varies inversely as the square of that diameter of
the conic which is parallel to its axis. It is to be noticed that the parameter c
of the cylindroid is unaltered if the two pitches h, k be increased by equal
amounts; the only change is that all the pitches are increased by the same
amount. It remains to show that a system of screws of the above type can be
constructed so as to contain any two given screws whatever. In the first
place, a cylindroid can be constructed so as to have its axis coincident with
the common perpendicular to the axes of the two given screws and to satisfy
three other conditions, for the position of the centre, the parameter, and the
orientation about the axis are still at our disposal. Hence we can adjust these
so that the surface shall contain the axes of the two given screws as
generators, and that the difference of the corresponding pitches shall have
the proper value. It follows that when a body has two degrees of freedom it
can twist about any one of a singly infinite system of screws whose axes lie
on a certain cylindroid. In particular cases the cylindroid may degenerate
into a plane, the pitches being then all equal.
§ 8. Three-dimensional Statics.—A system of parallel forces can be combined
two and two until they are replaced by a single resultant equal to their sum,
acting in a certain line. As special cases, the system may reduce to a couple, or it
may be in equilibrium.
In general, however, a three-dimensional system of forces cannot be replaced
by a single resultant force. But it may be reduced to simpler elements in a variety
of ways. For example, it may be reduced to two forces in perpendicular skew
lines. For consider any plane, and let each force, at its intersection with the plane,
be resolved into two components, one (P) normal to the plane, the other (Q) in
the plane. The assemblage of parallel forces P can be replaced in general by a
single force, and the coplanar system of forces Q by another single force.
If the plane in question be chosen perpendicular to the direction of the vector-
sum of the given forces, the vector-sum of the components Q is zero, and these
components are therefore equivalent to a couple (§ 4). Hence any three-
dimensional system can be reduced to a single force R acting in a certain line,
the conic which is parallel to its axis. It is to be noticed that the parameter c
of the cylindroid is unaltered if the two pitches h, k be increased by equal
amounts; the only change is that all the pitches are increased by the same
amount. It remains to show that a system of screws of the above type can be
constructed so as to contain any two given screws whatever. In the first
place, a cylindroid can be constructed so as to have its axis coincident with
the common perpendicular to the axes of the two given screws and to satisfy
three other conditions, for the position of the centre, the parameter, and the
orientation about the axis are still at our disposal. Hence we can adjust these
so that the surface shall contain the axes of the two given screws as
generators, and that the difference of the corresponding pitches shall have
the proper value. It follows that when a body has two degrees of freedom it
can twist about any one of a singly infinite system of screws whose axes lie
on a certain cylindroid. In particular cases the cylindroid may degenerate
into a plane, the pitches being then all equal.
§ 8. Three-dimensional Statics.—A system of parallel forces can be combined
two and two until they are replaced by a single resultant equal to their sum,
acting in a certain line. As special cases, the system may reduce to a couple, or it
may be in equilibrium.
In general, however, a three-dimensional system of forces cannot be replaced
by a single resultant force. But it may be reduced to simpler elements in a variety
of ways. For example, it may be reduced to two forces in perpendicular skew
lines. For consider any plane, and let each force, at its intersection with the plane,
be resolved into two components, one (P) normal to the plane, the other (Q) in
the plane. The assemblage of parallel forces P can be replaced in general by a
single force, and the coplanar system of forces Q by another single force.
If the plane in question be chosen perpendicular to the direction of the vector-
sum of the given forces, the vector-sum of the components Q is zero, and these
components are therefore equivalent to a couple (§ 4). Hence any three-
dimensional system can be reduced to a single force R acting in a certain line,
Page 383
together with a couple G in a plane perpendicular to the line. This theorem was
first given by L. Poinsot, and the line of action of R was called by him the
central axis of the system. The combination of a force and a couple in a
perpendicular plane is termed by Sir R. S. Ball a wrench. Its type, as
distinguished from its absolute magnitude, may be specified by a screw whose
axis is the line of action of R, and whose pitch is the ratio G/R.
The case of two forces may be
specially noticed. Let AB be the
shortest distance between the lines of
action, and let AA′, BB′ (fig. 42)
represent the forces. Let α, β be the
angles which AA′, BB′ make with
the direction of the vector-sum, on
opposite sides. Divide AB in O, so Fig. 42.
that
AA′ · cos α · AO = BB′ · cos β · OB,
(1)
and draw OC parallel to the vector-sum. Resolving AA′, BB′ each into two
components parallel and perpendicular to OC, we see that the former
components have a single resultant in OC, of amount
R = AA′ cos α + BB′ cos β,
(2)
whilst the latter components form a couple of moment
G = AA′ · AB · sin α = BB′ · AB · sin β.
(3)
Conversely it is seen that any wrench can be replaced in an infinite number
of ways by two forces, and that the line of action of one of these may be
chosen quite arbitrarily. Also, we find from (2) and (3) that
G · R = AA′ · BB′ · AB · sin (α + β).
first given by L. Poinsot, and the line of action of R was called by him the
central axis of the system. The combination of a force and a couple in a
perpendicular plane is termed by Sir R. S. Ball a wrench. Its type, as
distinguished from its absolute magnitude, may be specified by a screw whose
axis is the line of action of R, and whose pitch is the ratio G/R.
The case of two forces may be
specially noticed. Let AB be the
shortest distance between the lines of
action, and let AA′, BB′ (fig. 42)
represent the forces. Let α, β be the
angles which AA′, BB′ make with
the direction of the vector-sum, on
opposite sides. Divide AB in O, so Fig. 42.
that
AA′ · cos α · AO = BB′ · cos β · OB,
(1)
and draw OC parallel to the vector-sum. Resolving AA′, BB′ each into two
components parallel and perpendicular to OC, we see that the former
components have a single resultant in OC, of amount
R = AA′ cos α + BB′ cos β,
(2)
whilst the latter components form a couple of moment
G = AA′ · AB · sin α = BB′ · AB · sin β.
(3)
Conversely it is seen that any wrench can be replaced in an infinite number
of ways by two forces, and that the line of action of one of these may be
chosen quite arbitrarily. Also, we find from (2) and (3) that
G · R = AA′ · BB′ · AB · sin (α + β).
Page 384
(4)
The right-hand expression is six times the volume of the tetrahedron of
which the lines AA′, BB′ representing the forces are opposite edges; and we
infer that, in whatever way the wrench be resolved into two forces, the
volume of this tetrahedron is invariable.
To define the moment of a force about an axis HK, we project the force
orthogonally on a plane perpendicular to HK and take the moment of the
projection about the intersection of HK with the plane (see § 4). Some
convention as to sign is necessary; we shall reckon the moment to be positive
when the tendency of the force is right-handed as regards the direction from H to
K. Since two concurrent forces and their resultant obviously project into two
concurrent forces and their resultant, we see that the sum of the moments of two
concurrent forces about any axis HK is equal to the moment of their resultant.
Parallel forces may be included in this statement as a limiting case. Hence, in
whatever way one system of forces is by successive steps replaced by another, no
change is made in the sum of the moments about any assigned axis. By means of
this theorem we can show that the previous reduction of any system to a wrench
is unique.
From the analogy of couples to translations which was pointed out in § 7, we
may infer that a couple is sufficiently represented by a “free” (or non-localized)
vector perpendicular to its plane. The length of the vector must be proportional to
the moment of the couple, and its sense must be such that the sum of the
moments of the two forces of the couple about it is positive. In particular, we
infer that couples of the same moment in parallel planes are equivalent; and that
couples in any two planes may be compounded by geometrical addition of the
corresponding vectors. Independent statical proofs are of course easily given.
Thus, let the plane of the paper be perpendicular to the planes of two couples,
and therefore perpendicular to the line of intersection of these planes. By § 4,
each couple can be replaced by two forces ±P (fig. 43) perpendicular to the plane
of the paper, and so that one force of each couple is in the line of intersection (B);
the arms (AB, BC) will then be proportional to the respective moments. The two
The right-hand expression is six times the volume of the tetrahedron of
which the lines AA′, BB′ representing the forces are opposite edges; and we
infer that, in whatever way the wrench be resolved into two forces, the
volume of this tetrahedron is invariable.
To define the moment of a force about an axis HK, we project the force
orthogonally on a plane perpendicular to HK and take the moment of the
projection about the intersection of HK with the plane (see § 4). Some
convention as to sign is necessary; we shall reckon the moment to be positive
when the tendency of the force is right-handed as regards the direction from H to
K. Since two concurrent forces and their resultant obviously project into two
concurrent forces and their resultant, we see that the sum of the moments of two
concurrent forces about any axis HK is equal to the moment of their resultant.
Parallel forces may be included in this statement as a limiting case. Hence, in
whatever way one system of forces is by successive steps replaced by another, no
change is made in the sum of the moments about any assigned axis. By means of
this theorem we can show that the previous reduction of any system to a wrench
is unique.
From the analogy of couples to translations which was pointed out in § 7, we
may infer that a couple is sufficiently represented by a “free” (or non-localized)
vector perpendicular to its plane. The length of the vector must be proportional to
the moment of the couple, and its sense must be such that the sum of the
moments of the two forces of the couple about it is positive. In particular, we
infer that couples of the same moment in parallel planes are equivalent; and that
couples in any two planes may be compounded by geometrical addition of the
corresponding vectors. Independent statical proofs are of course easily given.
Thus, let the plane of the paper be perpendicular to the planes of two couples,
and therefore perpendicular to the line of intersection of these planes. By § 4,
each couple can be replaced by two forces ±P (fig. 43) perpendicular to the plane
of the paper, and so that one force of each couple is in the line of intersection (B);
the arms (AB, BC) will then be proportional to the respective moments. The two
Page 385
forces at B will cancel, and we are left with a couple of moment P·AC in the
plane AC. If we draw three vectors to represent these three couples, they will be
perpendicular and proportional to the respective sides of the triangle ABC; hence
the third vector is the geometric sum of the other two. Since, in this proof the
magnitude of P is arbitrary, It follows incidentally that couples of the same
moment in parallel planes, e.g. planes parallel to AC, are equivalent.
Fig. 43.
Hence a couple of moment G,
whose axis has the direction (l, m,
n) relative to a right-handed system
of rectangular axes, is equivalent to
three couples lG, mG, nG in the co-
ordinate planes. The analytical
reduction of a three-dimensional
system can now be conducted as
follows. Let (x1, y1, z1) be the co- Fig. 44.
ordinates of a point P1 on the line of
action of one of the forces, whose components are (say) X1, Y1, Z1. Draw P1H
normal to the plane zOx, and HK perpendicular to Oz. In KH introduce two
equal and opposite forces ±X1. The force X1 at P1 with −X1 in KH forms a
plane AC. If we draw three vectors to represent these three couples, they will be
perpendicular and proportional to the respective sides of the triangle ABC; hence
the third vector is the geometric sum of the other two. Since, in this proof the
magnitude of P is arbitrary, It follows incidentally that couples of the same
moment in parallel planes, e.g. planes parallel to AC, are equivalent.
Fig. 43.
Hence a couple of moment G,
whose axis has the direction (l, m,
n) relative to a right-handed system
of rectangular axes, is equivalent to
three couples lG, mG, nG in the co-
ordinate planes. The analytical
reduction of a three-dimensional
system can now be conducted as
follows. Let (x1, y1, z1) be the co- Fig. 44.
ordinates of a point P1 on the line of
action of one of the forces, whose components are (say) X1, Y1, Z1. Draw P1H
normal to the plane zOx, and HK perpendicular to Oz. In KH introduce two
equal and opposite forces ±X1. The force X1 at P1 with −X1 in KH forms a
Page 386
couple about Oz, of moment −y1X1. Next, introduce along Ox two equal and
opposite forces ±X1. The force X1 in KH with −X1 in Ox forms a couple about
Oy, of moment z1X1. Hence the force X1 can be transferred from P1 to O,
provided we introduce couples of moments z1X1 about Oy and −y1X1, about Oz.
Dealing in the same way with the forces Y1, Z1 at P1, we find that all three
components of the force at P1 can be transferred to O, provided we introduce
three couples L1, M1, N1 about Ox, Oy, Oz respectively, viz.
L1 = y1Z1 − z1Y1, M1 = z1X1 − x1Z1, N1 = x1Y1 − y1X1.
(5)
It is seen that L1, M1, N1 are the moments of the original force at P1 about the co-
ordinate axes. Summing up for all the forces of the given system, we obtain a
force R at O, whose components are
X = Σ(Xr), Y = Σ(Yr), Z = Σ(Zr),
(6)
and a couple G whose components are
L = Σ(Lr), M = Σ(Mr), N = Σ(Nr),
(7)
where r = 1, 2, 3 ... Since R2 = X2 + Y2 + Z2, G2 = L2 + M2 + N2, it is necessary
and sufficient for equilibrium that the six quantities X, Y, Z, L, M, N, should all
vanish. In words: the sum of the projections of the forces on each of the co-
ordinate axes must vanish; and, the sum of the moments of the forces about each
of these axes must vanish.
If any other point O′, whose co-ordinates are x, y, z, be chosen in place of O,
as the point to which the forces are transferred, we have to write x1 − x, y1 − y, z1
− z for x1, y1, z1, and so on, in the preceding process. The components of the
resultant force R are unaltered, but the new components of couple are found to be
L′ = L − yZ + zY,
M′ = M − zX + xZ,
opposite forces ±X1. The force X1 in KH with −X1 in Ox forms a couple about
Oy, of moment z1X1. Hence the force X1 can be transferred from P1 to O,
provided we introduce couples of moments z1X1 about Oy and −y1X1, about Oz.
Dealing in the same way with the forces Y1, Z1 at P1, we find that all three
components of the force at P1 can be transferred to O, provided we introduce
three couples L1, M1, N1 about Ox, Oy, Oz respectively, viz.
L1 = y1Z1 − z1Y1, M1 = z1X1 − x1Z1, N1 = x1Y1 − y1X1.
(5)
It is seen that L1, M1, N1 are the moments of the original force at P1 about the co-
ordinate axes. Summing up for all the forces of the given system, we obtain a
force R at O, whose components are
X = Σ(Xr), Y = Σ(Yr), Z = Σ(Zr),
(6)
and a couple G whose components are
L = Σ(Lr), M = Σ(Mr), N = Σ(Nr),
(7)
where r = 1, 2, 3 ... Since R2 = X2 + Y2 + Z2, G2 = L2 + M2 + N2, it is necessary
and sufficient for equilibrium that the six quantities X, Y, Z, L, M, N, should all
vanish. In words: the sum of the projections of the forces on each of the co-
ordinate axes must vanish; and, the sum of the moments of the forces about each
of these axes must vanish.
If any other point O′, whose co-ordinates are x, y, z, be chosen in place of O,
as the point to which the forces are transferred, we have to write x1 − x, y1 − y, z1
− z for x1, y1, z1, and so on, in the preceding process. The components of the
resultant force R are unaltered, but the new components of couple are found to be
L′ = L − yZ + zY,
M′ = M − zX + xZ,
Page 387
N′ = N − xY + yX.
(8)
By properly choosing O′ we can make the plane of the couple perpendicular to
the resultant force. The conditions for this are L′ : M′ : N′ = X : Y : Z, or
L − yZ + zY M − zX + xZ N − xY + yX
= =
X Y Z (9)
These are the equations of the central axis. Since the moment of the resultant
couple is now
X Y Z LX + MY + NZ
G′ = L′ + M′ + N′ = ,
R R R R (10)
the pitch of the equivalent wrench is
(LX + MY + NZ) / (X2 + Y2 + Z2).
It appears that X2 + Y2 + Z2 and LX + MY + NZ are absolute invariants (cf. § 7).
When the latter invariant, but not the former, vanishes, the system reduces to a
single force.
The analogy between the mathematical relations of infinitely small
displacements on the one hand and those of force-systems on the other enables us
immediately to convert any theorem in the one subject into a theorem in the
other. For example, we can assert without further proof that any infinitely small
displacement may be resolved into two rotations, and that the axis of one of these
can be chosen arbitrarily. Again, that wrenches of arbitrary amounts about two
given screws compound into a wrench the locus of whose axis is a cylindroid.
The mathematical properties of a twist or of a wrench have been the
subject of many remarkable investigations, which are, however, of
secondary importance from a physical point of view. In the “Null-System”
of A. F. Möbius (1790-1868), a line such that the moment of a given wrench
about it is zero is called a null-line. The triply infinite system of null-lines
form what is called in line-geometry a “complex.” As regards the
(8)
By properly choosing O′ we can make the plane of the couple perpendicular to
the resultant force. The conditions for this are L′ : M′ : N′ = X : Y : Z, or
L − yZ + zY M − zX + xZ N − xY + yX
= =
X Y Z (9)
These are the equations of the central axis. Since the moment of the resultant
couple is now
X Y Z LX + MY + NZ
G′ = L′ + M′ + N′ = ,
R R R R (10)
the pitch of the equivalent wrench is
(LX + MY + NZ) / (X2 + Y2 + Z2).
It appears that X2 + Y2 + Z2 and LX + MY + NZ are absolute invariants (cf. § 7).
When the latter invariant, but not the former, vanishes, the system reduces to a
single force.
The analogy between the mathematical relations of infinitely small
displacements on the one hand and those of force-systems on the other enables us
immediately to convert any theorem in the one subject into a theorem in the
other. For example, we can assert without further proof that any infinitely small
displacement may be resolved into two rotations, and that the axis of one of these
can be chosen arbitrarily. Again, that wrenches of arbitrary amounts about two
given screws compound into a wrench the locus of whose axis is a cylindroid.
The mathematical properties of a twist or of a wrench have been the
subject of many remarkable investigations, which are, however, of
secondary importance from a physical point of view. In the “Null-System”
of A. F. Möbius (1790-1868), a line such that the moment of a given wrench
about it is zero is called a null-line. The triply infinite system of null-lines
form what is called in line-geometry a “complex.” As regards the
Page 388
configuration of this complex, consider a line whose shortest distance from
the central axis is r, and whose inclination to the central axis is θ. The
moment of the resultant force R of the wrench about this line is − Rr sin θ,
and that of the couple G is G cos θ. Hence the line will be a null-line
provided
tan θ = k/r,
(11)
where k is the pitch of the wrench. The null-lines which are at a given
distance r from a point O of the central axis will therefore form one system
of generators of a hyperboloid of revolution; and by varying r we get a
series of such hyperboloids with a common centre and axis. By moving O
along the central axis we obtain the whole complex of null-lines. It appears
also from (11) that the null-lines whose distance from the central axis is r
are tangent lines to a system of helices of slope tan−1 (r/k); and it is to be
noticed that these helices are left-handed if the given wrench is right-
handed, and vice versa.
Since the given wrench can be replaced by a force acting through any
assigned point P, and a couple, the locus of the null-lines through P is a
plane, viz. a plane perpendicular to the vector which represents the couple.
The complex is therefore of the type called “linear” (in relation to the degree
of this locus). The plane in question is called the null-plane of P. If the null-
plane of P pass through Q, the null-plane of Q will pass through P, since PQ
is a null-line. Again, any plane ω is the locus of a system of null-lines
meeting in a point, called the null-point of ω. If a plane revolve about a
fixed straight line p in it, its null-point describes another straight line p′,
which is called the conjugate line of p. We have seen that the wrench may
be replaced by two forces, one of which may act in any arbitrary line p. It is
now evident that the second force must act in the conjugate line p′, since
every line meeting p, p′ is a null-line. Again, since the shortest distance
between any two conjugate lines cuts the central axis at right angles, the
orthogonal projections of two conjugate lines on a plane perpendicular to
the central axis is r, and whose inclination to the central axis is θ. The
moment of the resultant force R of the wrench about this line is − Rr sin θ,
and that of the couple G is G cos θ. Hence the line will be a null-line
provided
tan θ = k/r,
(11)
where k is the pitch of the wrench. The null-lines which are at a given
distance r from a point O of the central axis will therefore form one system
of generators of a hyperboloid of revolution; and by varying r we get a
series of such hyperboloids with a common centre and axis. By moving O
along the central axis we obtain the whole complex of null-lines. It appears
also from (11) that the null-lines whose distance from the central axis is r
are tangent lines to a system of helices of slope tan−1 (r/k); and it is to be
noticed that these helices are left-handed if the given wrench is right-
handed, and vice versa.
Since the given wrench can be replaced by a force acting through any
assigned point P, and a couple, the locus of the null-lines through P is a
plane, viz. a plane perpendicular to the vector which represents the couple.
The complex is therefore of the type called “linear” (in relation to the degree
of this locus). The plane in question is called the null-plane of P. If the null-
plane of P pass through Q, the null-plane of Q will pass through P, since PQ
is a null-line. Again, any plane ω is the locus of a system of null-lines
meeting in a point, called the null-point of ω. If a plane revolve about a
fixed straight line p in it, its null-point describes another straight line p′,
which is called the conjugate line of p. We have seen that the wrench may
be replaced by two forces, one of which may act in any arbitrary line p. It is
now evident that the second force must act in the conjugate line p′, since
every line meeting p, p′ is a null-line. Again, since the shortest distance
between any two conjugate lines cuts the central axis at right angles, the
orthogonal projections of two conjugate lines on a plane perpendicular to
Page 389
the central axis will be parallel (fig. 42). This property was employed by L.
Cremona to prove the existence under certain conditions of “reciprocal
figures” in a plane (§ 5). If we take any polyhedron with plane faces, the
null-planes of its vertices with respect to a given wrench will form another
polyhedron, and the edges of the latter will be conjugate (in the above sense)
to those of the former. Projecting orthogonally on a plane perpendicular to
the central axis we obtain two reciprocal figures.
In the analogous theory of infinitely small displacements of a solid, a
“null-line” is a line such that the lengthwise displacement of any point on it
is zero.
Since a wrench is defined by six independent quantities, it can in general
be replaced by any system of forces which involves six adjustable elements.
For instance, it can in general be replaced by six forces acting in six given
lines, e.g. in the six edges of a given tetrahedron. An exception to the
general statement occurs when the six lines are such that they are possible
lines of action of a system of six forces in equilibrium; they are then said to
be in involution. The theory of forces in involution has been studied by A.
Cayley, J. J. Sylvester and others. We have seen that a rigid structure may in
general be rigidly connected with the earth by six links, and it now appears
that any system of forces acting on the structure can in general be balanced
by six determinate forces exerted by the links. If, however, the links are in
involution, these forces become infinite or indeterminate. There is a
corresponding kinematic peculiarity, in that the connexion is now not
strictly rigid, an infinitely small relative displacement being possible. See §
9.
When parallel forces of given magnitudes act at given points, the resultant acts
through a definite point, or centre of parallel forces, which is independent of the
special direction of the forces. If Pr be the force at (xr, yr, zr), acting in the
direction (l, m, n), the formulae (6) and (7) reduce to
X = Σ(P)·l, Y = Σ(P)·m, Z = Σ(P)·n,
(12)
Cremona to prove the existence under certain conditions of “reciprocal
figures” in a plane (§ 5). If we take any polyhedron with plane faces, the
null-planes of its vertices with respect to a given wrench will form another
polyhedron, and the edges of the latter will be conjugate (in the above sense)
to those of the former. Projecting orthogonally on a plane perpendicular to
the central axis we obtain two reciprocal figures.
In the analogous theory of infinitely small displacements of a solid, a
“null-line” is a line such that the lengthwise displacement of any point on it
is zero.
Since a wrench is defined by six independent quantities, it can in general
be replaced by any system of forces which involves six adjustable elements.
For instance, it can in general be replaced by six forces acting in six given
lines, e.g. in the six edges of a given tetrahedron. An exception to the
general statement occurs when the six lines are such that they are possible
lines of action of a system of six forces in equilibrium; they are then said to
be in involution. The theory of forces in involution has been studied by A.
Cayley, J. J. Sylvester and others. We have seen that a rigid structure may in
general be rigidly connected with the earth by six links, and it now appears
that any system of forces acting on the structure can in general be balanced
by six determinate forces exerted by the links. If, however, the links are in
involution, these forces become infinite or indeterminate. There is a
corresponding kinematic peculiarity, in that the connexion is now not
strictly rigid, an infinitely small relative displacement being possible. See §
9.
When parallel forces of given magnitudes act at given points, the resultant acts
through a definite point, or centre of parallel forces, which is independent of the
special direction of the forces. If Pr be the force at (xr, yr, zr), acting in the
direction (l, m, n), the formulae (6) and (7) reduce to
X = Σ(P)·l, Y = Σ(P)·m, Z = Σ(P)·n,
(12)
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and
L = Σ(P)·(ny − mz), M = Σ(P)·(lz − nx), N = Σ(P)·(mx − ly),
(13)
provided
Σ(Px) Σ(Py) Σ(Pz)
x= , y= , z= .
Σ(P) Σ(P) Σ(P) (14)
These are the same as if we had a single force Σ(P) acting at the point (x, y, z),
which is the same for all directions (l, m, n). We can hence derive the theory of
the centre of gravity, as in § 4. An exceptional case occurs when Σ(P) = 0.
If we imagine a rigid body to be acted on at given points by forces of
given magnitudes in directions (not all parallel) which are fixed in space,
then as the body is turned about the resultant wrench will assume different
configurations in the body, and will in certain positions reduce to a single
force. The investigation of such questions forms the subject of “Astatics,”
which has been cultivated by Möbius, Minding, G. Darboux and others. As
it has no physical bearing it is passed over here.
Fig. 45.
§ 9. Work.—The work done by a force acting on a particle, in any infinitely
small displacement, is defined as the product of the force into the orthogonal
projection of the displacement on the direction of the force; i.e. it is equal to F·δs
cos θ, where F is the force, δs the displacement, and θ is the angle between the
directions of F and δs. In the language of vector analysis (q.v.) it is the “scalar
product” of the vector representing the force and the displacement. In the same
way, the work done by a force acting on a rigid body in any infinitely small
displacement of the body is the scalar product of the force into the displacement
L = Σ(P)·(ny − mz), M = Σ(P)·(lz − nx), N = Σ(P)·(mx − ly),
(13)
provided
Σ(Px) Σ(Py) Σ(Pz)
x= , y= , z= .
Σ(P) Σ(P) Σ(P) (14)
These are the same as if we had a single force Σ(P) acting at the point (x, y, z),
which is the same for all directions (l, m, n). We can hence derive the theory of
the centre of gravity, as in § 4. An exceptional case occurs when Σ(P) = 0.
If we imagine a rigid body to be acted on at given points by forces of
given magnitudes in directions (not all parallel) which are fixed in space,
then as the body is turned about the resultant wrench will assume different
configurations in the body, and will in certain positions reduce to a single
force. The investigation of such questions forms the subject of “Astatics,”
which has been cultivated by Möbius, Minding, G. Darboux and others. As
it has no physical bearing it is passed over here.
Fig. 45.
§ 9. Work.—The work done by a force acting on a particle, in any infinitely
small displacement, is defined as the product of the force into the orthogonal
projection of the displacement on the direction of the force; i.e. it is equal to F·δs
cos θ, where F is the force, δs the displacement, and θ is the angle between the
directions of F and δs. In the language of vector analysis (q.v.) it is the “scalar
product” of the vector representing the force and the displacement. In the same
way, the work done by a force acting on a rigid body in any infinitely small
displacement of the body is the scalar product of the force into the displacement
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of any point on the line of action. This product is the same whatever point on the
line of action be taken, since the lengthwise components of the displacements of
any two points A, B on a line AB are equal, to the first order of small quantities.
To see this, let A′, B′ be the displaced positions of A, B, and let φ be the infinitely
small angle between AB and A′B′. Then if α, β be the orthogonal projections of
A′, B′ on AB, we have
Aα − Bβ = AB − αβ = AB (1 − cos φ) = 1⁄2AB·φ2,
ultimately. Since this is of the second order, the products F·Aα and F·Bβ are
ultimately equal.
Fig. 46. Fig. 47.
The total work done by two concurrent forces acting on a particle, or on a rigid
body, in any infinitely small displacement, is equal to the work of their resultant.
Let AB, AC (fig. 46) represent the forces, AD their resultant, and let AH be the
direction of the displacement δs of the point A. The proposition follows at once
from the fact that the sum of orthogonal projections of AB>, AC> on AH is equal to
the projection of AD>. It is to be noticed that AH need not be in the same plane
with AB, AC.
It follows from the preceding statements that any two systems of forces which
are statically equivalent, according to the principles of §§ 4, 8, will (to the first
order of small quantities) do the same amount of work in any infinitely small
displacement of a rigid body to which they may be applied. It is also evident that
line of action be taken, since the lengthwise components of the displacements of
any two points A, B on a line AB are equal, to the first order of small quantities.
To see this, let A′, B′ be the displaced positions of A, B, and let φ be the infinitely
small angle between AB and A′B′. Then if α, β be the orthogonal projections of
A′, B′ on AB, we have
Aα − Bβ = AB − αβ = AB (1 − cos φ) = 1⁄2AB·φ2,
ultimately. Since this is of the second order, the products F·Aα and F·Bβ are
ultimately equal.
Fig. 46. Fig. 47.
The total work done by two concurrent forces acting on a particle, or on a rigid
body, in any infinitely small displacement, is equal to the work of their resultant.
Let AB, AC (fig. 46) represent the forces, AD their resultant, and let AH be the
direction of the displacement δs of the point A. The proposition follows at once
from the fact that the sum of orthogonal projections of AB>, AC> on AH is equal to
the projection of AD>. It is to be noticed that AH need not be in the same plane
with AB, AC.
It follows from the preceding statements that any two systems of forces which
are statically equivalent, according to the principles of §§ 4, 8, will (to the first
order of small quantities) do the same amount of work in any infinitely small
displacement of a rigid body to which they may be applied. It is also evident that
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the total work done in two or more successive infinitely small displacements is
equal to the work done in the resultant displacement.
The work of a couple in any infinitely small rotation of a rigid body about an
axis perpendicular to the plane of the couple is equal to the product of the
moment of the couple into the angle of rotation, proper conventions as to sign
being observed. Let the couple consist of two forces P, P (fig. 47) in the plane of
the paper, and let J be the point where this plane is met by the axis of rotation.
Draw JBA perpendicular to the lines of action, and let ε be the angle of rotation.
The work of the couple is
P·JA·ε − P·JB·ε = P·AB·ε = Gε,
if G be the moment of the couple.
The analytical calculation of the work done by a system of forces in any
infinitesimal displacement is as follows. For a two-dimensional system we have,
in the notation of §§ 3, 4,
Σ(Xδx + Yδy) = Σ{X(λ − yε) + Y(μ + xε)}
= Σ(X)·λ + Σ(Y)·μ + Σ(xY − yX) ε
= Xλ + Yμ + Nε.
(1)
Again, for a three-dimensional system, in the notation of §§ 7, 8,
Σ(Xδx + Yδy + Zδz)
= Σ{(X(λ + ηz − ζy) + Y(μ + ζx − ξx) + Z(ν + ξy − ηx)}
= Σ(X)·λ + Σ(Y)·μ + Σ(Z)·ν + Σ(yZ − zY)·ξ + Σ(zX − xZ)·η + Σ(xY − yX)·ζ
= Xλ + Yμ + Zν + Lξ + Mη + Nζ.
(2)
This expression gives the work done by a given wrench when the body receives a
given infinitely small twist; it must of course be an absolute invariant for all
equal to the work done in the resultant displacement.
The work of a couple in any infinitely small rotation of a rigid body about an
axis perpendicular to the plane of the couple is equal to the product of the
moment of the couple into the angle of rotation, proper conventions as to sign
being observed. Let the couple consist of two forces P, P (fig. 47) in the plane of
the paper, and let J be the point where this plane is met by the axis of rotation.
Draw JBA perpendicular to the lines of action, and let ε be the angle of rotation.
The work of the couple is
P·JA·ε − P·JB·ε = P·AB·ε = Gε,
if G be the moment of the couple.
The analytical calculation of the work done by a system of forces in any
infinitesimal displacement is as follows. For a two-dimensional system we have,
in the notation of §§ 3, 4,
Σ(Xδx + Yδy) = Σ{X(λ − yε) + Y(μ + xε)}
= Σ(X)·λ + Σ(Y)·μ + Σ(xY − yX) ε
= Xλ + Yμ + Nε.
(1)
Again, for a three-dimensional system, in the notation of §§ 7, 8,
Σ(Xδx + Yδy + Zδz)
= Σ{(X(λ + ηz − ζy) + Y(μ + ζx − ξx) + Z(ν + ξy − ηx)}
= Σ(X)·λ + Σ(Y)·μ + Σ(Z)·ν + Σ(yZ − zY)·ξ + Σ(zX − xZ)·η + Σ(xY − yX)·ζ
= Xλ + Yμ + Zν + Lξ + Mη + Nζ.
(2)
This expression gives the work done by a given wrench when the body receives a
given infinitely small twist; it must of course be an absolute invariant for all
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transformations of rectangular axes. The first three terms express the work done
by the components of a force (X, Y, Z) acting at O, and the remaining three terms
express the work of a couple (L, M, N).
Fig. 48.
The work done by a wrench about a given screw, when the body twists
about a second given screw, may be calculated directly as follows. In fig. 48
let R, G be the force and couple of the wrench, ε,τ the rotation and
translation in the twist. Let the axes of the wrench and the twist be inclined
at an angle θ, and let h be the shortest distance between them. The
displacement of the point H in the figure, resolved in the direction of R, is τ
cos θ − εh sin θ. The work is therefore
R (τ cos θ − εh sin θ) + G cos θ
= Rε {(p + p′) cos θ − h sin θ},
(3)
if G = pR, τ = p′ε, i.e. p, p′ are the pitches of the two screws. The factor (p +
p′) cos θ − h sin θ is called the virtual coefficient of the two screws which
define the types of the wrench and twist, respectively.
A screw is determined by its axis and its pitch, and therefore involves five
Independent elements. These may be, for instance, the five ratios ξ : η : ζ : λ
: μ : ν of the six quantities which specify an infinitesimal twist about the
screw. If the twist is a pure rotation, these quantities are subject to the
relation
λξ + μη + νζ = 0.
(4)
by the components of a force (X, Y, Z) acting at O, and the remaining three terms
express the work of a couple (L, M, N).
Fig. 48.
The work done by a wrench about a given screw, when the body twists
about a second given screw, may be calculated directly as follows. In fig. 48
let R, G be the force and couple of the wrench, ε,τ the rotation and
translation in the twist. Let the axes of the wrench and the twist be inclined
at an angle θ, and let h be the shortest distance between them. The
displacement of the point H in the figure, resolved in the direction of R, is τ
cos θ − εh sin θ. The work is therefore
R (τ cos θ − εh sin θ) + G cos θ
= Rε {(p + p′) cos θ − h sin θ},
(3)
if G = pR, τ = p′ε, i.e. p, p′ are the pitches of the two screws. The factor (p +
p′) cos θ − h sin θ is called the virtual coefficient of the two screws which
define the types of the wrench and twist, respectively.
A screw is determined by its axis and its pitch, and therefore involves five
Independent elements. These may be, for instance, the five ratios ξ : η : ζ : λ
: μ : ν of the six quantities which specify an infinitesimal twist about the
screw. If the twist is a pure rotation, these quantities are subject to the
relation
λξ + μη + νζ = 0.
(4)
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In the analytical investigations of line geometry, these six quantities,
supposed subject to the relation (4), are used to specify a line, and are called
the six “co-ordinates” of the line; they are of course equivalent to only four
independent quantities. If a line is a null-line with respect to the wrench (X,
Y, Z, L, M, N), the work done in an infinitely small rotation about it is zero,
and its co-ordinates are accordingly subject to the further relation
Lξ + Mη + Nζ + Xλ + Yμ + Zν = 0,
(5)
where the coefficients are constant. This is the equation of a “linear
complex” (cf. § 8).
Two screws are reciprocal when a wrench about one does no work on a
body which twists about the other. The condition for this is
λξ′ + μη′ + νζ′ + λ′ξ + μ′η + ν′ζ = 0,
(6)
if the screws be defined by the ratios ξ : η : ζ : λ : μ : ν and ξ′ : η′ : ζ′ : λ′ : μ′
: ν′, respectively. The theory of the screw-systems which are reciprocal to
one, two, three, four given screws respectively has been investigated by Sir
R. S. Ball.
Considering a rigid body in any given position, we may contemplate the whole
group of infinitesimal displacements which might be given to it. If the extraneous
forces are in equilibrium the total work which they would perform in any such
displacement would be zero, since they reduce to a zero force and a zero couple.
This is (in part) the celebrated principle of virtual velocities, now often described
as the principle of virtual work, enunciated by John Bernoulli (1667-1748). The
word “virtual” is used because the displacements in question are not regarded as
actually taking place, the body being in fact at rest. The “velocities” referred to
are the velocities of the various points of the body in any imagined motion of the
body through the position in question; they obviously bear to one another the
same ratios as the corresponding infinitesimal displacements. Conversely, we can
show that if the virtual work of the extraneous forces be zero for every
supposed subject to the relation (4), are used to specify a line, and are called
the six “co-ordinates” of the line; they are of course equivalent to only four
independent quantities. If a line is a null-line with respect to the wrench (X,
Y, Z, L, M, N), the work done in an infinitely small rotation about it is zero,
and its co-ordinates are accordingly subject to the further relation
Lξ + Mη + Nζ + Xλ + Yμ + Zν = 0,
(5)
where the coefficients are constant. This is the equation of a “linear
complex” (cf. § 8).
Two screws are reciprocal when a wrench about one does no work on a
body which twists about the other. The condition for this is
λξ′ + μη′ + νζ′ + λ′ξ + μ′η + ν′ζ = 0,
(6)
if the screws be defined by the ratios ξ : η : ζ : λ : μ : ν and ξ′ : η′ : ζ′ : λ′ : μ′
: ν′, respectively. The theory of the screw-systems which are reciprocal to
one, two, three, four given screws respectively has been investigated by Sir
R. S. Ball.
Considering a rigid body in any given position, we may contemplate the whole
group of infinitesimal displacements which might be given to it. If the extraneous
forces are in equilibrium the total work which they would perform in any such
displacement would be zero, since they reduce to a zero force and a zero couple.
This is (in part) the celebrated principle of virtual velocities, now often described
as the principle of virtual work, enunciated by John Bernoulli (1667-1748). The
word “virtual” is used because the displacements in question are not regarded as
actually taking place, the body being in fact at rest. The “velocities” referred to
are the velocities of the various points of the body in any imagined motion of the
body through the position in question; they obviously bear to one another the
same ratios as the corresponding infinitesimal displacements. Conversely, we can
show that if the virtual work of the extraneous forces be zero for every
Page 395
infinitesimal displacement of the body as rigid, these forces must be in
equilibrium. For by giving the body (in imagination) a displacement of
translation we learn that the sum of the resolved parts of the forces in any
assigned direction is zero, and by giving it a displacement of pure rotation we
learn that the sum of the moments about any assigned axis is zero. The same
thing follows of course from the analytical expression (2) for the virtual work. If
this vanishes for all values of λ, μ, ν, ξ, η, ζ we must have X, Y, Z, L, M, N = 0,
which are the conditions of equilibrium.
The principle can of course be extended to any system of particles or rigid
bodies, connected together in any way, provided we take into account the internal
stresses, or reactions, between the various parts. Each such reaction consists of
two equal and opposite forces, both of which may contribute to the equation of
virtual work.
The proper significance of the principle of virtual work, and of its converse,
will appear more clearly when we come to kinetics (§ 16); for the present it may
be regarded merely as a compact and (for many purposes) highly convenient
summary of the laws of equilibrium. Its special value lies in this, that by a
suitable adjustment of the hypothetical displacements we are often enabled to
eliminate unknown reactions. For example, in the case of a particle lying on a
smooth curve, or on a smooth surface, if it be displaced along the curve, or on the
surface, the virtual work of the normal component of the pressure may be
ignored, since it is of the second order. Again, if two bodies are connected by a
string or rod, and if the hypothetical displacements be adjusted so that the
distance between the points of attachment is unaltered, the corresponding stress
may be ignored. This is evident from fig. 45; if AB, A′B′ represent the two
positions of a string, and T be the tension, the virtual work of the two forces ±T
at A, B is T(Aα − Bβ), which was shown to be of the second order. Again, the
normal pressure between two surfaces disappears from the equation, provided the
displacements be such that one of these surfaces merely slides relatively to the
other. It is evident, in the first place, that in any displacement common to the two
surfaces, the work of the two equal and opposite normal pressures will cancel;
equilibrium. For by giving the body (in imagination) a displacement of
translation we learn that the sum of the resolved parts of the forces in any
assigned direction is zero, and by giving it a displacement of pure rotation we
learn that the sum of the moments about any assigned axis is zero. The same
thing follows of course from the analytical expression (2) for the virtual work. If
this vanishes for all values of λ, μ, ν, ξ, η, ζ we must have X, Y, Z, L, M, N = 0,
which are the conditions of equilibrium.
The principle can of course be extended to any system of particles or rigid
bodies, connected together in any way, provided we take into account the internal
stresses, or reactions, between the various parts. Each such reaction consists of
two equal and opposite forces, both of which may contribute to the equation of
virtual work.
The proper significance of the principle of virtual work, and of its converse,
will appear more clearly when we come to kinetics (§ 16); for the present it may
be regarded merely as a compact and (for many purposes) highly convenient
summary of the laws of equilibrium. Its special value lies in this, that by a
suitable adjustment of the hypothetical displacements we are often enabled to
eliminate unknown reactions. For example, in the case of a particle lying on a
smooth curve, or on a smooth surface, if it be displaced along the curve, or on the
surface, the virtual work of the normal component of the pressure may be
ignored, since it is of the second order. Again, if two bodies are connected by a
string or rod, and if the hypothetical displacements be adjusted so that the
distance between the points of attachment is unaltered, the corresponding stress
may be ignored. This is evident from fig. 45; if AB, A′B′ represent the two
positions of a string, and T be the tension, the virtual work of the two forces ±T
at A, B is T(Aα − Bβ), which was shown to be of the second order. Again, the
normal pressure between two surfaces disappears from the equation, provided the
displacements be such that one of these surfaces merely slides relatively to the
other. It is evident, in the first place, that in any displacement common to the two
surfaces, the work of the two equal and opposite normal pressures will cancel;
Page 396
moreover if, one of the surfaces being fixed, an infinitely small displacement
shifts the point of contact from A to B, and if A′ be the new position of that point
of the sliding body which was at A, the projection of AA′ on the normal at A is of
the second order. It is to be noticed, in this case, that the tangential reaction (if
any) between the two surfaces is not eliminated. Again, if the displacements be
such that one curved surface rolls without sliding on another, the reaction,
whether normal or tangential, at the point of contact may be ignored. For the
virtual work of two equal and opposite forces will cancel in any displacement
which is common to the two surfaces; whilst, if one surface be fixed, the
displacement of that point of the rolling surface which was in contact with the
other is of the second order. We are thus able to imagine a great variety of
mechanical systems to which the principle of virtual work can be applied without
any regard to the internal stresses, provided the hypothetical displacements be
such that none of the connexions of the system are violated.
If the system be subject to gravity, the corresponding part of the virtual work
can be calculated from the displacement of the centre of gravity. If W1, W2, ... be
the weights of a system of particles, whose depths below a fixed horizontal plane
of reference are z1, z2, ..., respectively, the virtual work of gravity is
W1δ·z1 + W2δz2 + ... = δ(W1z1 + W2z2 + ...) = (W1 + W2 + ...) δz,
where z is the depth of the centre of gravity (see § 8 (14) and § 11 (6)). This
expression is the same as if the whole mass were concentrated at the centre of
gravity, and displaced with this point. An important conclusion is that in any
displacement of a system of bodies in equilibrium, such that the virtual work of
all forces except gravity may be ignored, the depth of the centre of gravity is
“stationary.”
The question as to stability of equilibrium belongs essentially to kinetics; but
we may state by anticipation that in cases where gravity is the only force which
does work, the equilibrium of a body or system of bodies is stable only if the
depth of the centre of gravity be a maximum.
shifts the point of contact from A to B, and if A′ be the new position of that point
of the sliding body which was at A, the projection of AA′ on the normal at A is of
the second order. It is to be noticed, in this case, that the tangential reaction (if
any) between the two surfaces is not eliminated. Again, if the displacements be
such that one curved surface rolls without sliding on another, the reaction,
whether normal or tangential, at the point of contact may be ignored. For the
virtual work of two equal and opposite forces will cancel in any displacement
which is common to the two surfaces; whilst, if one surface be fixed, the
displacement of that point of the rolling surface which was in contact with the
other is of the second order. We are thus able to imagine a great variety of
mechanical systems to which the principle of virtual work can be applied without
any regard to the internal stresses, provided the hypothetical displacements be
such that none of the connexions of the system are violated.
If the system be subject to gravity, the corresponding part of the virtual work
can be calculated from the displacement of the centre of gravity. If W1, W2, ... be
the weights of a system of particles, whose depths below a fixed horizontal plane
of reference are z1, z2, ..., respectively, the virtual work of gravity is
W1δ·z1 + W2δz2 + ... = δ(W1z1 + W2z2 + ...) = (W1 + W2 + ...) δz,
where z is the depth of the centre of gravity (see § 8 (14) and § 11 (6)). This
expression is the same as if the whole mass were concentrated at the centre of
gravity, and displaced with this point. An important conclusion is that in any
displacement of a system of bodies in equilibrium, such that the virtual work of
all forces except gravity may be ignored, the depth of the centre of gravity is
“stationary.”
The question as to stability of equilibrium belongs essentially to kinetics; but
we may state by anticipation that in cases where gravity is the only force which
does work, the equilibrium of a body or system of bodies is stable only if the
depth of the centre of gravity be a maximum.
Page 397
Consider, for instance, the case of a bar resting with its ends on two
smooth inclines (fig. 18). If the bar be displaced in a vertical plane so that its
ends slide on the two inclines, the instantaneous centre is at the point J. The
displacement of G is at right angles to JG; this shows that for equilibrium JG
must be vertical. Again, the locus of G is an arc of an ellipse whose centre is
in the intersection of the planes; since this arc is convex upwards the
equilibrium is unstable. A general criterion for the case of a rigid body
movable in two dimensions, with one degree of freedom, can be obtained as
follows. We have seen (§ 3) that the sequence of possible positions is
obtained if we imagine the “body-centrode” to roll on the “space-centrode.”
For equilibrium, the altitude of the centre of gravity G must be stationary;
hence G must lie in the same vertical line with the point of contact J of the
two curves. Further, it is known from the theory of “roulettes” that the locus
of G will be concave or convex upwards according as
cos φ 1 1
= + ,
h ρ ρ′ (8)
where ρ, ρ′ are the radii of curvature of the
two curves at J, φ is the inclination of the
common tangent at J to the horizontal, and h
is the height of G above J. The signs of ρ, ρ′
are to be taken positive when the curvatures
are as in the standard case shown in fig. 49.
Hence for stability the upper sign must
obtain in (8). The same criterion may be
arrived at in a more intuitive manner as
follows. If the body be supposed to roll (say
to the right) until the curves touch at J′, and Fig. 49.
if JJ′ = δs, the angle through which the upper
figure rotates is δs/ρ + δs/ρ′, and the horizontal displacement of G is equal to
the product of this expression into h. If this displacement be less than the
horizontal projection of JJ′, viz. δs cosφ, the vertical through the new
smooth inclines (fig. 18). If the bar be displaced in a vertical plane so that its
ends slide on the two inclines, the instantaneous centre is at the point J. The
displacement of G is at right angles to JG; this shows that for equilibrium JG
must be vertical. Again, the locus of G is an arc of an ellipse whose centre is
in the intersection of the planes; since this arc is convex upwards the
equilibrium is unstable. A general criterion for the case of a rigid body
movable in two dimensions, with one degree of freedom, can be obtained as
follows. We have seen (§ 3) that the sequence of possible positions is
obtained if we imagine the “body-centrode” to roll on the “space-centrode.”
For equilibrium, the altitude of the centre of gravity G must be stationary;
hence G must lie in the same vertical line with the point of contact J of the
two curves. Further, it is known from the theory of “roulettes” that the locus
of G will be concave or convex upwards according as
cos φ 1 1
= + ,
h ρ ρ′ (8)
where ρ, ρ′ are the radii of curvature of the
two curves at J, φ is the inclination of the
common tangent at J to the horizontal, and h
is the height of G above J. The signs of ρ, ρ′
are to be taken positive when the curvatures
are as in the standard case shown in fig. 49.
Hence for stability the upper sign must
obtain in (8). The same criterion may be
arrived at in a more intuitive manner as
follows. If the body be supposed to roll (say
to the right) until the curves touch at J′, and Fig. 49.
if JJ′ = δs, the angle through which the upper
figure rotates is δs/ρ + δs/ρ′, and the horizontal displacement of G is equal to
the product of this expression into h. If this displacement be less than the
horizontal projection of JJ′, viz. δs cosφ, the vertical through the new
Page 398
position of G will fall to the left of J′ and gravity will tend to restore the
body to its former position. It is here assumed that the remaining forces
acting on the body in its displaced position have zero moment about J′; this
is evidently the case, for instance, in the problem of “rocking stones.”
The principle of virtual work is specially convenient in the theory of frames (§
6), since the reactions at smooth joints and the stresses in inextensible bars may
be left out of account. In particular, in the case of a frame which is just rigid, the
principle enables us to find the stress in any one bar independently of the rest. If
we imagine the bar in question to be removed, equilibrium will still persist if we
introduce two equal and opposite forces S, of suitable magnitude, at the joints
which it connected. In any infinitely small deformation of the frame as thus
modified, the virtual work of the forces S, together with that of the original
extraneous forces, must vanish; this determines S.
As a simple example, take the case of a light frame, whose bars form the
slides of a rhombus ABCD with the diagonal BD, suspended from A and
carrying a weight W at C; and let it be required to find the stress in BD. If
we remove the bar BD, and apply two equal and opposite forces S at B and
D, the equation is
W·δ(2l cosθ) + 2S·δ (l sin θ) = 0,
where l is the length of a side of the rhombus,
and θ its inclination to the vertical. Hence
S = W tan θ = W · BD/AC.
(8)
The method is specially appropriate when
the frame, although just rigid, is not “simple”
in the sense of § 6, and when accordingly the
method of reciprocal figures is not
immediately available. To avoid the intricate Fig. 50.
trigonometrical calculations which would
often be necessary, graphical devices have been introduced by H. Müller-
body to its former position. It is here assumed that the remaining forces
acting on the body in its displaced position have zero moment about J′; this
is evidently the case, for instance, in the problem of “rocking stones.”
The principle of virtual work is specially convenient in the theory of frames (§
6), since the reactions at smooth joints and the stresses in inextensible bars may
be left out of account. In particular, in the case of a frame which is just rigid, the
principle enables us to find the stress in any one bar independently of the rest. If
we imagine the bar in question to be removed, equilibrium will still persist if we
introduce two equal and opposite forces S, of suitable magnitude, at the joints
which it connected. In any infinitely small deformation of the frame as thus
modified, the virtual work of the forces S, together with that of the original
extraneous forces, must vanish; this determines S.
As a simple example, take the case of a light frame, whose bars form the
slides of a rhombus ABCD with the diagonal BD, suspended from A and
carrying a weight W at C; and let it be required to find the stress in BD. If
we remove the bar BD, and apply two equal and opposite forces S at B and
D, the equation is
W·δ(2l cosθ) + 2S·δ (l sin θ) = 0,
where l is the length of a side of the rhombus,
and θ its inclination to the vertical. Hence
S = W tan θ = W · BD/AC.
(8)
The method is specially appropriate when
the frame, although just rigid, is not “simple”
in the sense of § 6, and when accordingly the
method of reciprocal figures is not
immediately available. To avoid the intricate Fig. 50.
trigonometrical calculations which would
often be necessary, graphical devices have been introduced by H. Müller-
Page 399
Breslau and others. For this purpose the infinitesimal displacements of the
various joints are replaced by finite lengths proportional to them, and
therefore proportional to the velocities of the joints in some imagined
motion of the deformable frame through its actual configuration; this is
really (it may be remarked) a reversion to the original notion of “virtual
velocities.” Let J be the instantaneous centre for any bar CD (fig. 12), and
let s1, s2 represent the virtual velocities of C, D. If these lines be turned
through a right angle in the same sense, they take up positions such as CC′,
DD′, where C′, D′ are on JC, JD, respectively, and C′D′ is parallel to CD.
Further, if F1 (fig. 51) be any force acting on the joint C, its virtual work
will be equal to the moment of F1 about C′; the equation of virtual work is
thus transformed into an equation of moments.
Fig. 12. Fig. 51.
Consider, for example, a frame
whose sides form the six sides of a
hexagon ABCDEF and the three
diagonals AD, BE, CF; and suppose
that it is required to find the stress in
CF due to a given system of extraneous
forces in equilibrium, acting on the
Fig. 52.
joints. Imagine the bar CF to be
removed, and consider a deformation in
which AB is fixed. The instantaneous centre of CD will be at the
various joints are replaced by finite lengths proportional to them, and
therefore proportional to the velocities of the joints in some imagined
motion of the deformable frame through its actual configuration; this is
really (it may be remarked) a reversion to the original notion of “virtual
velocities.” Let J be the instantaneous centre for any bar CD (fig. 12), and
let s1, s2 represent the virtual velocities of C, D. If these lines be turned
through a right angle in the same sense, they take up positions such as CC′,
DD′, where C′, D′ are on JC, JD, respectively, and C′D′ is parallel to CD.
Further, if F1 (fig. 51) be any force acting on the joint C, its virtual work
will be equal to the moment of F1 about C′; the equation of virtual work is
thus transformed into an equation of moments.
Fig. 12. Fig. 51.
Consider, for example, a frame
whose sides form the six sides of a
hexagon ABCDEF and the three
diagonals AD, BE, CF; and suppose
that it is required to find the stress in
CF due to a given system of extraneous
forces in equilibrium, acting on the
Fig. 52.
joints. Imagine the bar CF to be
removed, and consider a deformation in
which AB is fixed. The instantaneous centre of CD will be at the
Page 400
intersection of AD, BC, and if C′D′ be drawn parallel to CD, the lines CC′,
DD′ may be taken to represent the virtual velocities of C, D turned each
through a right angle. Moreover, if we draw D′E′ parallel to DE, and E′F′
parallel to EF, the lines CC′, DD′, EE′, FF′ will represent on the same scale
the virtual velocities of the points C, D, E, F, respectively, turned each
through a right angle. The equation of virtual work is then formed by taking
moments about C′, D′, E′, F′ of the extraneous forces which act at C, D, E, F,
respectively. Amongst these forces we must include the two equal and
opposite forces S which take the place of the stress in the removed bar FC.
The above method lends itself naturally to the investigation of the critical
forms of a frame whose general structure is given. We have seen that the
stresses produced by an equilibrating system of extraneous forces in a frame
which is just rigid, according to the criterion of § 6, are in general uniquely
determinate; in particular, when there are no extraneous forces the bars are
in general free from stress. It may however happen that owing to some
special relation between the lengths of the bars the frame admits of an
infinitesimal deformation. The simplest case is that of a frame of three bars,
when the three joints A, B, C fall into a straight line; a small displacement of
the joint B at right angles to AC would involve changes in the lengths of
AB, BC which are only of the second order of small quantities. Another
example is shown in fig. 53. The graphical method leads at once to the
detection of such cases. Thus in the hexagonal frame of fig. 52, if an
infinitesimal deformation is possible without removing the bar CF, the
instantaneous centre of CF (when AB is fixed) will be at the intersection of
AF and BC, and since CC′, FF′ represent the virtual velocities of the points
C, F, turned each through a right angle, C′F′ must be parallel to CF.
Conversely, if this condition be satisfied, an infinitesimal deformation is
possible. The result may be generalized into the statement that a frame has a
critical form whenever a frame of the same structure can be designed with
corresponding bars parallel, but without complete geometric similarity. In
the case of fig. 52 it may be shown that an equivalent condition is that the
six points A, B, C, D, E, F should lie on a conic (M. W. Crofton). This is
DD′ may be taken to represent the virtual velocities of C, D turned each
through a right angle. Moreover, if we draw D′E′ parallel to DE, and E′F′
parallel to EF, the lines CC′, DD′, EE′, FF′ will represent on the same scale
the virtual velocities of the points C, D, E, F, respectively, turned each
through a right angle. The equation of virtual work is then formed by taking
moments about C′, D′, E′, F′ of the extraneous forces which act at C, D, E, F,
respectively. Amongst these forces we must include the two equal and
opposite forces S which take the place of the stress in the removed bar FC.
The above method lends itself naturally to the investigation of the critical
forms of a frame whose general structure is given. We have seen that the
stresses produced by an equilibrating system of extraneous forces in a frame
which is just rigid, according to the criterion of § 6, are in general uniquely
determinate; in particular, when there are no extraneous forces the bars are
in general free from stress. It may however happen that owing to some
special relation between the lengths of the bars the frame admits of an
infinitesimal deformation. The simplest case is that of a frame of three bars,
when the three joints A, B, C fall into a straight line; a small displacement of
the joint B at right angles to AC would involve changes in the lengths of
AB, BC which are only of the second order of small quantities. Another
example is shown in fig. 53. The graphical method leads at once to the
detection of such cases. Thus in the hexagonal frame of fig. 52, if an
infinitesimal deformation is possible without removing the bar CF, the
instantaneous centre of CF (when AB is fixed) will be at the intersection of
AF and BC, and since CC′, FF′ represent the virtual velocities of the points
C, F, turned each through a right angle, C′F′ must be parallel to CF.
Conversely, if this condition be satisfied, an infinitesimal deformation is
possible. The result may be generalized into the statement that a frame has a
critical form whenever a frame of the same structure can be designed with
corresponding bars parallel, but without complete geometric similarity. In
the case of fig. 52 it may be shown that an equivalent condition is that the
six points A, B, C, D, E, F should lie on a conic (M. W. Crofton). This is
Page 401
fulfilled when the opposite sides of the hexagon are parallel, and (as a still
more special case) when the hexagon is regular.
When a frame has a critical form it may
be in a state of stress independently of the
action of extraneous forces; moreover, the
stresses due to extraneous forces are
indeterminate, and may be infinite. For
suppose as before that one of the bars is
removed. If there are no extraneous forces
Fig. 53.
the equation of virtual work reduces to S·δs
= 0, where S is the stress in the removed bar,
and δs is the change in the distance between the joints which it connected. In
a critical form we have δs = 0, and the equation is satisfied by an arbitrary
value of S; a consistent system of stresses in the remaining bars can then be
found by preceding rules. Again, when extraneous forces P act on the joints,
the equation is
Σ(P·δp) + S·δs = 0,
where δp is the displacement of any joint in the direction of the
corresponding force P. If Σ(P·δp) = 0, the stresses are merely indeterminate
as before; but if Σ (P·δp) does not vanish, the equation cannot be satisfied
by any finite value of S, since δs = 0. This means that, if the material of the
frame were absolutely unyielding, no finite stresses in the bars would enable
it to withstand the extraneous forces. With actual materials, the frame would
yield elastically, until its configuration is no longer “critical.” The stresses in
the bars would then be comparatively very great, although finite. The use of
frames which approximate to a critical form is of course to be avoided in
practice.
A brief reference must suffice to the theory of three dimensional frames.
This is important from a technical point of view, since all structures are
practically three-dimensional. We may note that a frame of n joints which is
more special case) when the hexagon is regular.
When a frame has a critical form it may
be in a state of stress independently of the
action of extraneous forces; moreover, the
stresses due to extraneous forces are
indeterminate, and may be infinite. For
suppose as before that one of the bars is
removed. If there are no extraneous forces
Fig. 53.
the equation of virtual work reduces to S·δs
= 0, where S is the stress in the removed bar,
and δs is the change in the distance between the joints which it connected. In
a critical form we have δs = 0, and the equation is satisfied by an arbitrary
value of S; a consistent system of stresses in the remaining bars can then be
found by preceding rules. Again, when extraneous forces P act on the joints,
the equation is
Σ(P·δp) + S·δs = 0,
where δp is the displacement of any joint in the direction of the
corresponding force P. If Σ(P·δp) = 0, the stresses are merely indeterminate
as before; but if Σ (P·δp) does not vanish, the equation cannot be satisfied
by any finite value of S, since δs = 0. This means that, if the material of the
frame were absolutely unyielding, no finite stresses in the bars would enable
it to withstand the extraneous forces. With actual materials, the frame would
yield elastically, until its configuration is no longer “critical.” The stresses in
the bars would then be comparatively very great, although finite. The use of
frames which approximate to a critical form is of course to be avoided in
practice.
A brief reference must suffice to the theory of three dimensional frames.
This is important from a technical point of view, since all structures are
practically three-dimensional. We may note that a frame of n joints which is
Page 402
just rigid must have 3n − 6 bars; and that the stresses produced in such a
frame by a given system of extraneous forces in equilibrium are statically
determinate, subject to the exception of “critical forms.”
§ 10. Statics of Inextensible Chains.—The theory of bodies or structures which
are deformable in their smallest parts belongs properly to elasticity (q.v.). The
case of inextensible strings or chains is, however, so simple that it is generally
included in expositions of pure statics.
It is assumed that the form can be sufficiently represented by a plane curve,
that the stress (tension) at any point P of the curve, between the two portions
which meet there, is in the direction of the tangent at P, and that the forces on any
linear element δs must satisfy the conditions of equilibrium laid down in § 1. It
follows that the forces on any finite portion will satisfy the conditions of
equilibrium which apply to the case of a rigid body (§ 4).
We will suppose in the first instance that
the curve is plane. It is often convenient to
resolve the forces on an element PQ (= δs)
in the directions of the tangent and normal
respectively. If T, T + δT be the tensions at
P, Q, and δψ be the angle between the
directions of the curve at these points, the
components of the tensions along the
tangent at P give (T + δT) cos ψ − T, or δT, Fig. 54.
ultimately; whilst for the component along
the normal at P we have (T + δT) sin δψ, or Tδψ, or Tδs/ρ, where ρ is the radius
of curvature.
Suppose, for example, that we have a light string stretched over a smooth
curve; and let Rδs denote the normal pressure (outwards from the centre of
curvature) on δs. The two resolutions give δT = 0, Tδψ = Rδs, or
T = const., R = T/ρ.
(1)
frame by a given system of extraneous forces in equilibrium are statically
determinate, subject to the exception of “critical forms.”
§ 10. Statics of Inextensible Chains.—The theory of bodies or structures which
are deformable in their smallest parts belongs properly to elasticity (q.v.). The
case of inextensible strings or chains is, however, so simple that it is generally
included in expositions of pure statics.
It is assumed that the form can be sufficiently represented by a plane curve,
that the stress (tension) at any point P of the curve, between the two portions
which meet there, is in the direction of the tangent at P, and that the forces on any
linear element δs must satisfy the conditions of equilibrium laid down in § 1. It
follows that the forces on any finite portion will satisfy the conditions of
equilibrium which apply to the case of a rigid body (§ 4).
We will suppose in the first instance that
the curve is plane. It is often convenient to
resolve the forces on an element PQ (= δs)
in the directions of the tangent and normal
respectively. If T, T + δT be the tensions at
P, Q, and δψ be the angle between the
directions of the curve at these points, the
components of the tensions along the
tangent at P give (T + δT) cos ψ − T, or δT, Fig. 54.
ultimately; whilst for the component along
the normal at P we have (T + δT) sin δψ, or Tδψ, or Tδs/ρ, where ρ is the radius
of curvature.
Suppose, for example, that we have a light string stretched over a smooth
curve; and let Rδs denote the normal pressure (outwards from the centre of
curvature) on δs. The two resolutions give δT = 0, Tδψ = Rδs, or
T = const., R = T/ρ.
(1)
Page 403
The tension is constant, and the pressure per unit length varies as the curvature.
Next suppose that the curve is “rough”; and let Fδs be the tangential force of
friction on δs. We have δT ± Fδs = 0, Tδψ = Rδs, where the upper or lower sign
is to be taken according to the sense in which F acts. We assume that in limiting
equilibrium we have F = μR, everywhere, where μ is the coefficient of friction. If
the string be on the point of slipping in the direction in which ψ increases, the
lower sign is to be taken; hence δT = Fδs = μTδψ, whence
T = T0 eμψ,
(2)
if T0 be the tension corresponding to ψ = 0. This illustrates the resistance to
dragging of a rope coiled round a post; e.g. if we put μ = .3, ψ = 2π, we find for
the change of tension in one turn T/T0 = 6.5. In two turns this ratio is squared,
and so on.
Again, take the case of a string under gravity, in contact with a smooth curve
in a vertical plane. Let ψ denote the inclination to the horizontal, and wδs the
weight of an element δs. The tangential and normal components of wδs are −s
sinψ and −wδs cosψ. Hence
δT = wδs sin ψ, Tδψ = wδs cos ψ + Rδs.
(3)
If we take rectangular axes Ox, Oy, of which Oy is drawn vertically upwards, we
have δy = sin ψ δs, whence δT = wδy. If the string be uniform, w is constant, and
T = wy + const. = w (y − y0),
(4)
say; hence the tension varies as the height above some fixed level (y0). The
pressure is then given by the formula
dψ
R=T −w cos ψ.
ds (5)
Next suppose that the curve is “rough”; and let Fδs be the tangential force of
friction on δs. We have δT ± Fδs = 0, Tδψ = Rδs, where the upper or lower sign
is to be taken according to the sense in which F acts. We assume that in limiting
equilibrium we have F = μR, everywhere, where μ is the coefficient of friction. If
the string be on the point of slipping in the direction in which ψ increases, the
lower sign is to be taken; hence δT = Fδs = μTδψ, whence
T = T0 eμψ,
(2)
if T0 be the tension corresponding to ψ = 0. This illustrates the resistance to
dragging of a rope coiled round a post; e.g. if we put μ = .3, ψ = 2π, we find for
the change of tension in one turn T/T0 = 6.5. In two turns this ratio is squared,
and so on.
Again, take the case of a string under gravity, in contact with a smooth curve
in a vertical plane. Let ψ denote the inclination to the horizontal, and wδs the
weight of an element δs. The tangential and normal components of wδs are −s
sinψ and −wδs cosψ. Hence
δT = wδs sin ψ, Tδψ = wδs cos ψ + Rδs.
(3)
If we take rectangular axes Ox, Oy, of which Oy is drawn vertically upwards, we
have δy = sin ψ δs, whence δT = wδy. If the string be uniform, w is constant, and
T = wy + const. = w (y − y0),
(4)
say; hence the tension varies as the height above some fixed level (y0). The
pressure is then given by the formula
dψ
R=T −w cos ψ.
ds (5)
Page 404
In the case of a chain hanging freely under gravity it is usually convenient to
formulate the conditions of equilibrium of a finite portion PQ. The forces on this
reduce to three, viz. the weight of PQ and the tensions at P, Q. Hence these three
forces will be concurrent, and their ratios will be given by a triangle of forces. In
particular, if we consider a length AP beginning at the lowest point A, then
resolving horizontally and vertically we have
T cos ψ = T0, T sinψ = W,
(6)
where T0 is the tension at A, and W is the weight of PA. The former equation
expresses that the horizontal tension is constant.
Fig. 55.
If the chain be uniform we have W = ws, where s is the arc AP: hence ws = T0
tan ψ. If we write T0 = wa, so that a is the length of a portion of the chain whose
weight would equal the horizontal tension, this becomes
s = a tan ψ.
(7)
This is the “intrinsic” equation of the curve. If the axes of x and y be taken
horizontal and vertical (upwards), we derive
x = a log (sec ψ + tan ψ), y = a sec ψ.
(8)
formulate the conditions of equilibrium of a finite portion PQ. The forces on this
reduce to three, viz. the weight of PQ and the tensions at P, Q. Hence these three
forces will be concurrent, and their ratios will be given by a triangle of forces. In
particular, if we consider a length AP beginning at the lowest point A, then
resolving horizontally and vertically we have
T cos ψ = T0, T sinψ = W,
(6)
where T0 is the tension at A, and W is the weight of PA. The former equation
expresses that the horizontal tension is constant.
Fig. 55.
If the chain be uniform we have W = ws, where s is the arc AP: hence ws = T0
tan ψ. If we write T0 = wa, so that a is the length of a portion of the chain whose
weight would equal the horizontal tension, this becomes
s = a tan ψ.
(7)
This is the “intrinsic” equation of the curve. If the axes of x and y be taken
horizontal and vertical (upwards), we derive
x = a log (sec ψ + tan ψ), y = a sec ψ.
(8)
Page 405
Eliminating ψ we obtain the Cartesian equation
x
y = a cosh
a (9)
of the common catenary, as it is called (fig. 56). The omission of the additive
arbitrary constants of integration in (8) is equivalent to a special choice of the
origin O of co-ordinates; viz. O is at a distance a vertically below the lowest
point (ψ = 0) of the curve. The horizontal line through O is called the directrix.
The relations
x
s = a sinh , y2 = a2 + s2, T = T0 sec ψ = wy,
a (10)
which are involved in the preceding
formulae are also noteworthy. It is a
classical problem in the calculus of
variations to deduce the equation (9) from
the condition that the depth of the centre of
gravity of a chain of given length hanging
between fixed points must be stationary (§
9). The length a is called the parameter of
the catenary; it determines the scale of the
curve, all catenaries being geometrically
similar. If weights be suspended from
various points of a hanging chain, the Fig. 56.
intervening portions will form arcs of equal
catenaries, since the horizontal tension (wa) is the same for all. Again, if a chain
pass over a perfectly smooth peg, the catenaries in which it hangs on the two
sides, though usually of different parameters, will have the same directrix, since
by (10) y is the same for both at the peg.
As an example of the use of the formulae we may determine the
maximum span for a wire of given material. The condition is that the tension
x
y = a cosh
a (9)
of the common catenary, as it is called (fig. 56). The omission of the additive
arbitrary constants of integration in (8) is equivalent to a special choice of the
origin O of co-ordinates; viz. O is at a distance a vertically below the lowest
point (ψ = 0) of the curve. The horizontal line through O is called the directrix.
The relations
x
s = a sinh , y2 = a2 + s2, T = T0 sec ψ = wy,
a (10)
which are involved in the preceding
formulae are also noteworthy. It is a
classical problem in the calculus of
variations to deduce the equation (9) from
the condition that the depth of the centre of
gravity of a chain of given length hanging
between fixed points must be stationary (§
9). The length a is called the parameter of
the catenary; it determines the scale of the
curve, all catenaries being geometrically
similar. If weights be suspended from
various points of a hanging chain, the Fig. 56.
intervening portions will form arcs of equal
catenaries, since the horizontal tension (wa) is the same for all. Again, if a chain
pass over a perfectly smooth peg, the catenaries in which it hangs on the two
sides, though usually of different parameters, will have the same directrix, since
by (10) y is the same for both at the peg.
As an example of the use of the formulae we may determine the
maximum span for a wire of given material. The condition is that the tension
Page 406
must not exceed the weight of a certain length λ of the wire. At the ends we
shall have y = λ, or
x
λ = a cosh ,
a (11)
and the problem is to make x a maximum for variations of a. Differentiating
(11) we find that, if dx/da = 0,
x x
tanh = 1.
a a (12)
It is easily seen graphically, or from a table of hyperbolic tangents, that the
equation u tanh u = 1 has only one positive root (u = 1.200); the span is
therefore
2x = 2au = 2λ/sinh u = 1.326 λ,
and the length of wire is
2s = 2λ/u = 1.667 λ.
The tangents at the ends meet on the directrix, and their inclination to the
horizontal is 56° 30′.
Fig. 57.
The relation between the sag, the tension, and the span of a wire (e.g. a
telegraph wire) stretched nearly straight between two points A, B at the
same level is determined most simply from first principles. If T be the
tension, W the total weight, k the sag in the middle, and ψ the inclination to
shall have y = λ, or
x
λ = a cosh ,
a (11)
and the problem is to make x a maximum for variations of a. Differentiating
(11) we find that, if dx/da = 0,
x x
tanh = 1.
a a (12)
It is easily seen graphically, or from a table of hyperbolic tangents, that the
equation u tanh u = 1 has only one positive root (u = 1.200); the span is
therefore
2x = 2au = 2λ/sinh u = 1.326 λ,
and the length of wire is
2s = 2λ/u = 1.667 λ.
The tangents at the ends meet on the directrix, and their inclination to the
horizontal is 56° 30′.
Fig. 57.
The relation between the sag, the tension, and the span of a wire (e.g. a
telegraph wire) stretched nearly straight between two points A, B at the
same level is determined most simply from first principles. If T be the
tension, W the total weight, k the sag in the middle, and ψ the inclination to
Page 407
the horizontal at A or B, we have 2Tψ = W, AB = 2ρψ, approximately,
where ρ is the radius of curvature. Since 2kρ = (1⁄2AB)2, ultimately, we have
k = 1⁄8W · AB/T.
(13)
The same formula applies if A, B be at different levels, provided k be the
sag, measured vertically, half way between A and B.
In relation to the theory of suspension bridges the case where the weight of
any portion of the chain varies as its horizontal projection is of interest. The
vertical through the centre of gravity of the arc AP (see fig. 55) will then bisect
its horizontal projection AN; hence if PS be the tangent at P we shall have AS =
SN. This property is characteristic of a parabola whose axis is vertical. If we take
A as origin and AN as axis of x, the weight of AP may be denoted by wx, where
w is the weight per unit length at A. Since PNS is a triangle of forces for the
portion AP of the chain, we have wx/T0 = PN/NS, or
y = w · x2/2T0,
(14)
which is the equation of the parabola in question. The result might of course have
been inferred from the theory of the parabolic funicular in § 2.
Finally, we may refer to the catenary of uniform strength, where the
cross-section of the wire (or cable) is supposed to vary as the tension. Hence
w, the weight per foot, varies as T, and we may write T = wλ, where λ is a
constant length. Resolving along the normal the forces on an element δs, we
find Tδψ = wδs cos ψ, whence
ds
ρ= = λ sec ψ.
dψ (15)
From this we derive
x
x = λψ, y = λ log sec ,
λ
where ρ is the radius of curvature. Since 2kρ = (1⁄2AB)2, ultimately, we have
k = 1⁄8W · AB/T.
(13)
The same formula applies if A, B be at different levels, provided k be the
sag, measured vertically, half way between A and B.
In relation to the theory of suspension bridges the case where the weight of
any portion of the chain varies as its horizontal projection is of interest. The
vertical through the centre of gravity of the arc AP (see fig. 55) will then bisect
its horizontal projection AN; hence if PS be the tangent at P we shall have AS =
SN. This property is characteristic of a parabola whose axis is vertical. If we take
A as origin and AN as axis of x, the weight of AP may be denoted by wx, where
w is the weight per unit length at A. Since PNS is a triangle of forces for the
portion AP of the chain, we have wx/T0 = PN/NS, or
y = w · x2/2T0,
(14)
which is the equation of the parabola in question. The result might of course have
been inferred from the theory of the parabolic funicular in § 2.
Finally, we may refer to the catenary of uniform strength, where the
cross-section of the wire (or cable) is supposed to vary as the tension. Hence
w, the weight per foot, varies as T, and we may write T = wλ, where λ is a
constant length. Resolving along the normal the forces on an element δs, we
find Tδψ = wδs cos ψ, whence
ds
ρ= = λ sec ψ.
dψ (15)
From this we derive
x
x = λψ, y = λ log sec ,
λ
Page 408
(16)
where the directions of x and y are horizontal and vertical, and the origin is
taken at the lowest point. The curve (fig. 58) has two vertical asymptotes x =
± 1⁄2πλ; this shows that however the thickness of a cable be adjusted there is
a limit πλ to the horizontal span, where λ depends on the tensile strength of
the material. For a uniform catenary the limit was found above to be 1.326λ.
Fig. 58.
For investigations relating to the equilibrium of a string in three dimensions
we must refer to the textbooks. In the case of a string stretched over a smooth
surface, but in other respects free from extraneous force, the tensions at the ends
of a small element δs must be balanced by the normal reaction of the surface. It
follows that the osculating plane of the curve formed by the string must contain
the normal to the surface, i.e. the curve must be a “geodesic,” and that the normal
pressure per unit length must vary as the principal curvature of the curve.
§ 11. Theory of Mass-Systems.—This is a purely geometrical subject. We
consider a system of points P1, P2 ..., Pn, with which are associated certain
coefficients m1, m2, ... mn, respectively. In the application to mechanics these
coefficients are the masses of particles situate at the respective points, and are
therefore all positive. We shall make this supposition in what follows, but it
should be remarked that hardly any difference is made in the theory if some of
the coefficients have a different sign from the rest, except in the special case
where Σ(m) = 0. This has a certain interest in magnetism.
where the directions of x and y are horizontal and vertical, and the origin is
taken at the lowest point. The curve (fig. 58) has two vertical asymptotes x =
± 1⁄2πλ; this shows that however the thickness of a cable be adjusted there is
a limit πλ to the horizontal span, where λ depends on the tensile strength of
the material. For a uniform catenary the limit was found above to be 1.326λ.
Fig. 58.
For investigations relating to the equilibrium of a string in three dimensions
we must refer to the textbooks. In the case of a string stretched over a smooth
surface, but in other respects free from extraneous force, the tensions at the ends
of a small element δs must be balanced by the normal reaction of the surface. It
follows that the osculating plane of the curve formed by the string must contain
the normal to the surface, i.e. the curve must be a “geodesic,” and that the normal
pressure per unit length must vary as the principal curvature of the curve.
§ 11. Theory of Mass-Systems.—This is a purely geometrical subject. We
consider a system of points P1, P2 ..., Pn, with which are associated certain
coefficients m1, m2, ... mn, respectively. In the application to mechanics these
coefficients are the masses of particles situate at the respective points, and are
therefore all positive. We shall make this supposition in what follows, but it
should be remarked that hardly any difference is made in the theory if some of
the coefficients have a different sign from the rest, except in the special case
where Σ(m) = 0. This has a certain interest in magnetism.
Page 409
In a given mass-system there exists one and only one point G such that
Σ(m·GP>) = 0.
(1)
For, take any point O, and construct the vector
Σ(m·OP>)
OG> = .
Σ(m) (2)
Then
Σ(m·GP>) = Σ {m(GO> + OP>)} = Σ(m)·GO> + Σ(m)·OP> = 0.
(3)
Also there cannot be a distinct point G′ such that Σ(m·G′P) = 0, for we should
have, by subtraction,
Σ {m(GP> + PG>′)} = 0, or Σ(m)·GG′ = 0;
(4)
i.e. G′ must coincide with G. The point G determined by (1) is called the mass-
centre or centre of inertia of the given system. It is easily seen that, in the process
of determining the mass-centre, any group of particles may be replaced by a
single particle whose mass is equal to that of the group, situate at the mass-centre
of the group.
If through P1, P2, ... Pn we draw any system of parallel planes meeting a
straight line OX in the points M1, M2 ... Mn, the collinear vectors OM>1, OM>2 ...
OM>n may be called the “projections” of OP>1, OP>2, ... OP>n on OX. Let these
projections be denoted algebraically by x1, x2, ... xn, the sign being positive or
negative according as the direction is that of OX or the reverse. Since the
projection of a vector-sum is the sum of the projections of the several vectors, the
equation (2) gives
Σ(mx)
x= ,
Σ(m) (5)
Σ(m·GP>) = 0.
(1)
For, take any point O, and construct the vector
Σ(m·OP>)
OG> = .
Σ(m) (2)
Then
Σ(m·GP>) = Σ {m(GO> + OP>)} = Σ(m)·GO> + Σ(m)·OP> = 0.
(3)
Also there cannot be a distinct point G′ such that Σ(m·G′P) = 0, for we should
have, by subtraction,
Σ {m(GP> + PG>′)} = 0, or Σ(m)·GG′ = 0;
(4)
i.e. G′ must coincide with G. The point G determined by (1) is called the mass-
centre or centre of inertia of the given system. It is easily seen that, in the process
of determining the mass-centre, any group of particles may be replaced by a
single particle whose mass is equal to that of the group, situate at the mass-centre
of the group.
If through P1, P2, ... Pn we draw any system of parallel planes meeting a
straight line OX in the points M1, M2 ... Mn, the collinear vectors OM>1, OM>2 ...
OM>n may be called the “projections” of OP>1, OP>2, ... OP>n on OX. Let these
projections be denoted algebraically by x1, x2, ... xn, the sign being positive or
negative according as the direction is that of OX or the reverse. Since the
projection of a vector-sum is the sum of the projections of the several vectors, the
equation (2) gives
Σ(mx)
x= ,
Σ(m) (5)
Page 410
if x be the projection of OG>. Hence if the Cartesian co-ordinates of P1, P2, ... Pn
relative to any axes, rectangular or oblique be (x1, y1, z1), (x2, y2, z2), ..., (xn, yn,
zn), the mass-centre (x, y, z) is determined by the formulae
Σ(mx) Σ(my) Σ(mz)
x= , y= , z= .
Σ(m) Σ(m) Σ(m) (6)
If we write x = x + ξ, y = y + η, z = z + ζ, so that ξ, η, ζ denote co-ordinates
relative to the mass-centre G, we have from (6)
Σ(mξ) = 0, Σ(mη) = 0, Σ(mζ) = 0.
(7)
One or two special cases may be noticed. If three masses α, β, γ be situate
at the vertices of a triangle ABC, the mass-centre of β and γ is at a point A′
in BC, such that β·BA′ = γ·A′C. The mass-centre (G) of α, β, γ will then
divide AA′ so that α·AG = (β + γ) GA′. It is easily proved that
α : β : γ = ΔBGA : ΔGCA : ΔGAB;
also, by giving suitable values (positive or negative) to the ratios α : β : γ we
can make G assume any assigned position in the plane ABC. We have here
the origin of the “barycentric co-ordinates” of Möbius, now usually known
as “areal” co-ordinates. If α + β + γ = 0, G is at infinity; if α = β = γ, G is at
the intersection of the median lines of the triangle; if α : β : γ = a : b : c, G is
at the centre of the inscribed circle. Again, if G be the mass-centre of four
particles α, β, γ, δ situate at the vertices of a tetrahedron ABCD, we find
α : β : γ : δ = tetn GBCD : tetn GCDA : tetn GDAB : tetn GABC,
and by suitable determination of the ratios on the left hand we can make G
assume any assigned position in space. If α + β + γ + δ = O, G is at infinity;
if α = β = γ = δ, G bisects the lines joining the middle points of opposite
edges of the tetrahedron ABCD; if α : β : γ : δ = ΔBCD : ΔCDA : ΔDAB :
ΔABC, G is at the centre of the inscribed sphere.
relative to any axes, rectangular or oblique be (x1, y1, z1), (x2, y2, z2), ..., (xn, yn,
zn), the mass-centre (x, y, z) is determined by the formulae
Σ(mx) Σ(my) Σ(mz)
x= , y= , z= .
Σ(m) Σ(m) Σ(m) (6)
If we write x = x + ξ, y = y + η, z = z + ζ, so that ξ, η, ζ denote co-ordinates
relative to the mass-centre G, we have from (6)
Σ(mξ) = 0, Σ(mη) = 0, Σ(mζ) = 0.
(7)
One or two special cases may be noticed. If three masses α, β, γ be situate
at the vertices of a triangle ABC, the mass-centre of β and γ is at a point A′
in BC, such that β·BA′ = γ·A′C. The mass-centre (G) of α, β, γ will then
divide AA′ so that α·AG = (β + γ) GA′. It is easily proved that
α : β : γ = ΔBGA : ΔGCA : ΔGAB;
also, by giving suitable values (positive or negative) to the ratios α : β : γ we
can make G assume any assigned position in the plane ABC. We have here
the origin of the “barycentric co-ordinates” of Möbius, now usually known
as “areal” co-ordinates. If α + β + γ = 0, G is at infinity; if α = β = γ, G is at
the intersection of the median lines of the triangle; if α : β : γ = a : b : c, G is
at the centre of the inscribed circle. Again, if G be the mass-centre of four
particles α, β, γ, δ situate at the vertices of a tetrahedron ABCD, we find
α : β : γ : δ = tetn GBCD : tetn GCDA : tetn GDAB : tetn GABC,
and by suitable determination of the ratios on the left hand we can make G
assume any assigned position in space. If α + β + γ + δ = O, G is at infinity;
if α = β = γ = δ, G bisects the lines joining the middle points of opposite
edges of the tetrahedron ABCD; if α : β : γ : δ = ΔBCD : ΔCDA : ΔDAB :
ΔABC, G is at the centre of the inscribed sphere.
Page 411
If we have a continuous distribution of matter, instead of a system of
discrete particles, the summations in (6) are to be replaced by integrations.
Examples will be found in textbooks of the calculus and of analytical statics.
As particular cases: the mass-centre of a uniform thin triangular plate
coincides with that of three equal particles at the corners; and that of a
uniform solid tetrahedron coincides with that of four equal particles at the
vertices. Again, the mass-centre of a uniform solid right circular cone
divides the axis in the ratio 3 : 1; that of a uniform solid hemisphere divides
the axial radius in the ratio 3 : 5.
It is easily seen from (6) that if the configuration of a system of particles
be altered by “homogeneous strain” (see Elasticity) the new position of the
mass-centre will be at that point of the strained figure which corresponds to
the original mass-centre.
The formula (2) shows that a system of concurrent forces represented by
m1·OP>1, m2·OP>2, ... mn·OP>n will have a resultant represented hy Σ(m)·OG>. If
we imagine O to recede to infinity in any direction we learn that a system of
parallel forces proportional to m1, m2,... mn, acting at P1, P2 ... Pn have a resultant
proportional to Σ(m) which acts always through a point G fixed relatively to the
given mass-system. This contains the theory of the “centre of gravity” (§§ 4, 9).
We may note also that if P1, P2, ... Pn, and P1′, P2′, ... Pn′ represent two
configurations of the series of particles, then
Σ(m·PP>′) = Sigma(m)·GG>′,
(8)
where G, G′ are the two positions of the mass-centre. The forces m1·P>1P1′,
m2·P>2P2′, ... mn·P>nPn′, considered as localized vectors, do not, however, as a rule
reduce to a single resultant.
We proceed to the theory of the plane, axial and polar quadratic moments of
the system. The axial moments have alone a dynamical significance, but the
others are useful as subsidiary conceptions. If h1, h2, ... hn be the perpendicular
distances of the particles from any fixed plane, the sum Σ(mh2) is the quadratic
discrete particles, the summations in (6) are to be replaced by integrations.
Examples will be found in textbooks of the calculus and of analytical statics.
As particular cases: the mass-centre of a uniform thin triangular plate
coincides with that of three equal particles at the corners; and that of a
uniform solid tetrahedron coincides with that of four equal particles at the
vertices. Again, the mass-centre of a uniform solid right circular cone
divides the axis in the ratio 3 : 1; that of a uniform solid hemisphere divides
the axial radius in the ratio 3 : 5.
It is easily seen from (6) that if the configuration of a system of particles
be altered by “homogeneous strain” (see Elasticity) the new position of the
mass-centre will be at that point of the strained figure which corresponds to
the original mass-centre.
The formula (2) shows that a system of concurrent forces represented by
m1·OP>1, m2·OP>2, ... mn·OP>n will have a resultant represented hy Σ(m)·OG>. If
we imagine O to recede to infinity in any direction we learn that a system of
parallel forces proportional to m1, m2,... mn, acting at P1, P2 ... Pn have a resultant
proportional to Σ(m) which acts always through a point G fixed relatively to the
given mass-system. This contains the theory of the “centre of gravity” (§§ 4, 9).
We may note also that if P1, P2, ... Pn, and P1′, P2′, ... Pn′ represent two
configurations of the series of particles, then
Σ(m·PP>′) = Sigma(m)·GG>′,
(8)
where G, G′ are the two positions of the mass-centre. The forces m1·P>1P1′,
m2·P>2P2′, ... mn·P>nPn′, considered as localized vectors, do not, however, as a rule
reduce to a single resultant.
We proceed to the theory of the plane, axial and polar quadratic moments of
the system. The axial moments have alone a dynamical significance, but the
others are useful as subsidiary conceptions. If h1, h2, ... hn be the perpendicular
distances of the particles from any fixed plane, the sum Σ(mh2) is the quadratic
Page 412
moment with respect to the plane. If p1, p2, ... pn be the perpendicular distances
from any given axis, the sum Σ(mp2) is the quadratic moment with respect to the
axis; it is also called the moment of inertia about the axis. If r1, r2, ... rn be the
distances from a fixed point, the sum Σ(mr2) is the quadratic moment with
respect to that point (or pole). If we divide any of the above quadratic moments
by the total mass Σ(m), the result is called the mean square of the distances of the
particles from the respective plane, axis or pole. In the case of an axial moment,
the square root of the resulting mean square is called the radius of gyration of the
system about the axis in question. If we take rectangular axes through any point
O, the quadratic moments with respect to the co-ordinate planes are
Ix = Σ(mx2), Iy = Σ(my2), Iz = Σ(mz2);
(9)
those with respect to the co-ordinate axes are
Iyz = Σ {m (y2 + z2)}, Izx = Σ {m (z2 + x2)}, Ixy = Σ {m (x2 + y2)};
(10)
whilst the polar quadratic moment with respect to O is
I0 = Σ {m (x2 + y2 + z2)}.
(11)
We note that
Iyz = Iy + Iz, Izx = Iz + Ix, Ixy = Ix + Iy,
(12)
and
I0 = Ix + Iy + Iz = 1⁄2 (Iyz + Izx + Ixy).
(13)
In the case of continuous distributions of matter the summations in (9),
(10), (11) are of course to be replaced by integrations. For a uniform thin
circular plate, we find, taking the origin at its centre, and the axis of z
normal to its plane, I0 = 1⁄2Ma2, where M is the mass and a the radius. Since
from any given axis, the sum Σ(mp2) is the quadratic moment with respect to the
axis; it is also called the moment of inertia about the axis. If r1, r2, ... rn be the
distances from a fixed point, the sum Σ(mr2) is the quadratic moment with
respect to that point (or pole). If we divide any of the above quadratic moments
by the total mass Σ(m), the result is called the mean square of the distances of the
particles from the respective plane, axis or pole. In the case of an axial moment,
the square root of the resulting mean square is called the radius of gyration of the
system about the axis in question. If we take rectangular axes through any point
O, the quadratic moments with respect to the co-ordinate planes are
Ix = Σ(mx2), Iy = Σ(my2), Iz = Σ(mz2);
(9)
those with respect to the co-ordinate axes are
Iyz = Σ {m (y2 + z2)}, Izx = Σ {m (z2 + x2)}, Ixy = Σ {m (x2 + y2)};
(10)
whilst the polar quadratic moment with respect to O is
I0 = Σ {m (x2 + y2 + z2)}.
(11)
We note that
Iyz = Iy + Iz, Izx = Iz + Ix, Ixy = Ix + Iy,
(12)
and
I0 = Ix + Iy + Iz = 1⁄2 (Iyz + Izx + Ixy).
(13)
In the case of continuous distributions of matter the summations in (9),
(10), (11) are of course to be replaced by integrations. For a uniform thin
circular plate, we find, taking the origin at its centre, and the axis of z
normal to its plane, I0 = 1⁄2Ma2, where M is the mass and a the radius. Since
Page 413
Ix = Iy, Iz = 0, we deduce Izx = 1⁄2Ma2, Ixy = 1⁄2Ma2; hence the value of the
squared radius of gyration is for a diameter 1⁄4a2, and for the axis of
symmetry 1⁄2a2. Again, for a uniform solid sphere having its centre at the
origin we find I0 = 3⁄5Ma2, Ix = Iy = Iz = 1⁄5Ma2, Iyz = Izx = lxy = 3⁄5Ma2; i.e.
the square of the radius of gyration with respect to a diameter is 2⁄5a2. The
method of homogeneous strain can be applied to deduce the corresponding
results for an ellipsoid of semi-axes a, b, c. If the co-ordinate axes coincide
with the principal axes, we find Ix = 1⁄5Ma2, Iy = 1⁄5Mb2, Iz = 1⁄5Mc2, whence
Iyz = 1⁄5M (b2 + c2), &c.
If φ(x, y, z) be any homogeneous quadratic function of x, y, z, we have
Σ {mφ (x, y, z)} = Σ {mφ (x + ξ, y + η, z + ζ) }
= Σ {mφ (x, y, z)} + Σ {mφ (ξ, η, ζ)},
(14)
since the terms which are bilinear in respect to x, y, z, and ξ, η, ζ vanish, in virtue
of the relations (7). Thus
Ix = Iξ + Σ(m)x2,
(15)
Iyz = Iηζ + Σ(m) · (y2 + z2),
(16)
with similar relations, and
IO = IG + Σ(m) · OG2.
(17)
The formula (16) expresses that the squared radius of gyration about any axis
(Ox) exceeds the squared radius of gyration about a parallel axis through G by
the square of the distance between the two axes. The formula (17) is due to J. L.
Lagrange; it may be written
Σ(m · OP2) Σ(m · GP2)
= + OG2,
Σ(m) Σ(m) (18)
squared radius of gyration is for a diameter 1⁄4a2, and for the axis of
symmetry 1⁄2a2. Again, for a uniform solid sphere having its centre at the
origin we find I0 = 3⁄5Ma2, Ix = Iy = Iz = 1⁄5Ma2, Iyz = Izx = lxy = 3⁄5Ma2; i.e.
the square of the radius of gyration with respect to a diameter is 2⁄5a2. The
method of homogeneous strain can be applied to deduce the corresponding
results for an ellipsoid of semi-axes a, b, c. If the co-ordinate axes coincide
with the principal axes, we find Ix = 1⁄5Ma2, Iy = 1⁄5Mb2, Iz = 1⁄5Mc2, whence
Iyz = 1⁄5M (b2 + c2), &c.
If φ(x, y, z) be any homogeneous quadratic function of x, y, z, we have
Σ {mφ (x, y, z)} = Σ {mφ (x + ξ, y + η, z + ζ) }
= Σ {mφ (x, y, z)} + Σ {mφ (ξ, η, ζ)},
(14)
since the terms which are bilinear in respect to x, y, z, and ξ, η, ζ vanish, in virtue
of the relations (7). Thus
Ix = Iξ + Σ(m)x2,
(15)
Iyz = Iηζ + Σ(m) · (y2 + z2),
(16)
with similar relations, and
IO = IG + Σ(m) · OG2.
(17)
The formula (16) expresses that the squared radius of gyration about any axis
(Ox) exceeds the squared radius of gyration about a parallel axis through G by
the square of the distance between the two axes. The formula (17) is due to J. L.
Lagrange; it may be written
Σ(m · OP2) Σ(m · GP2)
= + OG2,
Σ(m) Σ(m) (18)
Page 414
and expresses that the mean square of the distances of the particles from O
exceeds the mean square of the distances from G by OG2. The mass-centre is
accordingly that point the mean square of whose distances from the several
particles is least. If in (18) we make O coincide with P1, P2, ... Pn in succession,
we obtain
0 + m2·P1P22 + ... + mn·P1Pn2 = Σ(m · GP2) + Σ(m) · GP12,
m1·P2P12 + 0 + ... + mn·P2Pn2 = Σ(m · GP2) + Σ(m) · GP22,
. . . . . . . . .
m1·PnP12 + m2·PnP22 + ... + 0 = Σ(m · GP2) + Σ(m) · GPn2.
exceeds the mean square of the distances from G by OG2. The mass-centre is
accordingly that point the mean square of whose distances from the several
particles is least. If in (18) we make O coincide with P1, P2, ... Pn in succession,
we obtain
0 + m2·P1P22 + ... + mn·P1Pn2 = Σ(m · GP2) + Σ(m) · GP12,
m1·P2P12 + 0 + ... + mn·P2Pn2 = Σ(m · GP2) + Σ(m) · GP22,
. . . . . . . . .
m1·PnP12 + m2·PnP22 + ... + 0 = Σ(m · GP2) + Σ(m) · GPn2.
Page 415
( )
If we multiply these equations by m1, m2 ... mn, respectively, and add, we find
ΣΣ (mrms · PrPs2) = Σ (m) · Σ (m · GP2),
(20)
provided the summation ΣΣ on the left hand be understood to include each pair of particles once only.
This theorem, also due to Lagrange, enables us to express the mean square of the distances of the
particles from the centre of mass in terms of the masses and mutual distances. For instance,
considering four equal particles at the vertices of a regular tetrahedron, we can infer that the radius R
of the circumscribing sphere is given by R2 = 3⁄8a2, if a be the length of an edge.
Another type of quadratic moment is supplied by the deviation-moments, or products of inertia of a
distribution of matter. Thus the sum Σ(m·yz) is called the “product of inertia” with respect to the
planes y = 0, z = 0. This may be expressed In terms of the product of inertia with respect to parallel
planes through G by means of the formula (14); viz.:—
Σ (m · yz) = Σ (m · ηζ) + Σ (m) · yz
(21)
The quadratic moments with respect to different planes through a fixed point O are related to one
another as follows. The moment with respect to the plane
λx + μy + νz = 0,
(22)
where λ, μ, ν are direction-cosines, is
Σ {m (λx + μy + νz)2} = Σ (mx2)·λ2 + Σ (my2)·μ2 + Σ (mz2)·ν2 + 2Σ (myz)·μν + 2Σ (mzx)·νλ + 2Σ
(mxy)·λμ,
(23)
and therefore varies as the square of the perpendicular drawn from O to a tangent plane of a certain
quadric surface, the tangent plane in question being parallel to (22). If the co-ordinate axes coincide
with the principal axes of this quadric, we shall have
Σ(myz) = 0, Σ(mzx) = 0, Σ(mxy) = 0;
(24)
and if we write
Σ(mx2) = Ma2, Σ(my2) = Mb2, Σ(mz2) = Mc2,
(25)
where M = Σ(m), the quadratic moment becomes M(a2λ2 + b2μ2 + c2ν2), or Mp2, where p is the
distance of the origin from that tangent plane of the ellipsoid
x2 y2 z2
+ + = 1,
a2 b2 c2 (26)
If we multiply these equations by m1, m2 ... mn, respectively, and add, we find
ΣΣ (mrms · PrPs2) = Σ (m) · Σ (m · GP2),
(20)
provided the summation ΣΣ on the left hand be understood to include each pair of particles once only.
This theorem, also due to Lagrange, enables us to express the mean square of the distances of the
particles from the centre of mass in terms of the masses and mutual distances. For instance,
considering four equal particles at the vertices of a regular tetrahedron, we can infer that the radius R
of the circumscribing sphere is given by R2 = 3⁄8a2, if a be the length of an edge.
Another type of quadratic moment is supplied by the deviation-moments, or products of inertia of a
distribution of matter. Thus the sum Σ(m·yz) is called the “product of inertia” with respect to the
planes y = 0, z = 0. This may be expressed In terms of the product of inertia with respect to parallel
planes through G by means of the formula (14); viz.:—
Σ (m · yz) = Σ (m · ηζ) + Σ (m) · yz
(21)
The quadratic moments with respect to different planes through a fixed point O are related to one
another as follows. The moment with respect to the plane
λx + μy + νz = 0,
(22)
where λ, μ, ν are direction-cosines, is
Σ {m (λx + μy + νz)2} = Σ (mx2)·λ2 + Σ (my2)·μ2 + Σ (mz2)·ν2 + 2Σ (myz)·μν + 2Σ (mzx)·νλ + 2Σ
(mxy)·λμ,
(23)
and therefore varies as the square of the perpendicular drawn from O to a tangent plane of a certain
quadric surface, the tangent plane in question being parallel to (22). If the co-ordinate axes coincide
with the principal axes of this quadric, we shall have
Σ(myz) = 0, Σ(mzx) = 0, Σ(mxy) = 0;
(24)
and if we write
Σ(mx2) = Ma2, Σ(my2) = Mb2, Σ(mz2) = Mc2,
(25)
where M = Σ(m), the quadratic moment becomes M(a2λ2 + b2μ2 + c2ν2), or Mp2, where p is the
distance of the origin from that tangent plane of the ellipsoid
x2 y2 z2
+ + = 1,
a2 b2 c2 (26)
Page 416
which is parallel to (22). It appears from (24) that through any assigned point O three rectangular
axes can be drawn such that the product of inertia with respect to each pair of co-ordinate planes
vanishes; these are called the principal axes of inertia at O. The ellipsoid (26) was first employed by
J. Binet (1811), and may be called “Binet’s Ellipsoid” for the point O. Evidently the quadratic
moment for a variable plane through O will have a “stationary” value when, and only when, the plane
coincides with a principal plane of (26). It may further be shown that if Binet’s ellipsoid be referred
to any system of conjugate diameters as co-ordinate axes, its equation will be
x′2 y′2 z′2
+ + = 1,
a′2 b′2 c′2 (27)
provided
Σ(mx′2) = Ma′2, Σ(my′2) Mb′2, Σ(mz′2) = Mc′2;
also that
Σ(my′z′) = 0, Σ(mz′x′) = 0, Σ(mx′y′) = 0.
(28)
Let us now take as co-ordinate axes the principal axes of inertia at the mass-centre G. If a, b, c be
the semi-axes of the Binet’s ellipsoid of G, the quadratic moment with respect to the plane λx + μy +
νz = 0 will be M(a2λ2 + b2μ2 + c2ν2), and that with respect to a parallel plane
λx + μy + νz = p
(29)
will be M (a2λ2 + b2μ2 + c2ν2 + p2), by (15). This will have a given value Mk2, provided
p2 = (k2 − a2) λ2 + (k2 − b2) μ2 + (k2 − c2) ν2.
(30)
Hence the planes of constant quadratic moment Mk2 will envelop the quadric
x2 y2 z2
+ + = 1,
k2 − a2 k2 − b2 k2 − c2 (31)
and the quadrics corresponding to different values of k2 will be confocal. If we write
k2 = a2 + b2 + c2 + θ,
b2 + c2 = α2, c2 + a2 = β2, a 2 + b2 = γ 2
(32)
the equation (31) becomes
x2 y2 z2
+ + = 1;
α2 + θ β2 + θ γ2 + θ (33)
for different values of θ this represents a system of quadrics confocal with the ellipsoid
axes can be drawn such that the product of inertia with respect to each pair of co-ordinate planes
vanishes; these are called the principal axes of inertia at O. The ellipsoid (26) was first employed by
J. Binet (1811), and may be called “Binet’s Ellipsoid” for the point O. Evidently the quadratic
moment for a variable plane through O will have a “stationary” value when, and only when, the plane
coincides with a principal plane of (26). It may further be shown that if Binet’s ellipsoid be referred
to any system of conjugate diameters as co-ordinate axes, its equation will be
x′2 y′2 z′2
+ + = 1,
a′2 b′2 c′2 (27)
provided
Σ(mx′2) = Ma′2, Σ(my′2) Mb′2, Σ(mz′2) = Mc′2;
also that
Σ(my′z′) = 0, Σ(mz′x′) = 0, Σ(mx′y′) = 0.
(28)
Let us now take as co-ordinate axes the principal axes of inertia at the mass-centre G. If a, b, c be
the semi-axes of the Binet’s ellipsoid of G, the quadratic moment with respect to the plane λx + μy +
νz = 0 will be M(a2λ2 + b2μ2 + c2ν2), and that with respect to a parallel plane
λx + μy + νz = p
(29)
will be M (a2λ2 + b2μ2 + c2ν2 + p2), by (15). This will have a given value Mk2, provided
p2 = (k2 − a2) λ2 + (k2 − b2) μ2 + (k2 − c2) ν2.
(30)
Hence the planes of constant quadratic moment Mk2 will envelop the quadric
x2 y2 z2
+ + = 1,
k2 − a2 k2 − b2 k2 − c2 (31)
and the quadrics corresponding to different values of k2 will be confocal. If we write
k2 = a2 + b2 + c2 + θ,
b2 + c2 = α2, c2 + a2 = β2, a 2 + b2 = γ 2
(32)
the equation (31) becomes
x2 y2 z2
+ + = 1;
α2 + θ β2 + θ γ2 + θ (33)
for different values of θ this represents a system of quadrics confocal with the ellipsoid
Page 417
x2 y2 z2
+ + = 1,
α2 β2 γ2 (34)
which we shall meet with presently as the “ellipsoid of gyration” at G. Now consider the tangent
plane ω at any point P of a confocal, the tangent plane ω′ at an adjacent point N′, and a plane ω″
through P parallel to ω′. The distance between the planes ω′ and ω″ will be of the second order of
small quantities, and the quadratic moments with respect to ω′ and ω″ will therefore be equal, to the
first order. Since the quadratic moments with respect to ω and ω′ are equal, it follows that ω is a plane
of stationary quadratic moment at P, and therefore a principal plane of inertia at P. In other words, the
principal axes of inertia at P arc the normals to the three confocals of the system (33) which pass
through P. Moreover if x, y, z be the co-ordinates of P, (33) is an equation to find the corresponding
values of θ; and if θ1, θ2, θ3 be the roots we find
θ1 + θ2 + θ3 = r2 − α2 − β2 − γ2,
(35)
where r2 = x2 + y2 + z2. The squares of the radii of gyration about the principal axes at P may be
denoted by k22 + k32, k32 + k12, k12 + k22; hence by (32) and (35) they are r2 −θ1, r2 − θ2, r2 − θ3,
respectively.
To find the relations between the moments of inertia about different axes through any assigned
point O, we take O as origin. Since the square of the distance of a point (x, y, z) from the axis
x y z
= =
λ μ ν (36)
is x2 + y2 + z2 − (λx + μy + νz)2, the moment of inertia about this axis is
I = Σ [m { (λ2 + μ2 + ν2) (x2 + y2 + z2) − (λx + μy + νz)2} ]
= Aλ2 + Bμ2 + Cν2 − 2Fμν − 2Gνλ − 2Hλμ,
(37)
provided
A = Σ {m (y2 + z2)}, B = Σ {m (z2 + x2)}, C = Σ {m (x2 + y2)},
F = Σ (myz), G = Σ (mzx), H = Σ (mxy);
(38)
i.e. A, B, C are the moments of inertia about the co-ordinate axes, and F, G, H are the products of
inertia with respect to the pairs of co-ordinate planes. If we construct the quadric
Ax2 + By2 + Cz2 − 2Fyz − 2Gzx − 2Hxy = Mε4
(39)
where ε is an arbitrary linear magnitude, the intercept r which it makes on a radius drawn in the
direction λ, μ, ν is found by putting x, y, z = λr, μr, νr. Hence, by comparison with (37),
+ + = 1,
α2 β2 γ2 (34)
which we shall meet with presently as the “ellipsoid of gyration” at G. Now consider the tangent
plane ω at any point P of a confocal, the tangent plane ω′ at an adjacent point N′, and a plane ω″
through P parallel to ω′. The distance between the planes ω′ and ω″ will be of the second order of
small quantities, and the quadratic moments with respect to ω′ and ω″ will therefore be equal, to the
first order. Since the quadratic moments with respect to ω and ω′ are equal, it follows that ω is a plane
of stationary quadratic moment at P, and therefore a principal plane of inertia at P. In other words, the
principal axes of inertia at P arc the normals to the three confocals of the system (33) which pass
through P. Moreover if x, y, z be the co-ordinates of P, (33) is an equation to find the corresponding
values of θ; and if θ1, θ2, θ3 be the roots we find
θ1 + θ2 + θ3 = r2 − α2 − β2 − γ2,
(35)
where r2 = x2 + y2 + z2. The squares of the radii of gyration about the principal axes at P may be
denoted by k22 + k32, k32 + k12, k12 + k22; hence by (32) and (35) they are r2 −θ1, r2 − θ2, r2 − θ3,
respectively.
To find the relations between the moments of inertia about different axes through any assigned
point O, we take O as origin. Since the square of the distance of a point (x, y, z) from the axis
x y z
= =
λ μ ν (36)
is x2 + y2 + z2 − (λx + μy + νz)2, the moment of inertia about this axis is
I = Σ [m { (λ2 + μ2 + ν2) (x2 + y2 + z2) − (λx + μy + νz)2} ]
= Aλ2 + Bμ2 + Cν2 − 2Fμν − 2Gνλ − 2Hλμ,
(37)
provided
A = Σ {m (y2 + z2)}, B = Σ {m (z2 + x2)}, C = Σ {m (x2 + y2)},
F = Σ (myz), G = Σ (mzx), H = Σ (mxy);
(38)
i.e. A, B, C are the moments of inertia about the co-ordinate axes, and F, G, H are the products of
inertia with respect to the pairs of co-ordinate planes. If we construct the quadric
Ax2 + By2 + Cz2 − 2Fyz − 2Gzx − 2Hxy = Mε4
(39)
where ε is an arbitrary linear magnitude, the intercept r which it makes on a radius drawn in the
direction λ, μ, ν is found by putting x, y, z = λr, μr, νr. Hence, by comparison with (37),
Page 418
I = Mε4 / r2.
(40)
The moment of inertia about any radius of the quadric (39) therefore varies inversely as the square of
the length of this radius. When referred to its principal axes, the equation of the quadric takes the
form
Ax2 + By2 + Cz2 = Mε4.
(41)
The directions of these axes are determined by the property (24), and therefore coincide with those
of the principal axes of inertia at O, as already defined in connexion with the theory of plane
quadratic moments. The new A, B, C are called the principal moments of inertia at O. Since they are
essentially positive the quadric is an ellipsoid; it is called the momental ellipsoid at O. Since, by (12),
B + C > A, &c., the sum of the two lesser principal moments must exceed the greatest principal
moment. A limitation is thus imposed on the possible forms of the momental ellipsoid; e.g. in the case
of symmetry about an axis it appears that the ratio of the polar to the equatorial diameter of the
ellipsoid cannot be less than 1/√2.
If we write A = Mα2, B = Mβ2, C = Mγ2, the formula (37), when referred to the principal axes at O,
becomes
I = M (α2λ2 + β2μ2 + γ2ν2) = Mp2,
(42)
if p denotes the perpendicular drawn from O in the direction (λ, μ, ν) to a tangent plane of the
ellipsoid
x2 y2 z2
+ + =1
α2 β2 γ2 (43)
This is called the ellipsoid of gyration at O; it was introduced into the theory by J. MacCullagh. The
ellipsoids (41) and (43) are reciprocal polars with respect to a sphere having O as centre.
If A = B = C, the momental ellipsoid becomes a sphere; all axes through O are then principal axes,
and the moment of inertia is the same for each. The mass-system is then said to possess kinetic
symmetry about O.
If all the masses lie in a plane (z = 0) we have, in the notation of (25), c2 = 0, and therefore A
= Mb2, B = Ma2, C = M(a2 + b2), so that the equation of the momental ellipsoid takes the form
b2x2 + a2y2 + (a2 + b2) z2 = ε4.
(44)
The section of this by the plane z = 0 is similar to
x2 + y2 = 1,
(40)
The moment of inertia about any radius of the quadric (39) therefore varies inversely as the square of
the length of this radius. When referred to its principal axes, the equation of the quadric takes the
form
Ax2 + By2 + Cz2 = Mε4.
(41)
The directions of these axes are determined by the property (24), and therefore coincide with those
of the principal axes of inertia at O, as already defined in connexion with the theory of plane
quadratic moments. The new A, B, C are called the principal moments of inertia at O. Since they are
essentially positive the quadric is an ellipsoid; it is called the momental ellipsoid at O. Since, by (12),
B + C > A, &c., the sum of the two lesser principal moments must exceed the greatest principal
moment. A limitation is thus imposed on the possible forms of the momental ellipsoid; e.g. in the case
of symmetry about an axis it appears that the ratio of the polar to the equatorial diameter of the
ellipsoid cannot be less than 1/√2.
If we write A = Mα2, B = Mβ2, C = Mγ2, the formula (37), when referred to the principal axes at O,
becomes
I = M (α2λ2 + β2μ2 + γ2ν2) = Mp2,
(42)
if p denotes the perpendicular drawn from O in the direction (λ, μ, ν) to a tangent plane of the
ellipsoid
x2 y2 z2
+ + =1
α2 β2 γ2 (43)
This is called the ellipsoid of gyration at O; it was introduced into the theory by J. MacCullagh. The
ellipsoids (41) and (43) are reciprocal polars with respect to a sphere having O as centre.
If A = B = C, the momental ellipsoid becomes a sphere; all axes through O are then principal axes,
and the moment of inertia is the same for each. The mass-system is then said to possess kinetic
symmetry about O.
If all the masses lie in a plane (z = 0) we have, in the notation of (25), c2 = 0, and therefore A
= Mb2, B = Ma2, C = M(a2 + b2), so that the equation of the momental ellipsoid takes the form
b2x2 + a2y2 + (a2 + b2) z2 = ε4.
(44)
The section of this by the plane z = 0 is similar to
x2 + y2 = 1,
Page 419
a2 b2 (45)
which may be called the momental ellipse at O. It possesses the property that the radius of
gyration about any diameter is half the distance between the two tangents which are parallel to
that diameter. In the case of a uniform triangular plate it may be shown that the momental
ellipse at G is concentric, similar and similarly situated to the ellipse which touches the sides of
the triangle at their middle points.
The graphical methods of determining the
moment of inertia of a plane system of particles
with respect to any line in its plane may be briefly
noticed. It appears from § 5 (fig. 31) that the linear
moment of each particle about the line may be
found by means of a funicular polygon. If we
replace the mass of each particle by its moment, as
thus found, we can in like manner obtain the
quadratic moment of the system with respect to the
line. For if the line in question be the axis of y, the
first process gives us the values of mx, and the
second the value of Σ(mx·x) or Σ(mx2). The
construction of a second funicular may be
dispensed with by the employment of a planimeter, Fig. 59.
as follows. In fig. 59 p is the line with respect to
which moments are to be taken, and the masses of the respective particles are indicated by the
corresponding segments of a line in the force-diagram, drawn parallel to p. The funicular
ZABCD ... corresponding to any pole O is constructed for a system of forces acting parallel to
p through the positions of the particles and proportional to the respective masses; and its
successive sides are produced to meet p in the points H, K, L, M, ... As explained in § 5, the
moment of the first particle is represented on a certain scale by HK, that of the second by KL,
and so on. The quadratic moment of the first particle will then be represented by twice the area
AHK, that of the second by twice the area BKL, and so on. The quadratic moment of the whole
system is therefore represented by twice the area AHEDCBA. Since a quadratic moment is
essentially positive, the various areas are to taken positive in all cases. If k be the radius of
gyration about p we find
k2 = 2 × area AHEDCBA × ON ÷ αβ,
which may be called the momental ellipse at O. It possesses the property that the radius of
gyration about any diameter is half the distance between the two tangents which are parallel to
that diameter. In the case of a uniform triangular plate it may be shown that the momental
ellipse at G is concentric, similar and similarly situated to the ellipse which touches the sides of
the triangle at their middle points.
The graphical methods of determining the
moment of inertia of a plane system of particles
with respect to any line in its plane may be briefly
noticed. It appears from § 5 (fig. 31) that the linear
moment of each particle about the line may be
found by means of a funicular polygon. If we
replace the mass of each particle by its moment, as
thus found, we can in like manner obtain the
quadratic moment of the system with respect to the
line. For if the line in question be the axis of y, the
first process gives us the values of mx, and the
second the value of Σ(mx·x) or Σ(mx2). The
construction of a second funicular may be
dispensed with by the employment of a planimeter, Fig. 59.
as follows. In fig. 59 p is the line with respect to
which moments are to be taken, and the masses of the respective particles are indicated by the
corresponding segments of a line in the force-diagram, drawn parallel to p. The funicular
ZABCD ... corresponding to any pole O is constructed for a system of forces acting parallel to
p through the positions of the particles and proportional to the respective masses; and its
successive sides are produced to meet p in the points H, K, L, M, ... As explained in § 5, the
moment of the first particle is represented on a certain scale by HK, that of the second by KL,
and so on. The quadratic moment of the first particle will then be represented by twice the area
AHK, that of the second by twice the area BKL, and so on. The quadratic moment of the whole
system is therefore represented by twice the area AHEDCBA. Since a quadratic moment is
essentially positive, the various areas are to taken positive in all cases. If k be the radius of
gyration about p we find
k2 = 2 × area AHEDCBA × ON ÷ αβ,
Page 420
Fig. 60.
where αβ is the line in the force-diagram which represents the sum of the masses, and ON is the
distance of the pole O from this line. If some of the particles lie on one side of p and some on
the other, the quadratic moment of each set may be found, and the results added. This is
illustrated in fig. 60, where the total quadratic moment is represented by the sum of the shaded
areas. It is seen that for a given direction of p this moment is least when p passes through the
intersection X of the first and last sides of the funicular; i.e. when p goes through the mass-
centre of the given system; cf. equation (15).
Part II.—Kinetics
§ 12. Rectilinear Motion.—Let x denote the distance OP of a moving point P at time t from a fixed
origin O on the line of motion, this distance being reckoned positive or negative according as it lies to
one side or the other of O. At time t + δt let the point be at Q, and let OQ = x + δx. The mean velocity
of the point in the interval δt is δx/δt. The limiting value of this when δt is infinitely small, viz. dx/dt,
is adopted as the definition of the velocity at the instant t. Again, let u be the velocity at time t, u + δu
that at time t + δt. The mean rate of increase of velocity, or the mean acceleration, in the interval δt is
then δu/δt. The limiting value of this when δt is infinitely small, viz., du/dt, is adopted as the
definition of the acceleration at the instant t. Since u = dx/dt, the acceleration is also denoted by
d2x/dt2. It is often convenient to use the “fluxional” notation for differential coefficients with respect
to time; thus the velocity may be represented by ẋ and the acceleration by u̇ or ẍ. There is another
formula for the acceleration, in which u is regarded as a function of the position; thus du/dt = (du/dx)
(dx/dt) = u(du/dx). The relation between x and t in any particular case may be illustrated by means of
a curve constructed with t as abscissa and x as ordinate. This is called the curve of positions or space-
time curve; its gradient represents the velocity. Such curves are often traced mechanically in
acoustical and other experiments. A, curve with t as abscissa and u as ordinate is called the curve of
velocities or velocity-time curve. Its gradient represents the acceleration, and the area (∫u dt) included
between any two ordinates represents the space described in the interval between the corresponding
instants (see fig. 62).
where αβ is the line in the force-diagram which represents the sum of the masses, and ON is the
distance of the pole O from this line. If some of the particles lie on one side of p and some on
the other, the quadratic moment of each set may be found, and the results added. This is
illustrated in fig. 60, where the total quadratic moment is represented by the sum of the shaded
areas. It is seen that for a given direction of p this moment is least when p passes through the
intersection X of the first and last sides of the funicular; i.e. when p goes through the mass-
centre of the given system; cf. equation (15).
Part II.—Kinetics
§ 12. Rectilinear Motion.—Let x denote the distance OP of a moving point P at time t from a fixed
origin O on the line of motion, this distance being reckoned positive or negative according as it lies to
one side or the other of O. At time t + δt let the point be at Q, and let OQ = x + δx. The mean velocity
of the point in the interval δt is δx/δt. The limiting value of this when δt is infinitely small, viz. dx/dt,
is adopted as the definition of the velocity at the instant t. Again, let u be the velocity at time t, u + δu
that at time t + δt. The mean rate of increase of velocity, or the mean acceleration, in the interval δt is
then δu/δt. The limiting value of this when δt is infinitely small, viz., du/dt, is adopted as the
definition of the acceleration at the instant t. Since u = dx/dt, the acceleration is also denoted by
d2x/dt2. It is often convenient to use the “fluxional” notation for differential coefficients with respect
to time; thus the velocity may be represented by ẋ and the acceleration by u̇ or ẍ. There is another
formula for the acceleration, in which u is regarded as a function of the position; thus du/dt = (du/dx)
(dx/dt) = u(du/dx). The relation between x and t in any particular case may be illustrated by means of
a curve constructed with t as abscissa and x as ordinate. This is called the curve of positions or space-
time curve; its gradient represents the velocity. Such curves are often traced mechanically in
acoustical and other experiments. A, curve with t as abscissa and u as ordinate is called the curve of
velocities or velocity-time curve. Its gradient represents the acceleration, and the area (∫u dt) included
between any two ordinates represents the space described in the interval between the corresponding
instants (see fig. 62).
Page 421
So far nothing has been said about the measurement of time. From the purely kinematic point of
view, the t of our formulae may be any continuous independent variable, suggested (it may be) by
some physical process. But from the dynamical standpoint it is obvious that equations which
represent the facts correctly on one system of time-measurement might become seriously defective
on another. It is found that for almost all purposes a system of measurement based ultimately on the
earth’s rotation is perfectly adequate. It is only when we come to consider such delicate questions as
the influence of tidal friction that other standards become necessary.
The most important conception in kinetics is that of “inertia.” It is a matter of ordinary observation
that different bodies acted on by the same force, or what is judged to be the same force, undergo
different changes of velocity in equal times. In our ideal representation of natural phenomena this is
allowed for by endowing each material particle with a suitable mass or inertia-coefficient m. The
product mu of the mass into the velocity is called the momentum or (in Newton’s phrase) the quantity
of motion. On the Newtonian system the motion of a particle entirely uninfluenced by other bodies,
when referred to a suitable base, would be rectilinear, with constant velocity. If the velocity changes,
this is attributed to the action of force; and if we agree to measure the force (X) by the rate of change
of momentum which it produces, we have the equation
d
(mu) = X.
dt (1)
From this point of view the equation is a mere truism, its real importance resting on the fact that by
attributing suitable values to the masses m, and by making simple assumptions as to the value of X in
each case, we are able to frame adequate representations of whole classes of phenomena as they
actually occur. The question remains, of course, as to how far the measurement of force here implied
is practically consistent with the gravitational method usually adopted in statics; this will be referred
to presently.
The practical unit or standard of mass must, from the nature of the case, be the mass of some
particular body, e.g. the imperial pound, or the kilogramme. In the “C.G.S.” system a subdivision of
the latter, viz. the gramme, is adopted, and is associated with the centimetre as the unit of length, and
the mean solar second as the unit of time. The unit of force implied in (1) is that which produces unit
momentum in unit time. On the C.G.S. system it is that force which acting on one gramme for one
second produces a velocity of one centimetre per second; this unit is known as the dyne. Units of this
kind are called absolute on account of their fundamental and invariable character as contrasted with
gravitational units, which (as we shall see presently) vary somewhat with the locality at which the
measurements are supposed to be made.
If we integrate the equation (1) with respect to t between the limits t, t′ we obtain
mu′ − mu = ∫tt′ X dt.
(2)
view, the t of our formulae may be any continuous independent variable, suggested (it may be) by
some physical process. But from the dynamical standpoint it is obvious that equations which
represent the facts correctly on one system of time-measurement might become seriously defective
on another. It is found that for almost all purposes a system of measurement based ultimately on the
earth’s rotation is perfectly adequate. It is only when we come to consider such delicate questions as
the influence of tidal friction that other standards become necessary.
The most important conception in kinetics is that of “inertia.” It is a matter of ordinary observation
that different bodies acted on by the same force, or what is judged to be the same force, undergo
different changes of velocity in equal times. In our ideal representation of natural phenomena this is
allowed for by endowing each material particle with a suitable mass or inertia-coefficient m. The
product mu of the mass into the velocity is called the momentum or (in Newton’s phrase) the quantity
of motion. On the Newtonian system the motion of a particle entirely uninfluenced by other bodies,
when referred to a suitable base, would be rectilinear, with constant velocity. If the velocity changes,
this is attributed to the action of force; and if we agree to measure the force (X) by the rate of change
of momentum which it produces, we have the equation
d
(mu) = X.
dt (1)
From this point of view the equation is a mere truism, its real importance resting on the fact that by
attributing suitable values to the masses m, and by making simple assumptions as to the value of X in
each case, we are able to frame adequate representations of whole classes of phenomena as they
actually occur. The question remains, of course, as to how far the measurement of force here implied
is practically consistent with the gravitational method usually adopted in statics; this will be referred
to presently.
The practical unit or standard of mass must, from the nature of the case, be the mass of some
particular body, e.g. the imperial pound, or the kilogramme. In the “C.G.S.” system a subdivision of
the latter, viz. the gramme, is adopted, and is associated with the centimetre as the unit of length, and
the mean solar second as the unit of time. The unit of force implied in (1) is that which produces unit
momentum in unit time. On the C.G.S. system it is that force which acting on one gramme for one
second produces a velocity of one centimetre per second; this unit is known as the dyne. Units of this
kind are called absolute on account of their fundamental and invariable character as contrasted with
gravitational units, which (as we shall see presently) vary somewhat with the locality at which the
measurements are supposed to be made.
If we integrate the equation (1) with respect to t between the limits t, t′ we obtain
mu′ − mu = ∫tt′ X dt.
(2)
Page 422
The time-integral on the right hand is called the impulse of the force on the interval t′ − t. The
statement that the increase of momentum is equal to the impulse is (it maybe remarked) equivalent to
Newton’s own formulation of his Second Law. The form (1) is deduced from it by putting t′ − t = δt,
and taking δt to be infinitely small. In problems of impact we have to deal with cases of practically
instantaneous impulse, where a very great and rapidly varying force produces an appreciable change
of momentum in an exceedingly minute interval of time.
In the case of a constant force, the acceleration u̇ or ẍ is, according to (1), constant, and we have
d2x
= α,
dt2 (3)
say, the general solution of which is
x = 1⁄2 αt2 + At + B.
(4)
The “arbitrary constants” A, B enable us to represent the circumstances of any particular case; thus if
the velocity ẋ and the position x be given for any one value of t, we have two conditions to determine
A, B. The curve of positions corresponding to (4) is a parabola, and that of velocities is a straight line.
We may take it as an experimental result, although the best evidence is indirect, that a particle falling
freely under gravity experiences a constant acceleration which at the same place is the same for all
bodies. This acceleration is denoted by g; its value at Greenwich is about 981 centimetre-second
units, or 32.2 feet per second. It increases somewhat with the latitude, the extreme variation from the
equator to the pole being about 1⁄2%. We infer that on our reckoning the force of gravity on a mass m
is to be measured by mg, the momentum produced per second when this force acts alone. Since this is
proportional to the mass, the relative masses to be attributed to various bodies can be determined
practically by means of the balance. We learn also that on account of the variation of g with the
locality a gravitational system of force-measurement is inapplicable when more than a moderate
degree of accuracy is desired.
We take next the case of a particle attracted towards a fixed point O in the line of motion with a
force varying as the distance from that point. If μ be the acceleration at unit distance, the equation of
motion becomes
d2x
= −μx,
dt2 (5)
the solution of which may be written in either of the forms
x = A cos σt + B sin σt, x = a cos (σt + ε),
(6)
where σ= √μ, and the two constants A, B or a, ε are arbitrary. The particle oscillates between the two
positions x = ±a, and the same point is passed through in the same direction with the same velocity at
equal intervals of time 2π/σ. The type of motion represented by (6) is of fundamental importance in
statement that the increase of momentum is equal to the impulse is (it maybe remarked) equivalent to
Newton’s own formulation of his Second Law. The form (1) is deduced from it by putting t′ − t = δt,
and taking δt to be infinitely small. In problems of impact we have to deal with cases of practically
instantaneous impulse, where a very great and rapidly varying force produces an appreciable change
of momentum in an exceedingly minute interval of time.
In the case of a constant force, the acceleration u̇ or ẍ is, according to (1), constant, and we have
d2x
= α,
dt2 (3)
say, the general solution of which is
x = 1⁄2 αt2 + At + B.
(4)
The “arbitrary constants” A, B enable us to represent the circumstances of any particular case; thus if
the velocity ẋ and the position x be given for any one value of t, we have two conditions to determine
A, B. The curve of positions corresponding to (4) is a parabola, and that of velocities is a straight line.
We may take it as an experimental result, although the best evidence is indirect, that a particle falling
freely under gravity experiences a constant acceleration which at the same place is the same for all
bodies. This acceleration is denoted by g; its value at Greenwich is about 981 centimetre-second
units, or 32.2 feet per second. It increases somewhat with the latitude, the extreme variation from the
equator to the pole being about 1⁄2%. We infer that on our reckoning the force of gravity on a mass m
is to be measured by mg, the momentum produced per second when this force acts alone. Since this is
proportional to the mass, the relative masses to be attributed to various bodies can be determined
practically by means of the balance. We learn also that on account of the variation of g with the
locality a gravitational system of force-measurement is inapplicable when more than a moderate
degree of accuracy is desired.
We take next the case of a particle attracted towards a fixed point O in the line of motion with a
force varying as the distance from that point. If μ be the acceleration at unit distance, the equation of
motion becomes
d2x
= −μx,
dt2 (5)
the solution of which may be written in either of the forms
x = A cos σt + B sin σt, x = a cos (σt + ε),
(6)
where σ= √μ, and the two constants A, B or a, ε are arbitrary. The particle oscillates between the two
positions x = ±a, and the same point is passed through in the same direction with the same velocity at
equal intervals of time 2π/σ. The type of motion represented by (6) is of fundamental importance in
Page 423
the theory of vibrations (§ 23); it is called a simple-harmonic or
(shortly) a simple vibration. If we imagine a point Q to describe a
circle of radius a with the angular velocity σ, its orthogonal projection
P on a fixed diameter AA′ will execute a vibration of this character.
The angle σt + ε (or AOQ) is called the phase; the arbitrary elements
a, ε are called the amplitude and epoch (or initial phase), respectively.
In the case of very rapid vibrations it is usual to specify, not the period
(2π/σ), but its reciprocal the frequency, i.e. the number of complete Fig. 61.
vibrations per unit time. Fig. 62 shows the curves of position and
velocity; they both have the form of the “curve of sines.” The numbers correspond to an amplitude of
10 centimetres and a period of two seconds.
The vertical oscillations of a weight which hangs from a fixed point by a spiral spring come under
this case. If M be the mass, and x the vertical displacement from the position of equilibrium, the
equation of motion is of the form
d2x
M = − Kx,
dt2 (7)
provided the inertia of the spring itself be neglected. This becomes identical with (5) if we put μ =
K/M; and the period is therefore 2π√(M/K), the same for all amplitudes. The period is increased by
an increase of the mass M, and diminished by an increase in the stiffness (K) of the spring. If c be the
statical increase of length which is produced by the gravity of the mass M, we have Kc = Mg, and the
period is 2π√(c/g).
Fig. 62.
The small oscillations of a simple pendulum in a vertical plane also come under equation (5).
According to the principles of § 13, the horizontal motion of the bob is affected only by the
horizontal component of the force acting upon it. If the inclination of the string to the vertical does
not exceed a few degrees, the vertical displacement of the particle is of the second order, so that the
vertical acceleration may be neglected, and the tension of the string may be equated to the gravity mg
(shortly) a simple vibration. If we imagine a point Q to describe a
circle of radius a with the angular velocity σ, its orthogonal projection
P on a fixed diameter AA′ will execute a vibration of this character.
The angle σt + ε (or AOQ) is called the phase; the arbitrary elements
a, ε are called the amplitude and epoch (or initial phase), respectively.
In the case of very rapid vibrations it is usual to specify, not the period
(2π/σ), but its reciprocal the frequency, i.e. the number of complete Fig. 61.
vibrations per unit time. Fig. 62 shows the curves of position and
velocity; they both have the form of the “curve of sines.” The numbers correspond to an amplitude of
10 centimetres and a period of two seconds.
The vertical oscillations of a weight which hangs from a fixed point by a spiral spring come under
this case. If M be the mass, and x the vertical displacement from the position of equilibrium, the
equation of motion is of the form
d2x
M = − Kx,
dt2 (7)
provided the inertia of the spring itself be neglected. This becomes identical with (5) if we put μ =
K/M; and the period is therefore 2π√(M/K), the same for all amplitudes. The period is increased by
an increase of the mass M, and diminished by an increase in the stiffness (K) of the spring. If c be the
statical increase of length which is produced by the gravity of the mass M, we have Kc = Mg, and the
period is 2π√(c/g).
Fig. 62.
The small oscillations of a simple pendulum in a vertical plane also come under equation (5).
According to the principles of § 13, the horizontal motion of the bob is affected only by the
horizontal component of the force acting upon it. If the inclination of the string to the vertical does
not exceed a few degrees, the vertical displacement of the particle is of the second order, so that the
vertical acceleration may be neglected, and the tension of the string may be equated to the gravity mg
Page 424
of the particle. Hence if l be the length of the string, and x the horizontal displacement of the bob
from the equilibrium position, the horizontal component of gravity is mgx/l, whence
d2x gx
=− ,
dt2 l (8)
The motion is therefore simple-harmonic, of period τ = 2π√(l/g). This indicates an experimental
method of determining g with considerable accuracy, using the formula g = 4π2l/τ2.
In the case of a repulsive force varying as the distance from the origin, the equation of
motion is of the type
d2x
= μx,
dt2 (9)
the solution of which is
x = Aent + Be−nt,
(10)
where n = √μ. Unless the initial conditions be adjusted so as to make A = 0 exactly, x will
ultimately increase indefinitely with t. The position x = 0 is one of equilibrium, but it is
unstable. This applies to the inverted pendulum, with μ = g/l, but the equation (9) is then only
approximate, and the solution therefore only serves to represent the initial stages of a motion in
the neighbourhood of the position of unstable equilibrium.
In acoustics we meet with the case where a body is urged towards a fixed point by a force varying
as the distance, and is also acted upon by an “extraneous” or “disturbing” force which is a given
function of the time. The most important case is where this function is simple-harmonic, so that the
equation (5) is replaced by
d2x
+ μx = ƒ cos (σ1t + α),
dt2 (11)
where σ1 is prescribed. A particular solution is
ƒ
x= cos (σ1t + α).
μ − σ12 (12)
This represents a forced oscillation whose period 2π/σ1, coincides with that of the disturbing force;
and the phase agrees with that of the force, or is opposed to it, according as σ12 < or > μ; i.e.
according as the imposed period is greater or less than the natural period 2π/√μ. The solution fails
when the two periods agree exactly; the formula (12) is then replaced by
ƒt
x= sin (σ1t + α),
2σ1 (13)
from the equilibrium position, the horizontal component of gravity is mgx/l, whence
d2x gx
=− ,
dt2 l (8)
The motion is therefore simple-harmonic, of period τ = 2π√(l/g). This indicates an experimental
method of determining g with considerable accuracy, using the formula g = 4π2l/τ2.
In the case of a repulsive force varying as the distance from the origin, the equation of
motion is of the type
d2x
= μx,
dt2 (9)
the solution of which is
x = Aent + Be−nt,
(10)
where n = √μ. Unless the initial conditions be adjusted so as to make A = 0 exactly, x will
ultimately increase indefinitely with t. The position x = 0 is one of equilibrium, but it is
unstable. This applies to the inverted pendulum, with μ = g/l, but the equation (9) is then only
approximate, and the solution therefore only serves to represent the initial stages of a motion in
the neighbourhood of the position of unstable equilibrium.
In acoustics we meet with the case where a body is urged towards a fixed point by a force varying
as the distance, and is also acted upon by an “extraneous” or “disturbing” force which is a given
function of the time. The most important case is where this function is simple-harmonic, so that the
equation (5) is replaced by
d2x
+ μx = ƒ cos (σ1t + α),
dt2 (11)
where σ1 is prescribed. A particular solution is
ƒ
x= cos (σ1t + α).
μ − σ12 (12)
This represents a forced oscillation whose period 2π/σ1, coincides with that of the disturbing force;
and the phase agrees with that of the force, or is opposed to it, according as σ12 < or > μ; i.e.
according as the imposed period is greater or less than the natural period 2π/√μ. The solution fails
when the two periods agree exactly; the formula (12) is then replaced by
ƒt
x= sin (σ1t + α),
2σ1 (13)
Page 425
which represents a vibration of continually increasing amplitude. Since the equation (12) is in
practice generally only an approximation (as in the case of the pendulum), this solution can only be
accepted as a representation of the initial stages of the forced oscillation. To obtain the complete
solution of (11) we must of course superpose the free vibration (6) with its arbitrary constants in
order to obtain a complete representation of the most general motion consequent on arbitrary initial
conditions.
A simple mechanical illustration is afforded by the pendulum. If the point of suspension have
an imposed simple vibration ξ = a cos σt in a horizontal line, the equation of small motion of
the bob is
x−ξ
mẍ = −mg ,
l
or
gx ξ
ẍ+ =g .
l l (14)
This is the same as if the point of suspension
were fixed, and a horizontal disturbing force
mgξ/l were to act on the bob. The difference of
phase of the forced vibration in the two cases is
illustrated and explained in the annexed fig. 63,
where the pendulum virtually oscillates about C
as a fixed point of suspension. This illustration
was given by T. Young in connexion with the
kinetic theory of the tides, where the same point
arises. Fig. 63.
We may notice also the case of an attractive
force varying inversely as the square of the distance from the origin. If μ be the acceleration at
unit distance, we have
du μ
u =−
dx x2 (15)
whence
2μ
u2 = + C.
x (16)
In the case of a particle falling directly towards the earth from rest at a very great distance we
have C = 0 and, by Newton’s Law of Gravitation, μ/a2 = g, where a is the earth’s radius. The
practice generally only an approximation (as in the case of the pendulum), this solution can only be
accepted as a representation of the initial stages of the forced oscillation. To obtain the complete
solution of (11) we must of course superpose the free vibration (6) with its arbitrary constants in
order to obtain a complete representation of the most general motion consequent on arbitrary initial
conditions.
A simple mechanical illustration is afforded by the pendulum. If the point of suspension have
an imposed simple vibration ξ = a cos σt in a horizontal line, the equation of small motion of
the bob is
x−ξ
mẍ = −mg ,
l
or
gx ξ
ẍ+ =g .
l l (14)
This is the same as if the point of suspension
were fixed, and a horizontal disturbing force
mgξ/l were to act on the bob. The difference of
phase of the forced vibration in the two cases is
illustrated and explained in the annexed fig. 63,
where the pendulum virtually oscillates about C
as a fixed point of suspension. This illustration
was given by T. Young in connexion with the
kinetic theory of the tides, where the same point
arises. Fig. 63.
We may notice also the case of an attractive
force varying inversely as the square of the distance from the origin. If μ be the acceleration at
unit distance, we have
du μ
u =−
dx x2 (15)
whence
2μ
u2 = + C.
x (16)
In the case of a particle falling directly towards the earth from rest at a very great distance we
have C = 0 and, by Newton’s Law of Gravitation, μ/a2 = g, where a is the earth’s radius. The
Page 426
deviation of the earth’s figure from sphericity, and the variation of g with latitude, are here
ignored. We find that the velocity with which the particle would arrive at the earth’s surface (x
= a) is √(2ga). If we take as rough values a = 21 × 106 feet, g = 32 foot-second units, we get a
velocity of 36,500 feet, or about seven miles, per second. If the particles start from rest at a
finite distance c, we have in (16), C = − 2μ/c, and therefore
dx 2μ (c − x)
dt = u = − √{ cx }, (17)
the minus sign indicating motion towards the origin. If we put x = c cos2 1⁄2φ, we find
c3/2
t= (φ + sin φ),
√(8μ) (18)
no additive constant being necessary if t be reckoned from the instant of starting, when φ = 0.
The time t of reaching the origin (φ = π) is
π c3/2
t1 = .
√(8μ) (19)
This may be compared with the period of revolution in a circular orbit of radius c about the
same centre of force, viz. 2πc3/2 / √μ (§ 14). We learn that if the orbital motion of a planet, or a
satellite, were arrested, the body would fall into the sun, or into its primary, in the fraction
0.1768 of its actual periodic time. Thus the moon would reach the earth in about five days. It
may be noticed that if the scales of x and t be properly adjusted, the curve of positions in the
present problem is the portion of a cycloid extending from a vertex to a cusp.
In any case of rectilinear motion, if we integrate both sides of the equation
du
mu = X,
dx (20)
which is equivalent to (1), with respect to x between the limits x0, x1, we obtain
⁄ mu12 − 1⁄2 mu02 = ∫x1
12 x0 X dx.
(21)
We recognize the right-hand member as the work done by the force X on the particle as the latter
moves from the position x0 to the position x1. If we construct a curve with x as abscissa and X as
ordinate, this work is represented, as in J. Watt’s “indicator-diagram,” by the area cut off by the
ordinates x = x0, x = x1. The product 1⁄2mu2 is called the kinetic energy of the particle, and the
equation (21) is therefore equivalent to the statement that the increment of the kinetic energy is equal
to the work done on the particle. If the force X be always the same in the same position, the particle
may be regarded as moving in a certain invariable “field of force.” The work which would have to be
ignored. We find that the velocity with which the particle would arrive at the earth’s surface (x
= a) is √(2ga). If we take as rough values a = 21 × 106 feet, g = 32 foot-second units, we get a
velocity of 36,500 feet, or about seven miles, per second. If the particles start from rest at a
finite distance c, we have in (16), C = − 2μ/c, and therefore
dx 2μ (c − x)
dt = u = − √{ cx }, (17)
the minus sign indicating motion towards the origin. If we put x = c cos2 1⁄2φ, we find
c3/2
t= (φ + sin φ),
√(8μ) (18)
no additive constant being necessary if t be reckoned from the instant of starting, when φ = 0.
The time t of reaching the origin (φ = π) is
π c3/2
t1 = .
√(8μ) (19)
This may be compared with the period of revolution in a circular orbit of radius c about the
same centre of force, viz. 2πc3/2 / √μ (§ 14). We learn that if the orbital motion of a planet, or a
satellite, were arrested, the body would fall into the sun, or into its primary, in the fraction
0.1768 of its actual periodic time. Thus the moon would reach the earth in about five days. It
may be noticed that if the scales of x and t be properly adjusted, the curve of positions in the
present problem is the portion of a cycloid extending from a vertex to a cusp.
In any case of rectilinear motion, if we integrate both sides of the equation
du
mu = X,
dx (20)
which is equivalent to (1), with respect to x between the limits x0, x1, we obtain
⁄ mu12 − 1⁄2 mu02 = ∫x1
12 x0 X dx.
(21)
We recognize the right-hand member as the work done by the force X on the particle as the latter
moves from the position x0 to the position x1. If we construct a curve with x as abscissa and X as
ordinate, this work is represented, as in J. Watt’s “indicator-diagram,” by the area cut off by the
ordinates x = x0, x = x1. The product 1⁄2mu2 is called the kinetic energy of the particle, and the
equation (21) is therefore equivalent to the statement that the increment of the kinetic energy is equal
to the work done on the particle. If the force X be always the same in the same position, the particle
may be regarded as moving in a certain invariable “field of force.” The work which would have to be
Page 427
supplied by other forces, extraneous to the field, in order to bring the particle from rest in some
standard position P0 to rest in any assigned position P, will depend only on the position of P; it is
called the statical or potential energy of the particle with respect to the field, in the position P.
Denoting this by V, we have δV − Xδx = 0, whence
dV
X=− .
dx (22)
The equation (21) may now be written
⁄ mu12 + V1 = 1⁄2 mu02 + V0,
12
(23)
which asserts that when no extraneous forces act the sum of the kinetic and potential energies is
constant. Thus in the case of a weight hanging by a spiral spring the work required to increase the
length by x is V = ∫0x Kx dx = 1⁄2Kx2, whence 1⁄2Mu2 + 1⁄2Kx2 = const., as is easily verified from
preceding results. It is easily seen that the effect of extraneous forces will be to increase the sum of
the kinetic and potential energies by an amount equal to the work done by them. If this amount be
negative the sum in question is diminished by a corresponding amount. It appears then that this sum
is a measure of the total capacity for doing work against extraneous resistances which the particle
possesses in virtue of its motion and its position; this is in fact the origin of the term “energy.” The
product mv2 had been called by G. W. Leibnitz the “vis viva”; the name “energy” was substituted by
T. Young; finally the name “actual energy” was appropriated to the expression 1⁄2mv2 by W. J. M.
Rankine.
The laws which regulate the resistance of a medium such as air to the motion of bodies
through it are only imperfectly known. We may briefly notice the case of resistance varying as
the square of the velocity, which is mathematically simple. If the positive direction of x be
downwards, the equation of motion of a falling particle will be of the form
du
= g − ku2;
dt (24)
this shows that the velocity u will send asymptotically to a certain limit V (called the terminal
velocity) such that kV2 = g. The solution is
gt V2 gt
u = V tanh , x= log cosh ,
V g V (25)
if the particle start from rest in the position x = 0 at the instant t = 0. In the case of a particle
projected vertically upwards we have
du
= −g − ku2,
dt (26)
standard position P0 to rest in any assigned position P, will depend only on the position of P; it is
called the statical or potential energy of the particle with respect to the field, in the position P.
Denoting this by V, we have δV − Xδx = 0, whence
dV
X=− .
dx (22)
The equation (21) may now be written
⁄ mu12 + V1 = 1⁄2 mu02 + V0,
12
(23)
which asserts that when no extraneous forces act the sum of the kinetic and potential energies is
constant. Thus in the case of a weight hanging by a spiral spring the work required to increase the
length by x is V = ∫0x Kx dx = 1⁄2Kx2, whence 1⁄2Mu2 + 1⁄2Kx2 = const., as is easily verified from
preceding results. It is easily seen that the effect of extraneous forces will be to increase the sum of
the kinetic and potential energies by an amount equal to the work done by them. If this amount be
negative the sum in question is diminished by a corresponding amount. It appears then that this sum
is a measure of the total capacity for doing work against extraneous resistances which the particle
possesses in virtue of its motion and its position; this is in fact the origin of the term “energy.” The
product mv2 had been called by G. W. Leibnitz the “vis viva”; the name “energy” was substituted by
T. Young; finally the name “actual energy” was appropriated to the expression 1⁄2mv2 by W. J. M.
Rankine.
The laws which regulate the resistance of a medium such as air to the motion of bodies
through it are only imperfectly known. We may briefly notice the case of resistance varying as
the square of the velocity, which is mathematically simple. If the positive direction of x be
downwards, the equation of motion of a falling particle will be of the form
du
= g − ku2;
dt (24)
this shows that the velocity u will send asymptotically to a certain limit V (called the terminal
velocity) such that kV2 = g. The solution is
gt V2 gt
u = V tanh , x= log cosh ,
V g V (25)
if the particle start from rest in the position x = 0 at the instant t = 0. In the case of a particle
projected vertically upwards we have
du
= −g − ku2,
dt (26)
Page 428
the positive direction being now upwards. This leads to
u u0 gt V2 V2 + u02
tan−1 = tan−1 − , x= log ,
V V V 2g V 2 + u2 (27)
where u0 is the velocity of projection. The particle comes to rest when
V u0 V2 u02
t=
g
tan−1
V
, x=
2g log ( 1 + V ).
2
(28)
For small velocities the resistance of the air is more nearly proportional to the first power of
the velocity. The effect of forces of this type on small vibratory motions may be investigated as
follows. The equation (5) when modified by the introduction of a frictional term becomes
ẍ = −μx − kẋ.
(29)
If k2 < 4μ the solution is
x = a e−t/τ cos (σt + ε),
(30)
where
τ = 2/k, σ = √(μ − 1⁄4k2),
(31)
and the constants a, ε are arbitrary. This may be described as a simple harmonic oscillation
whose amplitude diminishes asymptotically to zero according to the law e−t/τ. The constant τ is
called the modulus of decay of the oscillations; if it is large compared with 2π/σ the effect of
friction on the period is of the second order of small quantities and may in general be ignored.
We have seen that a true simple-harmonic vibration may be regarded as the orthogonal
projection of uniform circular motion; it was pointed out by P. G. Tait that a similar
representation of the type (30) is obtained if we replace the circle by an equiangular spiral
described, with a constant angular velocity about the pole, in the direction of diminishing
radius vector. When k2 > 4μ, the solution of (29) is, in real form,
x = a1e−t/τ1 + a2e−t/τ2,
(32)
where
1/τ1, 1/τ2 = 1⁄2k ± √(1⁄4k2 − μ).
(33)
The body now passes once (at most) through its equilibrium position, and the vibration is
therefore styled aperiodic.
u u0 gt V2 V2 + u02
tan−1 = tan−1 − , x= log ,
V V V 2g V 2 + u2 (27)
where u0 is the velocity of projection. The particle comes to rest when
V u0 V2 u02
t=
g
tan−1
V
, x=
2g log ( 1 + V ).
2
(28)
For small velocities the resistance of the air is more nearly proportional to the first power of
the velocity. The effect of forces of this type on small vibratory motions may be investigated as
follows. The equation (5) when modified by the introduction of a frictional term becomes
ẍ = −μx − kẋ.
(29)
If k2 < 4μ the solution is
x = a e−t/τ cos (σt + ε),
(30)
where
τ = 2/k, σ = √(μ − 1⁄4k2),
(31)
and the constants a, ε are arbitrary. This may be described as a simple harmonic oscillation
whose amplitude diminishes asymptotically to zero according to the law e−t/τ. The constant τ is
called the modulus of decay of the oscillations; if it is large compared with 2π/σ the effect of
friction on the period is of the second order of small quantities and may in general be ignored.
We have seen that a true simple-harmonic vibration may be regarded as the orthogonal
projection of uniform circular motion; it was pointed out by P. G. Tait that a similar
representation of the type (30) is obtained if we replace the circle by an equiangular spiral
described, with a constant angular velocity about the pole, in the direction of diminishing
radius vector. When k2 > 4μ, the solution of (29) is, in real form,
x = a1e−t/τ1 + a2e−t/τ2,
(32)
where
1/τ1, 1/τ2 = 1⁄2k ± √(1⁄4k2 − μ).
(33)
The body now passes once (at most) through its equilibrium position, and the vibration is
therefore styled aperiodic.
Page 429
To find the forced oscillation due to a periodic force we have
ẍ + kẋ + μx = ƒ cos (σ1t + ε).
(34)
The solution is
ƒ
x= cos (σ1t + ε − ε1),
R (35)
provided
kσ1
R = { (μ − σ12)2 + k2σ12}1/2, tan ε1 = .
μ − σ12 (36)
Hence the phase of the vibration lags behind that of the force by the amount ε1, which lies
between 0 and 1⁄2π or between 1⁄2π and π, according as σ12 ≶ μ. If the friction be comparatively
slight the amplitude is greatest when the imposed period coincides with the free period, being
then equal to ƒ/kσ1, and therefore very great compared with that due to a slowly varying force
of the same average intensity. We have here, in principle, the explanation of the phenomenon of
“resonance” in acoustics. The abnormal amplitude is greater, and is restricted to a narrower
range of frequency, the smaller the friction. For a complete solution of (34) we must of course
superpose the free vibration (30); but owing to the factor e−t/τ the influence of the initial
conditions gradually disappears.
For purposes of mathematical treatment a force which produces a finite change of velocity in a
time too short to be appreciated is regarded as infinitely great, and the time of action as infinitely
short. The whole effect is summed up in the value of the instantaneous impulse, which is the time-
integral of the force. Thus if an instantaneous impulse ξ changes the velocity of a mass m from u to u′
we have
mu′ − mu = ξ.
(37)
The effect of ordinary finite forces during the infinitely short duration of this impulse is of course
ignored.
We may apply this to the theory of impact. If two masses m1, m2 moving in the same straight line
impinge, with the result that the velocities are changed from u1, u2, to u1′, u2′, then, since the
impulses on the two bodies must be equal and opposite, the total momentum is unchanged, i.e.
m1u1′ + m2u2′ = m1u1 + m2u2.
(38)
The complete determination of the result of a collision under given circumstances is not a matter of
abstract dynamics alone, but requires some auxiliary assumption. If we assume that there is no loss of
ẍ + kẋ + μx = ƒ cos (σ1t + ε).
(34)
The solution is
ƒ
x= cos (σ1t + ε − ε1),
R (35)
provided
kσ1
R = { (μ − σ12)2 + k2σ12}1/2, tan ε1 = .
μ − σ12 (36)
Hence the phase of the vibration lags behind that of the force by the amount ε1, which lies
between 0 and 1⁄2π or between 1⁄2π and π, according as σ12 ≶ μ. If the friction be comparatively
slight the amplitude is greatest when the imposed period coincides with the free period, being
then equal to ƒ/kσ1, and therefore very great compared with that due to a slowly varying force
of the same average intensity. We have here, in principle, the explanation of the phenomenon of
“resonance” in acoustics. The abnormal amplitude is greater, and is restricted to a narrower
range of frequency, the smaller the friction. For a complete solution of (34) we must of course
superpose the free vibration (30); but owing to the factor e−t/τ the influence of the initial
conditions gradually disappears.
For purposes of mathematical treatment a force which produces a finite change of velocity in a
time too short to be appreciated is regarded as infinitely great, and the time of action as infinitely
short. The whole effect is summed up in the value of the instantaneous impulse, which is the time-
integral of the force. Thus if an instantaneous impulse ξ changes the velocity of a mass m from u to u′
we have
mu′ − mu = ξ.
(37)
The effect of ordinary finite forces during the infinitely short duration of this impulse is of course
ignored.
We may apply this to the theory of impact. If two masses m1, m2 moving in the same straight line
impinge, with the result that the velocities are changed from u1, u2, to u1′, u2′, then, since the
impulses on the two bodies must be equal and opposite, the total momentum is unchanged, i.e.
m1u1′ + m2u2′ = m1u1 + m2u2.
(38)
The complete determination of the result of a collision under given circumstances is not a matter of
abstract dynamics alone, but requires some auxiliary assumption. If we assume that there is no loss of
Page 430
apparent kinetic energy we have also
m1u1′2 + m2u2′2 = m1u12 + m2u22.
(39)
Hence, and from (38),
u2′ − u1′ = −(u2 − u1),
(40)
i.e. the relative velocity of the two bodies is reversed in direction, but unaltered in magnitude. This
appears to be the case very approximately with steel or glass balls; generally, however, there is some
appreciable loss of apparent energy; this is accounted for by vibrations produced in the balls and
imperfect elasticity of the materials. The usual empirical assumption is that
u2′ − u1′ = −e (u2 − u1),
(41)
where e is a proper fraction which is constant for the same two bodies. It follows from the formula §
15 (10) for the internal kinetic energy of a system of particles that as a result of the impact this energy
is diminished by the amount
m1m2
⁄ (1 − e2)
12 (u1 − u2)2.
m1 + m 2 (42)
The further theoretical discussion of the subject belongs to Elasticity.
This is perhaps the most suitable place for a few remarks on the theory of “dimensions.” (See also
Units, Dimensions of.) In any absolute system of dynamical measurement the fundamental units are
those of mass, length and time; we may denote them by the symbols M, L, T, respectively. They may
be chosen quite arbitrarily, e.g. on the C.G.S. system they are the gramme, centimetre and second. All
other units are derived from these. Thus the unit of velocity is that of a point describing the unit of
length in the unit of time; it may be denoted by LT−1, this symbol indicating that the magnitude of the
unit in question varies directly as the unit of length and inversely as the unit of time. The unit of
acceleration is the acceleration of a point which gains unit velocity in unit time; it is accordingly
denoted by LT−2. The unit of momentum is MLT−1; the unit force generates unit momentum in unit
time and is therefore denoted by MLT−2. The unit of work on the same principles is ML2T−2, and it is
to be noticed that this is identical with the unit of kinetic energy. Some of these derivative units have
special names assigned to them; thus on the C.G.S. system the unit of force is called the dyne, and the
unit of work or energy the erg. The number which expresses a physical quantity of any particular
kind will of course vary inversely as the magnitude of the corresponding unit. In any general
dynamical equation the dimensions of each term in the fundamental units must be the same, for a
change of units would otherwise alter the various terms in different ratios. This principle is often
useful as a check on the accuracy of an equation.
m1u1′2 + m2u2′2 = m1u12 + m2u22.
(39)
Hence, and from (38),
u2′ − u1′ = −(u2 − u1),
(40)
i.e. the relative velocity of the two bodies is reversed in direction, but unaltered in magnitude. This
appears to be the case very approximately with steel or glass balls; generally, however, there is some
appreciable loss of apparent energy; this is accounted for by vibrations produced in the balls and
imperfect elasticity of the materials. The usual empirical assumption is that
u2′ − u1′ = −e (u2 − u1),
(41)
where e is a proper fraction which is constant for the same two bodies. It follows from the formula §
15 (10) for the internal kinetic energy of a system of particles that as a result of the impact this energy
is diminished by the amount
m1m2
⁄ (1 − e2)
12 (u1 − u2)2.
m1 + m 2 (42)
The further theoretical discussion of the subject belongs to Elasticity.
This is perhaps the most suitable place for a few remarks on the theory of “dimensions.” (See also
Units, Dimensions of.) In any absolute system of dynamical measurement the fundamental units are
those of mass, length and time; we may denote them by the symbols M, L, T, respectively. They may
be chosen quite arbitrarily, e.g. on the C.G.S. system they are the gramme, centimetre and second. All
other units are derived from these. Thus the unit of velocity is that of a point describing the unit of
length in the unit of time; it may be denoted by LT−1, this symbol indicating that the magnitude of the
unit in question varies directly as the unit of length and inversely as the unit of time. The unit of
acceleration is the acceleration of a point which gains unit velocity in unit time; it is accordingly
denoted by LT−2. The unit of momentum is MLT−1; the unit force generates unit momentum in unit
time and is therefore denoted by MLT−2. The unit of work on the same principles is ML2T−2, and it is
to be noticed that this is identical with the unit of kinetic energy. Some of these derivative units have
special names assigned to them; thus on the C.G.S. system the unit of force is called the dyne, and the
unit of work or energy the erg. The number which expresses a physical quantity of any particular
kind will of course vary inversely as the magnitude of the corresponding unit. In any general
dynamical equation the dimensions of each term in the fundamental units must be the same, for a
change of units would otherwise alter the various terms in different ratios. This principle is often
useful as a check on the accuracy of an equation.
Page 431
The theory of dimensions often enables us to forecast, to some extent, the manner in which
the magnitudes involved in any particular problem will enter into the result. Thus, assuming
that the period of a small oscillation of a given pendulum at a given place is a definite quantity,
we see that it must vary as √(l/g). For it can only depend on the mass m of the bob, the length l
of the string, and the value of g at the place in question; and the above expression is the only
combination of these symbols whose dimensions are those of a time, simply. Again, the time of
falling from a distance a into a given centre of force varying inversely as the square of the
distance will depend only on a and on the constant μ of equation (15). The dimensions of μ/x2
are those of an acceleration; hence the dimensions of μ are L3T−2. Assuming that the time in
question varies as axμy, whose dimensions are Lx+3yT−2y, we must have x + 3y = 0, −2y = 1, so
that the time of falling will vary as a3/2/√μ, in agreement with (19).
The argument appears in a more demonstrative form in the theory of “similar” systems, or
(more precisely) of the similar motion of similar systems. Thus, considering the equations
d2x μ d2x′ μ′
=− , =− ,
dt2 x2 dt′2 x′2 (43)
which refer to two particles falling independently into two distinct centres of force, it is
obvious that it is possible to have x in a constant ratio to x′, and t in a constant ratio to t′,
provided that
x x′ μ μ′
: = : ,
t2 t′2 x2 x′2 (44)
and that there is a suitable correspondence between the initial conditions. The relation (44) is
equivalent to
x3/2 x′3/2
t : t′ = : ,
μ1/2 μ′1/2 (45)
where x, x′ are any two corresponding distances; e.g. they may be the initial distances, both
particles being supposed to start from rest. The consideration of dimensions was introduced by
J. B. Fourier (1822) in connexion with the conduction of heat.
Fig. 64.
the magnitudes involved in any particular problem will enter into the result. Thus, assuming
that the period of a small oscillation of a given pendulum at a given place is a definite quantity,
we see that it must vary as √(l/g). For it can only depend on the mass m of the bob, the length l
of the string, and the value of g at the place in question; and the above expression is the only
combination of these symbols whose dimensions are those of a time, simply. Again, the time of
falling from a distance a into a given centre of force varying inversely as the square of the
distance will depend only on a and on the constant μ of equation (15). The dimensions of μ/x2
are those of an acceleration; hence the dimensions of μ are L3T−2. Assuming that the time in
question varies as axμy, whose dimensions are Lx+3yT−2y, we must have x + 3y = 0, −2y = 1, so
that the time of falling will vary as a3/2/√μ, in agreement with (19).
The argument appears in a more demonstrative form in the theory of “similar” systems, or
(more precisely) of the similar motion of similar systems. Thus, considering the equations
d2x μ d2x′ μ′
=− , =− ,
dt2 x2 dt′2 x′2 (43)
which refer to two particles falling independently into two distinct centres of force, it is
obvious that it is possible to have x in a constant ratio to x′, and t in a constant ratio to t′,
provided that
x x′ μ μ′
: = : ,
t2 t′2 x2 x′2 (44)
and that there is a suitable correspondence between the initial conditions. The relation (44) is
equivalent to
x3/2 x′3/2
t : t′ = : ,
μ1/2 μ′1/2 (45)
where x, x′ are any two corresponding distances; e.g. they may be the initial distances, both
particles being supposed to start from rest. The consideration of dimensions was introduced by
J. B. Fourier (1822) in connexion with the conduction of heat.
Fig. 64.
Page 432
§ 13. General Motion of a Particle.—Let P, Q be the positions of a moving point at times t, t + δt
respectively. A vector OU> drawn parallel to PQ, of length proportional to PQ/δt on any convenient
scale, will represent the mean velocity in the interval δt, i.e. a point moving with a constant velocity
having the magnitude and direction indicated by this vector would experience the same resultant
displacement PQ> in the same time. As δt is indefinitely diminished, the vector OU> will tend to a
definite limit OV>; this is adopted as the definition of the velocity of the moving point at the instant t.
Obviously OV> is parallel to the tangent to the path at P, and its magnitude is ds/dt, where s is the arc.
If we project OV> on the co-ordinate axes (rectangular or oblique) in the usual manner, the projections
u, v, w are called the component velocities parallel to the axes. If x, y, z be the co-ordinates of P it is
easily proved that
dx dy dz
u= , v= , w= .
dt dt dt (1)
The momentum of a particle is the vector obtained by multiplying the velocity by the mass m. The
impulse of a force in any infinitely small interval of time δt is the product of the force into δt; it is to
be regarded as a vector. The total impulse in any finite interval of time is the integral of the impulses
corresponding to the infinitesimal elements δt into which the interval may be subdivided; the
summation of which the integral is the limit is of course to be understood in the vectorial sense.
Newton’s Second Law asserts that change of momentum is equal to the impulse; this is a statement
as to equality of vectors and so implies identity of direction as well as of magnitude. If X, Y, Z are the
components of force, then considering the changes in an infinitely short time δt we have, by
projection on the co-ordinate axes, δ(mu) = Xδt, and so on, or
du dv dw
m = X, m = Y, m = Z.
dt dt dt (2)
For example, the path of a particle projected anyhow under gravity will obviously be confined to
the vertical plane through the initial direction of motion. Taking this as the plane xy, with the axis of
x drawn horizontally, and that of y vertically upwards, we have X = 0, Y = −mg; so that
d2x d2y
= 0, = −g.
dt2 dt2 (3)
The solution is
x = At + B, y = −1⁄2gt2 + Ct + D.
(4)
If the initial values of x, y, ẋ, ẏ are given, we have four conditions to determine the four arbitrary
constants A, B, C, D. Thus if the particle start at time t = 0 from the origin, with the component
velocities u0, v0, we have
x = u0t, y = v0t − 1⁄2gt2.
respectively. A vector OU> drawn parallel to PQ, of length proportional to PQ/δt on any convenient
scale, will represent the mean velocity in the interval δt, i.e. a point moving with a constant velocity
having the magnitude and direction indicated by this vector would experience the same resultant
displacement PQ> in the same time. As δt is indefinitely diminished, the vector OU> will tend to a
definite limit OV>; this is adopted as the definition of the velocity of the moving point at the instant t.
Obviously OV> is parallel to the tangent to the path at P, and its magnitude is ds/dt, where s is the arc.
If we project OV> on the co-ordinate axes (rectangular or oblique) in the usual manner, the projections
u, v, w are called the component velocities parallel to the axes. If x, y, z be the co-ordinates of P it is
easily proved that
dx dy dz
u= , v= , w= .
dt dt dt (1)
The momentum of a particle is the vector obtained by multiplying the velocity by the mass m. The
impulse of a force in any infinitely small interval of time δt is the product of the force into δt; it is to
be regarded as a vector. The total impulse in any finite interval of time is the integral of the impulses
corresponding to the infinitesimal elements δt into which the interval may be subdivided; the
summation of which the integral is the limit is of course to be understood in the vectorial sense.
Newton’s Second Law asserts that change of momentum is equal to the impulse; this is a statement
as to equality of vectors and so implies identity of direction as well as of magnitude. If X, Y, Z are the
components of force, then considering the changes in an infinitely short time δt we have, by
projection on the co-ordinate axes, δ(mu) = Xδt, and so on, or
du dv dw
m = X, m = Y, m = Z.
dt dt dt (2)
For example, the path of a particle projected anyhow under gravity will obviously be confined to
the vertical plane through the initial direction of motion. Taking this as the plane xy, with the axis of
x drawn horizontally, and that of y vertically upwards, we have X = 0, Y = −mg; so that
d2x d2y
= 0, = −g.
dt2 dt2 (3)
The solution is
x = At + B, y = −1⁄2gt2 + Ct + D.
(4)
If the initial values of x, y, ẋ, ẏ are given, we have four conditions to determine the four arbitrary
constants A, B, C, D. Thus if the particle start at time t = 0 from the origin, with the component
velocities u0, v0, we have
x = u0t, y = v0t − 1⁄2gt2.
Page 433
(5)
Eliminating t we have the equation of the path, viz.
v0 gx2
y= x− .
u0 2u2 (6)
This is a parabola with vertical axis, of latus-rectum 2u02/g. The range on a horizontal plane through
O is got by putting y = 0, viz. it is 2u0v0/g. we denote the resultant velocity at any instant by ṡ we
have
ṡ2 = ẋ2 + ẏ2 = ṡ02 − 2gy.
(7)
Another important example is that of a particle subject to an acceleration which is directed always
towards a fixed point O and is proportional to the distance from O. The motion will evidently be in
one plane, which we take as the plane z = 0. If μ be the acceleration at unit distance, the component
accelerations parallel to axes of x and y through O as origin will be −μx, −μy, whence
d2x d2y
= −μx, = − μy.
dt2 dt2 (8)
The solution is
x = A cos nt + B sin nt, y = C cos nt + D sin nt,
(9)
where n = √μ. If P be the initial position of the particle, we may conveniently take OP as axis of x,
and draw Oy parallel to the direction of motion at P. If OP = a, and ṡ0 be the velocity at P, we have,
initially, x = a, y = 0, ẋ = 0, ẏ = ṡ0 whence
x = a cos nt, y = b sin nt,
(10)
if b = ṡ0/n. The path is therefore an ellipse of which a, b are conjugate semi-diameters, and is
described in the period 2π/√μ; moreover, the velocity at any point P is equal to √μ·OD, where OD is
the semi-diameter conjugate to OP. This type of motion is called elliptic harmonic. If the co-ordinate
axes are the principal axes of the ellipse, the angle nt in (10) is identical with the “excentric angle.”
The motion of the bob of a “spherical pendulum,” i.e. a simple pendulum whose oscillations are not
confined to one vertical plane, is of this character, provided the extreme inclination of the string to
the vertical be small. The acceleration is towards the vertical through the point of suspension, and is
equal to gr/l, approximately, if r denote distance from this vertical. Hence the path is approximately
an ellipse, and the period is 2π √(l/g).
Eliminating t we have the equation of the path, viz.
v0 gx2
y= x− .
u0 2u2 (6)
This is a parabola with vertical axis, of latus-rectum 2u02/g. The range on a horizontal plane through
O is got by putting y = 0, viz. it is 2u0v0/g. we denote the resultant velocity at any instant by ṡ we
have
ṡ2 = ẋ2 + ẏ2 = ṡ02 − 2gy.
(7)
Another important example is that of a particle subject to an acceleration which is directed always
towards a fixed point O and is proportional to the distance from O. The motion will evidently be in
one plane, which we take as the plane z = 0. If μ be the acceleration at unit distance, the component
accelerations parallel to axes of x and y through O as origin will be −μx, −μy, whence
d2x d2y
= −μx, = − μy.
dt2 dt2 (8)
The solution is
x = A cos nt + B sin nt, y = C cos nt + D sin nt,
(9)
where n = √μ. If P be the initial position of the particle, we may conveniently take OP as axis of x,
and draw Oy parallel to the direction of motion at P. If OP = a, and ṡ0 be the velocity at P, we have,
initially, x = a, y = 0, ẋ = 0, ẏ = ṡ0 whence
x = a cos nt, y = b sin nt,
(10)
if b = ṡ0/n. The path is therefore an ellipse of which a, b are conjugate semi-diameters, and is
described in the period 2π/√μ; moreover, the velocity at any point P is equal to √μ·OD, where OD is
the semi-diameter conjugate to OP. This type of motion is called elliptic harmonic. If the co-ordinate
axes are the principal axes of the ellipse, the angle nt in (10) is identical with the “excentric angle.”
The motion of the bob of a “spherical pendulum,” i.e. a simple pendulum whose oscillations are not
confined to one vertical plane, is of this character, provided the extreme inclination of the string to
the vertical be small. The acceleration is towards the vertical through the point of suspension, and is
equal to gr/l, approximately, if r denote distance from this vertical. Hence the path is approximately
an ellipse, and the period is 2π √(l/g).
Page 434
Fig. 65. Fig. 66.
The above problem is identical with that of the oscillation of a particle in a smooth spherical
bowl, in the neighbourhood of the lowest point. If the bowl has any other shape, the axes Ox,
Oy may be taken tangential to the lines of curvature at the lowest point O; the equations of
small motion then are
d2x x d2y y
= −g , = −g ,
dt2 ρ1 dt2 ρ2 (11)
where ρ1, ρ2, are the principal radii of curvature at O. The motion is therefore the resultant of
two simple vibrations in perpendicular directions, of periods 2π √(ρ1/g), 2π √(ρ2/g). The
circumstances are realized in “Blackburn’s pendulum,” which consists of a weight P hanging
from a point C of a string ACB whose ends A, B are fixed. If E be the point in which the line of
the string meets AB, we have ρ1 = CP, ρ2 = EP. Many contrivances for actually drawing the
resulting curves have been devised.
It is sometimes convenient to resolve the accelerations in directions having a more intrinsic
relation to the path. Thus, in a plane path, let P, Q be two consecutive positions, corresponding to the
times t, t + δt; and let the normals at P, Q meet in C, making an angle δψ. Let v (= ṡ) be the velocity at
P, v + δv that at Q. In the time δt the velocity parallel to the tangent at P changes from v to v + δv,
ultimately, and the tangential acceleration at P is therefore dv/dt or s̈ . Again, the velocity parallel to
the normal at P changes from 0 to vδψ, ultimately, so that the normal acceleration is v dψ/dt. Since
dv dv ds dv dψ dψ ds v2
= =v , v =v = ,
dt ds dt ds dt ds dt ρ (12)
where ρ is the radius of curvature of the path at P, the tangential and normal accelerations are also
expressed by v dv/ds and v2/ρ, respectively. Take, for example, the case of a particle moving on a
smooth curve in a vertical plane, under the action of gravity and the pressure R of the curve. If the
axes of x and y be drawn horizontal and vertical (upwards), and if ψ be the inclination of the tangent
to the horizontal, we have
mv dv = − mg sin ψ = − mg dy , mv2 = − mg cos ψ + R.
The above problem is identical with that of the oscillation of a particle in a smooth spherical
bowl, in the neighbourhood of the lowest point. If the bowl has any other shape, the axes Ox,
Oy may be taken tangential to the lines of curvature at the lowest point O; the equations of
small motion then are
d2x x d2y y
= −g , = −g ,
dt2 ρ1 dt2 ρ2 (11)
where ρ1, ρ2, are the principal radii of curvature at O. The motion is therefore the resultant of
two simple vibrations in perpendicular directions, of periods 2π √(ρ1/g), 2π √(ρ2/g). The
circumstances are realized in “Blackburn’s pendulum,” which consists of a weight P hanging
from a point C of a string ACB whose ends A, B are fixed. If E be the point in which the line of
the string meets AB, we have ρ1 = CP, ρ2 = EP. Many contrivances for actually drawing the
resulting curves have been devised.
It is sometimes convenient to resolve the accelerations in directions having a more intrinsic
relation to the path. Thus, in a plane path, let P, Q be two consecutive positions, corresponding to the
times t, t + δt; and let the normals at P, Q meet in C, making an angle δψ. Let v (= ṡ) be the velocity at
P, v + δv that at Q. In the time δt the velocity parallel to the tangent at P changes from v to v + δv,
ultimately, and the tangential acceleration at P is therefore dv/dt or s̈ . Again, the velocity parallel to
the normal at P changes from 0 to vδψ, ultimately, so that the normal acceleration is v dψ/dt. Since
dv dv ds dv dψ dψ ds v2
= =v , v =v = ,
dt ds dt ds dt ds dt ρ (12)
where ρ is the radius of curvature of the path at P, the tangential and normal accelerations are also
expressed by v dv/ds and v2/ρ, respectively. Take, for example, the case of a particle moving on a
smooth curve in a vertical plane, under the action of gravity and the pressure R of the curve. If the
axes of x and y be drawn horizontal and vertical (upwards), and if ψ be the inclination of the tangent
to the horizontal, we have
mv dv = − mg sin ψ = − mg dy , mv2 = − mg cos ψ + R.
Page 435
ds ds ρ (13)
The former equation gives
v2 = C − 2gy,
(14)
and the latter then determines R.
In the case of the pendulum the tension of the string takes the place of the pressure of the
curve. If l be the length of the string, ψ its inclination to the downward vertical, we have δs =
lδψ, so that v = ldψ/dt. The tangential resolution then gives
d2ψ
l = − g sin ψ.
dt2 (15)
If we multiply by 2dψ/dt and integrate, we obtain
( dψdt ) = 2gl cos ψ + const.,
2
(16)
which is seen to be equivalent to (14). If the pendulum oscillate between the limits ψ = ±α, we
have
δψ 2g 4g
( dt ) = l (cos ψ − cos α) = l (sin ⁄ α − sin ⁄ ψ);
2
2 12 2 12
(17)
and, putting sin 1⁄2ψ = sin 1⁄2α. sin φ, we find for the period (τ) of a complete oscillation
dt l dφ
τ = 4 ∫1⁄20π
dφ dφ = 4 √ g · ∫ √(1 − sin ⁄ α · sin φ)
⁄0π
12
2 12 2
l
= 4√ g · F (sin ⁄ α), 1 12
(18)
in the notation of elliptic integrals. The function F1 (sin β) was tabulated by A. M. Legendre
for values of β ranging from 0° to 90°. The following table gives the period, for various
amplitudes α, in terms of that of oscillation in an infinitely small arc [viz. 2π√(l/g)] as unit.
α/π τ α/π τ
.1 1.0062 .6 1.2817
.2 1.0253 .7 1.4283
.3 1.0585 .8 1.6551
.4 1.1087 .9 2.0724
.5 1.1804 1.0 ∞
The former equation gives
v2 = C − 2gy,
(14)
and the latter then determines R.
In the case of the pendulum the tension of the string takes the place of the pressure of the
curve. If l be the length of the string, ψ its inclination to the downward vertical, we have δs =
lδψ, so that v = ldψ/dt. The tangential resolution then gives
d2ψ
l = − g sin ψ.
dt2 (15)
If we multiply by 2dψ/dt and integrate, we obtain
( dψdt ) = 2gl cos ψ + const.,
2
(16)
which is seen to be equivalent to (14). If the pendulum oscillate between the limits ψ = ±α, we
have
δψ 2g 4g
( dt ) = l (cos ψ − cos α) = l (sin ⁄ α − sin ⁄ ψ);
2
2 12 2 12
(17)
and, putting sin 1⁄2ψ = sin 1⁄2α. sin φ, we find for the period (τ) of a complete oscillation
dt l dφ
τ = 4 ∫1⁄20π
dφ dφ = 4 √ g · ∫ √(1 − sin ⁄ α · sin φ)
⁄0π
12
2 12 2
l
= 4√ g · F (sin ⁄ α), 1 12
(18)
in the notation of elliptic integrals. The function F1 (sin β) was tabulated by A. M. Legendre
for values of β ranging from 0° to 90°. The following table gives the period, for various
amplitudes α, in terms of that of oscillation in an infinitely small arc [viz. 2π√(l/g)] as unit.
α/π τ α/π τ
.1 1.0062 .6 1.2817
.2 1.0253 .7 1.4283
.3 1.0585 .8 1.6551
.4 1.1087 .9 2.0724
.5 1.1804 1.0 ∞
Page 436
The value of τ can also be obtained as an infinite series, by expanding the integrand in (18) by
the binomial theorem, and integrating term by term. Thus
l 12 12 · 32
τ = 2π √ g ·{1+ 2 2
sin2 1⁄2α +
22 · 42 sin4 1⁄2α + ... }. (19)
If α be small, an approximation (usually sufficient) is
τ = 2π √(l/g) · (1 + 1⁄16α2).
In the extreme case of α = π, the equation (17) is immediately integrable; thus the time from the
lowest position is
t = √(l/g) · log tan (1⁄4π + 1⁄4ψ).
(20)
This becomes infinite for ψ = π, showing that the pendulum only tends asymptotically to the
highest position.
The variation of period with amplitude was at one time a hindrance to the accurate
performance of pendulum clocks, since the errors produced are cumulative. It was therefore
sought to replace the circular pendulum by some other contrivance free from this defect. The
equation of motion of a particle in any smooth path is
d2s
= −g sin ψ,
dt2 (21)
where ψ is the inclination of the tangent to the horizontal. If sin ψ were accurately and not
merely approximately proportional to the arc s, say
s = k sin ψ,
(22)
the equation (21) would assume the same form as § 12
(5). The motion along the arc would then be accurately
simple-harmonic, and the period 2π √(k/g) would be the
same for all amplitudes. Now equation (22) is the
intrinsic equation of a cycloid; viz. the curve is that traced
by a point on the circumference of a circle of radius 1⁄4k
which rolls on the under side of a horizontal straight line. Fig. 67.
Since the evolute of a cycloid is an equal cycloid the
object is attained by means of two metal cheeks, having the form of the evolute near the cusp,
on which the string wraps itself alternately as the pendulum swings. The device has long been
abandoned, the difficulty being met in other ways, but the problem, originally investigated by
C. Huygens, is important in the history of mathematics.
the binomial theorem, and integrating term by term. Thus
l 12 12 · 32
τ = 2π √ g ·{1+ 2 2
sin2 1⁄2α +
22 · 42 sin4 1⁄2α + ... }. (19)
If α be small, an approximation (usually sufficient) is
τ = 2π √(l/g) · (1 + 1⁄16α2).
In the extreme case of α = π, the equation (17) is immediately integrable; thus the time from the
lowest position is
t = √(l/g) · log tan (1⁄4π + 1⁄4ψ).
(20)
This becomes infinite for ψ = π, showing that the pendulum only tends asymptotically to the
highest position.
The variation of period with amplitude was at one time a hindrance to the accurate
performance of pendulum clocks, since the errors produced are cumulative. It was therefore
sought to replace the circular pendulum by some other contrivance free from this defect. The
equation of motion of a particle in any smooth path is
d2s
= −g sin ψ,
dt2 (21)
where ψ is the inclination of the tangent to the horizontal. If sin ψ were accurately and not
merely approximately proportional to the arc s, say
s = k sin ψ,
(22)
the equation (21) would assume the same form as § 12
(5). The motion along the arc would then be accurately
simple-harmonic, and the period 2π √(k/g) would be the
same for all amplitudes. Now equation (22) is the
intrinsic equation of a cycloid; viz. the curve is that traced
by a point on the circumference of a circle of radius 1⁄4k
which rolls on the under side of a horizontal straight line. Fig. 67.
Since the evolute of a cycloid is an equal cycloid the
object is attained by means of two metal cheeks, having the form of the evolute near the cusp,
on which the string wraps itself alternately as the pendulum swings. The device has long been
abandoned, the difficulty being met in other ways, but the problem, originally investigated by
C. Huygens, is important in the history of mathematics.
Page 437
The component accelerations of a point describing a tortuous curve, in the directions of the
tangent, the principal normal, and the binormal, respectively, are found as follows. If OV>, OV′> be
vectors representing the velocities at two consecutive points P, P′ of the path, the plane VOV′ is
ultimately parallel to the osculating plane of the path at P; the resultant acceleration is therefore in the
osculating plane. Also, the projections of VV′> on OV and on a perpendicular to OV in the plane
VOV′ are δv and vδε, where δε is the angle between the directions of the tangents at P, P′. Since δε =
δs/ρ, where δs = PP′ = vδt and ρ is the radius of principal curvature at P, the component accelerations
along the tangent and principal normal are dv/dt and vdε/dt, respectively, or vdv/ds and v2/ρ. For
example, if a particle moves on a smooth surface, under no forces except the reaction of the surface,
v is constant, and the principal normal to the path will coincide with the normal to the surface. Hence
the path is a “geodesic” on the surface.
If we resolve along the tangent to the path (whether plane or tortuous), the equation of motion of a
particle may be written
dv
mv = T,
ds (23)
where T is the tangential component of the force. Integrating with respect to s we find
⁄ mv12 − 1⁄2mv02 = ∫s1
12 s0 T ds;
(24)
i.e. the increase of kinetic energy between any two positions is equal to the work done by the forces.
The result follows also from the Cartesian equations (2); viz. we have
m (ẋẍ + ẏÿ + żz̈ ) = Xẋ + Yẏ + Zż,
(25)
whence, on integration with respect to t,
⁄ m (ẋ2 + ẏ2 + ż2)
12
= ∫ (Xẋ + Yẏ + Zż) dt + const.
= ∫ (X dx + Y dy + Z dz) + const.
(26)
If the axes be rectangular, this has the same interpretation as (24).
Suppose now that we have a constant field of force; i.e. the force acting on the particle is always
the same at the same place. The work which must be done by forces extraneous to the field in order to
bring the particle from rest in some standard position A to rest in any other position P will not
necessarily be the same for all paths between A and P. If it is different for different paths, then by
bringing the particle from A to P by one path, and back again from P to A by another, we might secure
a gain of work, and the process could be repeated indefinitely. If the work required is the same for all
tangent, the principal normal, and the binormal, respectively, are found as follows. If OV>, OV′> be
vectors representing the velocities at two consecutive points P, P′ of the path, the plane VOV′ is
ultimately parallel to the osculating plane of the path at P; the resultant acceleration is therefore in the
osculating plane. Also, the projections of VV′> on OV and on a perpendicular to OV in the plane
VOV′ are δv and vδε, where δε is the angle between the directions of the tangents at P, P′. Since δε =
δs/ρ, where δs = PP′ = vδt and ρ is the radius of principal curvature at P, the component accelerations
along the tangent and principal normal are dv/dt and vdε/dt, respectively, or vdv/ds and v2/ρ. For
example, if a particle moves on a smooth surface, under no forces except the reaction of the surface,
v is constant, and the principal normal to the path will coincide with the normal to the surface. Hence
the path is a “geodesic” on the surface.
If we resolve along the tangent to the path (whether plane or tortuous), the equation of motion of a
particle may be written
dv
mv = T,
ds (23)
where T is the tangential component of the force. Integrating with respect to s we find
⁄ mv12 − 1⁄2mv02 = ∫s1
12 s0 T ds;
(24)
i.e. the increase of kinetic energy between any two positions is equal to the work done by the forces.
The result follows also from the Cartesian equations (2); viz. we have
m (ẋẍ + ẏÿ + żz̈ ) = Xẋ + Yẏ + Zż,
(25)
whence, on integration with respect to t,
⁄ m (ẋ2 + ẏ2 + ż2)
12
= ∫ (Xẋ + Yẏ + Zż) dt + const.
= ∫ (X dx + Y dy + Z dz) + const.
(26)
If the axes be rectangular, this has the same interpretation as (24).
Suppose now that we have a constant field of force; i.e. the force acting on the particle is always
the same at the same place. The work which must be done by forces extraneous to the field in order to
bring the particle from rest in some standard position A to rest in any other position P will not
necessarily be the same for all paths between A and P. If it is different for different paths, then by
bringing the particle from A to P by one path, and back again from P to A by another, we might secure
a gain of work, and the process could be repeated indefinitely. If the work required is the same for all
Page 438
paths between A and P, and therefore zero for a closed circuit, the field is said to be conservative. In
this case the work required to bring the particle from rest at A to rest at P is called the potential
energy of the particle in the position P; we denote it by V. If PP′ be a linear element δs drawn in any
direction from P, and S be the force due to the field, resolved in the direction PP′, we have δV = −Sδs
or
∂V
S=− .
∂s (27)
In particular, by taking PP′ parallel to each of the (rectangular) co-ordinate axes in succession, we
find
∂V ∂V ∂V
X=− , Y=− , Z=− .
∂x ∂y ∂z (28)
The equation (24) or (26) now gives
⁄ mv12 + V1 = 1⁄2mv02 + V0;
12
(29)
i.e. the sum of the kinetic and potential energies is constant when no work is done by extraneous
forces. For example, if the field be that due to gravity we have V = ƒmg dy = mgy + const., if the axis
of y be drawn vertically upwards; hence
⁄ mv2 + mgy = const.
12
(30)
This applies to motion on a smooth curve, as well as to the free motion of a projectile; cf. (7), (14).
Again, in the case of a force Kr towards O, where r denotes distance from O we have V = ∫ Kr dr =
1⁄2Kr2 + const., whence
⁄ mv2 + 1⁄2Kr2 = const.
12
(31)
It has been seen that the orbit is in this case an ellipse; also that if we put μ = K/m the velocity at any
point P is v = √μ. OD, where OD is the semi-diameter conjugate to OP. Hence (31) is consistent with
the known property of the ellipse that OP2 + OD2 is constant.
The forms assumed by the dynamical equations when the axes of reference are themselves in
motion will be considered in § 21. At present we take only the case where the rectangular axes
Ox, Oy rotate in their own plane, with angular velocity ω about Oz, which is fixed. In the
interval δt the projections of the line joining the origin to any point (x, y, z) on the directions of
the co-ordinate axes at time t are changed from x, y, z to (x + δx) cos ω δt − (y + δy) sin ωδt, (x
+ δx) sin ω δt + (y + δy) cos ω δt, z respectively. Hence the component velocities parallel to the
instantaneous positions of the co-ordinate axes at time t are
u = ẋ − ωy, v = ẏ + ωz, ω = ż.
this case the work required to bring the particle from rest at A to rest at P is called the potential
energy of the particle in the position P; we denote it by V. If PP′ be a linear element δs drawn in any
direction from P, and S be the force due to the field, resolved in the direction PP′, we have δV = −Sδs
or
∂V
S=− .
∂s (27)
In particular, by taking PP′ parallel to each of the (rectangular) co-ordinate axes in succession, we
find
∂V ∂V ∂V
X=− , Y=− , Z=− .
∂x ∂y ∂z (28)
The equation (24) or (26) now gives
⁄ mv12 + V1 = 1⁄2mv02 + V0;
12
(29)
i.e. the sum of the kinetic and potential energies is constant when no work is done by extraneous
forces. For example, if the field be that due to gravity we have V = ƒmg dy = mgy + const., if the axis
of y be drawn vertically upwards; hence
⁄ mv2 + mgy = const.
12
(30)
This applies to motion on a smooth curve, as well as to the free motion of a projectile; cf. (7), (14).
Again, in the case of a force Kr towards O, where r denotes distance from O we have V = ∫ Kr dr =
1⁄2Kr2 + const., whence
⁄ mv2 + 1⁄2Kr2 = const.
12
(31)
It has been seen that the orbit is in this case an ellipse; also that if we put μ = K/m the velocity at any
point P is v = √μ. OD, where OD is the semi-diameter conjugate to OP. Hence (31) is consistent with
the known property of the ellipse that OP2 + OD2 is constant.
The forms assumed by the dynamical equations when the axes of reference are themselves in
motion will be considered in § 21. At present we take only the case where the rectangular axes
Ox, Oy rotate in their own plane, with angular velocity ω about Oz, which is fixed. In the
interval δt the projections of the line joining the origin to any point (x, y, z) on the directions of
the co-ordinate axes at time t are changed from x, y, z to (x + δx) cos ω δt − (y + δy) sin ωδt, (x
+ δx) sin ω δt + (y + δy) cos ω δt, z respectively. Hence the component velocities parallel to the
instantaneous positions of the co-ordinate axes at time t are
u = ẋ − ωy, v = ẏ + ωz, ω = ż.
Page 439
(32)
In the same way we find that the component accelerations are
u̇ − ωv, v̇ + ωu, ω̇
(33)
Hence if ω be constant the equations of motion take the forms
m (ẍ − 2ωẏ − ω2ẋ) = X, m (ÿ + 2ωẋ − ω2y) = Y, mz̈ = Z.
(34)
These become identical with the equations of motion relative to fixed axes provided we
introduce a fictitious force mω2r acting outwards from the axis of z, where r = √(x2 + y2), and a
second fictitious force 2mωv at right angles to the path, where v is the component of the
relative velocity parallel to the plane xy. The former force is called by French writers the force
centrifuge ordinaire, and the latter the force centrifuge composée, or force de Coriolis. As an
application of (34) we may take the case of a symmetrical Blackburn’s pendulum hanging from
a horizontal bar which is made to rotate about a vertical axis half-way between the points of
attachment of the upper string. The equations of small motion are then of the type
ẍ − 2ωẏ − ω2x = −p2x, ÿ + 2ωẋ − ω2y = −q2y.
(35)
This is satisfied by
ẍ = A cos (σt + ε), y = B sin (σt + ε),
(36)
provided
(σ2 + ω2 − p2) A + 2σωB = 0,
2σωA + (σ2 + ω2 − q2) B = 0.
(37)
Eliminating the ratio A : B we have
(σ2 + ω2 − p2) (σ2 + ω2 − q2) − 4σ2ω2 = 0.
(38)
It is easily proved that the roots of this quadratic in σ2 are always real, and that they are
moreover both positive unless ω2 lies between p2 and q2. The ratio B/A is determined in each
case by either of the equations (37); hence each root of the quadratic gives a solution of the
type (36), with two arbitrary constants A, ε. Since the equations (35) are linear, these two
solutions are to be superposed. If the quadratic (38) has a negative root, the trigonometrical
functions in (36) are to be replaced by real exponentials, and the position x = 0, y = 0 is
In the same way we find that the component accelerations are
u̇ − ωv, v̇ + ωu, ω̇
(33)
Hence if ω be constant the equations of motion take the forms
m (ẍ − 2ωẏ − ω2ẋ) = X, m (ÿ + 2ωẋ − ω2y) = Y, mz̈ = Z.
(34)
These become identical with the equations of motion relative to fixed axes provided we
introduce a fictitious force mω2r acting outwards from the axis of z, where r = √(x2 + y2), and a
second fictitious force 2mωv at right angles to the path, where v is the component of the
relative velocity parallel to the plane xy. The former force is called by French writers the force
centrifuge ordinaire, and the latter the force centrifuge composée, or force de Coriolis. As an
application of (34) we may take the case of a symmetrical Blackburn’s pendulum hanging from
a horizontal bar which is made to rotate about a vertical axis half-way between the points of
attachment of the upper string. The equations of small motion are then of the type
ẍ − 2ωẏ − ω2x = −p2x, ÿ + 2ωẋ − ω2y = −q2y.
(35)
This is satisfied by
ẍ = A cos (σt + ε), y = B sin (σt + ε),
(36)
provided
(σ2 + ω2 − p2) A + 2σωB = 0,
2σωA + (σ2 + ω2 − q2) B = 0.
(37)
Eliminating the ratio A : B we have
(σ2 + ω2 − p2) (σ2 + ω2 − q2) − 4σ2ω2 = 0.
(38)
It is easily proved that the roots of this quadratic in σ2 are always real, and that they are
moreover both positive unless ω2 lies between p2 and q2. The ratio B/A is determined in each
case by either of the equations (37); hence each root of the quadratic gives a solution of the
type (36), with two arbitrary constants A, ε. Since the equations (35) are linear, these two
solutions are to be superposed. If the quadratic (38) has a negative root, the trigonometrical
functions in (36) are to be replaced by real exponentials, and the position x = 0, y = 0 is
Page 440
unstable. This occurs only when the period (2π/ω) of revolution of the arm lies between the two
periods (2π/p, 2π/q) of oscillation when the arm is fixed.
§ 14. Central Forces. Hodograph.—The motion of a particle subject to a force which passes
always through a fixed point O is necessarily in a plane orbit. For its investigation we require two
equations; these may be obtained in a variety of forms.
Since the impulse of the force in any element of time δt has zero moment about O, the same will be
true of the additional momentum generated. Hence the moment of the momentum (considered as a
localized vector) about O will be constant. In symbols, if v be the velocity and p the perpendicular
from O to the tangent to the path,
pv = h,
(1)
where h is a constant. If δs be an element of the path, pδs is twice the area enclosed by δs and the
radii drawn to its extremities from O. Hence if δA be this area, we have δA = 1⁄2 pδs = 1⁄2 hδt, or
dA
= 1⁄2h
dt (2)
Hence equal areas are swept over by the radius vector in equal times.
If P be the acceleration towards O, we have
dv dr
v = −P
ds ds (3)
since dr/ds is the cosine of the angle between the directions of r and δs. We will suppose that P is a
function of r only; then integrating (3) we find
⁄ v2 = − ∫ P dr + const.,
12
(4)
which is recognized as the equation of energy. Combining this with (1) we have
h2
= C − 2 ∫ P dr,
p2 (5)
which completely determines the path except as to its orientation with respect to O.
If the law of attraction be that of the inverse square of the distance, we have P = μ/r2, and
h2 2μ
=C+ .
p2 τ (6)
Now in a conic whose focus is at O we have
l = 2 ± 1 ,
periods (2π/p, 2π/q) of oscillation when the arm is fixed.
§ 14. Central Forces. Hodograph.—The motion of a particle subject to a force which passes
always through a fixed point O is necessarily in a plane orbit. For its investigation we require two
equations; these may be obtained in a variety of forms.
Since the impulse of the force in any element of time δt has zero moment about O, the same will be
true of the additional momentum generated. Hence the moment of the momentum (considered as a
localized vector) about O will be constant. In symbols, if v be the velocity and p the perpendicular
from O to the tangent to the path,
pv = h,
(1)
where h is a constant. If δs be an element of the path, pδs is twice the area enclosed by δs and the
radii drawn to its extremities from O. Hence if δA be this area, we have δA = 1⁄2 pδs = 1⁄2 hδt, or
dA
= 1⁄2h
dt (2)
Hence equal areas are swept over by the radius vector in equal times.
If P be the acceleration towards O, we have
dv dr
v = −P
ds ds (3)
since dr/ds is the cosine of the angle between the directions of r and δs. We will suppose that P is a
function of r only; then integrating (3) we find
⁄ v2 = − ∫ P dr + const.,
12
(4)
which is recognized as the equation of energy. Combining this with (1) we have
h2
= C − 2 ∫ P dr,
p2 (5)
which completely determines the path except as to its orientation with respect to O.
If the law of attraction be that of the inverse square of the distance, we have P = μ/r2, and
h2 2μ
=C+ .
p2 τ (6)
Now in a conic whose focus is at O we have
l = 2 ± 1 ,
Page 441
p2 r a (7)
where l is half the latus-rectum, a is half the major axis, and the upper or lower sign is to be taken
according as the conic is an ellipse or hyperbola. In the intermediate case of the parabola we have a =
∞ and the last term disappears. The equations (6) and (7) are identified by putting
l = h2/μ, a = ± μ/C.
(8)
Since
h2
v2 =
p2 = μ ( 2r ± 1a ), (9)
it appears that the orbit is an ellipse, parabola or hyperbola, according as v2 is less than, equal to, or
greater than 2μ/r. Now it appears from (6) that 2μ/r is the square of the velocity which would be
acquired by a particle falling from rest at infinity to the distance r. Hence the character of the orbit
depends on whether the velocity at any point is less than, equal to, or greater than the velocity from
infinity, as it is called. In an elliptic orbit the area πab is swept over in the time
πab 2πa3/2
r= = ,
1⁄2h √μ (10)
since h = μ1/2l1/2 = μ1/2ba−1/2 by (8).
The converse problem, to determine the law of force under which a given orbit can be
described about a given pole, is solved by differentiating (5) with respect to r; thus
h2 dp
P= .
p3 dr (11)
In the case of an ellipse described about the centre as pole we have
a2b2
= a2 + b2 − r2;
p2 (12)
hence P = μr, if μ = h2/a2b2. This merely shows that a particular ellipse may be described under
the law of the direct distance provided the circumstances of projection be suitably adjusted. But
since an ellipse can always be constructed with a given centre so as to touch a given line at a
given point, and to have a given value of ab (= h/√μ) we infer that the orbit will be elliptic
whatever the initial circumstances. Also the period is 2πab/h = 2π/√μ, as previously found.
Again, in the equiangular spiral we have p = r sinα, and therefore P = μ/r3, if μ = h2/sin2 α.
But since an equiangular spiral having a given pole is completely determined by a given point
and a given tangent, this type of orbit is not a general one for the law of the inverse cube. In
order that the spiral may be described it is necessary that the velocity of projection should be
where l is half the latus-rectum, a is half the major axis, and the upper or lower sign is to be taken
according as the conic is an ellipse or hyperbola. In the intermediate case of the parabola we have a =
∞ and the last term disappears. The equations (6) and (7) are identified by putting
l = h2/μ, a = ± μ/C.
(8)
Since
h2
v2 =
p2 = μ ( 2r ± 1a ), (9)
it appears that the orbit is an ellipse, parabola or hyperbola, according as v2 is less than, equal to, or
greater than 2μ/r. Now it appears from (6) that 2μ/r is the square of the velocity which would be
acquired by a particle falling from rest at infinity to the distance r. Hence the character of the orbit
depends on whether the velocity at any point is less than, equal to, or greater than the velocity from
infinity, as it is called. In an elliptic orbit the area πab is swept over in the time
πab 2πa3/2
r= = ,
1⁄2h √μ (10)
since h = μ1/2l1/2 = μ1/2ba−1/2 by (8).
The converse problem, to determine the law of force under which a given orbit can be
described about a given pole, is solved by differentiating (5) with respect to r; thus
h2 dp
P= .
p3 dr (11)
In the case of an ellipse described about the centre as pole we have
a2b2
= a2 + b2 − r2;
p2 (12)
hence P = μr, if μ = h2/a2b2. This merely shows that a particular ellipse may be described under
the law of the direct distance provided the circumstances of projection be suitably adjusted. But
since an ellipse can always be constructed with a given centre so as to touch a given line at a
given point, and to have a given value of ab (= h/√μ) we infer that the orbit will be elliptic
whatever the initial circumstances. Also the period is 2πab/h = 2π/√μ, as previously found.
Again, in the equiangular spiral we have p = r sinα, and therefore P = μ/r3, if μ = h2/sin2 α.
But since an equiangular spiral having a given pole is completely determined by a given point
and a given tangent, this type of orbit is not a general one for the law of the inverse cube. In
order that the spiral may be described it is necessary that the velocity of projection should be
Page 442
adjusted to make h = √μ·sinα. Similarly, in the case of a
circle with the pole on the circumference we have p2 =
r2/2a, P = μ/r5, if μ = 8h2a2; but this orbit is not a
general one for the law of the inverse fifth power.
In astronomical and other investigations relating to
central forces it is often convenient to use polar co-
ordinates with the centre of force as pole. Let P, Q be the Fig. 68.
positions of a moving point at times t, t + δt, and write OP
= r, OQ = r + δr, ∠POQ = δθ, O being any fixed origin. If u, v be the component velocities at P along
and perpendicular to OP (in the direction of θ increasing), we have
δr dr r δθ dθ
u = lim. = , v = lim. =r .
δt dt δt dt (13)
Again, the velocities parallel and perpendicular to OP change in the time δt from u, v to u − v δθ, v +
u δθ, ultimately. The component accelerations at P in these directions are therefore
du dθ d2r dθ
dt
−v
dt
=
dt2 − r (
dt 2, )
dv dθ 1 d dθ
dt
+u
dt
=
r dt (
r2 dt , ) (14)
respectively.
In the case of a central force, with O as pole, the transverse acceleration vanishes, so that
r2 dθ / dt = h,
(15)
where h is constant; this shows (again) that the radius vector sweeps over equal areas in equal times.
The radial resolution gives
d2r
dt2 − r ( dθdt ) = −P,
2
(16)
where P, as before, denotes the acceleration towards O. If in this we put r = 1/u, and eliminate t by
means of (15), we obtain the general differential equation of central orbits, viz.
d2u P
+u= .
dθ2 h2u2 (17)
If, for example, the law be that of the inverse square, we have P = μu2, and the solution is of
the form
μ
u= {1 + e cos (θ − α)},
h2 (18)
circle with the pole on the circumference we have p2 =
r2/2a, P = μ/r5, if μ = 8h2a2; but this orbit is not a
general one for the law of the inverse fifth power.
In astronomical and other investigations relating to
central forces it is often convenient to use polar co-
ordinates with the centre of force as pole. Let P, Q be the Fig. 68.
positions of a moving point at times t, t + δt, and write OP
= r, OQ = r + δr, ∠POQ = δθ, O being any fixed origin. If u, v be the component velocities at P along
and perpendicular to OP (in the direction of θ increasing), we have
δr dr r δθ dθ
u = lim. = , v = lim. =r .
δt dt δt dt (13)
Again, the velocities parallel and perpendicular to OP change in the time δt from u, v to u − v δθ, v +
u δθ, ultimately. The component accelerations at P in these directions are therefore
du dθ d2r dθ
dt
−v
dt
=
dt2 − r (
dt 2, )
dv dθ 1 d dθ
dt
+u
dt
=
r dt (
r2 dt , ) (14)
respectively.
In the case of a central force, with O as pole, the transverse acceleration vanishes, so that
r2 dθ / dt = h,
(15)
where h is constant; this shows (again) that the radius vector sweeps over equal areas in equal times.
The radial resolution gives
d2r
dt2 − r ( dθdt ) = −P,
2
(16)
where P, as before, denotes the acceleration towards O. If in this we put r = 1/u, and eliminate t by
means of (15), we obtain the general differential equation of central orbits, viz.
d2u P
+u= .
dθ2 h2u2 (17)
If, for example, the law be that of the inverse square, we have P = μu2, and the solution is of
the form
μ
u= {1 + e cos (θ − α)},
h2 (18)
Page 443
where e, α are arbitrary constants. This is recognized as the polar equation of a conic referred to
the focus, the half latus-rectum being h2/μ.
The law of the inverse cube P = μu3 is interesting by way of contrast. The orbits may be
divided into two classes according as h2 ≷ μ, i.e. according as the transverse velocity (hu) is
greater or less than the velocity √μ·u appropriate to a circular orbit at the same distance. In the
former case the equation (17) takes the form
d2u
+ m2u = 0,
dθ2 (19)
the solution of which is
au = sin m (θ − α).
(20)
The orbit has therefore two asymptotes, inclined at an angle π/m. In the latter case the
differential equation is of the form
d2u
= m2u,
dθ2 (21)
so that
u = Aemθ + Be−mθ
(22)
If A, B have the same sign, this is equivalent to
au = cosh mθ,
(23)
if the origin of θ be suitably adjusted; hence r has a maximum value α, and the particle
ultimately approaches the pole asymptotically by an infinite number of convolutions. If A, B
have opposite signs the form is
au = sinh mθ,
(24)
this has an asymptote parallel to θ = 0, but the path near the origin has the same general form as
in the case of (23). If A or B vanish we have an equiangular spiral, and the velocity at infinity is
zero. In the critical case of h2 = μ, we have d2u/dθ2 = 0, and
u = Aθ + B;
(25)
the orbit is therefore a “reciprocal spiral,” except in the special case of A = 0, when it is a
circle. It will be seen that unless the conditions be exactly adjusted for a circular orbit the
the focus, the half latus-rectum being h2/μ.
The law of the inverse cube P = μu3 is interesting by way of contrast. The orbits may be
divided into two classes according as h2 ≷ μ, i.e. according as the transverse velocity (hu) is
greater or less than the velocity √μ·u appropriate to a circular orbit at the same distance. In the
former case the equation (17) takes the form
d2u
+ m2u = 0,
dθ2 (19)
the solution of which is
au = sin m (θ − α).
(20)
The orbit has therefore two asymptotes, inclined at an angle π/m. In the latter case the
differential equation is of the form
d2u
= m2u,
dθ2 (21)
so that
u = Aemθ + Be−mθ
(22)
If A, B have the same sign, this is equivalent to
au = cosh mθ,
(23)
if the origin of θ be suitably adjusted; hence r has a maximum value α, and the particle
ultimately approaches the pole asymptotically by an infinite number of convolutions. If A, B
have opposite signs the form is
au = sinh mθ,
(24)
this has an asymptote parallel to θ = 0, but the path near the origin has the same general form as
in the case of (23). If A or B vanish we have an equiangular spiral, and the velocity at infinity is
zero. In the critical case of h2 = μ, we have d2u/dθ2 = 0, and
u = Aθ + B;
(25)
the orbit is therefore a “reciprocal spiral,” except in the special case of A = 0, when it is a
circle. It will be seen that unless the conditions be exactly adjusted for a circular orbit the
Page 444
particle will either recede to infinity or approach the pole asymptotically. This problem was
investigated by R. Cotes (1682-1716), and the various curves obtained arc known as Coles’s
spirals.
A point on a central orbit where the radial velocity (dr/dt) vanishes is called an apse, and the
corresponding radius is called an apse-line. If the force is always the same at the same distance any
apse-line will divide the orbit symmetrically, as is seen by imagining the velocity at the apse to be
reversed. It follows that the angle between successive apse-lines is constant; it is called the apsidal
angle of the orbit.
If in a central orbit the velocity is equal to the velocity from infinity, we have, from (5),
h2
p2 = 2 ∫∞r P dr; (26)
this determines the form of the critical orbit, as it is called. If P = μ/rn, its polar equation is
rm cos mθ = am,
(27)
where m = 1⁄2(3 − n), except in the case n = 3, when the orbit is an equiangular spiral. The case n = 2
gives the parabola as before.
If we eliminate dθ/dt between (15) and (16) we obtain
d2r h2
− = −P = −ƒ(r),
dt2 r3
say. We may apply this to the investigation of the stability of a circular orbit. Assuming that r =
a + x, where x is small, we have, approximately,
d2x h2 3x
a3 ( 1 − a ) = −ƒ(a) − xƒ′(a).
−
dt2
Hence if h and a be connected by the relation h2 = a3ƒ(a) proper to a circular orbit, we have
d2x 3
dt2 + { ƒ′(a) + a ƒ(a) } x = 0 (28)
If the coefficient of x be positive the variations of x are simple-harmonic, and x can remain
permanently small; the circular orbit is then said to be stable. The condition for this may be
written
d
{ a3ƒ(a) } > 0,
da (29)
investigated by R. Cotes (1682-1716), and the various curves obtained arc known as Coles’s
spirals.
A point on a central orbit where the radial velocity (dr/dt) vanishes is called an apse, and the
corresponding radius is called an apse-line. If the force is always the same at the same distance any
apse-line will divide the orbit symmetrically, as is seen by imagining the velocity at the apse to be
reversed. It follows that the angle between successive apse-lines is constant; it is called the apsidal
angle of the orbit.
If in a central orbit the velocity is equal to the velocity from infinity, we have, from (5),
h2
p2 = 2 ∫∞r P dr; (26)
this determines the form of the critical orbit, as it is called. If P = μ/rn, its polar equation is
rm cos mθ = am,
(27)
where m = 1⁄2(3 − n), except in the case n = 3, when the orbit is an equiangular spiral. The case n = 2
gives the parabola as before.
If we eliminate dθ/dt between (15) and (16) we obtain
d2r h2
− = −P = −ƒ(r),
dt2 r3
say. We may apply this to the investigation of the stability of a circular orbit. Assuming that r =
a + x, where x is small, we have, approximately,
d2x h2 3x
a3 ( 1 − a ) = −ƒ(a) − xƒ′(a).
−
dt2
Hence if h and a be connected by the relation h2 = a3ƒ(a) proper to a circular orbit, we have
d2x 3
dt2 + { ƒ′(a) + a ƒ(a) } x = 0 (28)
If the coefficient of x be positive the variations of x are simple-harmonic, and x can remain
permanently small; the circular orbit is then said to be stable. The condition for this may be
written
d
{ a3ƒ(a) } > 0,
da (29)
Page 445
i.e. the intensity of the force in the region for which r = a, nearly, must diminish with increasing
distance less rapidly than according to the law of the inverse cube. Again, the half-period of x
is π / √{ƒ′(a) + 3−1ƒ(a)}, and since the angular velocity in the orbit is h/a2, approximately, the
apsidal angle is, ultimately,
ƒ(a)
π √{ aƒ′(a) + 3ƒ(a) }, (30)
or, in the case of ƒ(a) = μ/rn, π/√(3 − n). This is in agreement with the known results for n = 2, n
= −1.
We have seen that under the law of the inverse square all finite orbits are elliptical. The
question presents itself whether there then is any other law of force, giving a finite velocity
from infinity, under which all finite orbits are necessarily closed curves. If this is the case, the
apsidal angle must evidently be commensurable with π, and since it cannot vary
discontinuously the apsidal angle in a nearly circular orbit must be constant. Equating the
expression (30) to π/m, we find that ƒ(a) = C/an, where n = 3 − m2. The force must therefore
vary as a power of the distance, and n must be less than 3. Moreover, the case n = 2 is the only
one in which the critical orbit (27) can be regarded as the limiting form of a closed curve.
Hence the only law of force which satisfies the conditions is that of the inverse square.
At the beginning of § 13 the velocity of a moving point P was represented by a vector OV> drawn
from a fixed origin O. The locus of the point V is called the hodograph (q.v.); and it appears that the
velocity of the point V along the hodograph represents in magnitude and in direction the acceleration
in the original orbit. Thus in the case of a plane orbit, if v be the velocity of P, ψ the inclination of the
direction of motion to some fixed direction, the polar co-ordinates of V may be taken to be v, ψ;
hence the velocities of V along and perpendicular to OV will be dv/dt and v dψ/dt. These expressions
therefore give the tangential and normal accelerations of P; cf. § 13 (12).
In the motion of a projectile under gravity the hodograph is a vertical
line described with constant velocity. In elliptic harmonic motion the
velocity of P is parallel and proportional to the semi-diameter CD
which is conjugate to the radius CP; the hodograph is therefore an
ellipse similar to the actual orbit. In the case of a central orbit described
under the law of the inverse square we have v = h/SY = h. SZ/b2,
where S is the centre of force, SY is the perpendicular to the tangent at
P, and Z is the point where YS meets the auxiliary circle again. Hence Fig. 69.
the hodograph is similar and similarly situated to the locus of Z (the
auxiliary circle) turned about S through a right angle. This applies to an elliptic or hyperbolic
orbit; the case of the parabolic orbit may be examined separately or treated as a limiting case.
The annexed fig. 70 exhibits the various cases, with the hodograph in its proper orientation.
The pole O of the hodograph is inside on or outside the circle, according as the orbit is an
distance less rapidly than according to the law of the inverse cube. Again, the half-period of x
is π / √{ƒ′(a) + 3−1ƒ(a)}, and since the angular velocity in the orbit is h/a2, approximately, the
apsidal angle is, ultimately,
ƒ(a)
π √{ aƒ′(a) + 3ƒ(a) }, (30)
or, in the case of ƒ(a) = μ/rn, π/√(3 − n). This is in agreement with the known results for n = 2, n
= −1.
We have seen that under the law of the inverse square all finite orbits are elliptical. The
question presents itself whether there then is any other law of force, giving a finite velocity
from infinity, under which all finite orbits are necessarily closed curves. If this is the case, the
apsidal angle must evidently be commensurable with π, and since it cannot vary
discontinuously the apsidal angle in a nearly circular orbit must be constant. Equating the
expression (30) to π/m, we find that ƒ(a) = C/an, where n = 3 − m2. The force must therefore
vary as a power of the distance, and n must be less than 3. Moreover, the case n = 2 is the only
one in which the critical orbit (27) can be regarded as the limiting form of a closed curve.
Hence the only law of force which satisfies the conditions is that of the inverse square.
At the beginning of § 13 the velocity of a moving point P was represented by a vector OV> drawn
from a fixed origin O. The locus of the point V is called the hodograph (q.v.); and it appears that the
velocity of the point V along the hodograph represents in magnitude and in direction the acceleration
in the original orbit. Thus in the case of a plane orbit, if v be the velocity of P, ψ the inclination of the
direction of motion to some fixed direction, the polar co-ordinates of V may be taken to be v, ψ;
hence the velocities of V along and perpendicular to OV will be dv/dt and v dψ/dt. These expressions
therefore give the tangential and normal accelerations of P; cf. § 13 (12).
In the motion of a projectile under gravity the hodograph is a vertical
line described with constant velocity. In elliptic harmonic motion the
velocity of P is parallel and proportional to the semi-diameter CD
which is conjugate to the radius CP; the hodograph is therefore an
ellipse similar to the actual orbit. In the case of a central orbit described
under the law of the inverse square we have v = h/SY = h. SZ/b2,
where S is the centre of force, SY is the perpendicular to the tangent at
P, and Z is the point where YS meets the auxiliary circle again. Hence Fig. 69.
the hodograph is similar and similarly situated to the locus of Z (the
auxiliary circle) turned about S through a right angle. This applies to an elliptic or hyperbolic
orbit; the case of the parabolic orbit may be examined separately or treated as a limiting case.
The annexed fig. 70 exhibits the various cases, with the hodograph in its proper orientation.
The pole O of the hodograph is inside on or outside the circle, according as the orbit is an
Page 446
ellipse, parabola or hyperbola. In any case of a central orbit the hodograph (when turned
through a right angle) is similar and similarly situated to the “reciprocal polar” of the orbit with
respect to the centre of force. Thus for a circular orbit with the centre of force at an excentric
point, the hodograph is a conic with the pole as focus. In the case of a particle oscillating under
gravity on a smooth cycloid from rest at the cusp the hodograph is a circle through the pole,
described with constant velocity.
§ 15. Kinetics of a System of Discrete Particles.—The momenta of the several particles constitute a
system of localized vectors which, for purposes of resolving and taking moments, may be reduced
like a system of forces in statics (§ 8). Thus taking any point O as base, we have first a linear
momentum whose components referred to rectangular axes through O are
Σ(mẋ), Σ(mẏ), Σ(mż);
(1)
its representative vector is the same whatever point O be chosen. Secondly, we have an angular
momentum whose components are
Σ {m (yż − zẏ) }, Σ {m (zẋ − xż) }, Σ {m (xẏ − yẋ) },
(2)
these being the sums of the moments of the momenta of the several particles about the respective
axes. This is subject to the same relations as a couple in statics; it may be represented by a vector
which will, however, in general vary with the position of O.
The linear momentum is the same as if the whole mass were concentrated at the centre of mass G,
and endowed with the velocity of this point. This follows at once from equation (8) of § 11, if we
imagine the two configurations of the system there referred to to be those corresponding to the
instants t, t + δt. Thus
Σ ( m· PPδt ) = Σ(m)· GG′
>
δt
.
>
(3)
Analytically we have
d dx
Σ(mẋ) = Σ(mx) = Σ(m)· ,
dt dt (4)
with two similar formulae.
through a right angle) is similar and similarly situated to the “reciprocal polar” of the orbit with
respect to the centre of force. Thus for a circular orbit with the centre of force at an excentric
point, the hodograph is a conic with the pole as focus. In the case of a particle oscillating under
gravity on a smooth cycloid from rest at the cusp the hodograph is a circle through the pole,
described with constant velocity.
§ 15. Kinetics of a System of Discrete Particles.—The momenta of the several particles constitute a
system of localized vectors which, for purposes of resolving and taking moments, may be reduced
like a system of forces in statics (§ 8). Thus taking any point O as base, we have first a linear
momentum whose components referred to rectangular axes through O are
Σ(mẋ), Σ(mẏ), Σ(mż);
(1)
its representative vector is the same whatever point O be chosen. Secondly, we have an angular
momentum whose components are
Σ {m (yż − zẏ) }, Σ {m (zẋ − xż) }, Σ {m (xẏ − yẋ) },
(2)
these being the sums of the moments of the momenta of the several particles about the respective
axes. This is subject to the same relations as a couple in statics; it may be represented by a vector
which will, however, in general vary with the position of O.
The linear momentum is the same as if the whole mass were concentrated at the centre of mass G,
and endowed with the velocity of this point. This follows at once from equation (8) of § 11, if we
imagine the two configurations of the system there referred to to be those corresponding to the
instants t, t + δt. Thus
Σ ( m· PPδt ) = Σ(m)· GG′
>
δt
.
>
(3)
Analytically we have
d dx
Σ(mẋ) = Σ(mx) = Σ(m)· ,
dt dt (4)
with two similar formulae.
Page 447
Fig. 70.
Again, if the instantaneous position of G be taken as base, the angular momentum of the absolute
motion is the same as the angular momentum of the motion relative to G. For the velocity of a
particle m at P may be replaced by two components one of which (v) is identical in magnitude and
direction with the velocity of G, whilst the other (v) is the velocity relative to G. The aggregate of the
components mv of momentum is equivalent to a single localized vector Σ(m)·v in a line through G,
and has therefore zero moment about any axis through G; hence in taking moments about such an
axis we need only regard the velocities relative to G. In symbols, we have
Again, if the instantaneous position of G be taken as base, the angular momentum of the absolute
motion is the same as the angular momentum of the motion relative to G. For the velocity of a
particle m at P may be replaced by two components one of which (v) is identical in magnitude and
direction with the velocity of G, whilst the other (v) is the velocity relative to G. The aggregate of the
components mv of momentum is equivalent to a single localized vector Σ(m)·v in a line through G,
and has therefore zero moment about any axis through G; hence in taking moments about such an
axis we need only regard the velocities relative to G. In symbols, we have
Page 448
Σ { m(yż − zẏ) } = Σ(m)· ( y dzdt − z dydt ) + Σ { m (ηζ − ζ̇η̇) }. (5)
since Σ(mξ) = 0, Σ(mξ̇) = 0, and so on, the notation being as in § 11.
This expresses that the moment of momentum about any fixed axis
(e.g. Ox) is equal to the moment of momentum of the motion
relative to G about a parallel axis through G, together with the
moment of momentum of the whole mass supposed concentrated at
G and moving with this point. If in (5) we make O coincide with the Fig. 71.
instantaneous position of G, we have x, z, z = 0, and the theorem
follows.
Finally, the rates of change of the components of the angular momentum of the motion relative to
G referred to G as a moving base, are equal to the rates of change of the corresponding components
of angular momentum relative to a fixed base coincident with the instantaneous position of G. For let
G′ be a consecutive position of G. At the instant t + δt the momenta of the system are equivalent to a
linear momentum represented by a localized vector Σ(m)·(v + δv) in a line through G′ tangential to
the path of G′, together with a certain angular momentum. Now the moment of this localized vector
with respect to any axis through G is zero, to the first order of δt, since the perpendicular distance of
G from the tangent line at G′ is of the order (δt)2. Analytically we have from (5),
d dz2 d2y d
dt Σ { m (yż − zẏ) } = Σ(m)· ( y dt − z dt ) + dt Σ { m(ηζ − ζη̇) }
2 2 (6)
If we put x, y, z = 0, the theorem is proved as regards axes parallel to Ox.
Next consider the kinetic energy of the system. If from a fixed point O we draw vectors OV1>, OV2>
to represent the velocities of the several particles m1, m2 ..., and if we construct the vector
Σ ( m·OV> )
OK> =
Σ(m) (7)
this will represent the velocity of the mass-centre, by (3). We find, exactly as in the proof of
Lagrange’s First Theorem (§ 11), that
⁄ Σ (m·OV2) = 1⁄2Σ (m)·OK2 + 1⁄2Σ (m·KV2);
12
(8)
i.e. the total kinetic energy is equal to the kinetic energy of the whole mass supposed concentrated at
G and moving with this point, together with the kinetic energy of the motion relative to G. The latter
may be called the internal kinetic energy of the system. Analytically we have
⁄ Σ { m (ẋ2 + ẏ2 + ż2) } = 1⁄2Σ(m)·
12 { ( dxdt ) + ( dydt ) + ( dzdt ) } + ⁄ Σ{ m(ζ + η̇ + (9)
2 2 2 12 ζ̇ ) }.2 2 2
There is also an analogue to Lagrange’s Second Theorem, viz.
since Σ(mξ) = 0, Σ(mξ̇) = 0, and so on, the notation being as in § 11.
This expresses that the moment of momentum about any fixed axis
(e.g. Ox) is equal to the moment of momentum of the motion
relative to G about a parallel axis through G, together with the
moment of momentum of the whole mass supposed concentrated at
G and moving with this point. If in (5) we make O coincide with the Fig. 71.
instantaneous position of G, we have x, z, z = 0, and the theorem
follows.
Finally, the rates of change of the components of the angular momentum of the motion relative to
G referred to G as a moving base, are equal to the rates of change of the corresponding components
of angular momentum relative to a fixed base coincident with the instantaneous position of G. For let
G′ be a consecutive position of G. At the instant t + δt the momenta of the system are equivalent to a
linear momentum represented by a localized vector Σ(m)·(v + δv) in a line through G′ tangential to
the path of G′, together with a certain angular momentum. Now the moment of this localized vector
with respect to any axis through G is zero, to the first order of δt, since the perpendicular distance of
G from the tangent line at G′ is of the order (δt)2. Analytically we have from (5),
d dz2 d2y d
dt Σ { m (yż − zẏ) } = Σ(m)· ( y dt − z dt ) + dt Σ { m(ηζ − ζη̇) }
2 2 (6)
If we put x, y, z = 0, the theorem is proved as regards axes parallel to Ox.
Next consider the kinetic energy of the system. If from a fixed point O we draw vectors OV1>, OV2>
to represent the velocities of the several particles m1, m2 ..., and if we construct the vector
Σ ( m·OV> )
OK> =
Σ(m) (7)
this will represent the velocity of the mass-centre, by (3). We find, exactly as in the proof of
Lagrange’s First Theorem (§ 11), that
⁄ Σ (m·OV2) = 1⁄2Σ (m)·OK2 + 1⁄2Σ (m·KV2);
12
(8)
i.e. the total kinetic energy is equal to the kinetic energy of the whole mass supposed concentrated at
G and moving with this point, together with the kinetic energy of the motion relative to G. The latter
may be called the internal kinetic energy of the system. Analytically we have
⁄ Σ { m (ẋ2 + ẏ2 + ż2) } = 1⁄2Σ(m)·
12 { ( dxdt ) + ( dydt ) + ( dzdt ) } + ⁄ Σ{ m(ζ + η̇ + (9)
2 2 2 12 ζ̇ ) }.2 2 2
There is also an analogue to Lagrange’s Second Theorem, viz.
Page 449
ΣΣ (mpmq · VpVq2)
⁄ Σ (m·KV2) = 1⁄2
12 ,
Σm (10)
which expresses the internal kinetic energy in terms of the relative velocities of the several pairs of
particles. This formula is due to Möbius.
The preceding theorems are purely kinematical. We have now to consider the effect of the forces
acting on the particles. These may be divided into two categories; we have first, the extraneous forces
exerted on the various particles from without, and, secondly, the mutual or internal forces between
the various pairs of particles. It is assumed that these latter are subject to the law of equality of action
and reaction. If the equations of motion of each particle be formed separately, each such internal
force will appear twice over, with opposite signs for its components, viz. as affecting the motion of
each of the two particles between which it acts. The full working out is in general difficult, the
comparatively simple problem of “three bodies,” for instance, in gravitational astronomy being still
unsolved, but some general theorems can be formulated.
The first of these may be called the Principle of Linear Momentum. If there are no extraneous
forces, the resultant linear momentum is constant in every respect. For consider any two particles at P
and Q, acting on one another with equal and opposite forces in the line PQ. In the time δt a certain
impulse is given to the first particle in the direction (say) from P to Q, whilst an equal and opposite
impulse is given to the second in the direction from Q to P. Since these impulses produce equal and
opposite momenta in the two particles, the resultant linear momentum of the system is unaltered. If
extraneous forces act, it is seen in like manner that the resultant linear momentum of the system is in
any given time modified by the geometric addition of the total impulse of the extraneous forces. It
follows, by the preceding kinematic theory, that the mass-centre G of the system will move exactly as
if the whole mass were concentrated there and were acted on by the extraneous forces applied parallel
to their original directions. For example, the mass-centre of a system free from extraneous force will
describe a straight line with constant velocity. Again, the mass-centre of a chain of particles
connected by strings, projected anyhow under gravity, will describe a parabola.
The second general result is the Principle of Angular Momentum. If there are no extraneous forces,
the moment of momentum about any fixed axis is constant. For in time δt the mutual action between
two particles at P and Q produces equal and opposite momenta in the line PQ, and these will have
equal and opposite moments about the fixed axis. If extraneous forces act, the total angular
momentum about any fixed axis is in time δt increased by the total extraneous impulse about that
axis. The kinematical relations above explained now lead to the conclusion that in calculating the
effect of extraneous forces in an infinitely short time δt we may take moments about an axis passing
through the instantaneous position of G exactly as if G were fixed; moreover, the result will be the
same whether in this process we employ the true velocities of the particles or merely their velocities
relative to G. If there are no extraneous forces, or if the extraneous forces have zero moment about
any axis through G, the vector which represents the resultant angular momentum relative to G is
constant in every respect. A plane through G perpendicular to this vector has a fixed direction in
⁄ Σ (m·KV2) = 1⁄2
12 ,
Σm (10)
which expresses the internal kinetic energy in terms of the relative velocities of the several pairs of
particles. This formula is due to Möbius.
The preceding theorems are purely kinematical. We have now to consider the effect of the forces
acting on the particles. These may be divided into two categories; we have first, the extraneous forces
exerted on the various particles from without, and, secondly, the mutual or internal forces between
the various pairs of particles. It is assumed that these latter are subject to the law of equality of action
and reaction. If the equations of motion of each particle be formed separately, each such internal
force will appear twice over, with opposite signs for its components, viz. as affecting the motion of
each of the two particles between which it acts. The full working out is in general difficult, the
comparatively simple problem of “three bodies,” for instance, in gravitational astronomy being still
unsolved, but some general theorems can be formulated.
The first of these may be called the Principle of Linear Momentum. If there are no extraneous
forces, the resultant linear momentum is constant in every respect. For consider any two particles at P
and Q, acting on one another with equal and opposite forces in the line PQ. In the time δt a certain
impulse is given to the first particle in the direction (say) from P to Q, whilst an equal and opposite
impulse is given to the second in the direction from Q to P. Since these impulses produce equal and
opposite momenta in the two particles, the resultant linear momentum of the system is unaltered. If
extraneous forces act, it is seen in like manner that the resultant linear momentum of the system is in
any given time modified by the geometric addition of the total impulse of the extraneous forces. It
follows, by the preceding kinematic theory, that the mass-centre G of the system will move exactly as
if the whole mass were concentrated there and were acted on by the extraneous forces applied parallel
to their original directions. For example, the mass-centre of a system free from extraneous force will
describe a straight line with constant velocity. Again, the mass-centre of a chain of particles
connected by strings, projected anyhow under gravity, will describe a parabola.
The second general result is the Principle of Angular Momentum. If there are no extraneous forces,
the moment of momentum about any fixed axis is constant. For in time δt the mutual action between
two particles at P and Q produces equal and opposite momenta in the line PQ, and these will have
equal and opposite moments about the fixed axis. If extraneous forces act, the total angular
momentum about any fixed axis is in time δt increased by the total extraneous impulse about that
axis. The kinematical relations above explained now lead to the conclusion that in calculating the
effect of extraneous forces in an infinitely short time δt we may take moments about an axis passing
through the instantaneous position of G exactly as if G were fixed; moreover, the result will be the
same whether in this process we employ the true velocities of the particles or merely their velocities
relative to G. If there are no extraneous forces, or if the extraneous forces have zero moment about
any axis through G, the vector which represents the resultant angular momentum relative to G is
constant in every respect. A plane through G perpendicular to this vector has a fixed direction in
Page 450
space, and is called the invariable plane; it may sometimes be conveniently used as a plane of
reference.
For example, if we have two particles connected by a string, the invariable plane passes
through the string, and if ω be the angular velocity in this plane, the angular momentum
relative to G is
m1ω1r1·r1 + m2ωr2·r2 = (m1r12 + m2r22) ω,
where r1, r2 are the distances of m1, m2 from their mass-centre G. Hence if the extraneous
forces (e.g. gravity) have zero moment about G, ω will be constant. Again, the tension R of the
string is given by
m1m2
R = m1ω2r1 = ω2a,
m1 + m 2
where a = r1 + r2. Also by (10) the internal kinetic energy is
m1m2
⁄
12 ω2a2
m1 + m 2
The increase of the kinetic energy of the system in any interval of time will of course be equal to
the total work done by all the forces acting on the particles. In many questions relating to systems of
discrete particles the internal force Rpq (which we will reckon positive when attractive) between any
two particles mp, mq is a function only of the distance rpq between them. In this case the work done
by the internal forces will be represented by
−Σ ∫ Rpg drpq,
when the summation includes every pair of particles, and each integral is to be taken between the
proper limits. If we write
V = Σ ∫ Rpq drpq,
(11)
when rpq ranges from its value in some standard configuration A of the system to its value in any
other configuration P, it is plain that V represents the work which would have to be done in order to
bring the system from rest in the configuration A to rest in the configuration P. Hence V is a definite
function of the configuration P; it is called the internal potential energy. If T denote the kinetic
energy, we may say then that the sum T + V is in any interval of time increased by an amount equal to
the work done by the extraneous forces. In particular, if there are no extraneous forces T + V is
constant. Again, if some of the extraneous forces are due to a conservative field of force, the work
which they do may be reckoned as a diminution of the potential energy relative to the field as in § 13.
reference.
For example, if we have two particles connected by a string, the invariable plane passes
through the string, and if ω be the angular velocity in this plane, the angular momentum
relative to G is
m1ω1r1·r1 + m2ωr2·r2 = (m1r12 + m2r22) ω,
where r1, r2 are the distances of m1, m2 from their mass-centre G. Hence if the extraneous
forces (e.g. gravity) have zero moment about G, ω will be constant. Again, the tension R of the
string is given by
m1m2
R = m1ω2r1 = ω2a,
m1 + m 2
where a = r1 + r2. Also by (10) the internal kinetic energy is
m1m2
⁄
12 ω2a2
m1 + m 2
The increase of the kinetic energy of the system in any interval of time will of course be equal to
the total work done by all the forces acting on the particles. In many questions relating to systems of
discrete particles the internal force Rpq (which we will reckon positive when attractive) between any
two particles mp, mq is a function only of the distance rpq between them. In this case the work done
by the internal forces will be represented by
−Σ ∫ Rpg drpq,
when the summation includes every pair of particles, and each integral is to be taken between the
proper limits. If we write
V = Σ ∫ Rpq drpq,
(11)
when rpq ranges from its value in some standard configuration A of the system to its value in any
other configuration P, it is plain that V represents the work which would have to be done in order to
bring the system from rest in the configuration A to rest in the configuration P. Hence V is a definite
function of the configuration P; it is called the internal potential energy. If T denote the kinetic
energy, we may say then that the sum T + V is in any interval of time increased by an amount equal to
the work done by the extraneous forces. In particular, if there are no extraneous forces T + V is
constant. Again, if some of the extraneous forces are due to a conservative field of force, the work
which they do may be reckoned as a diminution of the potential energy relative to the field as in § 13.
Page 451
§ 16. Kinetics of a Rigid Body. Fundamental Principles.—When we pass from the consideration of
discrete particles to that of continuous distributions of matter, we require some physical postulate
over and above what is contained in the Laws of Motion, in their original formulation. This additional
postulate may be introduced under various forms. One plan is to assume that any body whatever may
be treated as if it were composed of material particles, i.e. mathematical points endowed with inertia
coefficients, separated by finite intervals, and acting on one another with forces in the lines joining
them subject to the law of equality of action and reaction. In the case of a rigid body we must
suppose that those forces adjust themselves so as to preserve the mutual distances of the various
particles unaltered. On this basis we can predicate the principles of linear and angular momentum, as
in § 15.
An alternative procedure is to adopt the principle first formally enunciated by J. Le R. d’Alembert
and since known by his name. If x, y, z be the rectangular co-ordinates of a mass-element m, the
expressions mẍ, mÿ, mz̈ must be equal to the components of the total force on m, these forces being
partly extraneous and partly forces exerted on m by other mass-elements of the system. Hence (mẍ,
mÿ, mz̈ ) is called the actual or effective force on m. According to d’Alembert’s formulation, the
extraneous forces together with the effective forces reversed fulfil the statical conditions of
equilibrium. In other words, the whole assemblage of effective forces is statically equivalent to the
extraneous forces. This leads, by the principles of § 8, to the equations
Σ(mẍ) = X, Σ(mÿ) = Y, Σ(mz̈ ) = Z,
Σ {m (yz̈ − zÿ) } = L, Σ {m (zẍ − xz̈ ) } = M, Σ{m (xÿ − yẍ) } = N,
(1)
where (X, Y, Z) and (L, M, N) are the force—and couple—constituents of the system of extraneous
forces, referred to O as base, and the summations extend over all the mass-elements of the system.
These equations may be written
d d d
Σ(mẋ) = X, Σ(mẏ) = Y, Σ(mż) = Z,
dt dt dt
d d d
Σ {m (yż − zẏ) } = L, Σ {m (zẋ − xż) } = M, Σ {m (xẏ − yẋ) } = N,
dt dt dt (2)
and so express that the rate of change of the linear momentum in any fixed direction (e.g. that of Ox)
is equal to the total extraneous force in that direction, and that the rate of change of the angular
momentum about any fixed axis is equal to the moment of the extraneous forces about that axis. If we
integrate with respect to t between fixed limits, we obtain the principles of linear and angular
momentum in the form previously given. Hence, whichever form of postulate we adopt, we are led to
the principles of linear and angular momentum, which form in fact the basis of all our subsequent
work. It is to be noticed that the preceding statements are not intended to be restricted to rigid bodies;
they are assumed to hold for all material systems whatever. The peculiar status of rigid bodies is that
the principles in question are in most cases sufficient for the complete determination of the motion,
the dynamical equations (1 or 2) being equal in number to the degrees of freedom (six) of a rigid
discrete particles to that of continuous distributions of matter, we require some physical postulate
over and above what is contained in the Laws of Motion, in their original formulation. This additional
postulate may be introduced under various forms. One plan is to assume that any body whatever may
be treated as if it were composed of material particles, i.e. mathematical points endowed with inertia
coefficients, separated by finite intervals, and acting on one another with forces in the lines joining
them subject to the law of equality of action and reaction. In the case of a rigid body we must
suppose that those forces adjust themselves so as to preserve the mutual distances of the various
particles unaltered. On this basis we can predicate the principles of linear and angular momentum, as
in § 15.
An alternative procedure is to adopt the principle first formally enunciated by J. Le R. d’Alembert
and since known by his name. If x, y, z be the rectangular co-ordinates of a mass-element m, the
expressions mẍ, mÿ, mz̈ must be equal to the components of the total force on m, these forces being
partly extraneous and partly forces exerted on m by other mass-elements of the system. Hence (mẍ,
mÿ, mz̈ ) is called the actual or effective force on m. According to d’Alembert’s formulation, the
extraneous forces together with the effective forces reversed fulfil the statical conditions of
equilibrium. In other words, the whole assemblage of effective forces is statically equivalent to the
extraneous forces. This leads, by the principles of § 8, to the equations
Σ(mẍ) = X, Σ(mÿ) = Y, Σ(mz̈ ) = Z,
Σ {m (yz̈ − zÿ) } = L, Σ {m (zẍ − xz̈ ) } = M, Σ{m (xÿ − yẍ) } = N,
(1)
where (X, Y, Z) and (L, M, N) are the force—and couple—constituents of the system of extraneous
forces, referred to O as base, and the summations extend over all the mass-elements of the system.
These equations may be written
d d d
Σ(mẋ) = X, Σ(mẏ) = Y, Σ(mż) = Z,
dt dt dt
d d d
Σ {m (yż − zẏ) } = L, Σ {m (zẋ − xż) } = M, Σ {m (xẏ − yẋ) } = N,
dt dt dt (2)
and so express that the rate of change of the linear momentum in any fixed direction (e.g. that of Ox)
is equal to the total extraneous force in that direction, and that the rate of change of the angular
momentum about any fixed axis is equal to the moment of the extraneous forces about that axis. If we
integrate with respect to t between fixed limits, we obtain the principles of linear and angular
momentum in the form previously given. Hence, whichever form of postulate we adopt, we are led to
the principles of linear and angular momentum, which form in fact the basis of all our subsequent
work. It is to be noticed that the preceding statements are not intended to be restricted to rigid bodies;
they are assumed to hold for all material systems whatever. The peculiar status of rigid bodies is that
the principles in question are in most cases sufficient for the complete determination of the motion,
the dynamical equations (1 or 2) being equal in number to the degrees of freedom (six) of a rigid
Page 452
solid, whereas in cases where the freedom is greater we have to invoke the aid of other
supplementary physical hypotheses (cf. Elasticity; Hydromechanics).
The increase of the kinetic energy of a rigid body in any interval of time is equal to the work done
by the extraneous forces acting on the body. This is an immediate consequence of the fundamental
postulate, in either of the forms above stated, since the internal forces do on the whole no work. The
statement may be extended to a system of rigid bodies, provided the mutual reactions consist of the
stresses in inextensible links, or the pressures between smooth surfaces, or the reactions at rolling
contacts (§ 9).
§ 17. Two-dimensional Problems.—In the case of rotation about a fixed axis, the principles take a
very simple form. The position of the body is specified by a single co-ordinate, viz. the angle θ
through which some plane passing through the axis and fixed in the body has turned from a standard
position in space. Then dθ/dt, = ω say, is the angular velocity of the body. The angular momentum of
a particle m at a distance r from the axis is mωr·r, and the total angular momentum is Σ(mr2)·ω, or
Iω, if I denote the moment of inertia (§ 11) about the axis. Hence if N be the moment of the
extraneous forces about the axis, we have
d
(Iω) = N.
dt (1)
This may be compared with the equation of rectilinear motion of a particle, viz. d/dt·(Mu) = X; it
shows that I measures the inertia of the body as regards rotation, just as M measures its inertia as
regards translation. If N = 0, ω is constant.
Fig. 72. Fig. 73.
As a first example, suppose we have a flywheel free to rotate about a horizontal axis, and
that a weight m hangs by a vertical string from the circumferences of an axle of radius b (fig.
72). Neglecting frictional resistance we have, if R be the tension of the string,
Iω̇ = Rb, mu̇ = mg − R,
whence
supplementary physical hypotheses (cf. Elasticity; Hydromechanics).
The increase of the kinetic energy of a rigid body in any interval of time is equal to the work done
by the extraneous forces acting on the body. This is an immediate consequence of the fundamental
postulate, in either of the forms above stated, since the internal forces do on the whole no work. The
statement may be extended to a system of rigid bodies, provided the mutual reactions consist of the
stresses in inextensible links, or the pressures between smooth surfaces, or the reactions at rolling
contacts (§ 9).
§ 17. Two-dimensional Problems.—In the case of rotation about a fixed axis, the principles take a
very simple form. The position of the body is specified by a single co-ordinate, viz. the angle θ
through which some plane passing through the axis and fixed in the body has turned from a standard
position in space. Then dθ/dt, = ω say, is the angular velocity of the body. The angular momentum of
a particle m at a distance r from the axis is mωr·r, and the total angular momentum is Σ(mr2)·ω, or
Iω, if I denote the moment of inertia (§ 11) about the axis. Hence if N be the moment of the
extraneous forces about the axis, we have
d
(Iω) = N.
dt (1)
This may be compared with the equation of rectilinear motion of a particle, viz. d/dt·(Mu) = X; it
shows that I measures the inertia of the body as regards rotation, just as M measures its inertia as
regards translation. If N = 0, ω is constant.
Fig. 72. Fig. 73.
As a first example, suppose we have a flywheel free to rotate about a horizontal axis, and
that a weight m hangs by a vertical string from the circumferences of an axle of radius b (fig.
72). Neglecting frictional resistance we have, if R be the tension of the string,
Iω̇ = Rb, mu̇ = mg − R,
whence
Page 453
mb2
bω̇ = g.
1 + mb2 (2)
This gives the acceleration of m as modified by the inertia of the wheel.
A “compound pendulum” is a body of any form which is free to rotate about a fixed
horizontal axis, the only extraneous force (other than the pressures of the axis) being that of
gravity. If M be the total mass, k the radius of gyration (§ 11) about the axis, we have
d dθ
dt ( Mk2 dt ) = −Mgh sin θ, (3)
where θ is the angle which the plane containing the axis and the centre of gravity G makes with
the vertical, and h is the distance of G from the axis. This coincides with the equation of motion
of a simple pendulum [§ 13 (15)] of length l, provided l = k2/h. The plane of the diagram (fig.
73) is supposed to be a plane through G perpendicular to the axis, which it meets in O. If we
produce OG to P, making OP = l, the point P is called the centre of oscillation; the bob of a
simple pendulum of length OP suspended from O will keep step with the motion of P, if
properly started. If κ be the radius of gyration about a parallel axis through G, we have k2 = κ2
+ h2 by § 11 (16), and therefore l = h + κ2/h, whence
GO · GP = κ2.
(4)
This shows that if the body were swung from a parallel axis through P the new centre of
oscillation would be at O. For different parallel axes, the period of a small oscillation varies as
√l, or √(GO + OP); this is least, subject to the condition (4), when GO = GP = κ. The reciprocal
relation between the centres of suspension and oscillation is the basis of Kater’s method of
determining g experimentally. A pendulum is constructed with two parallel knife-edges as
nearly as possible in the same plane with G, the position of one of them being adjustable. If it
could be arranged that the period of a small oscillation should be exactly the same about either
edge, the two knife-edges would in general occupy the positions of conjugate centres of
suspension and oscillation; and the distances between them would be the length l of the
equivalent simple pendulum. For if h1 + κ2/h1 = h2 + κ2/h2, then unless h1 = h2, we must have
κ2 = h1h2, l = h1 + h2. Exact equality of the two observed periods (τ1, τ2, say) cannot of course
be secured in practice, and a modification is necessary. If we write l1 = h1 + κ2/h1, l2 = h2 +
κ2/h2, we find, on elimination of κ,
l1 + l 2 l1 − l 2
⁄
12 + 1⁄2 = 1,
h1 + h2 h1 − h2
whence
bω̇ = g.
1 + mb2 (2)
This gives the acceleration of m as modified by the inertia of the wheel.
A “compound pendulum” is a body of any form which is free to rotate about a fixed
horizontal axis, the only extraneous force (other than the pressures of the axis) being that of
gravity. If M be the total mass, k the radius of gyration (§ 11) about the axis, we have
d dθ
dt ( Mk2 dt ) = −Mgh sin θ, (3)
where θ is the angle which the plane containing the axis and the centre of gravity G makes with
the vertical, and h is the distance of G from the axis. This coincides with the equation of motion
of a simple pendulum [§ 13 (15)] of length l, provided l = k2/h. The plane of the diagram (fig.
73) is supposed to be a plane through G perpendicular to the axis, which it meets in O. If we
produce OG to P, making OP = l, the point P is called the centre of oscillation; the bob of a
simple pendulum of length OP suspended from O will keep step with the motion of P, if
properly started. If κ be the radius of gyration about a parallel axis through G, we have k2 = κ2
+ h2 by § 11 (16), and therefore l = h + κ2/h, whence
GO · GP = κ2.
(4)
This shows that if the body were swung from a parallel axis through P the new centre of
oscillation would be at O. For different parallel axes, the period of a small oscillation varies as
√l, or √(GO + OP); this is least, subject to the condition (4), when GO = GP = κ. The reciprocal
relation between the centres of suspension and oscillation is the basis of Kater’s method of
determining g experimentally. A pendulum is constructed with two parallel knife-edges as
nearly as possible in the same plane with G, the position of one of them being adjustable. If it
could be arranged that the period of a small oscillation should be exactly the same about either
edge, the two knife-edges would in general occupy the positions of conjugate centres of
suspension and oscillation; and the distances between them would be the length l of the
equivalent simple pendulum. For if h1 + κ2/h1 = h2 + κ2/h2, then unless h1 = h2, we must have
κ2 = h1h2, l = h1 + h2. Exact equality of the two observed periods (τ1, τ2, say) cannot of course
be secured in practice, and a modification is necessary. If we write l1 = h1 + κ2/h1, l2 = h2 +
κ2/h2, we find, on elimination of κ,
l1 + l 2 l1 − l 2
⁄
12 + 1⁄2 = 1,
h1 + h2 h1 − h2
whence
Page 454
4π2 ⁄ (τ12 + τ22)
12 ⁄ (τ12 − τ22)
12
= + .
g h1 + h2 h1 − h2 (5)
The distance h1 + h2, which occurs in the first term on the right hand can be measured directly.
For the second term we require the values of h1, h2 separately, but if τ1, τ2 are nearly equal
whilst h1, h2 are distinctly unequal this term will be relatively small, so that an approximate
knowledge of h1, h2 is sufficient.
As a final example we may note the arrangement, often employed in physical measurements,
where a body performs small oscillations about a vertical axis through its mass-centre G, under
the influence of a couple whose moment varies as the angle of rotation from the equilibrium
position. The equation of motion is of the type
I θ̈ = −Kθ,
(6)
and the period is therefore τ = 2π√(I/K). If by the attachment of another body of known
moment of inertia I′, the period is altered from τ to τ′, we have τ′ = 2π√{ (I + I′)/K }. We are
thus enabled to determine both I and K, viz.
I / I′ = τ2 / (τ′2 − τ2), K = 4π2τ2I / (τ′2 − τ2).
(7)
The couple may be due to the earth’s magnetism, or to the torsion of a suspending wire, or to a
“bifilar” suspension. In the latter case, the body hangs by two vertical threads of equal length l
in a plane through G. The motion being assumed to be small, the tensions of the two strings
may be taken to have their statical values Mgb/(a + b), Mga/(a + b), where a, b are the distances
of G from the two threads. When the body is twisted through an angle θ the threads make
angles aθ/l, bθ/l with the vertical, and the moment of the tensions about the vertical through G
is accordingly −Kθ, where K = M gab/l.
For the determination of the motion it has only been necessary to use one of the dynamical
equations. The remaining equations serve to determine the reactions of the rotating body on its
bearings. Suppose, for example, that there are no extraneous forces. Take rectangular axes, of which
Oz coincides with the axis of rotation. The angular velocity being constant, the effective force on a
particle m at a distance r from Oz is mω2r towards this axis, and its components are accordingly
−ω2mx, −ω2my, O. Since the reactions on the bearings must be statically equivalent to the whole
system of effective forces, they will reduce to a force (X Y Z) at O and a couple (L M N) given by
X = −ω2Σ(mx) = −ω2Σ(m)x, Y = −ω2Σ(my) = −ω2Σ(m)y, Z = 0,
L = ω2Σ(myz), M = −ω2Σ(mzx), N = 0,
12 ⁄ (τ12 − τ22)
12
= + .
g h1 + h2 h1 − h2 (5)
The distance h1 + h2, which occurs in the first term on the right hand can be measured directly.
For the second term we require the values of h1, h2 separately, but if τ1, τ2 are nearly equal
whilst h1, h2 are distinctly unequal this term will be relatively small, so that an approximate
knowledge of h1, h2 is sufficient.
As a final example we may note the arrangement, often employed in physical measurements,
where a body performs small oscillations about a vertical axis through its mass-centre G, under
the influence of a couple whose moment varies as the angle of rotation from the equilibrium
position. The equation of motion is of the type
I θ̈ = −Kθ,
(6)
and the period is therefore τ = 2π√(I/K). If by the attachment of another body of known
moment of inertia I′, the period is altered from τ to τ′, we have τ′ = 2π√{ (I + I′)/K }. We are
thus enabled to determine both I and K, viz.
I / I′ = τ2 / (τ′2 − τ2), K = 4π2τ2I / (τ′2 − τ2).
(7)
The couple may be due to the earth’s magnetism, or to the torsion of a suspending wire, or to a
“bifilar” suspension. In the latter case, the body hangs by two vertical threads of equal length l
in a plane through G. The motion being assumed to be small, the tensions of the two strings
may be taken to have their statical values Mgb/(a + b), Mga/(a + b), where a, b are the distances
of G from the two threads. When the body is twisted through an angle θ the threads make
angles aθ/l, bθ/l with the vertical, and the moment of the tensions about the vertical through G
is accordingly −Kθ, where K = M gab/l.
For the determination of the motion it has only been necessary to use one of the dynamical
equations. The remaining equations serve to determine the reactions of the rotating body on its
bearings. Suppose, for example, that there are no extraneous forces. Take rectangular axes, of which
Oz coincides with the axis of rotation. The angular velocity being constant, the effective force on a
particle m at a distance r from Oz is mω2r towards this axis, and its components are accordingly
−ω2mx, −ω2my, O. Since the reactions on the bearings must be statically equivalent to the whole
system of effective forces, they will reduce to a force (X Y Z) at O and a couple (L M N) given by
X = −ω2Σ(mx) = −ω2Σ(m)x, Y = −ω2Σ(my) = −ω2Σ(m)y, Z = 0,
L = ω2Σ(myz), M = −ω2Σ(mzx), N = 0,
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( )
where x, y refer to the mass-centre G. The reactions do not therefore reduce to a single force at O unless
Σ(myz) = 0, Σ(msx) = 0, i.e. unless the axis of rotation be a principal axis of inertia (§ 11) at O. In order that
the force may vanish we must also have x, y = 0, i.e. the mass-centre must lie in the axis of rotation. These
considerations are important in the “balancing” of machinery. We note further that if a body be free to turn
about a fixed point O, there are three mutually perpendicular lines through this point about which it can
rotate steadily, without further constraint. The theory of principal or “permanent” axes was first investigated
from this point of view by J. A. Segner (1755). The origin of the name “deviation moment” sometimes
applied to a product of inertia is also now apparent.
Proceeding to the general motion of a rigid body in two dimensions
we may take as the three co-ordinates of the body the rectangular
Cartesian co-ordinates x, y of the mass-centre G and the angle θ
through which the body has turned from some standard position. The
components of linear momentum are then Mẋ, Mẏ, and the angular
momentum relative to G as base is Iθ̇, where M is the mass and I the
moment of inertia about G. If the extraneous forces be reduced to a
force (X, Y) at G and a couple N, we have
Mẍ = X, Mÿ = Y, Iθ̈ = N.
(9)
Fig. 74.
If the extraneous forces have zero moment about G the angular
velocity θ̇ is constant. Thus a circular disk projected under gravity in a vertical plane spins with constant
angular velocity, whilst its centre describes a parabola.
We may apply the equations (9) to the case of a solid of revolution rolling with its axis horizontal
on a plane of inclination α. If the axis of x be taken parallel to the slope of the plane, with x increasing
downwards, we have
Mẍ = Mg sin α − F, 0 = Mg cos α − R, Mκ2θ̈ = Fa,
(10)
where κ is the radius of gyration about the axis of symmetry, a is the constant distance of G from the
plane, and R, F are the normal and tangential components of the reaction of the plane, as shown in fig.
74. We have also the kinematical relation ẋ = aθ̇. Hence
a2 κ2
ẍ= g sin α, R = Mg cos α, F= Mg sin α.
κ2 + a2 κ2 + a2 (11)
The acceleration of G is therefore less than in the case of frictionless sliding in the ratio a2/(κ2 + a2).
For a homogeneous sphere this ratio is 5⁄7, for a uniform circular cylinder or disk 2⁄3, for a circular hoop
or a thin cylindrical shell 1⁄2.
The equation of energy for a rigid body has already been stated (in effect) as a corollary from fundamental
assumptions. It may also be deduced from the principles of linear and angular momentum as embodied in the
equations (9). We have
where x, y refer to the mass-centre G. The reactions do not therefore reduce to a single force at O unless
Σ(myz) = 0, Σ(msx) = 0, i.e. unless the axis of rotation be a principal axis of inertia (§ 11) at O. In order that
the force may vanish we must also have x, y = 0, i.e. the mass-centre must lie in the axis of rotation. These
considerations are important in the “balancing” of machinery. We note further that if a body be free to turn
about a fixed point O, there are three mutually perpendicular lines through this point about which it can
rotate steadily, without further constraint. The theory of principal or “permanent” axes was first investigated
from this point of view by J. A. Segner (1755). The origin of the name “deviation moment” sometimes
applied to a product of inertia is also now apparent.
Proceeding to the general motion of a rigid body in two dimensions
we may take as the three co-ordinates of the body the rectangular
Cartesian co-ordinates x, y of the mass-centre G and the angle θ
through which the body has turned from some standard position. The
components of linear momentum are then Mẋ, Mẏ, and the angular
momentum relative to G as base is Iθ̇, where M is the mass and I the
moment of inertia about G. If the extraneous forces be reduced to a
force (X, Y) at G and a couple N, we have
Mẍ = X, Mÿ = Y, Iθ̈ = N.
(9)
Fig. 74.
If the extraneous forces have zero moment about G the angular
velocity θ̇ is constant. Thus a circular disk projected under gravity in a vertical plane spins with constant
angular velocity, whilst its centre describes a parabola.
We may apply the equations (9) to the case of a solid of revolution rolling with its axis horizontal
on a plane of inclination α. If the axis of x be taken parallel to the slope of the plane, with x increasing
downwards, we have
Mẍ = Mg sin α − F, 0 = Mg cos α − R, Mκ2θ̈ = Fa,
(10)
where κ is the radius of gyration about the axis of symmetry, a is the constant distance of G from the
plane, and R, F are the normal and tangential components of the reaction of the plane, as shown in fig.
74. We have also the kinematical relation ẋ = aθ̇. Hence
a2 κ2
ẍ= g sin α, R = Mg cos α, F= Mg sin α.
κ2 + a2 κ2 + a2 (11)
The acceleration of G is therefore less than in the case of frictionless sliding in the ratio a2/(κ2 + a2).
For a homogeneous sphere this ratio is 5⁄7, for a uniform circular cylinder or disk 2⁄3, for a circular hoop
or a thin cylindrical shell 1⁄2.
The equation of energy for a rigid body has already been stated (in effect) as a corollary from fundamental
assumptions. It may also be deduced from the principles of linear and angular momentum as embodied in the
equations (9). We have
Page 456
M (ẋẍ + ẏÿ) + lθ̇θ̈ + Xẋ + Yẏ + Nθ̇,
(12)
whence, integrating with respect to t,
⁄ M (ẋ2 + ẏ2) + 1⁄2Iθ̇2 = ∫ (X dx + Y dy + N dθ) + const.
12
(13)
The left-hand side is the kinetic energy of the whole mass, supposed concentrated at G and moving with this
point, together with the kinetic energy of the motion relative to G (§ 15); and the right-hand member
represents the integral work done by the extraneous forces in the successive infinitesimal displacements into
which the motion may be resolved.
The formula (13) may be easily verified in the case of the
compound pendulum, or of the solid rolling down an incline. As
another example, suppose we have a circular cylinder whose mass-
centre is at an excentric point, rolling on a horizontal plane. This
includes the case of a compound pendulum in which the knife-
edge is replaced by a cylindrical pin. If α be the radius of the
cylinder, h the distance of G from its axis (O), κ the radius of
gyration about a longitudinal axis through G, and θ the inclination
of OG to the vertical, the kinetic energy is 1⁄2Mκ2θ̇2 + Fig. 75.
⁄ M·CG2·thetȧ2, by § 3, since the body is turning about the line of
12
contact (C) as instantaneous axis, and the potential energy is −Mgh cos θ. The equation of energy is
therefore
⁄ M (κ2 + α2 + h2 − 2 ah cos θ) θ̇2 − Mgh cos θ − const.
12
(14)
Whenever, as in the preceding examples, a body or a system of bodies, is subject to constraints which
leave it virtually only one degree of freedom, the equation of energy is sufficient for the complete
determination of the motion. If q be any variable co-ordinate defining the position or (in the case of a system
of bodies) the configuration, the velocity of each particle at any instant will be proportional to q̇ , and the total
kinetic energy may be expressed in the form 1⁄2Aq̇ 2, where A is in general a function of q [cf. equation (14)].
This coefficient A is called the coefficient of inertia, or the reduced inertia of the system, referred to the co-
ordinate q.
Thus in the case of a railway truck travelling with velocity u
the kinetic energy is 1⁄2 (M + mκ2/α2)u2, where M is the total
mass, α the radius and κ the radius of gyration of each wheel,
and m is the sum of the masses of the wheels; the reduced
inertia is therefore M + mκ2/α2. Again, take the system
composed of the flywheel, connecting rod, and piston of a
steam-engine. We have here a limiting case of three-bar
motion (§ 3), and the instantaneous centre J of the connecting- Fig. 76.
rod PQ will have the position shown in the figure. The
velocities of P and Q will be in the ratio of JP to JQ, or OR to OQ; the velocity of the piston is
(12)
whence, integrating with respect to t,
⁄ M (ẋ2 + ẏ2) + 1⁄2Iθ̇2 = ∫ (X dx + Y dy + N dθ) + const.
12
(13)
The left-hand side is the kinetic energy of the whole mass, supposed concentrated at G and moving with this
point, together with the kinetic energy of the motion relative to G (§ 15); and the right-hand member
represents the integral work done by the extraneous forces in the successive infinitesimal displacements into
which the motion may be resolved.
The formula (13) may be easily verified in the case of the
compound pendulum, or of the solid rolling down an incline. As
another example, suppose we have a circular cylinder whose mass-
centre is at an excentric point, rolling on a horizontal plane. This
includes the case of a compound pendulum in which the knife-
edge is replaced by a cylindrical pin. If α be the radius of the
cylinder, h the distance of G from its axis (O), κ the radius of
gyration about a longitudinal axis through G, and θ the inclination
of OG to the vertical, the kinetic energy is 1⁄2Mκ2θ̇2 + Fig. 75.
⁄ M·CG2·thetȧ2, by § 3, since the body is turning about the line of
12
contact (C) as instantaneous axis, and the potential energy is −Mgh cos θ. The equation of energy is
therefore
⁄ M (κ2 + α2 + h2 − 2 ah cos θ) θ̇2 − Mgh cos θ − const.
12
(14)
Whenever, as in the preceding examples, a body or a system of bodies, is subject to constraints which
leave it virtually only one degree of freedom, the equation of energy is sufficient for the complete
determination of the motion. If q be any variable co-ordinate defining the position or (in the case of a system
of bodies) the configuration, the velocity of each particle at any instant will be proportional to q̇ , and the total
kinetic energy may be expressed in the form 1⁄2Aq̇ 2, where A is in general a function of q [cf. equation (14)].
This coefficient A is called the coefficient of inertia, or the reduced inertia of the system, referred to the co-
ordinate q.
Thus in the case of a railway truck travelling with velocity u
the kinetic energy is 1⁄2 (M + mκ2/α2)u2, where M is the total
mass, α the radius and κ the radius of gyration of each wheel,
and m is the sum of the masses of the wheels; the reduced
inertia is therefore M + mκ2/α2. Again, take the system
composed of the flywheel, connecting rod, and piston of a
steam-engine. We have here a limiting case of three-bar
motion (§ 3), and the instantaneous centre J of the connecting- Fig. 76.
rod PQ will have the position shown in the figure. The
velocities of P and Q will be in the ratio of JP to JQ, or OR to OQ; the velocity of the piston is
Page 457
therefore yθ̇, where y = OR. Hence if, for simplicity, we neglect the inertia of the connecting-rod, the
kinetic energy will be 1⁄2 (I + My2)thetȧ2, where I is the moment of inertia of the flywheel, and M is
the mass of the piston. The effect of the mass of the piston is therefore to increase the apparent
moment of inertia of the flywheel by the variable amount My2. If, on the other hand, we take OP (= x)
as our variable, the kinetic energy is 1⁄2 (M + I/y2)ẋ2. We may also say, therefore, that the effect of the
flywheel is to increase the apparent mass of the piston by the amount I/y2; this becomes infinite at the
“dead-points” where the crank is in line with the connecting-rod.
If the system be “conservative,” we have
⁄ Aq2 + V = const.,
12
(15)
where V is the potential energy. If we differentiate this with respect to t, and divide out by q̇ , we obtain
dA dV
Aq̈ + 1⁄2 q̇ 2 + =0
dq dq (16)
as the equation of motion of the system with the unknown reactions (if any) eliminated. For equilibrium this
must be satisfied by q̇ = O; this requires that dV/dq = 0, i.e. the potential energy must be “stationary.” To
examine the effect of a small disturbance from equilibrium we put V = ƒ(q), and write q = q0 + η, where q0 is
a root of ƒ′ (q0) = 0 and η is small. Neglecting terms of the second order in η we have dV/dq = ƒ′(q) = ƒ″
(q0)·η, and the equation (16) reduces to
Aη̈ + ƒ″ (q0)η = 0,
(17)
where A may be supposed to be constant and to have the value corresponding to q = q0. Hence if ƒ″ (q0) > 0,
i.e. if V is a minimum in the configuration of equilibrium, the variation of η is simple-harmonic, and the
period is 2π √{A/ƒ″(q0) }. This depends only on the constitution of the system, whereas the amplitude and
epoch will vary with the initial circumstances. If ƒ″ (q0) < 0, the solution of (17) will involve real
exponentials, and η will in general increase until the neglect of the terms of the second order is no longer
justified. The configuration q = q0, is then unstable.
As an example of the method, we may take the problem to which equation (14) relates. If we
differentiate, and divide by θ, and retain only the terms of the first order in θ, we obtain
{x2 + (h − α)2} θ̈ + ghθ = 0,
(18)
as the equation of small oscillations about the position θ = 0. The length of the equivalent simple
pendulum is {κ2 + (h − α)2}/h.
The equations which express the change of motion (in two dimensions) due to an instantaneous impulse
are of the forms
M (u′ − u) = ξ, M (ν′ − ν) = η, I (ω′ − ω) = ν.
(19)
kinetic energy will be 1⁄2 (I + My2)thetȧ2, where I is the moment of inertia of the flywheel, and M is
the mass of the piston. The effect of the mass of the piston is therefore to increase the apparent
moment of inertia of the flywheel by the variable amount My2. If, on the other hand, we take OP (= x)
as our variable, the kinetic energy is 1⁄2 (M + I/y2)ẋ2. We may also say, therefore, that the effect of the
flywheel is to increase the apparent mass of the piston by the amount I/y2; this becomes infinite at the
“dead-points” where the crank is in line with the connecting-rod.
If the system be “conservative,” we have
⁄ Aq2 + V = const.,
12
(15)
where V is the potential energy. If we differentiate this with respect to t, and divide out by q̇ , we obtain
dA dV
Aq̈ + 1⁄2 q̇ 2 + =0
dq dq (16)
as the equation of motion of the system with the unknown reactions (if any) eliminated. For equilibrium this
must be satisfied by q̇ = O; this requires that dV/dq = 0, i.e. the potential energy must be “stationary.” To
examine the effect of a small disturbance from equilibrium we put V = ƒ(q), and write q = q0 + η, where q0 is
a root of ƒ′ (q0) = 0 and η is small. Neglecting terms of the second order in η we have dV/dq = ƒ′(q) = ƒ″
(q0)·η, and the equation (16) reduces to
Aη̈ + ƒ″ (q0)η = 0,
(17)
where A may be supposed to be constant and to have the value corresponding to q = q0. Hence if ƒ″ (q0) > 0,
i.e. if V is a minimum in the configuration of equilibrium, the variation of η is simple-harmonic, and the
period is 2π √{A/ƒ″(q0) }. This depends only on the constitution of the system, whereas the amplitude and
epoch will vary with the initial circumstances. If ƒ″ (q0) < 0, the solution of (17) will involve real
exponentials, and η will in general increase until the neglect of the terms of the second order is no longer
justified. The configuration q = q0, is then unstable.
As an example of the method, we may take the problem to which equation (14) relates. If we
differentiate, and divide by θ, and retain only the terms of the first order in θ, we obtain
{x2 + (h − α)2} θ̈ + ghθ = 0,
(18)
as the equation of small oscillations about the position θ = 0. The length of the equivalent simple
pendulum is {κ2 + (h − α)2}/h.
The equations which express the change of motion (in two dimensions) due to an instantaneous impulse
are of the forms
M (u′ − u) = ξ, M (ν′ − ν) = η, I (ω′ − ω) = ν.
(19)
Page 458
Here u′, ν′ are the values of the component velocities of G just before, and u, ν
their values just after, the impulse, whilst ω′, ω denote the corresponding angular
velocities. Further, ξ, η are the time-integrals of the forces parallel to the co-
ordinate axes, and ν is the time-integral of their moment about G. Suppose, for
example, that a rigid lamina at rest, but free to move, is struck by an
instantaneous impulse F in a given line. Evidently G will begin to move parallel
to the line of F; let its initial velocity be u′, and let ω′ be the initial angular
Fig. 77.
velocity. Then Mu′ = F, Iω′ = F·GP, where GP is the perpendicular from G to the
line of F. If PG be produced to any point C, the initial velocity of the point C of the lamina will be
u′ − ω′·GC = (F/M) · (I − GC·CP/κ2),
where κ2 is the radius of gyration about G. The initial centre of rotation will therefore be at C, provided
GC·GP = κ2. If this condition be satisfied there would be no impulsive reaction at C even if this point were
fixed. The point P is therefore called the centre of percussion for the axis at C. It will be noted that the
relation between C and P is the same as that which connects the centres of suspension and oscillation in the
compound pendulum.
§ 18. Equations of Motion in Three Dimensions.—It was proved in § 7 that a body moving about a fixed
point O can be brought from its position at time t to its position at time t + δt by an infinitesimal rotation ε
about some axis through O; and the limiting position of this axis, when δt is infinitely small, was called the
“instantaneous axis.” The limiting value of the ratio ε/δt is called the angular velocity of the body; we denote
it by ω. If ξ, η, ζ are the components of ε about rectangular co-ordinate axes through O, the limiting values of
ξ/δt, η/δt, ζ/δt are called the component angular velocities; we denote them by p, q, r. If l, m, n be the
direction-cosines of the instantaneous axis we have
p = lω, q = mω, r = nω,
(1)
p2 + q2 + r2 = ω2.
(2)
If we draw a vector OJ to represent the angular velocity, then J traces out a certain curve in the body, called
the polhode, and a certain curve in space, called the herpolhode. The cones generated by the instantaneous
axis in the body and in space are called the polhode and herpolhode cones, respectively; in the actual motion
the former cone rolls on the latter (§ 7).
Fig. 78.
their values just after, the impulse, whilst ω′, ω denote the corresponding angular
velocities. Further, ξ, η are the time-integrals of the forces parallel to the co-
ordinate axes, and ν is the time-integral of their moment about G. Suppose, for
example, that a rigid lamina at rest, but free to move, is struck by an
instantaneous impulse F in a given line. Evidently G will begin to move parallel
to the line of F; let its initial velocity be u′, and let ω′ be the initial angular
Fig. 77.
velocity. Then Mu′ = F, Iω′ = F·GP, where GP is the perpendicular from G to the
line of F. If PG be produced to any point C, the initial velocity of the point C of the lamina will be
u′ − ω′·GC = (F/M) · (I − GC·CP/κ2),
where κ2 is the radius of gyration about G. The initial centre of rotation will therefore be at C, provided
GC·GP = κ2. If this condition be satisfied there would be no impulsive reaction at C even if this point were
fixed. The point P is therefore called the centre of percussion for the axis at C. It will be noted that the
relation between C and P is the same as that which connects the centres of suspension and oscillation in the
compound pendulum.
§ 18. Equations of Motion in Three Dimensions.—It was proved in § 7 that a body moving about a fixed
point O can be brought from its position at time t to its position at time t + δt by an infinitesimal rotation ε
about some axis through O; and the limiting position of this axis, when δt is infinitely small, was called the
“instantaneous axis.” The limiting value of the ratio ε/δt is called the angular velocity of the body; we denote
it by ω. If ξ, η, ζ are the components of ε about rectangular co-ordinate axes through O, the limiting values of
ξ/δt, η/δt, ζ/δt are called the component angular velocities; we denote them by p, q, r. If l, m, n be the
direction-cosines of the instantaneous axis we have
p = lω, q = mω, r = nω,
(1)
p2 + q2 + r2 = ω2.
(2)
If we draw a vector OJ to represent the angular velocity, then J traces out a certain curve in the body, called
the polhode, and a certain curve in space, called the herpolhode. The cones generated by the instantaneous
axis in the body and in space are called the polhode and herpolhode cones, respectively; in the actual motion
the former cone rolls on the latter (§ 7).
Fig. 78.
Page 459
The special case where both cones are right circular and ω is constant is important in astronomy and
also in mechanism (theory of bevel wheels). The “precession of the equinoxes” is due to the fact that
the earth performs a motion of this kind about its centre, and the whole class of such motions has
therefore been termed precessional. In fig. 78, which shows the various cases, OZ is the axis of the
fixed and OC that of the rolling cone, and J is the point of contact of the polhode and herpolhode,
which are of course both circles. If αbe the semi-angle of the rolling cone, β the constant inclination of
OC to OZ, and ψ̇ the angular velocity with which the plane ZOC revolves about OZ, then, considering
the velocity of a point in OC at unit distance from O, we have
ω sin α = ±ψ̇ sin β,
(3)
where the lower sign belongs to the third case. The earth’s precessional motion is of this latter type,
the angles being α = .0087″, β = 23° 28′.
If m be the mass of a particle at P, and PN the perpendicular to the instantaneous axis, the kinetic energy T
is given by
2T = Σ {m (ω·PN)2 } = ω2·Σ (m·PN2) = Iω2,
(4)
where I is the moment of inertia about the instantaneous axis. With the same notation for moments and
products of inertia as in § 11 (38), we have
I = Al2 + Bm2 + Cn2 − 2Fmn − 2Gnl − 2Hlm,
and therefore by (1),
2T = Ap2 + Bq2 + Cr2 − 2Fqr − 2Grp − 2Hpq.
(5)
Again, if x, y, z be the co-ordinates of P, the component velocities of m are
qz − ry, rx − pz, py − qx,
(6)
by § 7 (5); hence, if λ, μ, ν be now used to denote the component angular momenta about the co-ordinate
axes, we have λ = Σ {m (py − qx)y − m(rx − pz) z }, with two similar formulae, or
∂T
λ = Ap −Hq − Gr = ,
∂p
∂T
μ = −Hp + Bq − Fr = ,
∂q
∂T
ν = −Gp − Fq + Cr = .
∂r (7)
If the co-ordinate axes be taken to coincide with the principal axes of inertia at O, at the instant under
consideration, we have the simpler formulae
2T = Ap2 + Bq2 + Cr2,
(8)
also in mechanism (theory of bevel wheels). The “precession of the equinoxes” is due to the fact that
the earth performs a motion of this kind about its centre, and the whole class of such motions has
therefore been termed precessional. In fig. 78, which shows the various cases, OZ is the axis of the
fixed and OC that of the rolling cone, and J is the point of contact of the polhode and herpolhode,
which are of course both circles. If αbe the semi-angle of the rolling cone, β the constant inclination of
OC to OZ, and ψ̇ the angular velocity with which the plane ZOC revolves about OZ, then, considering
the velocity of a point in OC at unit distance from O, we have
ω sin α = ±ψ̇ sin β,
(3)
where the lower sign belongs to the third case. The earth’s precessional motion is of this latter type,
the angles being α = .0087″, β = 23° 28′.
If m be the mass of a particle at P, and PN the perpendicular to the instantaneous axis, the kinetic energy T
is given by
2T = Σ {m (ω·PN)2 } = ω2·Σ (m·PN2) = Iω2,
(4)
where I is the moment of inertia about the instantaneous axis. With the same notation for moments and
products of inertia as in § 11 (38), we have
I = Al2 + Bm2 + Cn2 − 2Fmn − 2Gnl − 2Hlm,
and therefore by (1),
2T = Ap2 + Bq2 + Cr2 − 2Fqr − 2Grp − 2Hpq.
(5)
Again, if x, y, z be the co-ordinates of P, the component velocities of m are
qz − ry, rx − pz, py − qx,
(6)
by § 7 (5); hence, if λ, μ, ν be now used to denote the component angular momenta about the co-ordinate
axes, we have λ = Σ {m (py − qx)y − m(rx − pz) z }, with two similar formulae, or
∂T
λ = Ap −Hq − Gr = ,
∂p
∂T
μ = −Hp + Bq − Fr = ,
∂q
∂T
ν = −Gp − Fq + Cr = .
∂r (7)
If the co-ordinate axes be taken to coincide with the principal axes of inertia at O, at the instant under
consideration, we have the simpler formulae
2T = Ap2 + Bq2 + Cr2,
(8)
Page 460
λ = Ap, μ = Bq, ν = Cr.
(9)
It is to be carefully noticed that the axis of resultant angular momentum about O does not in general
coincide with the instantaneous axis of rotation. The relation between these axes may be expressed by means
of the momental ellipsoid at O. The equation of the latter, referred to its principal axes, being as in § 11 (41),
the co-ordinates of the point J where it is met by the instantaneous axis are proportional to p, q, r, and the
direction-cosines of the normal at J are therefore proportional to Ap, Bq, Cr, or λ, μ, ν. The axis of resultant
angular momentum is therefore normal to the tangent plane at J, and does not coincide with OJ unless the
latter be a principal axis. Again, if Γ be the resultant angular momentum, so that
λ2 + μ2 + ν2 = Γ2,
(10)
the length of the perpendicular OH on the tangent plane at J is
Ap p Bq q Cr r 2T ρ
OH = · ρ+ · ρ+ · ρ= · ,
Γ ω Γ ω Γ ω Γ ω (11)
where ρ = OJ. This relation will be of use to us presently (§ 19).
The motion of a rigid body in the most general case may be specified by means of the component
velocities u, v, w of any point O of it which is taken as base, and the component angular velocities p, q, r.
The component velocities of any point whose co-ordinates relative to O are x, y, z are then
u + qz − ry, v + rx − pz, w + py − qx
(12)
by § 7 (6). It is usually convenient to take as our base-point the mass-centre of the body. In this case the
kinetic energy is given by
2T = M0 (u2 + v2 + w2) + Ap2 + Bq2 + Cr2 − 2Fqr − 2Grp − 2Hpg,
(13)
where M0 is the mass, and A, B, C, F, G, H are the moments and products of inertia with respect to the mass-
centre; cf. § 15 (9).
The components ξ, η, ζ of linear momentum are
∂T ∂T ∂T
ξ = M 0u = , η = M0v = , ζ = M0w =
∂u ∂v ∂w (14)
whilst those of the relative angular momentum are given by (7). The preceding formulae are sufficient for the
treatment of instantaneous impulses. Thus if an impulse (ξ, η, ζ, λ, μ, ν) change the motion from (u, v, w, p,
q, r) to (u′, v′, w′, p′, q′, r′) we have
M0 (u′ − u) = ξ, M0 (v′ − v) = η, M0(w′ − w) = ζ,
A (p′ − p) = λ, B (q′ − q) = μ, C (r′ − r) = ν,
(15)
(9)
It is to be carefully noticed that the axis of resultant angular momentum about O does not in general
coincide with the instantaneous axis of rotation. The relation between these axes may be expressed by means
of the momental ellipsoid at O. The equation of the latter, referred to its principal axes, being as in § 11 (41),
the co-ordinates of the point J where it is met by the instantaneous axis are proportional to p, q, r, and the
direction-cosines of the normal at J are therefore proportional to Ap, Bq, Cr, or λ, μ, ν. The axis of resultant
angular momentum is therefore normal to the tangent plane at J, and does not coincide with OJ unless the
latter be a principal axis. Again, if Γ be the resultant angular momentum, so that
λ2 + μ2 + ν2 = Γ2,
(10)
the length of the perpendicular OH on the tangent plane at J is
Ap p Bq q Cr r 2T ρ
OH = · ρ+ · ρ+ · ρ= · ,
Γ ω Γ ω Γ ω Γ ω (11)
where ρ = OJ. This relation will be of use to us presently (§ 19).
The motion of a rigid body in the most general case may be specified by means of the component
velocities u, v, w of any point O of it which is taken as base, and the component angular velocities p, q, r.
The component velocities of any point whose co-ordinates relative to O are x, y, z are then
u + qz − ry, v + rx − pz, w + py − qx
(12)
by § 7 (6). It is usually convenient to take as our base-point the mass-centre of the body. In this case the
kinetic energy is given by
2T = M0 (u2 + v2 + w2) + Ap2 + Bq2 + Cr2 − 2Fqr − 2Grp − 2Hpg,
(13)
where M0 is the mass, and A, B, C, F, G, H are the moments and products of inertia with respect to the mass-
centre; cf. § 15 (9).
The components ξ, η, ζ of linear momentum are
∂T ∂T ∂T
ξ = M 0u = , η = M0v = , ζ = M0w =
∂u ∂v ∂w (14)
whilst those of the relative angular momentum are given by (7). The preceding formulae are sufficient for the
treatment of instantaneous impulses. Thus if an impulse (ξ, η, ζ, λ, μ, ν) change the motion from (u, v, w, p,
q, r) to (u′, v′, w′, p′, q′, r′) we have
M0 (u′ − u) = ξ, M0 (v′ − v) = η, M0(w′ − w) = ζ,
A (p′ − p) = λ, B (q′ − q) = μ, C (r′ − r) = ν,
(15)
Page 461
where, for simplicity, the co-ordinate axes are supposed to coincide with the principal axes at the mass-
centre. Hence the change of kinetic energy is
T′ − T = ξ · 1⁄2 (u + u′) + η · 1⁄2 (v + v′) + ζ · 1⁄2 (w + w′),
+ λ · 1⁄2 (p + p′) + μ · 1⁄2 (q + q′) + ν · 1⁄2 (r + r′).
(16)
The factors of ξ, η, ζ, λ, μ, ν on the right-hand side are proportional to the constituents of a possible
infinitesimal displacement of the solid, and the whole expression is proportional (on the same scale) to the
work done by the given system of impulsive forces in such a displacement. As in § 9 this must be equal to
the total work done in such a displacement by the several forces, whatever they are, which make up the
impulse. We are thus led to the following statement: the change of kinetic energy due to any system of
impulsive forces is equal to the sum of the products of the several forces into the semi-sum of the initial and
final velocities of their respective points of application, resolved in the directions of the forces. Thus in the
problem of fig. 77 the kinetic energy generated is 1⁄2M (κ2 + Cq2)ω′2, if C be the instantaneous centre; this is
seen to be equal to 1⁄2F·ω′·CP, where ω′·CP represents the initial velocity of P.
The equations of continuous motion of a solid are obtained by substituting the values of ξ, η, ζ, λ, μ, ν
from (14) and (7) in the general equations
dξ dη dζ
= X, = Y, = Z,
dt dt dt
dλ dμ dν
= L, = M, = N,
dt dt dt (17)
where (X, Y, Z, L, M, N) denotes the system of extraneous forces
referred (like the momenta) to the mass-centre as base, the co-ordinate
axes being of course fixed in direction. The resulting equations are not as
a rule easy of application, owing to the fact that the moments and
products of inertia A, B, C, F, G, H are not constants but vary in
consequence of the changing orientation of the body with respect to the
co-ordinate axes.
An exception occurs, however, in the case of a solid which is
kinetically symmetrical (§ 11) about the mass-centre, e.g. a uniform
Fig. 79.
sphere. The equations then take the forms
M0u̇ = X, M0v̇ = Y, M0ẇ = Z,
Cṗ = L, Cq̇ = M, Cṙ = N,
(18)
where C is the constant moment of inertia about any axis through the mass-centre. Take, for example,
the case of a sphere rolling on a plane; and let the axes Ox, Oy be drawn through the centre parallel to
the plane, so that the equation of the latter is z = −a. We will suppose that the extraneous forces consist
of a known force (X, Y, Z) at the centre, and of the reactions (F1, F2, R) at the point of contact. Hence
centre. Hence the change of kinetic energy is
T′ − T = ξ · 1⁄2 (u + u′) + η · 1⁄2 (v + v′) + ζ · 1⁄2 (w + w′),
+ λ · 1⁄2 (p + p′) + μ · 1⁄2 (q + q′) + ν · 1⁄2 (r + r′).
(16)
The factors of ξ, η, ζ, λ, μ, ν on the right-hand side are proportional to the constituents of a possible
infinitesimal displacement of the solid, and the whole expression is proportional (on the same scale) to the
work done by the given system of impulsive forces in such a displacement. As in § 9 this must be equal to
the total work done in such a displacement by the several forces, whatever they are, which make up the
impulse. We are thus led to the following statement: the change of kinetic energy due to any system of
impulsive forces is equal to the sum of the products of the several forces into the semi-sum of the initial and
final velocities of their respective points of application, resolved in the directions of the forces. Thus in the
problem of fig. 77 the kinetic energy generated is 1⁄2M (κ2 + Cq2)ω′2, if C be the instantaneous centre; this is
seen to be equal to 1⁄2F·ω′·CP, where ω′·CP represents the initial velocity of P.
The equations of continuous motion of a solid are obtained by substituting the values of ξ, η, ζ, λ, μ, ν
from (14) and (7) in the general equations
dξ dη dζ
= X, = Y, = Z,
dt dt dt
dλ dμ dν
= L, = M, = N,
dt dt dt (17)
where (X, Y, Z, L, M, N) denotes the system of extraneous forces
referred (like the momenta) to the mass-centre as base, the co-ordinate
axes being of course fixed in direction. The resulting equations are not as
a rule easy of application, owing to the fact that the moments and
products of inertia A, B, C, F, G, H are not constants but vary in
consequence of the changing orientation of the body with respect to the
co-ordinate axes.
An exception occurs, however, in the case of a solid which is
kinetically symmetrical (§ 11) about the mass-centre, e.g. a uniform
Fig. 79.
sphere. The equations then take the forms
M0u̇ = X, M0v̇ = Y, M0ẇ = Z,
Cṗ = L, Cq̇ = M, Cṙ = N,
(18)
where C is the constant moment of inertia about any axis through the mass-centre. Take, for example,
the case of a sphere rolling on a plane; and let the axes Ox, Oy be drawn through the centre parallel to
the plane, so that the equation of the latter is z = −a. We will suppose that the extraneous forces consist
of a known force (X, Y, Z) at the centre, and of the reactions (F1, F2, R) at the point of contact. Hence
Page 462
M0u̇ = X + F1, M0v̇ = Y + F2, 0 = Z + R,
Cṗ = F2a, Cq̇ = −F1a, Cṙ = 0.
(19)
The last equation shows that the angular velocity about the normal to the plane is constant. Again,
since the point of the sphere which is in contact with the plane is instantaneously at rest, we have the
geometrical relations
u + qa = 0, v + pa = 0, w = 0,
(20)
by (12). Eliminating p, q, we get
(M0 + Ca−2) u̇ = X, (M0 + Ca−2) v̇ = Y.
(21)
The acceleration of the centre is therefore the same as if the plane were smooth and the mass of the
sphere were increased by C/α2. Thus the centre of a sphere rolling under gravity on a plane of
inclination a describes a parabola with an acceleration
g sin α/(1 + C/Ma2)
parallel to the lines of greatest slope.
Take next the case of a sphere rolling on a fixed spherical surface. Let a be the radius of the rolling
sphere, c that of the spherical surface which is the locus of its centre, and let x, y, z be the co-ordinates
of this centre relative to axes through O, the centre of the fixed sphere. If the only extraneous forces
are the reactions (P, Q, R) at the point of contact, we have
M0ẍ = P, M0ÿ = Q, M0z̈ = R,
a a a
Cṗ = − (yR − zQ), Cq̇ = − (zP − xR), Cṙ = − (xQ − yP),
c c c (22)
the standard case being that where the rolling sphere is outside the fixed surface. The opposite case is
obtained by reversing the sign of a. We have also the geometrical relations
ẋ = (a/c) (qz − ry), ẏ = (a/c) (rx − pz), ż = (a/c) (py − gx),
(23)
If we eliminate P, Q, R from (22), the resulting equations are integrable with respect to t; thus
M0a M0a M0a
p=− (yż − zẏ) + α, q=− (zẋ − xż) + β, r=− (xẏ − yẋ) + γ,
Cc Cc Cc (24)
where α, β, γ are arbitrary constants. Substituting in (23) we find
M0a2 a M0a2 a M0a2 a
(1+ C
) ẋ = c (βz − γy), ( 1 + C
) ẏ = c (γx − αz), ( 1 + C
) ż = c (25)
(αy − βx).
Hence αẋ + βẏ + γż = 0, or
Cṗ = F2a, Cq̇ = −F1a, Cṙ = 0.
(19)
The last equation shows that the angular velocity about the normal to the plane is constant. Again,
since the point of the sphere which is in contact with the plane is instantaneously at rest, we have the
geometrical relations
u + qa = 0, v + pa = 0, w = 0,
(20)
by (12). Eliminating p, q, we get
(M0 + Ca−2) u̇ = X, (M0 + Ca−2) v̇ = Y.
(21)
The acceleration of the centre is therefore the same as if the plane were smooth and the mass of the
sphere were increased by C/α2. Thus the centre of a sphere rolling under gravity on a plane of
inclination a describes a parabola with an acceleration
g sin α/(1 + C/Ma2)
parallel to the lines of greatest slope.
Take next the case of a sphere rolling on a fixed spherical surface. Let a be the radius of the rolling
sphere, c that of the spherical surface which is the locus of its centre, and let x, y, z be the co-ordinates
of this centre relative to axes through O, the centre of the fixed sphere. If the only extraneous forces
are the reactions (P, Q, R) at the point of contact, we have
M0ẍ = P, M0ÿ = Q, M0z̈ = R,
a a a
Cṗ = − (yR − zQ), Cq̇ = − (zP − xR), Cṙ = − (xQ − yP),
c c c (22)
the standard case being that where the rolling sphere is outside the fixed surface. The opposite case is
obtained by reversing the sign of a. We have also the geometrical relations
ẋ = (a/c) (qz − ry), ẏ = (a/c) (rx − pz), ż = (a/c) (py − gx),
(23)
If we eliminate P, Q, R from (22), the resulting equations are integrable with respect to t; thus
M0a M0a M0a
p=− (yż − zẏ) + α, q=− (zẋ − xż) + β, r=− (xẏ − yẋ) + γ,
Cc Cc Cc (24)
where α, β, γ are arbitrary constants. Substituting in (23) we find
M0a2 a M0a2 a M0a2 a
(1+ C
) ẋ = c (βz − γy), ( 1 + C
) ẏ = c (γx − αz), ( 1 + C
) ż = c (25)
(αy − βx).
Hence αẋ + βẏ + γż = 0, or
Page 463
αx + βy + γz = const.;
(26)
which shows that the centre of the rolling sphere describes a circle. If the axis of z be taken normal to
the plane of this circle we have α = 0, β = 0, and
M0a2 a M0a2 a
(1+ C
) ẋ = −γ c y, ( 1 + C
) ẏ = γ c x. (27)
The solution of these equations is of the type
x = b cos (στ + ε), y = b sin (σt + ε),
(28)
where b, ε are arbitrary, and
γa/c
σ= .
1 + M0a2/C (29)
The circle is described with the constant angular velocity σ.
When the gravity of the rolling sphere is to be taken into account the preceding method is not in
general convenient, unless the whole motion of G is small. As an example of this latter type, suppose
that a sphere is placed on the highest point of a fixed sphere and set spinning about the vertical
diameter with the angular velocity n; it will appear that under a certain condition the motion of G
consequent on a slight disturbance will be oscillatory. If Oz be drawn vertically upwards, then in the
beginning of the disturbed motion the quantities x, y, p, q, P, Q will all be small. Hence, omitting terms
of the second order, we find
M0ẍ = P, M0ẏ = Q, R = M0g,
Cṗ = −(M0ga/c) y + aQ, Cq̇ = (M0ga/c) x − aP, Cṙ = 0.
(30)
The last equation shows that the component r of the angular velocity retains (to the first order) the
constant value n. The geometrical relations reduce to
ẋ = aq − (na/c) y, ẏ = −ap + (na/c) x.
(31)
Eliminating p, g, P, Q, we obtain the equations
(C + M0a2) ẍ + (Cna/c) y − (M0ga2/c) x = 0,
(C + M0a2) ÿ − (Cna/c) x − (M0ga2/c) y = 0,
(32)
which are both contained in
d2 Cna d M0ga2
{ (C + M0a2) dt2 − i c dt
−
c
} (x + iy) = 0. (33)
(26)
which shows that the centre of the rolling sphere describes a circle. If the axis of z be taken normal to
the plane of this circle we have α = 0, β = 0, and
M0a2 a M0a2 a
(1+ C
) ẋ = −γ c y, ( 1 + C
) ẏ = γ c x. (27)
The solution of these equations is of the type
x = b cos (στ + ε), y = b sin (σt + ε),
(28)
where b, ε are arbitrary, and
γa/c
σ= .
1 + M0a2/C (29)
The circle is described with the constant angular velocity σ.
When the gravity of the rolling sphere is to be taken into account the preceding method is not in
general convenient, unless the whole motion of G is small. As an example of this latter type, suppose
that a sphere is placed on the highest point of a fixed sphere and set spinning about the vertical
diameter with the angular velocity n; it will appear that under a certain condition the motion of G
consequent on a slight disturbance will be oscillatory. If Oz be drawn vertically upwards, then in the
beginning of the disturbed motion the quantities x, y, p, q, P, Q will all be small. Hence, omitting terms
of the second order, we find
M0ẍ = P, M0ẏ = Q, R = M0g,
Cṗ = −(M0ga/c) y + aQ, Cq̇ = (M0ga/c) x − aP, Cṙ = 0.
(30)
The last equation shows that the component r of the angular velocity retains (to the first order) the
constant value n. The geometrical relations reduce to
ẋ = aq − (na/c) y, ẏ = −ap + (na/c) x.
(31)
Eliminating p, g, P, Q, we obtain the equations
(C + M0a2) ẍ + (Cna/c) y − (M0ga2/c) x = 0,
(C + M0a2) ÿ − (Cna/c) x − (M0ga2/c) y = 0,
(32)
which are both contained in
d2 Cna d M0ga2
{ (C + M0a2) dt2 − i c dt
−
c
} (x + iy) = 0. (33)
Page 464
This has two solutions of the type x + iy = αei(σt + ε), where α, ε are arbitrary, and σ is a root of the
quadratic
(C + M0a2) σ2 − (Cna/c) σ + M0ga2/c = 0.
(34)
If
n2 > (4Mgc/C) (1 + M0a2/C),
(35)
both roots are real, and have the same sign as n. The motion of G then consists of two superposed
circular vibrations of the type
x = α cos (σt + ε), y = α sin (σt + ε),
(36)
in each of which the direction of revolution is the same as that of the initial spin of the sphere. It
follows therefore that the original position is stable provided the spin n exceed the limit defined by
(35). The case of a sphere spinning about a vertical axis at the lowest point of a spherical bowl is
obtained by reversing the signs of α and c. It appears that this position is always stable.
It is to be remarked, however, that in the first form of the problem the stability above investigated is
practically of a limited or temporary kind. The slightest frictional forces—such as the resistance of the
air—even if they act in lines through the centre of the rolling sphere, and so do not directly affect its
angular momentum, will cause the centre gradually to descend in an ever-widening spiral path.
§ 19. Free Motion of a Solid.—Before proceeding to further problems of motion under extraneous forces it
is convenient to investigate the free motion of a solid relative to its mass-centre O, in the most general case.
This is the same as the motion about a fixed point under the action of extraneous forces which have zero
moment about that point. The question was first discussed by Euler (1750); the geometrical representation to
be given is due to Poinsot (1851).
The kinetic energy T of the motion relative to O will be constant. Now T = 1⁄2Iω2, where ω is the angular
velocity and I is the moment of inertia about the instantaneous axis. If ρ be the radius-vector OJ of the
momental ellipsoid
Ax2 + By2 + Cz2 = Mε4
(1)
drawn in the direction of the instantaneous axis, we have I = Mε4/ρ2 (§ 11); hence ω varies as ρ. The locus of
J may therefore be taken as the “polhode” (§ 18). Again, the vector which represents the angular momentum
with respect to O will be constant in every respect. We have seen (§ 18) that this vector coincides in direction
with the perpendicular OH to the tangent plane of the momental ellipsoid at J; also that
2T ρ
OH = · ,
Γ ω (2)
where Γ is the resultant angular momentum about O. Since ω varies as ρ, it follows that OH is constant, and
the tangent plane at J is therefore fixed in space. The motion of the body relative to O is therefore completely
quadratic
(C + M0a2) σ2 − (Cna/c) σ + M0ga2/c = 0.
(34)
If
n2 > (4Mgc/C) (1 + M0a2/C),
(35)
both roots are real, and have the same sign as n. The motion of G then consists of two superposed
circular vibrations of the type
x = α cos (σt + ε), y = α sin (σt + ε),
(36)
in each of which the direction of revolution is the same as that of the initial spin of the sphere. It
follows therefore that the original position is stable provided the spin n exceed the limit defined by
(35). The case of a sphere spinning about a vertical axis at the lowest point of a spherical bowl is
obtained by reversing the signs of α and c. It appears that this position is always stable.
It is to be remarked, however, that in the first form of the problem the stability above investigated is
practically of a limited or temporary kind. The slightest frictional forces—such as the resistance of the
air—even if they act in lines through the centre of the rolling sphere, and so do not directly affect its
angular momentum, will cause the centre gradually to descend in an ever-widening spiral path.
§ 19. Free Motion of a Solid.—Before proceeding to further problems of motion under extraneous forces it
is convenient to investigate the free motion of a solid relative to its mass-centre O, in the most general case.
This is the same as the motion about a fixed point under the action of extraneous forces which have zero
moment about that point. The question was first discussed by Euler (1750); the geometrical representation to
be given is due to Poinsot (1851).
The kinetic energy T of the motion relative to O will be constant. Now T = 1⁄2Iω2, where ω is the angular
velocity and I is the moment of inertia about the instantaneous axis. If ρ be the radius-vector OJ of the
momental ellipsoid
Ax2 + By2 + Cz2 = Mε4
(1)
drawn in the direction of the instantaneous axis, we have I = Mε4/ρ2 (§ 11); hence ω varies as ρ. The locus of
J may therefore be taken as the “polhode” (§ 18). Again, the vector which represents the angular momentum
with respect to O will be constant in every respect. We have seen (§ 18) that this vector coincides in direction
with the perpendicular OH to the tangent plane of the momental ellipsoid at J; also that
2T ρ
OH = · ,
Γ ω (2)
where Γ is the resultant angular momentum about O. Since ω varies as ρ, it follows that OH is constant, and
the tangent plane at J is therefore fixed in space. The motion of the body relative to O is therefore completely
Page 465
represented if we imagine the momental ellipsoid at O to roll without sliding on a plane fixed in space, with
an angular velocity proportional at each instant to the radius-vector of the point of contact. The fixed plane is
parallel to the invariable plane at O, and the line OH is called the invariable line. The trace of the point of
contact J on the fixed plane is the “herpolhode.”
If p, q, r be the component angular velocities about the principal axes at O, we have
(A2p2 + B2q2 + C2r2) / Γ2 = (Ap2 + Bq2 + Cr2) / 2T,
(3)
each side being in fact equal to unity. At a point on the polhode cone x : y : z = p : q : r, and the equation of
this cone is therefore
Γ Γ Γ
A2 ( 1 − 2AT
2
) x + B ( 1 − 2BT
2 2 ) y + C ( 1 − 2CT
2
) z = 0.
2 2
2
2
(4)
Since 2AT − Γ2 = B (A − B)q2 + C(A − C)r2, it appears that if A > B > C the coefficient of x2 in (4) is
positive, that of z2 is negative, whilst that of y2 is positive or negative according as 2BT ≷ Γ2. Hence the
polhode cone surrounds the axis of greatest or least moment according as 2BT ≷ Γ2. In the critical case of
2BT = Γ2 it breaks up into two planes through the axis of mean moment (Oy). The herpolhode curve in the
fixed plane is obviously confined between two concentric circles which it alternately touches; it is not in
general a re-entrant curve. It has been shown by De Sparre that, owing to the limitation imposed on the
possible forms of the momental ellipsoid by the relation B + C > A, the curve has no points of inflexion. The
invariable line OH describes another cone in the body, called the invariable cone. At any point of this we
have x : y : z = Ap · Bq : Cr, and the equation is therefore
Γ2 Γ2 Γ2
( 1 − 2AT ) x + ( 1 − 2BT ) y + ( 1 − 2CT ) z = 0.
2 2 2
(5)
The signs of the coefficients follow the same rule as in the case of
(4). The possible forms of the invariable cone are indicated in fig. 80
by means of the intersections with a concentric spherical surface. In
the critical case of 2BT = Γ2 the cone degenerates into two planes. It
appears that if the body be sightly disturbed from a state of rotation
about the principal axis of greatest or least moment, the invariable
cone will closely surround this axis, which will therefore never
deviate far from the invariable line. If, on the other hand, the body be
slightly disturbed from a state of rotation about the mean axis a wide
deviation will take place. Hence a rotation about the axis of greatest
Fig. 80.
or least moment is reckoned as stable, a rotation about the mean axis
as unstable. The question is greatly simplified when two of the
principal moments are equal, say A = B. The polhode and herpolhode cones are then right circular, and the
motion is “precessional” according to the definition of § 18. If α be the inclination of the instantaneous axis
to the axis of symmetry, β the inclination of the latter axis to the invariable line, we have
Γ cos β = C ω cos α, Γ sin β = A ω sin α,
(6)
an angular velocity proportional at each instant to the radius-vector of the point of contact. The fixed plane is
parallel to the invariable plane at O, and the line OH is called the invariable line. The trace of the point of
contact J on the fixed plane is the “herpolhode.”
If p, q, r be the component angular velocities about the principal axes at O, we have
(A2p2 + B2q2 + C2r2) / Γ2 = (Ap2 + Bq2 + Cr2) / 2T,
(3)
each side being in fact equal to unity. At a point on the polhode cone x : y : z = p : q : r, and the equation of
this cone is therefore
Γ Γ Γ
A2 ( 1 − 2AT
2
) x + B ( 1 − 2BT
2 2 ) y + C ( 1 − 2CT
2
) z = 0.
2 2
2
2
(4)
Since 2AT − Γ2 = B (A − B)q2 + C(A − C)r2, it appears that if A > B > C the coefficient of x2 in (4) is
positive, that of z2 is negative, whilst that of y2 is positive or negative according as 2BT ≷ Γ2. Hence the
polhode cone surrounds the axis of greatest or least moment according as 2BT ≷ Γ2. In the critical case of
2BT = Γ2 it breaks up into two planes through the axis of mean moment (Oy). The herpolhode curve in the
fixed plane is obviously confined between two concentric circles which it alternately touches; it is not in
general a re-entrant curve. It has been shown by De Sparre that, owing to the limitation imposed on the
possible forms of the momental ellipsoid by the relation B + C > A, the curve has no points of inflexion. The
invariable line OH describes another cone in the body, called the invariable cone. At any point of this we
have x : y : z = Ap · Bq : Cr, and the equation is therefore
Γ2 Γ2 Γ2
( 1 − 2AT ) x + ( 1 − 2BT ) y + ( 1 − 2CT ) z = 0.
2 2 2
(5)
The signs of the coefficients follow the same rule as in the case of
(4). The possible forms of the invariable cone are indicated in fig. 80
by means of the intersections with a concentric spherical surface. In
the critical case of 2BT = Γ2 the cone degenerates into two planes. It
appears that if the body be sightly disturbed from a state of rotation
about the principal axis of greatest or least moment, the invariable
cone will closely surround this axis, which will therefore never
deviate far from the invariable line. If, on the other hand, the body be
slightly disturbed from a state of rotation about the mean axis a wide
deviation will take place. Hence a rotation about the axis of greatest
Fig. 80.
or least moment is reckoned as stable, a rotation about the mean axis
as unstable. The question is greatly simplified when two of the
principal moments are equal, say A = B. The polhode and herpolhode cones are then right circular, and the
motion is “precessional” according to the definition of § 18. If α be the inclination of the instantaneous axis
to the axis of symmetry, β the inclination of the latter axis to the invariable line, we have
Γ cos β = C ω cos α, Γ sin β = A ω sin α,
(6)
Page 466
whence
A
tan β tan α.
C (7)
Fig. 81.
Hence β ≷ α, and the circumstances are therefore those of the first or second case in fig. 78, according as A
≷ C. If ψ be the rate at which the plane HOJ revolves about OH, we have
sin α C cos α
ψ= ω= ω,
sin β A cos β (8)
by § 18 (3). Also if χ̇ be the rate at which J describes the polhode, we have ψ̇ sin (β − α) = χ̇ sin β, whence
sin (α − β)
χ̇ = ω.
sin α (9)
If the instantaneous axis only deviate slightly from the axis of symmetry the angles α, β are small, and χ̇ = (A
− C) A·ω; the instantaneous axis therefore completes its revolution in the body in the period
2π A−C
= ω.
χ̇ A (10)
In the case of the earth it is inferred from the independent phenomenon of luni-solar precession that
(C − A)/A = .00313. Hence if the earth’s axis of rotation deviates slightly from the axis of figure, it
should describe a cone about the latter in 320 sidereal days. This would cause a periodic variation in
the latitude of any place on the earth’s surface, as determined by astronomical methods. There appears
to be evidence of a slight periodic variation of latitude, but the period would seem to be about fourteen
months. The discrepancy is attributed to a defect of rigidity in the earth. The phenomenon is known as
the Eulerian nutation, since it is supposed to come under the free rotations first discussed by Euler.
§ 20. Motion of a Solid of Revolution.—In the case of a solid of revolution, or (more generally) whenever
there is kinetic symmetry about an axis through the mass-centre, or through a fixed point O, a number of
interesting problems can be treated almost directly from first principles. It frequently happens that the
extraneous forces have zero moment about the axis of symmetry, as e.g. in the case of the flywheel of a
gyroscope if we neglect the friction at the bearings. The angular velocity (r) about this axis is then constant.
For we have seen that r is constant when there are no extraneous forces; and r is evidently not affected by an
instantaneous impulse which leaves the angular momentum Cr, about the axis of symmetry, unaltered. And a
continuous force may be regarded as the limit of a succession of infinitesimal instantaneous impulses.
A
tan β tan α.
C (7)
Fig. 81.
Hence β ≷ α, and the circumstances are therefore those of the first or second case in fig. 78, according as A
≷ C. If ψ be the rate at which the plane HOJ revolves about OH, we have
sin α C cos α
ψ= ω= ω,
sin β A cos β (8)
by § 18 (3). Also if χ̇ be the rate at which J describes the polhode, we have ψ̇ sin (β − α) = χ̇ sin β, whence
sin (α − β)
χ̇ = ω.
sin α (9)
If the instantaneous axis only deviate slightly from the axis of symmetry the angles α, β are small, and χ̇ = (A
− C) A·ω; the instantaneous axis therefore completes its revolution in the body in the period
2π A−C
= ω.
χ̇ A (10)
In the case of the earth it is inferred from the independent phenomenon of luni-solar precession that
(C − A)/A = .00313. Hence if the earth’s axis of rotation deviates slightly from the axis of figure, it
should describe a cone about the latter in 320 sidereal days. This would cause a periodic variation in
the latitude of any place on the earth’s surface, as determined by astronomical methods. There appears
to be evidence of a slight periodic variation of latitude, but the period would seem to be about fourteen
months. The discrepancy is attributed to a defect of rigidity in the earth. The phenomenon is known as
the Eulerian nutation, since it is supposed to come under the free rotations first discussed by Euler.
§ 20. Motion of a Solid of Revolution.—In the case of a solid of revolution, or (more generally) whenever
there is kinetic symmetry about an axis through the mass-centre, or through a fixed point O, a number of
interesting problems can be treated almost directly from first principles. It frequently happens that the
extraneous forces have zero moment about the axis of symmetry, as e.g. in the case of the flywheel of a
gyroscope if we neglect the friction at the bearings. The angular velocity (r) about this axis is then constant.
For we have seen that r is constant when there are no extraneous forces; and r is evidently not affected by an
instantaneous impulse which leaves the angular momentum Cr, about the axis of symmetry, unaltered. And a
continuous force may be regarded as the limit of a succession of infinitesimal instantaneous impulses.
Page 467
Suppose, for example, that a flywheel is rotating with angular
velocity n about its axis, which is (say) horizontal, and that this axis is
made to rotate with the angular velocity ψ̇ in the horizontal plane. The
components of angular momentum about the axis of the flywheel and
about the vertical will be Cn and A ψ̇ respectively, where A is the
moment of inertia about any axis through the mass-centre (or through Fig. 82.
the fixed point O) perpendicular to that of symmetry. If OK> be the
vector representing the former component at time t, the vector which represents it at time t + δt will be
OK′>, equal to OK> in magnitude and making with it an angle δψ. Hence KK′> (= Cn δψ) will represent
the change in this component due to the extraneous forces. Hence, so far as this component is
concerned, the extraneous forces must supply a couple of moment Cnψ̇ in a vertical plane through the
axis of the flywheel. If this couple be absent, the axis will be tilted out of the horizontal plane in such
a sense that the direction of the spin n approximates to that of the azimuthal rotation ψ̇. The remaining
constituent of the extraneous forces is a couple Aψ̈ about the vertical; this vanishes if ψ̇ is constant. If
the axis of the flywheel make an angle θ with the vertical, it is seen in like manner that the required
couple in the vertical plane through the axis is Cn sin θ ψ̇. This matter can be strikingly illustrated with
an ordinary gyroscope, e.g. by making the larger movable ring in fig. 37 rotate about its vertical
diameter.
If the direction of the axis of kinetic symmetry be specified by means of the
angular co-ordinates θ, ψ of § 7, then considering the component velocities of the
point C in fig. 83, which are θ̇ and sin θψ̇ along and perpendicular to the meridian
ZC, we see that the component angular velocities about the lines OA′, OB′ are
−sin θ ψ̇ and θ̇ respectively. Hence if the principal moments of inertia at O be A,
A, C, and if n be the constant angular velocity about the axis OC, the kinetic
energy is given by
2T = A (θ̇2 + sin2 θψ̇2) + Cn2. Fig. 83.
(1)
Again, the components of angular momentum about OC, OA′ are Cn, −A sin θ ψ̇, and therefore the angular
momentum (μ, say) about OZ is
μ = A sin2 θψ̇ + Cn cos θ.
(2)
We can hence deduce the condition of steady precessional motion in a top. A solid of revolution is
supposed to be free to turn about a fixed point O on its axis of symmetry, its mass-centre G being in this axis
at a distance h from O. In fig. 83 OZ is supposed to be vertical, and OC is the axis of the solid drawn in the
direction OG. If θ is constant the points C, A′ will in time δt come to positions C″, A″ such that CC″ = sin θ
δψ, A′A″ = cos θ δψ, and the angular momentum about OB′ will become Cn sin θ δψ − A sin θ ψ̇ · cos θ δψ.
Equating this to Mgh sin θ δt, and dividing out by sin θ, we obtain
A cos θ ψ̇2 − Cnψ̇ + Mgh = 0,
(3)
velocity n about its axis, which is (say) horizontal, and that this axis is
made to rotate with the angular velocity ψ̇ in the horizontal plane. The
components of angular momentum about the axis of the flywheel and
about the vertical will be Cn and A ψ̇ respectively, where A is the
moment of inertia about any axis through the mass-centre (or through Fig. 82.
the fixed point O) perpendicular to that of symmetry. If OK> be the
vector representing the former component at time t, the vector which represents it at time t + δt will be
OK′>, equal to OK> in magnitude and making with it an angle δψ. Hence KK′> (= Cn δψ) will represent
the change in this component due to the extraneous forces. Hence, so far as this component is
concerned, the extraneous forces must supply a couple of moment Cnψ̇ in a vertical plane through the
axis of the flywheel. If this couple be absent, the axis will be tilted out of the horizontal plane in such
a sense that the direction of the spin n approximates to that of the azimuthal rotation ψ̇. The remaining
constituent of the extraneous forces is a couple Aψ̈ about the vertical; this vanishes if ψ̇ is constant. If
the axis of the flywheel make an angle θ with the vertical, it is seen in like manner that the required
couple in the vertical plane through the axis is Cn sin θ ψ̇. This matter can be strikingly illustrated with
an ordinary gyroscope, e.g. by making the larger movable ring in fig. 37 rotate about its vertical
diameter.
If the direction of the axis of kinetic symmetry be specified by means of the
angular co-ordinates θ, ψ of § 7, then considering the component velocities of the
point C in fig. 83, which are θ̇ and sin θψ̇ along and perpendicular to the meridian
ZC, we see that the component angular velocities about the lines OA′, OB′ are
−sin θ ψ̇ and θ̇ respectively. Hence if the principal moments of inertia at O be A,
A, C, and if n be the constant angular velocity about the axis OC, the kinetic
energy is given by
2T = A (θ̇2 + sin2 θψ̇2) + Cn2. Fig. 83.
(1)
Again, the components of angular momentum about OC, OA′ are Cn, −A sin θ ψ̇, and therefore the angular
momentum (μ, say) about OZ is
μ = A sin2 θψ̇ + Cn cos θ.
(2)
We can hence deduce the condition of steady precessional motion in a top. A solid of revolution is
supposed to be free to turn about a fixed point O on its axis of symmetry, its mass-centre G being in this axis
at a distance h from O. In fig. 83 OZ is supposed to be vertical, and OC is the axis of the solid drawn in the
direction OG. If θ is constant the points C, A′ will in time δt come to positions C″, A″ such that CC″ = sin θ
δψ, A′A″ = cos θ δψ, and the angular momentum about OB′ will become Cn sin θ δψ − A sin θ ψ̇ · cos θ δψ.
Equating this to Mgh sin θ δt, and dividing out by sin θ, we obtain
A cos θ ψ̇2 − Cnψ̇ + Mgh = 0,
(3)
Page 468
as the condition in question. For given values of n and θ we have two possible values of ψ̇ provided n
exceed a certain limit. With a very rapid spin, or (more precisely) with Cn large in comparison with
√(4AMgh cos θ), one value of ψ̇ is small and the other large, viz. the two values are Mgh/Cn and Cn/A cos θ
approximately. The absence of g from the latter expression indicates that the circumstances of the rapid
precession are very nearly those of a free Eulerian rotation (§ 19), gravity playing only a subordinate part.
Again, take the case of a circular disk rolling in steady motion
on a horizontal plane. The centre O of the disk is supposed to
describe a horizontal circle of radius c with the constant angular
velocity ψ̇, whilst its plane preserves a constant inclination θ to the
horizontal. The components of the reaction of the horizontal lane
will be Mcψ̇2 at right angles to the tangent line at the point of
contact and Mg vertically upwards, and the moment of these about Fig. 84.
the horizontal diameter of the disk, which corresponds to OB′ in
fig. 83, is Mcψ̇2. α sin θ − Mgα cos θ, where α is the radius of the disk. Equating this to the rate of
increase of the angular momentum about OB′, investigated as above, we find
a a2
( C + Ma2 + A c cos θ ) ψ̇2 = Mg c cot θ, (4)
where use has been made of the obvious relation nα = cψ̇. If c and θ be given this formula determines
the value of ψ̇ for which the motion will be steady.
In the case of the top, the equation of energy and the condition of constant angular momentum (μ) about
the vertical OZ are sufficient to determine the motion of the axis. Thus, we have
⁄ A (θ̇2 + sin2 θψ̇2) + 1⁄2Cn2 + Mgh cos θ = const.,
12
(5)
A sin2 θψ̇ + ν cos θ = μ,
(6)
where ν is written for Cn. From these ψ̇ may be eliminated, and on differentiating the resulting equation
with respect to t we obtain
(μ − ν cos θ) (μ cos θ − ν)
Aθ̈ − − Mgh sin θ = 0.
A sin3 θ (7)
If we put θ̈ = 0 we get the condition of steady precessional motion in a form equivalent to (3). To find the
small oscillation about a state of steady precession in which the axis makes a constant angle α with the
vertical, we write θ = α + χ, and neglect terms of the second order in χ. The result is of the form
χ̈ + σ2χ = 0,
(8)
where
σ2 = { (μ − ν cos α)2 + 2 (μ − ν cos α) (μ cos α − ν) cos α +
(μ cos α − ν)2 } / A2 sin4 α.
(9)
exceed a certain limit. With a very rapid spin, or (more precisely) with Cn large in comparison with
√(4AMgh cos θ), one value of ψ̇ is small and the other large, viz. the two values are Mgh/Cn and Cn/A cos θ
approximately. The absence of g from the latter expression indicates that the circumstances of the rapid
precession are very nearly those of a free Eulerian rotation (§ 19), gravity playing only a subordinate part.
Again, take the case of a circular disk rolling in steady motion
on a horizontal plane. The centre O of the disk is supposed to
describe a horizontal circle of radius c with the constant angular
velocity ψ̇, whilst its plane preserves a constant inclination θ to the
horizontal. The components of the reaction of the horizontal lane
will be Mcψ̇2 at right angles to the tangent line at the point of
contact and Mg vertically upwards, and the moment of these about Fig. 84.
the horizontal diameter of the disk, which corresponds to OB′ in
fig. 83, is Mcψ̇2. α sin θ − Mgα cos θ, where α is the radius of the disk. Equating this to the rate of
increase of the angular momentum about OB′, investigated as above, we find
a a2
( C + Ma2 + A c cos θ ) ψ̇2 = Mg c cot θ, (4)
where use has been made of the obvious relation nα = cψ̇. If c and θ be given this formula determines
the value of ψ̇ for which the motion will be steady.
In the case of the top, the equation of energy and the condition of constant angular momentum (μ) about
the vertical OZ are sufficient to determine the motion of the axis. Thus, we have
⁄ A (θ̇2 + sin2 θψ̇2) + 1⁄2Cn2 + Mgh cos θ = const.,
12
(5)
A sin2 θψ̇ + ν cos θ = μ,
(6)
where ν is written for Cn. From these ψ̇ may be eliminated, and on differentiating the resulting equation
with respect to t we obtain
(μ − ν cos θ) (μ cos θ − ν)
Aθ̈ − − Mgh sin θ = 0.
A sin3 θ (7)
If we put θ̈ = 0 we get the condition of steady precessional motion in a form equivalent to (3). To find the
small oscillation about a state of steady precession in which the axis makes a constant angle α with the
vertical, we write θ = α + χ, and neglect terms of the second order in χ. The result is of the form
χ̈ + σ2χ = 0,
(8)
where
σ2 = { (μ − ν cos α)2 + 2 (μ − ν cos α) (μ cos α − ν) cos α +
(μ cos α − ν)2 } / A2 sin4 α.
(9)
Page 469
When ν is large we have, for the “slow” precession σ = ν/A, and for the “rapid” precession σ = A/ν cos α =
ψ̇, approximately. Further, on examining the small variation in ψ̇, it appears that in a slightly disturbed slow
precession the motion of any point of the axis consists of a rapid circular vibration superposed on the steady
precession, so that the resultant path has a trochoidal character. This is a type of motion commonly observed
in a top spun in the ordinary way, although the successive undulations of the trochoid may be too small to be
easily observed. In a slightly disturbed rapid precession the superposed vibration is elliptic-harmonic, with a
period equal to that of the precession itself. The ratio of the axes of the ellipse is sec α, the longer axis being
in the plane of θ. The result is that the axis of the top describes a circular cone about a fixed line making a
small angle with the vertical. This is, in fact, the “invariable line” of the free Eulerian rotation with which (as
already remarked) we are here virtually concerned. For the more general discussion of the motion of a top
see Gyroscope.
§ 21. Moving Axes of Reference.—For the more general treatment of the kinetics of a rigid body it is
usually convenient to adopt a system of moving axes. In order that the moments and products of inertia with
respect to these axes may be constant, it is in general necessary to suppose them fixed in the solid.
We will assume for the present that the origin O is fixed. The moving axes Ox, Oy, Oz form a rigid frame
of reference whose motion at time t may be specified by the three component angular velocities p, q, r. The
components of angular momentum about Ox, Oy, Oz will be denoted as usual by λ, μ, ν. Now consider a
system of fixed axes Ox′, Oy′, Oz′ chosen so as to coincide at the instant t with the moving system Ox, Oy,
Oz. At the instant t + δt, Ox, Oy, Oz will no longer coincide with Ox′, Oy′, Oz′; in particular they will make
with Ox′ angles whose cosines are, to the first order, 1, −rδt, qδt, respectively. Hence the altered angular
momentum about Ox′ will be λ + δλ + (μ + δμ) (−rδt) + (ν + δν) qδt. If L, M, N be the moments of the
extraneous forces about Ox, Oy, Oz this must be equal to λ + Lδt. Hence, and by symmetry, we obtain
dλ
− rν + qν = L,
dt
dμ
− pν + rλ = M,
dt
dν
− qλ + pν = N.
dt (1)
These equations are applicable to any dynamical system whatever. If we now apply them to the case of a
rigid body moving about a fixed point O, and make Ox, Oy, Oz coincide with the principal axes of inertia at
O, we have λ, μ, ν = Ap, Bq, Cr, whence
dp
A − (B − C) qr = L,
dt
B dq − (C − A) rp = M,
dr
C − (A − B) pq = N.
dt (2)
If we multiply these by p, q, r and add, we get
d
· 1⁄2 (Ap2 + Bq2 + Cr2) = Lp + Mq + Nr,
dt (3)
which is (virtually) the equation of energy.
ψ̇, approximately. Further, on examining the small variation in ψ̇, it appears that in a slightly disturbed slow
precession the motion of any point of the axis consists of a rapid circular vibration superposed on the steady
precession, so that the resultant path has a trochoidal character. This is a type of motion commonly observed
in a top spun in the ordinary way, although the successive undulations of the trochoid may be too small to be
easily observed. In a slightly disturbed rapid precession the superposed vibration is elliptic-harmonic, with a
period equal to that of the precession itself. The ratio of the axes of the ellipse is sec α, the longer axis being
in the plane of θ. The result is that the axis of the top describes a circular cone about a fixed line making a
small angle with the vertical. This is, in fact, the “invariable line” of the free Eulerian rotation with which (as
already remarked) we are here virtually concerned. For the more general discussion of the motion of a top
see Gyroscope.
§ 21. Moving Axes of Reference.—For the more general treatment of the kinetics of a rigid body it is
usually convenient to adopt a system of moving axes. In order that the moments and products of inertia with
respect to these axes may be constant, it is in general necessary to suppose them fixed in the solid.
We will assume for the present that the origin O is fixed. The moving axes Ox, Oy, Oz form a rigid frame
of reference whose motion at time t may be specified by the three component angular velocities p, q, r. The
components of angular momentum about Ox, Oy, Oz will be denoted as usual by λ, μ, ν. Now consider a
system of fixed axes Ox′, Oy′, Oz′ chosen so as to coincide at the instant t with the moving system Ox, Oy,
Oz. At the instant t + δt, Ox, Oy, Oz will no longer coincide with Ox′, Oy′, Oz′; in particular they will make
with Ox′ angles whose cosines are, to the first order, 1, −rδt, qδt, respectively. Hence the altered angular
momentum about Ox′ will be λ + δλ + (μ + δμ) (−rδt) + (ν + δν) qδt. If L, M, N be the moments of the
extraneous forces about Ox, Oy, Oz this must be equal to λ + Lδt. Hence, and by symmetry, we obtain
dλ
− rν + qν = L,
dt
dμ
− pν + rλ = M,
dt
dν
− qλ + pν = N.
dt (1)
These equations are applicable to any dynamical system whatever. If we now apply them to the case of a
rigid body moving about a fixed point O, and make Ox, Oy, Oz coincide with the principal axes of inertia at
O, we have λ, μ, ν = Ap, Bq, Cr, whence
dp
A − (B − C) qr = L,
dt
B dq − (C − A) rp = M,
dr
C − (A − B) pq = N.
dt (2)
If we multiply these by p, q, r and add, we get
d
· 1⁄2 (Ap2 + Bq2 + Cr2) = Lp + Mq + Nr,
dt (3)
which is (virtually) the equation of energy.
Page 470
As a first application of the equations (2) take the case of a solid constrained to rotate with constant
angular velocity ω about a fixed axis (l, m, n). Since p, q, r are then constant, the requisite constraining
couple is
L = (C − B) mnω2, M = (A − C) nlω2, N = (B − A) lmω2.
(4)
If we reverse the signs, we get the “centrifugal couple” exerted by the solid on its bearings. This couple
vanishes when the axis of rotation is a principal axis at O, and in no other case (cf. § 17).
If in (2) we put, L, M, N = O we get the case of free rotation; thus
dp
A (B − C) qr,
dt
dq
B (C − A) rp,
dt
dr
C (A − B) pq.
dt (5)
These equations are due to Euler, with whom the conception of moving axes, and the application to the
problem of free rotation, originated. If we multiply them by p, q, r, respectively, or again by Ap, Bq, Cr
respectively, and add, we verify that the expressions Ap2 + Bq2 + Cr2 and A2p2 + B2q2 + C2r2 are both
constant. The former is, in fact, equal to 2T, and the latter to Γ2, where T is the kinetic energy and Γ the
resultant angular momentum.
To complete the solution of (2) a third integral is required; this involves in general the use of elliptic
functions. The problem has been the subject of numerous memoirs; we will here notice only the form
of solution given by Rueb (1834), and at a later period by G. Kirchhoff (1875), If we write
dφ
u = ∫φ0 , Δφ = √(1 − k2 sin2 φ),
Δφ
we have, in the notation of elliptic functions, φ = am u. If we assume
p = p0 cos am (σt + ε), q = q0sin am (σt + ε), r = r0Δ am (σt + ε),
(7)
we find
σp0 σq0 k2σr0
ṗ=− qr, q̇ = rp, ṙ= pq.
q0r0 r0p0 p0q0 (8)
Hence (5) will be satisfied, provided
−σp0 B−C σq0 C−A −k2σr0 A−B
= , = , = .
q0r0 A r0p0 B p0q0 C (9)
These equations, together with the arbitrary initial values of p, q, r, determine the six constants which
we have denoted by p0, q0, r0, k2, σ, ε. We will suppose that A > B > C. From the form of the polhode
curves referred to in § 19 it appears that the angular velocity q about the axis of mean moment must
angular velocity ω about a fixed axis (l, m, n). Since p, q, r are then constant, the requisite constraining
couple is
L = (C − B) mnω2, M = (A − C) nlω2, N = (B − A) lmω2.
(4)
If we reverse the signs, we get the “centrifugal couple” exerted by the solid on its bearings. This couple
vanishes when the axis of rotation is a principal axis at O, and in no other case (cf. § 17).
If in (2) we put, L, M, N = O we get the case of free rotation; thus
dp
A (B − C) qr,
dt
dq
B (C − A) rp,
dt
dr
C (A − B) pq.
dt (5)
These equations are due to Euler, with whom the conception of moving axes, and the application to the
problem of free rotation, originated. If we multiply them by p, q, r, respectively, or again by Ap, Bq, Cr
respectively, and add, we verify that the expressions Ap2 + Bq2 + Cr2 and A2p2 + B2q2 + C2r2 are both
constant. The former is, in fact, equal to 2T, and the latter to Γ2, where T is the kinetic energy and Γ the
resultant angular momentum.
To complete the solution of (2) a third integral is required; this involves in general the use of elliptic
functions. The problem has been the subject of numerous memoirs; we will here notice only the form
of solution given by Rueb (1834), and at a later period by G. Kirchhoff (1875), If we write
dφ
u = ∫φ0 , Δφ = √(1 − k2 sin2 φ),
Δφ
we have, in the notation of elliptic functions, φ = am u. If we assume
p = p0 cos am (σt + ε), q = q0sin am (σt + ε), r = r0Δ am (σt + ε),
(7)
we find
σp0 σq0 k2σr0
ṗ=− qr, q̇ = rp, ṙ= pq.
q0r0 r0p0 p0q0 (8)
Hence (5) will be satisfied, provided
−σp0 B−C σq0 C−A −k2σr0 A−B
= , = , = .
q0r0 A r0p0 B p0q0 C (9)
These equations, together with the arbitrary initial values of p, q, r, determine the six constants which
we have denoted by p0, q0, r0, k2, σ, ε. We will suppose that A > B > C. From the form of the polhode
curves referred to in § 19 it appears that the angular velocity q about the axis of mean moment must
Page 471
vanish periodically. If we adopt one of these epochs as the origin of t, we have ε = 0, and p0, r0 will
become identical with the initial values of p, r. The conditions (9) then lead to
A (A − C) (A − C) (B − C) A (A − B) p02
q02 = p02, σ2 = r02, k2 = · .
B (B − C) AB C (B − C) r02 (10)
For a real solution we must have k2 < 1, which is equivalent to 2BT > Γ2. If the initial conditions are
such as to make 2BT < Γ2, we must interchange the forms of p and r in (7). In the present case the
instantaneous axis returns to its initial position in the body whenever φ increases by 2π, i.e. whenever t
increases by 4K/σ, when K is the “complete” elliptic integral of the first kind with respect to the
modulus k.
The elliptic functions degenerate into simpler forms when k2 = 0 or k2 = 1. The former case arises
when two of the principal moments are equal; this has been sufficiently dealt with in § 19. If k2 = 1,
we must have 2BT = Γ2. We have seen that the alternative 2BT ≷ Γ2 determines whether the polhode
cone surrounds the principal axis of least or greatest moment. The case of 2BT = Γ2, exactly, is
therefore a critical case; it may be shown that the instantaneous axis either coincides permanently with
the axis of mean moment or approaches it asymptotically.
When the origin of the moving axes is also in motion with a velocity whose components are u, v, w, the
dynamical equations are
dξ dη dζ
− rη + qζ = X, − pζ + rχ = Y, − qχ + pη = Z,
dt dt dt (11)
dλ dμ
− rμ + qν − wη + vζ = L, − pν + rλ- uζ + wξ = M,
dt dt
dν
− qλ + pμ − vξ + uη = N.
dt (12)
To prove these, we may take fixed axes O′x′, O′y′, O′z′ coincident with the moving axes at time t, and
compare the linear and angular momenta ξ + δξ, η + δη, ζ + δζ, λ + δλ, μ + δμ, ν + δν relative to the new
position of the axes, Ox, Oy, Oz at time t + δt with the original momenta ξ, η, ζ, λ, μ, ν relative to O′x′, O′y′,
O′z′ at time t. As in the case of (2), the equations are applicable to any dynamical system whatever. If the
moving origin coincide always with the mass-centre, we have ξ, η, ζ = M0u, M0v, M0w, where M0 is the total
mass, and the equations simplify.
When, in any problem, the values of u, v, w, p, q, r have been determined as functions of t, it still remains
to connect the moving axes with some fixed frame of reference. It will be sufficient to take the case of
motion about a fixed point O; the angular co-ordinates θ, φ, ψ of Euler may then be used for the purpose.
Referring to fig. 36 we see that the angular velocities p, q, r of the moving lines, OA, OB, OC about their
instantaneous positions are
p = θ̇ sin φ − sin θ cos φψ̇, q = θ̇ cos φ + sin θ sin φψ̇,
r = φ̇ + cos θψ̇,
(13)
become identical with the initial values of p, r. The conditions (9) then lead to
A (A − C) (A − C) (B − C) A (A − B) p02
q02 = p02, σ2 = r02, k2 = · .
B (B − C) AB C (B − C) r02 (10)
For a real solution we must have k2 < 1, which is equivalent to 2BT > Γ2. If the initial conditions are
such as to make 2BT < Γ2, we must interchange the forms of p and r in (7). In the present case the
instantaneous axis returns to its initial position in the body whenever φ increases by 2π, i.e. whenever t
increases by 4K/σ, when K is the “complete” elliptic integral of the first kind with respect to the
modulus k.
The elliptic functions degenerate into simpler forms when k2 = 0 or k2 = 1. The former case arises
when two of the principal moments are equal; this has been sufficiently dealt with in § 19. If k2 = 1,
we must have 2BT = Γ2. We have seen that the alternative 2BT ≷ Γ2 determines whether the polhode
cone surrounds the principal axis of least or greatest moment. The case of 2BT = Γ2, exactly, is
therefore a critical case; it may be shown that the instantaneous axis either coincides permanently with
the axis of mean moment or approaches it asymptotically.
When the origin of the moving axes is also in motion with a velocity whose components are u, v, w, the
dynamical equations are
dξ dη dζ
− rη + qζ = X, − pζ + rχ = Y, − qχ + pη = Z,
dt dt dt (11)
dλ dμ
− rμ + qν − wη + vζ = L, − pν + rλ- uζ + wξ = M,
dt dt
dν
− qλ + pμ − vξ + uη = N.
dt (12)
To prove these, we may take fixed axes O′x′, O′y′, O′z′ coincident with the moving axes at time t, and
compare the linear and angular momenta ξ + δξ, η + δη, ζ + δζ, λ + δλ, μ + δμ, ν + δν relative to the new
position of the axes, Ox, Oy, Oz at time t + δt with the original momenta ξ, η, ζ, λ, μ, ν relative to O′x′, O′y′,
O′z′ at time t. As in the case of (2), the equations are applicable to any dynamical system whatever. If the
moving origin coincide always with the mass-centre, we have ξ, η, ζ = M0u, M0v, M0w, where M0 is the total
mass, and the equations simplify.
When, in any problem, the values of u, v, w, p, q, r have been determined as functions of t, it still remains
to connect the moving axes with some fixed frame of reference. It will be sufficient to take the case of
motion about a fixed point O; the angular co-ordinates θ, φ, ψ of Euler may then be used for the purpose.
Referring to fig. 36 we see that the angular velocities p, q, r of the moving lines, OA, OB, OC about their
instantaneous positions are
p = θ̇ sin φ − sin θ cos φψ̇, q = θ̇ cos φ + sin θ sin φψ̇,
r = φ̇ + cos θψ̇,
(13)
Page 472
by § 7 (3), (4). If OA, OB, OC be principal axes of inertia of a solid, and if A, B, C denote the corresponding
moments of inertia, the kinetic energy is given by
2T = A (θ̇ sin φ − sin θ cos φψ̇)2 + B (θ̇ cos φ + sin θ sin θψ)2
+ C (φ̇ + cos θψ̇)2.
(14)
If A = B this reduces to
2T = A (θ̇2 + sin2 θ ψ̇2) + C (φ̇ + cos θ ψ̇)2;
(15)
cf. § 20 (1).
§ 22. Equations of Motion in Generalized Co-ordinates.—Suppose we have a dynamical system composed
of a finite number of material particles or rigid bodies, whether free or constrained in any way, which are
subject to mutual forces and also to the action of any given extraneous forces. The configuration of such a
system can be completely specified by means of a certain number (n) of independent quantities, called the
generalized co-ordinates of the system. These co-ordinates may be chosen in an endless variety of ways, but
their number is determinate, and expresses the number of degrees of freedom of the system. We denote these
co-ordinates by q1, q2, ... qn. It is implied in the above description of the system that the Cartesian co-
ordinates x, y, z of any particle of the system are known functions of the q’s, varying in form (of course)
from particle to particle. Hence the kinetic energy T is given by
2T = Σ {m (ẋ2 + ẏ2 + ż2) }
= a11q̇ 12 + a22q̇ 22 + ... + 2a12q̇ 1q̇ 2 + ...,
(1)
where
∂x ∂y ∂z
[ m { ( ∂q ) + ( ∂q ) + ( ∂q ) } ],
arr = Σ r 2 r 2 r 2
∂x ∂x ∂y ∂y ∂z ∂z
a = Σ { m ( ∂q
rs ∂q
+
∂q
r ∂q
+
∂q
s ∂q ) } = a .
r s r s sr
(2)
Thus T is expressed as a homogeneous quadratic function of the quantities q̇ 1, q̇ 2, ... q̇ n, which are called the
generalized components of velocity. The coefficients arr, ars are called the coefficients of inertia; they are not
in general constants, being functions of the q’s and so variable with the configuration. Again, If (X, Y, Z) be
the force on m, the work done in an infinitesimal change of configuration is
Σ (Xδx + Yδy + Zδz) = Q1δq1 + Q2δq2 + ... + Qnδqn,
(3)
where
∂x ∂y ∂z
Qr = Σ ( X ∂q + Y ∂q + Z ∂q ).
r r r (4)
The quantities Qr are called the generalized components of force.
moments of inertia, the kinetic energy is given by
2T = A (θ̇ sin φ − sin θ cos φψ̇)2 + B (θ̇ cos φ + sin θ sin θψ)2
+ C (φ̇ + cos θψ̇)2.
(14)
If A = B this reduces to
2T = A (θ̇2 + sin2 θ ψ̇2) + C (φ̇ + cos θ ψ̇)2;
(15)
cf. § 20 (1).
§ 22. Equations of Motion in Generalized Co-ordinates.—Suppose we have a dynamical system composed
of a finite number of material particles or rigid bodies, whether free or constrained in any way, which are
subject to mutual forces and also to the action of any given extraneous forces. The configuration of such a
system can be completely specified by means of a certain number (n) of independent quantities, called the
generalized co-ordinates of the system. These co-ordinates may be chosen in an endless variety of ways, but
their number is determinate, and expresses the number of degrees of freedom of the system. We denote these
co-ordinates by q1, q2, ... qn. It is implied in the above description of the system that the Cartesian co-
ordinates x, y, z of any particle of the system are known functions of the q’s, varying in form (of course)
from particle to particle. Hence the kinetic energy T is given by
2T = Σ {m (ẋ2 + ẏ2 + ż2) }
= a11q̇ 12 + a22q̇ 22 + ... + 2a12q̇ 1q̇ 2 + ...,
(1)
where
∂x ∂y ∂z
[ m { ( ∂q ) + ( ∂q ) + ( ∂q ) } ],
arr = Σ r 2 r 2 r 2
∂x ∂x ∂y ∂y ∂z ∂z
a = Σ { m ( ∂q
rs ∂q
+
∂q
r ∂q
+
∂q
s ∂q ) } = a .
r s r s sr
(2)
Thus T is expressed as a homogeneous quadratic function of the quantities q̇ 1, q̇ 2, ... q̇ n, which are called the
generalized components of velocity. The coefficients arr, ars are called the coefficients of inertia; they are not
in general constants, being functions of the q’s and so variable with the configuration. Again, If (X, Y, Z) be
the force on m, the work done in an infinitesimal change of configuration is
Σ (Xδx + Yδy + Zδz) = Q1δq1 + Q2δq2 + ... + Qnδqn,
(3)
where
∂x ∂y ∂z
Qr = Σ ( X ∂q + Y ∂q + Z ∂q ).
r r r (4)
The quantities Qr are called the generalized components of force.
Page 473
The equations of motion of m being
mẍ = X, mÿ = Y, mz̈ = Z,
(5)
we have
Σ { m ( ẍ ∂q∂x + ÿ ∂q∂y + z̈ ∂q∂z ) } = Q .
r r r r
(6)
Now
(7)
∂x ∂x ∂x
ẋ= q̇ 1 + q̇ 2 + ... + q̇ n,
∂q1 ∂q2 ∂qn
whence
(8)
∂ẋ ∂x
= .
∂q̇ r ∂qr
Also
(9)
d ∂x ∂2x ∂2x ∂2x ∂ẋ
dt ( ∂qr ) = ∂q1∂qr q̇ 1 + ∂q2∂qr q̇ 2 + ... + ∂qn∂qr q̇ r = ∂qr .
Hence
(10)
∂x d ∂x d ∂x d ∂ẋ ∂ẋ
ẍ
∂qr
=
dt ( ẋ ∂qr ) − ẋ dt ( ∂qr ) = dt ( ẋ ∂q̇ r ) − ẋ ∂qr .
By these and the similar transformations relating to y and z the equation (6) takes the form
d
dt ( ∂q̇∂T ) − ∂q∂T = Q .
r r
r
(11)
If we put r = 1, 2, ... n in succession, we get the n independent equations of motion of the system. These
equations are due to Lagrange, with whom indeed the first conception, as well as the establishment, of a
general dynamical method applicable to all systems whatever appears to have originated. The above proof
was given by Sir W. R. Hamilton (1835). Lagrange’s own proof will be found under Dynamics, § Analytical.
In a conservative system free from extraneous force we have
Σ (X δx + Y δy + Z δz) = −δV,
(12)
where V is the potential energy. Hence
∂V
Qr = − ,
∂qr (13)
and
d
dt ( ∂q̇∂T ) − ∂q∂T = − ∂q∂V .
r r r (14)
mẍ = X, mÿ = Y, mz̈ = Z,
(5)
we have
Σ { m ( ẍ ∂q∂x + ÿ ∂q∂y + z̈ ∂q∂z ) } = Q .
r r r r
(6)
Now
(7)
∂x ∂x ∂x
ẋ= q̇ 1 + q̇ 2 + ... + q̇ n,
∂q1 ∂q2 ∂qn
whence
(8)
∂ẋ ∂x
= .
∂q̇ r ∂qr
Also
(9)
d ∂x ∂2x ∂2x ∂2x ∂ẋ
dt ( ∂qr ) = ∂q1∂qr q̇ 1 + ∂q2∂qr q̇ 2 + ... + ∂qn∂qr q̇ r = ∂qr .
Hence
(10)
∂x d ∂x d ∂x d ∂ẋ ∂ẋ
ẍ
∂qr
=
dt ( ẋ ∂qr ) − ẋ dt ( ∂qr ) = dt ( ẋ ∂q̇ r ) − ẋ ∂qr .
By these and the similar transformations relating to y and z the equation (6) takes the form
d
dt ( ∂q̇∂T ) − ∂q∂T = Q .
r r
r
(11)
If we put r = 1, 2, ... n in succession, we get the n independent equations of motion of the system. These
equations are due to Lagrange, with whom indeed the first conception, as well as the establishment, of a
general dynamical method applicable to all systems whatever appears to have originated. The above proof
was given by Sir W. R. Hamilton (1835). Lagrange’s own proof will be found under Dynamics, § Analytical.
In a conservative system free from extraneous force we have
Σ (X δx + Y δy + Z δz) = −δV,
(12)
where V is the potential energy. Hence
∂V
Qr = − ,
∂qr (13)
and
d
dt ( ∂q̇∂T ) − ∂q∂T = − ∂q∂V .
r r r (14)
Page 474
If we imagine any given state of motion (q̇ 1, q̇ 2 ... q̇ n) through the configuration (q1, q2, ... qn) to be
generated instantaneously from rest by the action of suitable impulsive forces, we find on integrating (11)
with respect to t over the infinitely short duration of the impulse
∂T
= Qr′,
∂q̇ r (15)
where Qr′ is the time integral of Qr and so represents a generalized component of impulse. By an obvious
analogy, the expressions ∂T/∂q̇ r may be called the generalized components of momentum; they are usually
denoted by pr thus
pr = ∂T / ∂q̇ r = a1rq̇ 1 + a2rq̇ 2 + ... + anrq̇ n.
(16)
Since T is a homogeneous quadratic function of the velocities q̇ 1, q̇ 2, ... q̇ n, we have
∂T ∂T ∂T
2T = q̇ 1 + q̇ 2 + ... + q̇ n = p1q̇ 2 + p2q̇ 2 + ... + pnq̇ n.
∂q̇ 1 ∂q̇ 2 ∂q̇ n (17)
Hence
dT
2 = ṗ1q̇ 1 + ṗ2q̇ 2 + ... + ṗnq̇ n + ṗ1q̈ 1 + ṗ2q̈ 2 + ... + ṗnq̈ n
dt
= ( ∂q̇∂T + Q ) q̇ + (
1 1 1
∂T
)
∂q̇ 2 + Q2 q̇ 2 + ... + ( ∂T
)
∂q̇ n + Qn q̇ n + ∂q̇ 1
∂T
q̈ 1 +
∂T
∂q̇ 2
q̈ 2 + ... +
∂T
∂q̇ n
q̈ n
dT
= + Q1q̇ 1 + Q2q̇ 2 + ... + Qnq̇ n,
dt (18)
or
dT
= Q1q̇ 1 + Q2q̇ 2 + ... + Qnq̇ n.
dt (19)
This equation expresses that the kinetic energy is increasing at a rate equal to that at which work is being
done by the forces. In the case of a conservative system free from extraneous force it becomes the equation
of energy
d
(T + V) = 0, or T + V = const.,
dt (20)
in virtue of (13).
As a first application of Lagrange’s formula (11) we may form the equations of motion of a particle
in spherical polar co-ordinates. Let r be the distance of a point P from a fixed origin O, θ the angle
which OP makes with a fixed direction OZ, ψ the azimuth of the plane ZOP relative to some fixed
plane through OZ. The displacements of P due to small variations of these co-ordinates are ∂r along
OP, r δθ perpendicular to OP in the plane ZOP, and r sin θ δψ perpendicular to this plane. The
component velocities in these directions are therefore ṙ, rθ̇, r sin θψ̇, and if m be the mass of a moving
particle at P we have
2T = m (ṙ2 + r2θ;̇2 + r2 sin2 θψ;̇2).
generated instantaneously from rest by the action of suitable impulsive forces, we find on integrating (11)
with respect to t over the infinitely short duration of the impulse
∂T
= Qr′,
∂q̇ r (15)
where Qr′ is the time integral of Qr and so represents a generalized component of impulse. By an obvious
analogy, the expressions ∂T/∂q̇ r may be called the generalized components of momentum; they are usually
denoted by pr thus
pr = ∂T / ∂q̇ r = a1rq̇ 1 + a2rq̇ 2 + ... + anrq̇ n.
(16)
Since T is a homogeneous quadratic function of the velocities q̇ 1, q̇ 2, ... q̇ n, we have
∂T ∂T ∂T
2T = q̇ 1 + q̇ 2 + ... + q̇ n = p1q̇ 2 + p2q̇ 2 + ... + pnq̇ n.
∂q̇ 1 ∂q̇ 2 ∂q̇ n (17)
Hence
dT
2 = ṗ1q̇ 1 + ṗ2q̇ 2 + ... + ṗnq̇ n + ṗ1q̈ 1 + ṗ2q̈ 2 + ... + ṗnq̈ n
dt
= ( ∂q̇∂T + Q ) q̇ + (
1 1 1
∂T
)
∂q̇ 2 + Q2 q̇ 2 + ... + ( ∂T
)
∂q̇ n + Qn q̇ n + ∂q̇ 1
∂T
q̈ 1 +
∂T
∂q̇ 2
q̈ 2 + ... +
∂T
∂q̇ n
q̈ n
dT
= + Q1q̇ 1 + Q2q̇ 2 + ... + Qnq̇ n,
dt (18)
or
dT
= Q1q̇ 1 + Q2q̇ 2 + ... + Qnq̇ n.
dt (19)
This equation expresses that the kinetic energy is increasing at a rate equal to that at which work is being
done by the forces. In the case of a conservative system free from extraneous force it becomes the equation
of energy
d
(T + V) = 0, or T + V = const.,
dt (20)
in virtue of (13).
As a first application of Lagrange’s formula (11) we may form the equations of motion of a particle
in spherical polar co-ordinates. Let r be the distance of a point P from a fixed origin O, θ the angle
which OP makes with a fixed direction OZ, ψ the azimuth of the plane ZOP relative to some fixed
plane through OZ. The displacements of P due to small variations of these co-ordinates are ∂r along
OP, r δθ perpendicular to OP in the plane ZOP, and r sin θ δψ perpendicular to this plane. The
component velocities in these directions are therefore ṙ, rθ̇, r sin θψ̇, and if m be the mass of a moving
particle at P we have
2T = m (ṙ2 + r2θ;̇2 + r2 sin2 θψ;̇2).
Page 475
(21)
Hence the formula (11) gives
m (r̈ − rθ̇2 − r sin2 θψ̇2) = R,
d/dt (mr2θ̇) − mr2 · sin θ cos θψ̇2 = Θ,
d/dt (mr2 sin2 θψ̇) = Ψ.
(22)
The quantities R, Θ, Ψ are the coefficients in the expression R δr + Θ δθ + Ψ δψ for the work done in
an infinitely small displacement; viz. R is the radial component of force, Θ is the moment about a line
through O perpendicular to the plane ZOP, and Ψ is the moment about OZ. In the case of the spherical
pendulum we have r = l, Θ = − mgl sin θ, Ψ = 0, if OZ be drawn vertically downwards, and therefore
θ̈ − sin θ cos θψ̇2 = − (g/l) sin θ,
sin2 θψ̇ = h,
(23)
where h is a constant. The latter equation expresses that the angular momentum ml2 sin2 θψ̇ about the
vertical OZ is constant. By elimination of ψ̇ we obtain
g
θ̈ − h2 cos2 θ / sin3 θ = − sin θ.
l (24)
If the particle describes a horizontal circle of angular radius α with constant angular velocity Ω, we
have ω̇ = 0, h = Ω2 sin α, and therefore
g
Ω2 = cos α,
l (25)
as is otherwise evident from the elementary theory of uniform circular motion. To investigate the
small oscillations about this state of steady motion we write θ = α + χ in (24) and neglect terms of the
second order in χ. We find, after some reductions,
χ̈ + (1 + 3 cos2 α) Ω2χ = 0;
(26)
this shows that the variation of χ is simple-harmonic, with the period
2π / √(1 + 3 cos2 α)·Ω
As regards the most general motion of a spherical pendulum, it is obvious that a particle moving
under gravity on a smooth sphere cannot pass through the highest or lowest point unless it describes a
vertical circle. In all other cases there must be an upper and a lower limit to the altitude. Again, a
vertical plane passing through O and a point where the motion is horizontal is evidently a plane of
symmetry as regards the path. Hence the path will be confined between two horizontal circles which it
Hence the formula (11) gives
m (r̈ − rθ̇2 − r sin2 θψ̇2) = R,
d/dt (mr2θ̇) − mr2 · sin θ cos θψ̇2 = Θ,
d/dt (mr2 sin2 θψ̇) = Ψ.
(22)
The quantities R, Θ, Ψ are the coefficients in the expression R δr + Θ δθ + Ψ δψ for the work done in
an infinitely small displacement; viz. R is the radial component of force, Θ is the moment about a line
through O perpendicular to the plane ZOP, and Ψ is the moment about OZ. In the case of the spherical
pendulum we have r = l, Θ = − mgl sin θ, Ψ = 0, if OZ be drawn vertically downwards, and therefore
θ̈ − sin θ cos θψ̇2 = − (g/l) sin θ,
sin2 θψ̇ = h,
(23)
where h is a constant. The latter equation expresses that the angular momentum ml2 sin2 θψ̇ about the
vertical OZ is constant. By elimination of ψ̇ we obtain
g
θ̈ − h2 cos2 θ / sin3 θ = − sin θ.
l (24)
If the particle describes a horizontal circle of angular radius α with constant angular velocity Ω, we
have ω̇ = 0, h = Ω2 sin α, and therefore
g
Ω2 = cos α,
l (25)
as is otherwise evident from the elementary theory of uniform circular motion. To investigate the
small oscillations about this state of steady motion we write θ = α + χ in (24) and neglect terms of the
second order in χ. We find, after some reductions,
χ̈ + (1 + 3 cos2 α) Ω2χ = 0;
(26)
this shows that the variation of χ is simple-harmonic, with the period
2π / √(1 + 3 cos2 α)·Ω
As regards the most general motion of a spherical pendulum, it is obvious that a particle moving
under gravity on a smooth sphere cannot pass through the highest or lowest point unless it describes a
vertical circle. In all other cases there must be an upper and a lower limit to the altitude. Again, a
vertical plane passing through O and a point where the motion is horizontal is evidently a plane of
symmetry as regards the path. Hence the path will be confined between two horizontal circles which it
Page 476
touches alternately, and the direction of motion is never horizontal except at these circles. In the case
of disturbed steady motion, just considered, these circles are nearly coincident. When both are near the
lowest point the horizontal projection of the path is approximately an ellipse, as shown in § 13; a
closer investigation shows that the ellipse is to be regarded as revolving about its centre with the
angular velocity 2⁄3 abΩ/l2, where a, b are the semi-axes.
To apply the equations (11) to the case of the top we start with the expression (15) of § 21 for the
kinetic energy, the simplified form (1) of § 20 being for the present purpose inadmissible, since it is
essential that the generalized co-ordinates employed should be competent to specify the position of
every particle. If λ, μ, ν be the components of momentum, we have
λ = ∂T / ∂θ̇ = Aθ̇,
μ = ∂T / ∂ψ̇ = A sin2 θψ̇ + C (φ̇ + cos θψ̇) cos θ,
ν = ∂T / ∂φ̇ = C (θ̇ + cos θψ̇).
(27)
The meaning of these quantities is easily recognized; thus λ is the angular momentum about a
horizontal axis normal to the plane of θ, μ is the angular momentum about the vertical OZ, and ν is the
angular momentum about the axis of symmetry. If M be the total mass, the potential energy is V =
Mgh cos θ, if OZ be drawn vertically upwards. Hence the equations (11) become
Aθ̇ − A sin θ cos θψ̇2 + C (φ̇ + cos θψ̇) ψ̇ sin θ = Mgh sin θ,
d/dt · { A sin2 θψ̇ + C(φ̇ + cos θψ̇) cos θ } = 0,
d/dt · { C (φ̇ + cos θψ̇) } = 0,
(28)
of which the last two express the constancy of the momenta μ, ν. Hence
Aθ̈ − A sin θ cos θψ̇2 + ν sin θψ̇ = Mgh sin θ,
A sin2 θψ̇ + ν cosθ = μ.
(29)
If we eliminate ψ̇ we obtain the equation (7) of § 20. The theory of disturbed precessional motion
there outlined does not give a convenient view of the oscillations of the axis about the vertical
position. If θ be small the equations (29) may be written
ν2 − 4AMgh
θ̈ − θω̇2 = − θ,
4A2
θ2ω̇ = const.,
(30)
where
ω = ψ − ν t.
of disturbed steady motion, just considered, these circles are nearly coincident. When both are near the
lowest point the horizontal projection of the path is approximately an ellipse, as shown in § 13; a
closer investigation shows that the ellipse is to be regarded as revolving about its centre with the
angular velocity 2⁄3 abΩ/l2, where a, b are the semi-axes.
To apply the equations (11) to the case of the top we start with the expression (15) of § 21 for the
kinetic energy, the simplified form (1) of § 20 being for the present purpose inadmissible, since it is
essential that the generalized co-ordinates employed should be competent to specify the position of
every particle. If λ, μ, ν be the components of momentum, we have
λ = ∂T / ∂θ̇ = Aθ̇,
μ = ∂T / ∂ψ̇ = A sin2 θψ̇ + C (φ̇ + cos θψ̇) cos θ,
ν = ∂T / ∂φ̇ = C (θ̇ + cos θψ̇).
(27)
The meaning of these quantities is easily recognized; thus λ is the angular momentum about a
horizontal axis normal to the plane of θ, μ is the angular momentum about the vertical OZ, and ν is the
angular momentum about the axis of symmetry. If M be the total mass, the potential energy is V =
Mgh cos θ, if OZ be drawn vertically upwards. Hence the equations (11) become
Aθ̇ − A sin θ cos θψ̇2 + C (φ̇ + cos θψ̇) ψ̇ sin θ = Mgh sin θ,
d/dt · { A sin2 θψ̇ + C(φ̇ + cos θψ̇) cos θ } = 0,
d/dt · { C (φ̇ + cos θψ̇) } = 0,
(28)
of which the last two express the constancy of the momenta μ, ν. Hence
Aθ̈ − A sin θ cos θψ̇2 + ν sin θψ̇ = Mgh sin θ,
A sin2 θψ̇ + ν cosθ = μ.
(29)
If we eliminate ψ̇ we obtain the equation (7) of § 20. The theory of disturbed precessional motion
there outlined does not give a convenient view of the oscillations of the axis about the vertical
position. If θ be small the equations (29) may be written
ν2 − 4AMgh
θ̈ − θω̇2 = − θ,
4A2
θ2ω̇ = const.,
(30)
where
ω = ψ − ν t.
Page 477
2A (31)
Since θ, ω are the polar co-ordinates (in a horizontal plane) of a point on the axis of symmetry, relative
to an initial line which revolves with constant angular velocity ν/2A, we see by comparison with § 14
(15) (16) that the motion of such a point will be elliptic-harmonic superposed on a uniform rotation
ν/2A, provided ν2 > 4AMgh. This gives (in essentials) the theory of the “gyroscopic pendulum.”
§ 23. Stability of Equilibrium. Theory of Vibrations.—If, in a conservative system, the configuration (q1,
q2, ... qn) be one of equilibrium, the equations (14) of § 22 must be satisfied by q̇ 1, q̇ 2 ... q̇ n = 0, whence
∂V / ∂qr = 0.
(1)
A necessary and sufficient condition of equilibrium is therefore that the value of the potential energy should
be stationary for infinitesimal variations of the co-ordinates. If, further, V be a minimum, the equilibrium is
necessarily stable, as was shown by P. G. L. Dirichlet (1846). In the motion consequent on any slight
disturbance the total energy T + V is constant, and since T is essentially positive it follows that V can never
exceed its equilibrium value by more than a slight amount, depending on the energy of the disturbance. This
implies, on the present hypothesis, that there is an upper limit to the deviation of each co-ordinate from its
equilibrium value; moreover, this limit diminishes indefinitely with the energy of the original disturbance.
No such simple proof is available to show without qualification that the above condition is necessary. If,
however, we recognize the existence of dissipative forces called into play by any motion whatever of the
system, the conclusion can be drawn as follows. However slight these forces may be, the total energy T + V
must continually diminish so long as the velocities q̇ 1, q̇ 2, ... q̇ n differ from zero. Hence if the system be
started from rest in a configuration for which V is less than in the equilibrium configuration considered, this
quantity must still further decrease (since T cannot be negative), and it is evident that either the system will
finally come to rest in some other equilibrium configuration, or V will in the long run diminish indefinitely.
This argument is due to Lord Kelvin and P. G. Tait (1879).
In discussing the small oscillations of a system about a configuration of stable equilibrium it is convenient
so to choose the generalized cc-ordinates q1, q2, ... qn that they shall vanish in the configuration in question.
The potential energy is then given with sufficient approximation by an expression of the form
2V = c11q12 + c22q22 + ... + 2c12q1q2 + ...,
(2)
a constant term being irrelevant, and the terms of the first order being absent since the equilibrium value of V
is stationary. The coefficients crr, crs are called coefficients of stability. We may further treat the coefficients
of inertia arr, ars of § 22 (1) as constants. The Lagrangian equations of motion are then of the type
a1rq̈ 1 + a2rq̈ 2 + ... + anrq̈ n + c1rq1 + c2rq2 + ... + cnrqn = Qr,
(3)
where Qr now stands for a component of extraneous force. In a free oscillation we have Q1, Q2, ... Qn = 0,
and if we assume
qr = Ar eiσt,
(4)
Since θ, ω are the polar co-ordinates (in a horizontal plane) of a point on the axis of symmetry, relative
to an initial line which revolves with constant angular velocity ν/2A, we see by comparison with § 14
(15) (16) that the motion of such a point will be elliptic-harmonic superposed on a uniform rotation
ν/2A, provided ν2 > 4AMgh. This gives (in essentials) the theory of the “gyroscopic pendulum.”
§ 23. Stability of Equilibrium. Theory of Vibrations.—If, in a conservative system, the configuration (q1,
q2, ... qn) be one of equilibrium, the equations (14) of § 22 must be satisfied by q̇ 1, q̇ 2 ... q̇ n = 0, whence
∂V / ∂qr = 0.
(1)
A necessary and sufficient condition of equilibrium is therefore that the value of the potential energy should
be stationary for infinitesimal variations of the co-ordinates. If, further, V be a minimum, the equilibrium is
necessarily stable, as was shown by P. G. L. Dirichlet (1846). In the motion consequent on any slight
disturbance the total energy T + V is constant, and since T is essentially positive it follows that V can never
exceed its equilibrium value by more than a slight amount, depending on the energy of the disturbance. This
implies, on the present hypothesis, that there is an upper limit to the deviation of each co-ordinate from its
equilibrium value; moreover, this limit diminishes indefinitely with the energy of the original disturbance.
No such simple proof is available to show without qualification that the above condition is necessary. If,
however, we recognize the existence of dissipative forces called into play by any motion whatever of the
system, the conclusion can be drawn as follows. However slight these forces may be, the total energy T + V
must continually diminish so long as the velocities q̇ 1, q̇ 2, ... q̇ n differ from zero. Hence if the system be
started from rest in a configuration for which V is less than in the equilibrium configuration considered, this
quantity must still further decrease (since T cannot be negative), and it is evident that either the system will
finally come to rest in some other equilibrium configuration, or V will in the long run diminish indefinitely.
This argument is due to Lord Kelvin and P. G. Tait (1879).
In discussing the small oscillations of a system about a configuration of stable equilibrium it is convenient
so to choose the generalized cc-ordinates q1, q2, ... qn that they shall vanish in the configuration in question.
The potential energy is then given with sufficient approximation by an expression of the form
2V = c11q12 + c22q22 + ... + 2c12q1q2 + ...,
(2)
a constant term being irrelevant, and the terms of the first order being absent since the equilibrium value of V
is stationary. The coefficients crr, crs are called coefficients of stability. We may further treat the coefficients
of inertia arr, ars of § 22 (1) as constants. The Lagrangian equations of motion are then of the type
a1rq̈ 1 + a2rq̈ 2 + ... + anrq̈ n + c1rq1 + c2rq2 + ... + cnrqn = Qr,
(3)
where Qr now stands for a component of extraneous force. In a free oscillation we have Q1, Q2, ... Qn = 0,
and if we assume
qr = Ar eiσt,
(4)
Page 478
we obtain n equations of the type
(c1r − σ2a1r) A1 + (c2r − σ2a2r) A2 + ... + (cnr − σ2anr) An = 0.
(5)
Eliminating the n − 1 ratios A1 : A2 : ... : An we obtain the determinantal equation
Δ (σ2) = 0,
(6)
where
Δ(σ2) = c11 − σ2a11, c21 − σ2a21, ..., Cn1 − σ2anl
c12 − σ2a12, c22 − σ2a22, ..., Cn2 − σ2an2
. . ... .
. . ... .
. . ... .
c1n − σ2a1n, c2n − σ2a2n, ..., Cnn − σ2ann
(7)
The quadratic expression for T is essentially positive, and the same holds with regard to V in virtue of the
assumed stability. It may be shown algebraically that under these conditions the n roots of the above equation
in σ2 are all real and positive. For any particular root, the equations (5) determine the ratios of the quantities
A1, A2, ... An, the absolute values being alone arbitrary; these quantities are in fact proportional to the minors
of any one row in the determinate Δ(σ2). By combining the solutions corresponding to a pair of equal and
opposite values of σ we obtain a solution in real form:
qr = Car cos (σt + ε),
(8)
where a1, a2 ... ar are a determinate series of quantities having to one another the above-
mentioned ratios, whilst the constants C, ε are arbitrary. This solution, taken by itself,
represents a motion in which each particle of the system (since its displacements parallel
to Cartesian co-ordinate axes are linear functions of the q’s) executes a simple vibration of
period 2π/σ. The amplitudes of oscillation of the various particles have definite ratios to
one another, and the phases are in agreement, the absolute amplitude (depending on C)
and the phase-constant (ε) being alone arbitrary. A vibration of this character is called a
normal mode of vibration of the system; the number n of such modes is equal to that of
the degrees of freedom possessed by the system. These statements require some
modification when two or more of the roots of the equation (6) are equal. In the case of a
multiple root the minors of Δ(σ2) all vanish, and the basis for the determination of the
quantities ar disappears. Two or more normal modes then become to some extent
indeterminate, and elliptic vibrations of the individual particles are possible. An example Fig. 85.
is furnished by the spherical pendulum (§ 13).
As an example of the method of determination of the normal modes we may take the “double
pendulum.” A mass M hangs from a fixed point by a string of length a, and a second mass m hangs
(c1r − σ2a1r) A1 + (c2r − σ2a2r) A2 + ... + (cnr − σ2anr) An = 0.
(5)
Eliminating the n − 1 ratios A1 : A2 : ... : An we obtain the determinantal equation
Δ (σ2) = 0,
(6)
where
Δ(σ2) = c11 − σ2a11, c21 − σ2a21, ..., Cn1 − σ2anl
c12 − σ2a12, c22 − σ2a22, ..., Cn2 − σ2an2
. . ... .
. . ... .
. . ... .
c1n − σ2a1n, c2n − σ2a2n, ..., Cnn − σ2ann
(7)
The quadratic expression for T is essentially positive, and the same holds with regard to V in virtue of the
assumed stability. It may be shown algebraically that under these conditions the n roots of the above equation
in σ2 are all real and positive. For any particular root, the equations (5) determine the ratios of the quantities
A1, A2, ... An, the absolute values being alone arbitrary; these quantities are in fact proportional to the minors
of any one row in the determinate Δ(σ2). By combining the solutions corresponding to a pair of equal and
opposite values of σ we obtain a solution in real form:
qr = Car cos (σt + ε),
(8)
where a1, a2 ... ar are a determinate series of quantities having to one another the above-
mentioned ratios, whilst the constants C, ε are arbitrary. This solution, taken by itself,
represents a motion in which each particle of the system (since its displacements parallel
to Cartesian co-ordinate axes are linear functions of the q’s) executes a simple vibration of
period 2π/σ. The amplitudes of oscillation of the various particles have definite ratios to
one another, and the phases are in agreement, the absolute amplitude (depending on C)
and the phase-constant (ε) being alone arbitrary. A vibration of this character is called a
normal mode of vibration of the system; the number n of such modes is equal to that of
the degrees of freedom possessed by the system. These statements require some
modification when two or more of the roots of the equation (6) are equal. In the case of a
multiple root the minors of Δ(σ2) all vanish, and the basis for the determination of the
quantities ar disappears. Two or more normal modes then become to some extent
indeterminate, and elliptic vibrations of the individual particles are possible. An example Fig. 85.
is furnished by the spherical pendulum (§ 13).
As an example of the method of determination of the normal modes we may take the “double
pendulum.” A mass M hangs from a fixed point by a string of length a, and a second mass m hangs
Page 479
from M by a string of length b. For simplicity we will suppose that the motion is confined to one
vertical plane. If θ, φ be the inclinations of the two strings to the vertical, we have, approximately,
2T = Ma2θ̇2 + m (aθ̇ + bψ̇)2
2V = Mgaθ2 + mg (aθ2 + bψ2).
(9)
The equations (3) take the forms
aθ ̈ + μbφ̈ + gθ = 0,
aθ ̈ + bφ̈ + gφ = 0.
(10)
where μ = m/(M + m). Hence
(σ2 − g/a) aθ + μσ2bφ = 0,
σ2aθ + (σ2 − g/b) bφ = 0.
(11)
The frequency equation is therefore
(σ2 − g/a) (σ2 − g/b) − μσ4 = 0.
(12)
The roots of this quadratic in σ2 are easily seen to be real and positive. If M be large compared with m,
μ is small, and the roots are g/a and g/b, approximately. In the normal mode corresponding to the
former root, M swings almost like the bob of a simple pendulum of length a, being comparatively
uninfluenced by the presence of m, whilst m executes a “forced” vibration (§ 12) of the corresponding
period. In the second mode, M is nearly at rest [as appears from the second of equations (11)], whilst
m swings almost like the bob of a simple pendulum of length b. Whatever the ratio M/m, the two
values of σ2 can never be exactly equal, but they are approximately equal if a, b are nearly equal and μ
is very small. A curious phenomenon is then to be observed; the motion of each particle, being made
up (in general) of two superposed simple vibrations of nearly equal period, is seen to fluctuate greatly
in extent, and if the amplitudes be equal we have periods of approximate rest, as in the case of “beats”
in acoustics. The vibration then appears to be transferred alternately from m to M at regular intervals.
If, on the other hand, M is small compared with m, μ is nearly equal to unity, and the roots of (12) are
σ2 = g/(a + b) and σ2 = mg/M·(a + b)/ab, approximately. The former root makes θ = φ, nearly; in the
corresponding normal mode m oscillates like the bob of a simple pendulum of length a + b. In the
second mode aθ + bφ = 0, nearly, so that m is approximately at rest. The oscillation of M then
resembles that of a particle at a distance a from one end of a string of length a + b fixed at the ends
and subject to a tension mg.
vertical plane. If θ, φ be the inclinations of the two strings to the vertical, we have, approximately,
2T = Ma2θ̇2 + m (aθ̇ + bψ̇)2
2V = Mgaθ2 + mg (aθ2 + bψ2).
(9)
The equations (3) take the forms
aθ ̈ + μbφ̈ + gθ = 0,
aθ ̈ + bφ̈ + gφ = 0.
(10)
where μ = m/(M + m). Hence
(σ2 − g/a) aθ + μσ2bφ = 0,
σ2aθ + (σ2 − g/b) bφ = 0.
(11)
The frequency equation is therefore
(σ2 − g/a) (σ2 − g/b) − μσ4 = 0.
(12)
The roots of this quadratic in σ2 are easily seen to be real and positive. If M be large compared with m,
μ is small, and the roots are g/a and g/b, approximately. In the normal mode corresponding to the
former root, M swings almost like the bob of a simple pendulum of length a, being comparatively
uninfluenced by the presence of m, whilst m executes a “forced” vibration (§ 12) of the corresponding
period. In the second mode, M is nearly at rest [as appears from the second of equations (11)], whilst
m swings almost like the bob of a simple pendulum of length b. Whatever the ratio M/m, the two
values of σ2 can never be exactly equal, but they are approximately equal if a, b are nearly equal and μ
is very small. A curious phenomenon is then to be observed; the motion of each particle, being made
up (in general) of two superposed simple vibrations of nearly equal period, is seen to fluctuate greatly
in extent, and if the amplitudes be equal we have periods of approximate rest, as in the case of “beats”
in acoustics. The vibration then appears to be transferred alternately from m to M at regular intervals.
If, on the other hand, M is small compared with m, μ is nearly equal to unity, and the roots of (12) are
σ2 = g/(a + b) and σ2 = mg/M·(a + b)/ab, approximately. The former root makes θ = φ, nearly; in the
corresponding normal mode m oscillates like the bob of a simple pendulum of length a + b. In the
second mode aθ + bφ = 0, nearly, so that m is approximately at rest. The oscillation of M then
resembles that of a particle at a distance a from one end of a string of length a + b fixed at the ends
and subject to a tension mg.
Page 480
The motion of the system consequent on arbitrary initial conditions may be obtained by superposition of
the n normal modes with suitable amplitudes and phases. We have then
qr = αrθ + αr′θ′ + αr″θ″ + ...,
(13)
where
θ = C cos (σt + ε), θ′ = C′ cos (σ′t + ε), θ″ = C″ cos (σ″t + ε), ...
(14)
provided σ2, σ′2, σ″2, ... are the n roots of (6). The coefficients of θ, θ′, θ″, ... in (13) satisfy the conjugate or
orthogonal relations
a11α1α1′ + a22α2α2′ + ... + a12 (α1α2′ + α2α1′) + ... = 0,
(15)
c11α1α1′ + c22α2α2′ + ... + c12 (α1α2′ + α2α1′) + ... = 0,
(16)
provided the symbols αr, αr′ correspond to two distinct roots σ2, σ′2 of (6). To prove these relations, we
replace the symbols A1, A2, ... An in (5) by α1, α2, ... αn respectively, multiply the resulting equations by a′1,
a′2, ... a′n, in order, and add. The result, owing to its symmetry, must still hold if we interchange accented and
unaccented Greek letters, and by comparison we deduce (15) and (16), provided σ2 and σ′2 are unequal. The
actual determination of C, C′, C″, ... and ε, ε′, ε″, ... in terms of the initial conditions is as follows. If we write
C cos ε = H, −C sin ε = K,
(17)
we must have
αrH + αr′H′ + αr″H″ + ... = [qr]0,
σαrH + σ′αr′H′ + σ″αr″H″ + ... = [q̇ r]0,
(18)
where the zero suffix indicates initial values. These equations can be at once solved for H, H′, H″, ... and K,
K′, K″, ... by means of the orthogonal relations (15).
By a suitable choice of the generalized co-ordinates it is possible to reduce T and V simultaneously to
sums of squares. The transformation is in fact effected by the assumption (13), in virtue of the relations (15)
(16), and we may write
2T = aθ̇2 + a′θ̇′2 + a″θ̇″2 + ...,
2V = cθ2 + c′θ′2 + c″θ″2 + ....
(19)
The new co-ordinates θ, θ′, θ″ ... are called the normal co-ordinates of the system; in a normal mode of
vibration one of these varies alone. The physical characteristics of a normal mode are that an impulse of a
particular normal type generates an initial velocity of that type only, and that a constant extraneous force of a
particular normal type maintains a displacement of that type only. The normal modes are further
the n normal modes with suitable amplitudes and phases. We have then
qr = αrθ + αr′θ′ + αr″θ″ + ...,
(13)
where
θ = C cos (σt + ε), θ′ = C′ cos (σ′t + ε), θ″ = C″ cos (σ″t + ε), ...
(14)
provided σ2, σ′2, σ″2, ... are the n roots of (6). The coefficients of θ, θ′, θ″, ... in (13) satisfy the conjugate or
orthogonal relations
a11α1α1′ + a22α2α2′ + ... + a12 (α1α2′ + α2α1′) + ... = 0,
(15)
c11α1α1′ + c22α2α2′ + ... + c12 (α1α2′ + α2α1′) + ... = 0,
(16)
provided the symbols αr, αr′ correspond to two distinct roots σ2, σ′2 of (6). To prove these relations, we
replace the symbols A1, A2, ... An in (5) by α1, α2, ... αn respectively, multiply the resulting equations by a′1,
a′2, ... a′n, in order, and add. The result, owing to its symmetry, must still hold if we interchange accented and
unaccented Greek letters, and by comparison we deduce (15) and (16), provided σ2 and σ′2 are unequal. The
actual determination of C, C′, C″, ... and ε, ε′, ε″, ... in terms of the initial conditions is as follows. If we write
C cos ε = H, −C sin ε = K,
(17)
we must have
αrH + αr′H′ + αr″H″ + ... = [qr]0,
σαrH + σ′αr′H′ + σ″αr″H″ + ... = [q̇ r]0,
(18)
where the zero suffix indicates initial values. These equations can be at once solved for H, H′, H″, ... and K,
K′, K″, ... by means of the orthogonal relations (15).
By a suitable choice of the generalized co-ordinates it is possible to reduce T and V simultaneously to
sums of squares. The transformation is in fact effected by the assumption (13), in virtue of the relations (15)
(16), and we may write
2T = aθ̇2 + a′θ̇′2 + a″θ̇″2 + ...,
2V = cθ2 + c′θ′2 + c″θ″2 + ....
(19)
The new co-ordinates θ, θ′, θ″ ... are called the normal co-ordinates of the system; in a normal mode of
vibration one of these varies alone. The physical characteristics of a normal mode are that an impulse of a
particular normal type generates an initial velocity of that type only, and that a constant extraneous force of a
particular normal type maintains a displacement of that type only. The normal modes are further
Page 481
distinguished by an important “stationary” property, as regards the frequency. If we imagine the system
reduced by frictionless constraints to one degree of freedom, so that the co-ordinates θ, θ′, θ″, ... have
prescribed ratios to one another, we have, from (19),
cθ2 + c′θ′2 = c″θ″2 + ...
σ2 = ,
aθ2 + a′θ′2 + a″θ″2 + ... (20)
This shows that the value of σ2 for the constrained mode is intermediate to the greatest and least of the
values c/a, c′/a′, c″/a″, ... proper to the several normal modes. Also that if the constrained mode differs little
from a normal mode of free vibration (e.g. if θ′, θ″, ... are small compared with θ), the change in the
frequency is of the second order. This property can often be utilized to estimate the frequency of the gravest
normal mode of a system, by means of an assumed approximate type, when the exact determination would
be difficult. It also appears that an estimate thus obtained is necessarily too high.
From another point of view it is easily recognized that the equations (5) are exactly those to which we are
led in the ordinary process of finding the stationary values of the function
V (q1, q2, ... qn)
,
T (q1, q2, ... qn)
where the denominator stands for the same homogeneous quadratic function of the q’s that T is for the q̇ ’s. It
is easy to construct in this connexion a proof that the n values of σ2 are all real and positive.
The case of three degrees of freedom is instructive on account of the geometrical analogies. With a
view to these we may write
2T = aẋ2 + bẏ2 + cż2 + 2fẏż + 2gżẋ + 2hẋẏ,
2V = Ax2 + By2 + Cz2 + 2Fyz + 2Gzx + 2Hxy.
(21)
It is obvious that the ratio
V (x, y, z)
T (x, y, z) (22)
must have a least value, which is moreover positive, since the numerator and denominator are both
essentially positive. Denoting this value by σ12, we have
Ax1 + Hy1 + Gz1 = σ12 (ax1 + hy1 + ∂gz1),
Hx1 + By1 + Fz1 = σ12 (hx1 + by1 + fz1),
Gx1 + Fy1 + Cz1 = σ12 (gx1 + fy1 + cz1),
(23)
provided x1 : y1 : z1 be the corresponding values of the ratios x:y:z. Again, the expression (22) will
also have a least value when the ratios x : y : z are subject to the condition
reduced by frictionless constraints to one degree of freedom, so that the co-ordinates θ, θ′, θ″, ... have
prescribed ratios to one another, we have, from (19),
cθ2 + c′θ′2 = c″θ″2 + ...
σ2 = ,
aθ2 + a′θ′2 + a″θ″2 + ... (20)
This shows that the value of σ2 for the constrained mode is intermediate to the greatest and least of the
values c/a, c′/a′, c″/a″, ... proper to the several normal modes. Also that if the constrained mode differs little
from a normal mode of free vibration (e.g. if θ′, θ″, ... are small compared with θ), the change in the
frequency is of the second order. This property can often be utilized to estimate the frequency of the gravest
normal mode of a system, by means of an assumed approximate type, when the exact determination would
be difficult. It also appears that an estimate thus obtained is necessarily too high.
From another point of view it is easily recognized that the equations (5) are exactly those to which we are
led in the ordinary process of finding the stationary values of the function
V (q1, q2, ... qn)
,
T (q1, q2, ... qn)
where the denominator stands for the same homogeneous quadratic function of the q’s that T is for the q̇ ’s. It
is easy to construct in this connexion a proof that the n values of σ2 are all real and positive.
The case of three degrees of freedom is instructive on account of the geometrical analogies. With a
view to these we may write
2T = aẋ2 + bẏ2 + cż2 + 2fẏż + 2gżẋ + 2hẋẏ,
2V = Ax2 + By2 + Cz2 + 2Fyz + 2Gzx + 2Hxy.
(21)
It is obvious that the ratio
V (x, y, z)
T (x, y, z) (22)
must have a least value, which is moreover positive, since the numerator and denominator are both
essentially positive. Denoting this value by σ12, we have
Ax1 + Hy1 + Gz1 = σ12 (ax1 + hy1 + ∂gz1),
Hx1 + By1 + Fz1 = σ12 (hx1 + by1 + fz1),
Gx1 + Fy1 + Cz1 = σ12 (gx1 + fy1 + cz1),
(23)
provided x1 : y1 : z1 be the corresponding values of the ratios x:y:z. Again, the expression (22) will
also have a least value when the ratios x : y : z are subject to the condition
Page 482
∂V ∂V ∂V
x1 + y1 + z1 = 0;
∂x ∂y ∂z (24)
and if this be denoted by σ22 we have a second system of equations similar to (23). The remaining
value σ22 is the value of (22) when x : y : z arc chosen so as to satisfy (24) and
∂V ∂V ∂V
x2 + y2 + z2 = 0;
∂x ∂y ∂z (25)
The problem is identical with that of finding the common conjugate diameters of the ellipsoids T(x, y,
z) = const., V(x, y, z) = const. If in (21) we imagine that x, y, z denote infinitesimal rotations of a solid
free to turn about a fixed point in a given field of force, it appears that the three normal modes consist
each of a rotation about one of the three diameters aforesaid, and that the values of σ are proportional
to the ratios of the lengths of corresponding diameters of the two quadrics.
We proceed to the forced vibrations of the system. The typical case is where the extraneous forces are of
the simple-harmonic type cos (σt + ε); the most general law of variation with time can be derived from this
by superposition, in virtue of Fourier’s theorem. Analytically, it is convenient to put Qr, equal to eiσt
multiplied by a complex coefficient; owing to the linearity of the equations the factor eiσt will run through
them all, and need not always be exhibited. For a system of one degree of freedom we have
aq̈ + cq = Q,
(26)
and therefore on the present supposition as to the nature of Q
Q
q= .
c − σ2a (27)
This solution has been discussed to some extent in § 12, in connexion with the forced oscillations of a
pendulum. We may note further that when σ is small the displacement q has the “equilibrium value” Q/c, the
same as would be produced by a steady force equal to the instantaneous value of the actual force, the inertia
of the system being inoperative. On the other hand, when σ2 is great q tends to the value −Q/σ2a, the same as
if the potential energy were ignored. When there are n degrees of freedom we have from
(3)
(c1r − σ2a2r) q1 + (c22r − σ2a2r) q2 + ... + (cnr − σ2anr) qn = Qr,
(28)
and therefore
Δ(σ2) · qr = a1rQ1 + a2rQ2 + ... + anrQn,
(29)
where a1r, a2r, ... anr are the minors of the rth row of the determinant (7). Every particle of the system
executes in general a simple vibration of the imposed period 2π/σ, and all the particles pass simultaneously
through their equilibrium positions. The amplitude becomes very great when σ2 approximates to a root of
(6), i.e. when the imposed period nearly coincides with one of the free periods. Since ars = asr, the coefficient
of Qs in the expression for qr is identical with that of Qr in the expression for qs. Various important
x1 + y1 + z1 = 0;
∂x ∂y ∂z (24)
and if this be denoted by σ22 we have a second system of equations similar to (23). The remaining
value σ22 is the value of (22) when x : y : z arc chosen so as to satisfy (24) and
∂V ∂V ∂V
x2 + y2 + z2 = 0;
∂x ∂y ∂z (25)
The problem is identical with that of finding the common conjugate diameters of the ellipsoids T(x, y,
z) = const., V(x, y, z) = const. If in (21) we imagine that x, y, z denote infinitesimal rotations of a solid
free to turn about a fixed point in a given field of force, it appears that the three normal modes consist
each of a rotation about one of the three diameters aforesaid, and that the values of σ are proportional
to the ratios of the lengths of corresponding diameters of the two quadrics.
We proceed to the forced vibrations of the system. The typical case is where the extraneous forces are of
the simple-harmonic type cos (σt + ε); the most general law of variation with time can be derived from this
by superposition, in virtue of Fourier’s theorem. Analytically, it is convenient to put Qr, equal to eiσt
multiplied by a complex coefficient; owing to the linearity of the equations the factor eiσt will run through
them all, and need not always be exhibited. For a system of one degree of freedom we have
aq̈ + cq = Q,
(26)
and therefore on the present supposition as to the nature of Q
Q
q= .
c − σ2a (27)
This solution has been discussed to some extent in § 12, in connexion with the forced oscillations of a
pendulum. We may note further that when σ is small the displacement q has the “equilibrium value” Q/c, the
same as would be produced by a steady force equal to the instantaneous value of the actual force, the inertia
of the system being inoperative. On the other hand, when σ2 is great q tends to the value −Q/σ2a, the same as
if the potential energy were ignored. When there are n degrees of freedom we have from
(3)
(c1r − σ2a2r) q1 + (c22r − σ2a2r) q2 + ... + (cnr − σ2anr) qn = Qr,
(28)
and therefore
Δ(σ2) · qr = a1rQ1 + a2rQ2 + ... + anrQn,
(29)
where a1r, a2r, ... anr are the minors of the rth row of the determinant (7). Every particle of the system
executes in general a simple vibration of the imposed period 2π/σ, and all the particles pass simultaneously
through their equilibrium positions. The amplitude becomes very great when σ2 approximates to a root of
(6), i.e. when the imposed period nearly coincides with one of the free periods. Since ars = asr, the coefficient
of Qs in the expression for qr is identical with that of Qr in the expression for qs. Various important
Page 483
“reciprocal theorems” formulated by H. Helmholtz and Lord Rayleigh are founded on this relation. Free
vibrations must of course be superposed on the forced vibrations given by (29) in order to obtain the
complete solution of the dynamical equations.
In practice the vibrations of a system are more or less affected by dissipative forces. In order to obtain at
all events a qualitative representation of these it is usual to introduce into the equations frictional terms
proportional to the velocities. Thus in the case of one degree of freedom we have, in place of (26),
aq̈ + bq̇ + cq = Q,
(30)
where a, b, c are positive. The solution of this has been sufficiently discussed in § 12. In the case of multiple
freedom, the equations of small motion when modified by the introduction of terms proportional to the
velocities are of the type
d ∂T ∂V
+ B1rq̇ 1 + B2rq̇ 2 + ... + Bnrq̇ n + = Q r.
dt ∂q̇ r ∂qr (31)
If we put
brs = bsr = 1⁄2 (Brs + Bsr), βrs = −βsr = 1⁄2 (Brs − Bsr),
(32)
this may be written
d ∂T ∂F ∂V
+ + β1rq̇ 1 + β2rq̇ 2 + ... + βnrq̇ r + = Q r,
dt ∂q̇ r ∂q̇ r ∂qr (33)
provided
2F = b11q̇ 12 + b22q̇ 22 + ... + 2b12q̇ 1q̇ 2 + ...
(34)
The terms due to F in (33) are such as would arise from frictional resistances proportional to the absolute
velocities of the particles, or to mutual forces of resistance proportional to the relative velocities; they are
therefore classed as frictional or dissipative forces. The terms affected with the coefficients βrs on the other
hand are such as occur in “cyclic” systems with latent motion (Dynamics, § Analytical); they are called the
gyrostatic terms. If we multiply (33) by q̇ r and sum with respect to r from 1 to n, we obtain, in virtue of the
relations βrs = −βsr, βrr = 0,
d
(T + V) = 2F + Q1q̇ 1 + Q2q̇ 2 + ... + Qnq̇ n.
dt (35)
This shows that mechanical energy is lost at the rate 2F per unit time. The function F is therefore called by
Lord Rayleigh the dissipation function.
If we omit the gyrostatic terms, and write qr = Creλt, we find, for a free vibration,
(a1rλ2 + b1rλ + c1r) C1 + (a2rλ2 + b2rλ + c2r) C2 + ...
+ (anrλ2 + bnrλ + cnr) Cn = 0.
(36)
vibrations must of course be superposed on the forced vibrations given by (29) in order to obtain the
complete solution of the dynamical equations.
In practice the vibrations of a system are more or less affected by dissipative forces. In order to obtain at
all events a qualitative representation of these it is usual to introduce into the equations frictional terms
proportional to the velocities. Thus in the case of one degree of freedom we have, in place of (26),
aq̈ + bq̇ + cq = Q,
(30)
where a, b, c are positive. The solution of this has been sufficiently discussed in § 12. In the case of multiple
freedom, the equations of small motion when modified by the introduction of terms proportional to the
velocities are of the type
d ∂T ∂V
+ B1rq̇ 1 + B2rq̇ 2 + ... + Bnrq̇ n + = Q r.
dt ∂q̇ r ∂qr (31)
If we put
brs = bsr = 1⁄2 (Brs + Bsr), βrs = −βsr = 1⁄2 (Brs − Bsr),
(32)
this may be written
d ∂T ∂F ∂V
+ + β1rq̇ 1 + β2rq̇ 2 + ... + βnrq̇ r + = Q r,
dt ∂q̇ r ∂q̇ r ∂qr (33)
provided
2F = b11q̇ 12 + b22q̇ 22 + ... + 2b12q̇ 1q̇ 2 + ...
(34)
The terms due to F in (33) are such as would arise from frictional resistances proportional to the absolute
velocities of the particles, or to mutual forces of resistance proportional to the relative velocities; they are
therefore classed as frictional or dissipative forces. The terms affected with the coefficients βrs on the other
hand are such as occur in “cyclic” systems with latent motion (Dynamics, § Analytical); they are called the
gyrostatic terms. If we multiply (33) by q̇ r and sum with respect to r from 1 to n, we obtain, in virtue of the
relations βrs = −βsr, βrr = 0,
d
(T + V) = 2F + Q1q̇ 1 + Q2q̇ 2 + ... + Qnq̇ n.
dt (35)
This shows that mechanical energy is lost at the rate 2F per unit time. The function F is therefore called by
Lord Rayleigh the dissipation function.
If we omit the gyrostatic terms, and write qr = Creλt, we find, for a free vibration,
(a1rλ2 + b1rλ + c1r) C1 + (a2rλ2 + b2rλ + c2r) C2 + ...
+ (anrλ2 + bnrλ + cnr) Cn = 0.
(36)
Page 484
This leads to a determinantal equation in λ whose 2n roots are either real and negative, or complex with
negative real parts, on the present hypothesis that the functions T, V, F are all essentially positive. If we
combine the solutions corresponding to a pair of conjugate complex roots, we obtain, in real form,
qr = Cαr e−t/τ cos (σt + ε − εr),
(37)
where σ, τ, αr, εr are determined by the constitution of the system, whilst C, ε are arbitrary, and independent
of r. The n formulae of this type represent a normal mode of free vibration: the individual particles revolve as
a rule in elliptic orbits which gradually contract according to the law indicated by the exponential factor. If
the friction be relatively small, all the normal modes are of this character, and unless two or more values of σ
are nearly equal the elliptic orbits are very elongated. The effect of friction on the period is moreover of the
second order.
In a forced vibration of eiσt the variation of each co-ordinate is simple-harmonic, with the prescribed
period, but there is a retardation of phase as compared with the force. If the friction be small the amplitude
becomes relatively very great if the imposed period approximate to a free period. The validity of the
“reciprocal theorems” of Helmholtz and Lord Rayleigh, already referred to, is not affected by frictional
forces of the kind here considered.
The most important applications of the theory of vibrations are to the case of continuous systems
such as strings, bars, membranes, plates, columns of air, where the number of degrees of freedom is
infinite. The series of equations of the type (3) is then replaced by a single linear partial differential
equation, or by a set of two or three such equations, according to the number of dependent variables.
These variables represent the whole assemblage of generalized co-ordinates qr; they are continuous
functions of the independent variables x, y, z whose range of variation corresponds to that of the index
r, and of t. For example, in a one-dimensional system such as a string or a bar, we have one dependent
variable, and two independent variables x and t. To determine the free oscillations we assume a time
factor eiσt; the equations then become linear differential equations between the dependent variables of
the problem and the independent variables x, or x, y, or x, y, z as the case may be. If the range of the
independent variable or variables is unlimited, the value of σ is at our disposal, and the solution gives
us the laws of wave-propagation (see Wave). If, on the other hand, the body is finite, certain terminal
conditions have to be satisfied. These limit the admissible values of σ, which are in general
determined by a transcendental equation corresponding to the determinantal equation (6).
Numerous examples of this procedure, and of the corresponding treatment of forced oscillations,
present themselves in theoretical acoustics. It must suffice here to consider the small oscillations of a
chain hanging vertically from a fixed extremity. If x be measured upwards from the lower end, the
horizontal component of the tension P at any point will be Pδy/δx, approximately, if y denote the
lateral displacement. Hence, forming the equation of motion of a mass-element, ρδx, we have
ρ δx · ÿ = δ (P · ∂y/∂x).
(38)
Neglecting the vertical acceleration we have P = gρx, whence
∂2y = g ∂
( x ∂y ).
negative real parts, on the present hypothesis that the functions T, V, F are all essentially positive. If we
combine the solutions corresponding to a pair of conjugate complex roots, we obtain, in real form,
qr = Cαr e−t/τ cos (σt + ε − εr),
(37)
where σ, τ, αr, εr are determined by the constitution of the system, whilst C, ε are arbitrary, and independent
of r. The n formulae of this type represent a normal mode of free vibration: the individual particles revolve as
a rule in elliptic orbits which gradually contract according to the law indicated by the exponential factor. If
the friction be relatively small, all the normal modes are of this character, and unless two or more values of σ
are nearly equal the elliptic orbits are very elongated. The effect of friction on the period is moreover of the
second order.
In a forced vibration of eiσt the variation of each co-ordinate is simple-harmonic, with the prescribed
period, but there is a retardation of phase as compared with the force. If the friction be small the amplitude
becomes relatively very great if the imposed period approximate to a free period. The validity of the
“reciprocal theorems” of Helmholtz and Lord Rayleigh, already referred to, is not affected by frictional
forces of the kind here considered.
The most important applications of the theory of vibrations are to the case of continuous systems
such as strings, bars, membranes, plates, columns of air, where the number of degrees of freedom is
infinite. The series of equations of the type (3) is then replaced by a single linear partial differential
equation, or by a set of two or three such equations, according to the number of dependent variables.
These variables represent the whole assemblage of generalized co-ordinates qr; they are continuous
functions of the independent variables x, y, z whose range of variation corresponds to that of the index
r, and of t. For example, in a one-dimensional system such as a string or a bar, we have one dependent
variable, and two independent variables x and t. To determine the free oscillations we assume a time
factor eiσt; the equations then become linear differential equations between the dependent variables of
the problem and the independent variables x, or x, y, or x, y, z as the case may be. If the range of the
independent variable or variables is unlimited, the value of σ is at our disposal, and the solution gives
us the laws of wave-propagation (see Wave). If, on the other hand, the body is finite, certain terminal
conditions have to be satisfied. These limit the admissible values of σ, which are in general
determined by a transcendental equation corresponding to the determinantal equation (6).
Numerous examples of this procedure, and of the corresponding treatment of forced oscillations,
present themselves in theoretical acoustics. It must suffice here to consider the small oscillations of a
chain hanging vertically from a fixed extremity. If x be measured upwards from the lower end, the
horizontal component of the tension P at any point will be Pδy/δx, approximately, if y denote the
lateral displacement. Hence, forming the equation of motion of a mass-element, ρδx, we have
ρ δx · ÿ = δ (P · ∂y/∂x).
(38)
Neglecting the vertical acceleration we have P = gρx, whence
∂2y = g ∂
( x ∂y ).
Page 485
∂t2 ∂x ∂x (39)
Assuming that y varies as eiσt we have
∂ ∂y
∂x ( x ∂x ) + ky = 0. (40)
provided k = σ2/g. The solution of (40) which is finite for x = 0 is readily obtained in the form of a
series, thus
kx k2x2
y=C ( 1 − 12 + 1222 − ... ) = CJ0(z),
(41)
in the notation of Bessel’s functions, if z2 = 4kx. Since y must vanish at the upper end (x = l), the
admissible values of σ are determined by
σ2 = gz2/4l, J0(z) = 0.
(42)
The function J0(z) has been tabulated; its lower roots are given by
z/π= .7655, 1.7571, 2.7546,...,
approximately, where the numbers tend to the form s − 1⁄4. The frequency of the gravest mode is to that
of a uniform bar in the ratio .9815 That this ratio should be less than unity agrees with the theory of
“constrained types” already given. In the higher normal modes there are nodes or points of rest (y =
0); thus in the second mode there is a node at a distance .190l from the lower end.
Authorities.—For indications as to the earlier history of the subject see W. W. R. Ball, Short
Account of the History of Mathematics; M. Cantor, Geschichte der Mathematik (Leipzig, 1880 ... ); J.
Cox, Mechanics (Cambridge, 1904); E. Mach, Die Mechanik in ihrer Entwickelung (4th ed., Leipzig,
1901; Eng. trans.). Of the classical treatises which have had a notable influence on the development of
the subject, and which may still be consulted with advantage, we may note particularly, Sir I. Newton,
Philosophiae naturalis Principia Mathematica (1st ed., London, 1687); J. L. Lagrange, Mécanique
analytique (2nd ed., Paris, 1811-1815); P. S. Laplace, Mécanique céleste (Paris, 1799-1825); A. F.
Möbius, Lehrbuch der Statik (Leipzig, 1837), and Mechanik des Himmels; L. Poinsot, Éléments de
statique (Paris, 1804), and Théorie nouvelle de la rotation des corps (Paris, 1834).
Of the more recent general treatises we may mention Sir W. Thomson (Lord Kelvin) and P. G. Tait,
Natural Philosophy (2nd ed., Cambridge, 1879-1883); E. J. Routh, Analytical Statics (2nd ed.,
Cambridge, 1896), Dynamics of a Particle (Cambridge, 1898), Rigid Dynamics (6th ed., Cambridge
1905); G. Minchin, Statics (4th ed., Oxford, 1888); A. E. H. Love, Theoretical Mechanics (2nd ed.,
Cambridge, 1909); A. G. Webster, Dynamics of Particles, &c. (1904); E. T. Whittaker, Analytical
Dynamics (Cambridge, 1904); L. Arnal, Traitê de mécanique (1888-1898); P. Appell, Mécanique
rationelle (Paris, vols. i. and ii., 2nd ed., 1902 and 1904; vol. iii., 1st ed., 1896); G. Kirchhoff,
Vorlesungen über Mechanik (Leipzig, 1896); H. Helmholtz, Vorlesungen über theoretische Physik,
vol. i. (Leipzig, 1898); J. Somoff, Theoretische Mechanik (Leipzig, 1878-1879).
Assuming that y varies as eiσt we have
∂ ∂y
∂x ( x ∂x ) + ky = 0. (40)
provided k = σ2/g. The solution of (40) which is finite for x = 0 is readily obtained in the form of a
series, thus
kx k2x2
y=C ( 1 − 12 + 1222 − ... ) = CJ0(z),
(41)
in the notation of Bessel’s functions, if z2 = 4kx. Since y must vanish at the upper end (x = l), the
admissible values of σ are determined by
σ2 = gz2/4l, J0(z) = 0.
(42)
The function J0(z) has been tabulated; its lower roots are given by
z/π= .7655, 1.7571, 2.7546,...,
approximately, where the numbers tend to the form s − 1⁄4. The frequency of the gravest mode is to that
of a uniform bar in the ratio .9815 That this ratio should be less than unity agrees with the theory of
“constrained types” already given. In the higher normal modes there are nodes or points of rest (y =
0); thus in the second mode there is a node at a distance .190l from the lower end.
Authorities.—For indications as to the earlier history of the subject see W. W. R. Ball, Short
Account of the History of Mathematics; M. Cantor, Geschichte der Mathematik (Leipzig, 1880 ... ); J.
Cox, Mechanics (Cambridge, 1904); E. Mach, Die Mechanik in ihrer Entwickelung (4th ed., Leipzig,
1901; Eng. trans.). Of the classical treatises which have had a notable influence on the development of
the subject, and which may still be consulted with advantage, we may note particularly, Sir I. Newton,
Philosophiae naturalis Principia Mathematica (1st ed., London, 1687); J. L. Lagrange, Mécanique
analytique (2nd ed., Paris, 1811-1815); P. S. Laplace, Mécanique céleste (Paris, 1799-1825); A. F.
Möbius, Lehrbuch der Statik (Leipzig, 1837), and Mechanik des Himmels; L. Poinsot, Éléments de
statique (Paris, 1804), and Théorie nouvelle de la rotation des corps (Paris, 1834).
Of the more recent general treatises we may mention Sir W. Thomson (Lord Kelvin) and P. G. Tait,
Natural Philosophy (2nd ed., Cambridge, 1879-1883); E. J. Routh, Analytical Statics (2nd ed.,
Cambridge, 1896), Dynamics of a Particle (Cambridge, 1898), Rigid Dynamics (6th ed., Cambridge
1905); G. Minchin, Statics (4th ed., Oxford, 1888); A. E. H. Love, Theoretical Mechanics (2nd ed.,
Cambridge, 1909); A. G. Webster, Dynamics of Particles, &c. (1904); E. T. Whittaker, Analytical
Dynamics (Cambridge, 1904); L. Arnal, Traitê de mécanique (1888-1898); P. Appell, Mécanique
rationelle (Paris, vols. i. and ii., 2nd ed., 1902 and 1904; vol. iii., 1st ed., 1896); G. Kirchhoff,
Vorlesungen über Mechanik (Leipzig, 1896); H. Helmholtz, Vorlesungen über theoretische Physik,
vol. i. (Leipzig, 1898); J. Somoff, Theoretische Mechanik (Leipzig, 1878-1879).
Page 486
The literature of graphical statics and its technical applications is very extensive. We may mention
K. Culmann, Graphische Statik (2nd ed., Zürich, 1895); A. Föppl, Technische Mechanik, vol. ii.
(Leipzig, 1900); L. Henneberg, Statik des starren Systems (Darmstadt, 1886); M. Lévy, La statique
graphique (2nd ed., Paris, 1886-1888); H. Müller-Breslau, Graphische Statik (3rd ed., Berlin, 1901).
Sir R. S. Ball’s highly original investigations in kinematics and dynamics were published in collected
form under the title Theory of Screws (Cambridge, 1900).
Detailed accounts of the developments of the various branches of the subject from the beginning of
the 19th century to the present time, with full bibliographical references, are given in the fourth
volume (edited by Professor F. Klein) of the Encyclopädie der mathematischen Wissenschaften
(Leipzig). There is a French translation of this work. (See also Dynamics.) (H. Lb.)
II.—Applied Mechanics1
§ 1. The practical application of mechanics may be divided into two classes, according as the assemblages
of material objects to which they relate are intended to remain fixed or to move relatively to each other—the
former class being comprehended under the term “Theory of Structures” and the latter under the term
“Theory of Machines.”
PART I.—OUTLINE OF THE THEORY OF STRUCTURES
K. Culmann, Graphische Statik (2nd ed., Zürich, 1895); A. Föppl, Technische Mechanik, vol. ii.
(Leipzig, 1900); L. Henneberg, Statik des starren Systems (Darmstadt, 1886); M. Lévy, La statique
graphique (2nd ed., Paris, 1886-1888); H. Müller-Breslau, Graphische Statik (3rd ed., Berlin, 1901).
Sir R. S. Ball’s highly original investigations in kinematics and dynamics were published in collected
form under the title Theory of Screws (Cambridge, 1900).
Detailed accounts of the developments of the various branches of the subject from the beginning of
the 19th century to the present time, with full bibliographical references, are given in the fourth
volume (edited by Professor F. Klein) of the Encyclopädie der mathematischen Wissenschaften
(Leipzig). There is a French translation of this work. (See also Dynamics.) (H. Lb.)
II.—Applied Mechanics1
§ 1. The practical application of mechanics may be divided into two classes, according as the assemblages
of material objects to which they relate are intended to remain fixed or to move relatively to each other—the
former class being comprehended under the term “Theory of Structures” and the latter under the term
“Theory of Machines.”
PART I.—OUTLINE OF THE THEORY OF STRUCTURES
Page 487
§ 2. Support of Structures.—Every structure, as a whole, is maintained in equilibrium by
the joint action of its own weight, of the external load or pressure applied to it from
without and tending to displace it, and of the resistance of the material which supports it. A
structure is supported either by resting on the solid crust of the earth, as buildings do, or by
floating in a fluid, as ships do in water and balloons in air. The principles of the support of
a floating structure form an important part of Hydromechanics (q.v.). The principles of the
support, as a whole, of a structure resting on the land, are so far identical with those which
regulate the equilibrium and stability of the several parts of that structure that the only
principle which seems to require special mention here is one which comprehends in one
statement the power both of liquids and of loose earth to support structures. This was first
demonstrated in a paper “On the Stability of Loose Earth,” read to the Royal Society on the
19th of June 1856 (Phil. Trans. 1856), as follows:—
Let E represent the weight of the portion of a horizontal stratum of earth which is
displaced by the foundation of a structure, S the utmost weight of that structure consistently
with the power of the earth to resist displacement, φ the angle of repose of the earth; then
S 1 + sin φ
E = ( 1 − sin φ )2.
To apply this to liquids φ must be made zero, and then S/E = 1, as is well known. For a
proof of this expression see Rankine’s Applied Mechanics, 17th ed., p. 219.
§ 3. Composition of a Structure, and Connexion of its Pieces.—A structure is composed
of pieces,—such as the stones of a building in masonry, the beams of a timber framework,
the bars, plates and bolts of an iron bridge. Those pieces are connected at their joints or
surfaces of mutual contact, either by simple pressure and friction (as in masonry with moist
mortar or without mortar), by pressure and adhesion (as in masonry with cement or with
hardened mortar, and timber with glue), or by the resistance of fastenings of different
kinds, whether made by means of the form of the joint (as dovetails, notches, mortices and
tenons) or by separate fastening pieces (as trenails, pins, spikes, nails, holdfasts, screws,
bolts, rivets, hoops, straps and sockets.)
§ 4. Stability, Stiffness and Strength.—A structure may be damaged or destroyed in three
ways:—first, by displacement of its pieces from their proper positions relatively to each
other or to the earth; secondly by disfigurement of one or more of those pieces, owing to
their being unable to preserve their proper shapes under the pressures to which they are
subjected; thirdly, by breaking of one or more of those pieces. The power of resisting
displacement constitutes stability, the power of each piece to resist disfigurement is its
stiffness; and its power to resist breaking, its strength.
the joint action of its own weight, of the external load or pressure applied to it from
without and tending to displace it, and of the resistance of the material which supports it. A
structure is supported either by resting on the solid crust of the earth, as buildings do, or by
floating in a fluid, as ships do in water and balloons in air. The principles of the support of
a floating structure form an important part of Hydromechanics (q.v.). The principles of the
support, as a whole, of a structure resting on the land, are so far identical with those which
regulate the equilibrium and stability of the several parts of that structure that the only
principle which seems to require special mention here is one which comprehends in one
statement the power both of liquids and of loose earth to support structures. This was first
demonstrated in a paper “On the Stability of Loose Earth,” read to the Royal Society on the
19th of June 1856 (Phil. Trans. 1856), as follows:—
Let E represent the weight of the portion of a horizontal stratum of earth which is
displaced by the foundation of a structure, S the utmost weight of that structure consistently
with the power of the earth to resist displacement, φ the angle of repose of the earth; then
S 1 + sin φ
E = ( 1 − sin φ )2.
To apply this to liquids φ must be made zero, and then S/E = 1, as is well known. For a
proof of this expression see Rankine’s Applied Mechanics, 17th ed., p. 219.
§ 3. Composition of a Structure, and Connexion of its Pieces.—A structure is composed
of pieces,—such as the stones of a building in masonry, the beams of a timber framework,
the bars, plates and bolts of an iron bridge. Those pieces are connected at their joints or
surfaces of mutual contact, either by simple pressure and friction (as in masonry with moist
mortar or without mortar), by pressure and adhesion (as in masonry with cement or with
hardened mortar, and timber with glue), or by the resistance of fastenings of different
kinds, whether made by means of the form of the joint (as dovetails, notches, mortices and
tenons) or by separate fastening pieces (as trenails, pins, spikes, nails, holdfasts, screws,
bolts, rivets, hoops, straps and sockets.)
§ 4. Stability, Stiffness and Strength.—A structure may be damaged or destroyed in three
ways:—first, by displacement of its pieces from their proper positions relatively to each
other or to the earth; secondly by disfigurement of one or more of those pieces, owing to
their being unable to preserve their proper shapes under the pressures to which they are
subjected; thirdly, by breaking of one or more of those pieces. The power of resisting
displacement constitutes stability, the power of each piece to resist disfigurement is its
stiffness; and its power to resist breaking, its strength.
Page 488
§ 5. Conditions of Stability.—The principles of the stability of a structure can be to a
certain extent investigated independently of the stiffness and strength, by assuming, in the
first instance, that each piece has strength sufficient to be safe against being broken, and
stiffness sufficient to prevent its being disfigured to an extent inconsistent with the
purposes of the structure, by the greatest forces which are to be applied to it. The condition
that each piece of the structure is to be maintained in equilibrium by having its gross load,
consisting of its own weight and of the external pressure applied to it, balanced by the
resistances or pressures exerted between it and the contiguous pieces, furnishes the means
of determining the magnitude, position and direction of the resistances required at each
joint in order to produce equilibrium; and the conditions of stability are, first, that the
position, and, secondly, that the direction, of the resistance required at each joint shall,
under all the variations to which the load is subject, be such as the joint is capable of
exerting—conditions which are fulfilled by suitably adjusting the figures and positions of
the joints, and the ratios of the gross loads of the pieces. As for the magnitude of the
resistance, it is limited by conditions, not of stability, but of strength and stiffness.
§ 6. Principle of Least Resistance.—Where more than one system of resistances are alike
capable of balancing the same system of loads applied to a given structure, the smallest of
those alternative systems, as was demonstrated by the Rev. Henry Moseley in his
Mechanics of Engineering and Architecture, is that which will actually be exerted—
because the resistances to displacement are the effect of a strained state of the pieces,
which strained state is the effect of the load, and when the load is applied the strained state
and the resistances produced by it increase until the resistances acquire just those
magnitudes which are sufficient to balance the load, after which they increase no further.
This principle of least resistance renders determinate many problems in the statics of
structures which were formerly considered indeterminate.
§ 7. Relations between Polygons of Loads and of Resistances.—In a structure in which
each piece is supported at two joints only, the well-known laws of statics show that the
directions of the gross load on each piece and of the two resistances by which it is
supported must lie in one plane, must either be parallel or meet in one point, and must bear
to each other, if not parallel, the proportions of the sides of a triangle respectively parallel
to their directions, and, if parallel, such proportions that each of the three forces shall be
proportional to the distance between the other two,—all the three distances being measured
along one direction.
Considering, in the first place, the case in which the load and the two resistances by
which each piece is balanced meet in one point, which may be called the centre of load,
there will be as many such points of intersection, or centres of load, as there are pieces in
the structure; and the directions and positions of the resistances or mutual pressures exerted
certain extent investigated independently of the stiffness and strength, by assuming, in the
first instance, that each piece has strength sufficient to be safe against being broken, and
stiffness sufficient to prevent its being disfigured to an extent inconsistent with the
purposes of the structure, by the greatest forces which are to be applied to it. The condition
that each piece of the structure is to be maintained in equilibrium by having its gross load,
consisting of its own weight and of the external pressure applied to it, balanced by the
resistances or pressures exerted between it and the contiguous pieces, furnishes the means
of determining the magnitude, position and direction of the resistances required at each
joint in order to produce equilibrium; and the conditions of stability are, first, that the
position, and, secondly, that the direction, of the resistance required at each joint shall,
under all the variations to which the load is subject, be such as the joint is capable of
exerting—conditions which are fulfilled by suitably adjusting the figures and positions of
the joints, and the ratios of the gross loads of the pieces. As for the magnitude of the
resistance, it is limited by conditions, not of stability, but of strength and stiffness.
§ 6. Principle of Least Resistance.—Where more than one system of resistances are alike
capable of balancing the same system of loads applied to a given structure, the smallest of
those alternative systems, as was demonstrated by the Rev. Henry Moseley in his
Mechanics of Engineering and Architecture, is that which will actually be exerted—
because the resistances to displacement are the effect of a strained state of the pieces,
which strained state is the effect of the load, and when the load is applied the strained state
and the resistances produced by it increase until the resistances acquire just those
magnitudes which are sufficient to balance the load, after which they increase no further.
This principle of least resistance renders determinate many problems in the statics of
structures which were formerly considered indeterminate.
§ 7. Relations between Polygons of Loads and of Resistances.—In a structure in which
each piece is supported at two joints only, the well-known laws of statics show that the
directions of the gross load on each piece and of the two resistances by which it is
supported must lie in one plane, must either be parallel or meet in one point, and must bear
to each other, if not parallel, the proportions of the sides of a triangle respectively parallel
to their directions, and, if parallel, such proportions that each of the three forces shall be
proportional to the distance between the other two,—all the three distances being measured
along one direction.
Considering, in the first place, the case in which the load and the two resistances by
which each piece is balanced meet in one point, which may be called the centre of load,
there will be as many such points of intersection, or centres of load, as there are pieces in
the structure; and the directions and positions of the resistances or mutual pressures exerted
Page 489
between the pieces will be represented
by the sides of a polygon joining those
points, as in fig. 86 where P1, P2, P3, P4
represent the centres of load in a
structure of four pieces, and the sides
of the polygon of resistances P1 P2 P3
P4 represent respectively the directions
and positions of the resistances exerted Fig. 86.
at the joints. Further, at any one of the
centres of load let PL represent the magnitude and direction of the gross load, and Pa, Pb
the two resistances by which the piece to which that load is applied is supported; then will
those three lines be respectively the diagonal and sides of a parallelogram; or, what is the
same thing, they will be equal to the three sides of a triangle; and they must be in the same
plane, although the sides of the polygon of resistances may be in different planes.
According to a well-known principle of statics, because the
loads or external pressures P1L1, &c., balance each other, they
must be proportional to the sides of a closed polygon drawn
respectively parallel to their directions. In fig. 87 construct
such a polygon of loads by drawing the lines L1, &c., parallel
and proportional to, and joined end to end in the order of, the
gross loads on the pieces of the structure. Then from the
proportionality and parallelism of the load and the two
resistances applied to each piece of the structure to the three
Fig. 87.
sides of a triangle, there results the following theorem
(originally due to Rankine):—
If from the angles of the polygon of loads there be drawn lines (R1, R2, &c.), each of
which is parallel to the resistance (as P1P2, &c.) exerted at the joint between the pieces to
which the two loads represented by the contiguous sides of the polygon of loads (such as
L1, L2, &c.) are applied; then will all those lines meet in one point (O), and their lengths,
measured from that point to the angles of the polygon, will represent the magnitudes of the
resistances to which they are respectively parallel.
When the load on one of the pieces is parallel to the resistances which balance it, the
polygon of resistances ceases to be closed, two of the sides becoming parallel to each other
and to the load in question, and extending indefinitely. In the polygon of loads the direction
of a load sustained by parallel resistances traverses the point O.2
by the sides of a polygon joining those
points, as in fig. 86 where P1, P2, P3, P4
represent the centres of load in a
structure of four pieces, and the sides
of the polygon of resistances P1 P2 P3
P4 represent respectively the directions
and positions of the resistances exerted Fig. 86.
at the joints. Further, at any one of the
centres of load let PL represent the magnitude and direction of the gross load, and Pa, Pb
the two resistances by which the piece to which that load is applied is supported; then will
those three lines be respectively the diagonal and sides of a parallelogram; or, what is the
same thing, they will be equal to the three sides of a triangle; and they must be in the same
plane, although the sides of the polygon of resistances may be in different planes.
According to a well-known principle of statics, because the
loads or external pressures P1L1, &c., balance each other, they
must be proportional to the sides of a closed polygon drawn
respectively parallel to their directions. In fig. 87 construct
such a polygon of loads by drawing the lines L1, &c., parallel
and proportional to, and joined end to end in the order of, the
gross loads on the pieces of the structure. Then from the
proportionality and parallelism of the load and the two
resistances applied to each piece of the structure to the three
Fig. 87.
sides of a triangle, there results the following theorem
(originally due to Rankine):—
If from the angles of the polygon of loads there be drawn lines (R1, R2, &c.), each of
which is parallel to the resistance (as P1P2, &c.) exerted at the joint between the pieces to
which the two loads represented by the contiguous sides of the polygon of loads (such as
L1, L2, &c.) are applied; then will all those lines meet in one point (O), and their lengths,
measured from that point to the angles of the polygon, will represent the magnitudes of the
resistances to which they are respectively parallel.
When the load on one of the pieces is parallel to the resistances which balance it, the
polygon of resistances ceases to be closed, two of the sides becoming parallel to each other
and to the load in question, and extending indefinitely. In the polygon of loads the direction
of a load sustained by parallel resistances traverses the point O.2
Page 490
§ 8. How the Earth’s Resistance is to be treated.... When the pressure exerted by a
structure on the earth (to which the earth’s resistance is equal and opposite) consists either
of one pressure, which is necessarily the resultant of the weight of the structure and of all
the other forces applied to it, or of two or more parallel vertical forces, whose amount can
be determined at the outset of the investigation, the resistance of the earth can be treated as
one or more upward loads applied to the structure. But in other cases the earth is to be
treated as one of the pieces of the structure, loaded with a force equal and opposite in
direction and position to the resultant of the weight of the structure and of the other
pressures applied to it.
§ 9. Partial Polygons of Resistance.—In a structure in which there are pieces supported
at more than two joints, let a polygon be constructed of lines connecting the centres of load
of any continuous series of pieces. This may be called a partial polygon of resistances. In
considering its properties, the load at each centre of load is to be held to include the
resistances of those joints which are not comprehended in the partial polygon of
resistances, to which the theorem of § 7 will then apply in every respect. By constructing
several partial polygons, and computing the relations between the loads and resistances
which are determined by the application of that theorem to each of them, with the aid, if
necessary, of Moseley’s principle of the least resistance, the whole of the relations amongst
the loads and resistances may be found.
§ 10. Line of Pressures—Centres and Line of Resistance.—The line of pressures is a line
to which the directions of all the resistances in one polygon are tangents. The centre of
resistance at any joint is the point where the line representing the total resistance exerted at
that joint intersects the joint. The line of resistance is a line traversing all the centres of
resistance of a series of joints,—its form, in the positions intermediate between the actual
joints of the structure, being determined by supposing the pieces and their loads to be
subdivided by the introduction of intermediate joints ad infinitum, and finding the
continuous line, curved or straight, in which the intermediate centres of resistance are all
situated, however great their number. The difference between the line of resistance and the
line of pressures was first pointed out by Moseley.
§ 11.* The principles of the two preceding sections may be illustrated by the
consideration of a particular case of a buttress of blocks forming a continuous series of
pieces (fig. 88), where aa, bb, cc, dd represent plane joints. Let the centre of pressure C at
the first joint aa be known, and also the pressure P acting at C in direction and magnitude.
Find R1 the resultant of this pressure, the weight of the block aabb acting through its centre
of gravity, and any other external force which may be acting on the block, and produce its
line of action to cut the joint bb in C1. C1 is then the centre of pressure for the joint bb, and
R1 is the total force acting there. Repeating this process for each block in succession there
structure on the earth (to which the earth’s resistance is equal and opposite) consists either
of one pressure, which is necessarily the resultant of the weight of the structure and of all
the other forces applied to it, or of two or more parallel vertical forces, whose amount can
be determined at the outset of the investigation, the resistance of the earth can be treated as
one or more upward loads applied to the structure. But in other cases the earth is to be
treated as one of the pieces of the structure, loaded with a force equal and opposite in
direction and position to the resultant of the weight of the structure and of the other
pressures applied to it.
§ 9. Partial Polygons of Resistance.—In a structure in which there are pieces supported
at more than two joints, let a polygon be constructed of lines connecting the centres of load
of any continuous series of pieces. This may be called a partial polygon of resistances. In
considering its properties, the load at each centre of load is to be held to include the
resistances of those joints which are not comprehended in the partial polygon of
resistances, to which the theorem of § 7 will then apply in every respect. By constructing
several partial polygons, and computing the relations between the loads and resistances
which are determined by the application of that theorem to each of them, with the aid, if
necessary, of Moseley’s principle of the least resistance, the whole of the relations amongst
the loads and resistances may be found.
§ 10. Line of Pressures—Centres and Line of Resistance.—The line of pressures is a line
to which the directions of all the resistances in one polygon are tangents. The centre of
resistance at any joint is the point where the line representing the total resistance exerted at
that joint intersects the joint. The line of resistance is a line traversing all the centres of
resistance of a series of joints,—its form, in the positions intermediate between the actual
joints of the structure, being determined by supposing the pieces and their loads to be
subdivided by the introduction of intermediate joints ad infinitum, and finding the
continuous line, curved or straight, in which the intermediate centres of resistance are all
situated, however great their number. The difference between the line of resistance and the
line of pressures was first pointed out by Moseley.
§ 11.* The principles of the two preceding sections may be illustrated by the
consideration of a particular case of a buttress of blocks forming a continuous series of
pieces (fig. 88), where aa, bb, cc, dd represent plane joints. Let the centre of pressure C at
the first joint aa be known, and also the pressure P acting at C in direction and magnitude.
Find R1 the resultant of this pressure, the weight of the block aabb acting through its centre
of gravity, and any other external force which may be acting on the block, and produce its
line of action to cut the joint bb in C1. C1 is then the centre of pressure for the joint bb, and
R1 is the total force acting there. Repeating this process for each block in succession there
Page 491
will be found the centres of pressure C2, C3,
&c., and also the resultant pressures R2, R3,
&c., acting at these respective centres. The
centres of pressure at the joints are also called
centres of resistance, and the curve passing
through these points is called a line of
resistance. Let all the resultants acting at the
several centres of resistance be produced until
they cut one another in a series of points so as
to form an unclosed polygon. This polygon is
the partial polygon of resistance. A curve
tangential to all the sides of the polygon is the
line of pressures.
§ 12. Stability of Position, and Stability of Fig. 88.
Friction.—The resistances at the several joints
having been determined by the principles set forth in §§ 6, 7, 8, 9 and 10, not only under
the ordinary load of the structure, but under all the variations to which the load is subject as
to amount and distribution, the joints are now to be placed and shaped so that the pieces
shall not suffer relative displacement under any of those loads. The relative displacement of
the two pieces which abut against each other at a joint may take place either by turning or
by sliding. Safety against displacement by turning is called stability of position; safety
against displacement by sliding, stability of friction.
§ 13. Condition of Stability of Position.—If the materials of a structure were infinitely
stiff and strong, stability of position at any joint would be insured simply by making the
centre of resistance fall within the joint under all possible variations of load. In order to
allow for the finite stiffness and strength of materials, the least distance of the centre of
resistance inward from the nearest edge of the joint is made to bear a definite proportion to
the depth of the joint measured in the same direction, which proportion is fixed, sometimes
empirically, sometimes by theoretical deduction from the laws of the strength of materials.
That least distance is called by Moseley the modulus of stability. The following are some of
the ratios of the modulus of stability to the depth of the joint which occur in practice:—
Retaining walls, as designed by British engineers 1:8
Retaining walls, as designed by French engineers 1:5
Rectangular piers of bridges and other buildings, and arch-stones 1:3
Rectangular foundations, firm ground 1:3
Rectangular foundations, very soft ground 1:2
&c., and also the resultant pressures R2, R3,
&c., acting at these respective centres. The
centres of pressure at the joints are also called
centres of resistance, and the curve passing
through these points is called a line of
resistance. Let all the resultants acting at the
several centres of resistance be produced until
they cut one another in a series of points so as
to form an unclosed polygon. This polygon is
the partial polygon of resistance. A curve
tangential to all the sides of the polygon is the
line of pressures.
§ 12. Stability of Position, and Stability of Fig. 88.
Friction.—The resistances at the several joints
having been determined by the principles set forth in §§ 6, 7, 8, 9 and 10, not only under
the ordinary load of the structure, but under all the variations to which the load is subject as
to amount and distribution, the joints are now to be placed and shaped so that the pieces
shall not suffer relative displacement under any of those loads. The relative displacement of
the two pieces which abut against each other at a joint may take place either by turning or
by sliding. Safety against displacement by turning is called stability of position; safety
against displacement by sliding, stability of friction.
§ 13. Condition of Stability of Position.—If the materials of a structure were infinitely
stiff and strong, stability of position at any joint would be insured simply by making the
centre of resistance fall within the joint under all possible variations of load. In order to
allow for the finite stiffness and strength of materials, the least distance of the centre of
resistance inward from the nearest edge of the joint is made to bear a definite proportion to
the depth of the joint measured in the same direction, which proportion is fixed, sometimes
empirically, sometimes by theoretical deduction from the laws of the strength of materials.
That least distance is called by Moseley the modulus of stability. The following are some of
the ratios of the modulus of stability to the depth of the joint which occur in practice:—
Retaining walls, as designed by British engineers 1:8
Retaining walls, as designed by French engineers 1:5
Rectangular piers of bridges and other buildings, and arch-stones 1:3
Rectangular foundations, firm ground 1:3
Rectangular foundations, very soft ground 1:2
Page 492
Rectangular foundations, intermediate kinds of ground 1 : 3 to 1 : 2
Thin, hollow towers (such as furnace chimneys exposed to high winds), square 1:6
Thin, hollow towers, circular 1:4
Frames of timber or metal, under their ordinary or average distribution of load 1:3
Frames of timber or metal, under the greatest irregularities of load 1:3
In the case of the towers, the depth of the joint is to be understood to mean the diameter
of the tower.
§ 14. Condition of Stability of Friction.—If the resistance
to be exerted at a joint is always perpendicular to the
surfaces which abut at and form that joint, there is no
tendency of the pieces to be displaced by sliding. If the
resistance be oblique, let JK (fig. 89) be the joint, C its centre
of resistance, CR a line representing the resistance, CN a
perpendicular to the joint at the centre of resistance. The
angle NCR is the obliquity of the resistance. From R draw
Fig. 89.
RP parallel and RQ perpendicular to the joint; then, by the
principles of statics, the component of the resistance normal to the joint is—
CP = CR · cos PCR;
and the component tangential to the joint is—
CQ = CR · sin PCR = CP · tan PCR.
If the joint be provided either with projections and recesses, such as mortises and tenons, or
with fastenings, such as pins or bolts, so as to resist displacement by sliding, the question
of the utmost amount of the tangential resistance CQ which it is capable of exerting
depends on the strength of such projections, recesses, or fastenings; and belongs to the
subject of strength, and not to that of stability. In other cases the safety of the joint against
displacement by sliding depends on its power of exerting friction, and that power depends
on the law, known by experiment, that the friction between two surfaces bears a constant
ratio, depending on the nature of the surfaces, to the force by which they are pressed
together. In order that the surfaces which abut at the joint JK may be pressed together, the
resistance required by the conditions of equilibrium CR, must be a thrust and not a pull;
and in that case the force by which the surfaces are pressed together is equal and opposite
to the normal component CP of the resistance. The condition of stability of friction is that
the tangential component CQ of the resistance required shall not exceed the friction due to
the normal component; that is, that
Thin, hollow towers (such as furnace chimneys exposed to high winds), square 1:6
Thin, hollow towers, circular 1:4
Frames of timber or metal, under their ordinary or average distribution of load 1:3
Frames of timber or metal, under the greatest irregularities of load 1:3
In the case of the towers, the depth of the joint is to be understood to mean the diameter
of the tower.
§ 14. Condition of Stability of Friction.—If the resistance
to be exerted at a joint is always perpendicular to the
surfaces which abut at and form that joint, there is no
tendency of the pieces to be displaced by sliding. If the
resistance be oblique, let JK (fig. 89) be the joint, C its centre
of resistance, CR a line representing the resistance, CN a
perpendicular to the joint at the centre of resistance. The
angle NCR is the obliquity of the resistance. From R draw
Fig. 89.
RP parallel and RQ perpendicular to the joint; then, by the
principles of statics, the component of the resistance normal to the joint is—
CP = CR · cos PCR;
and the component tangential to the joint is—
CQ = CR · sin PCR = CP · tan PCR.
If the joint be provided either with projections and recesses, such as mortises and tenons, or
with fastenings, such as pins or bolts, so as to resist displacement by sliding, the question
of the utmost amount of the tangential resistance CQ which it is capable of exerting
depends on the strength of such projections, recesses, or fastenings; and belongs to the
subject of strength, and not to that of stability. In other cases the safety of the joint against
displacement by sliding depends on its power of exerting friction, and that power depends
on the law, known by experiment, that the friction between two surfaces bears a constant
ratio, depending on the nature of the surfaces, to the force by which they are pressed
together. In order that the surfaces which abut at the joint JK may be pressed together, the
resistance required by the conditions of equilibrium CR, must be a thrust and not a pull;
and in that case the force by which the surfaces are pressed together is equal and opposite
to the normal component CP of the resistance. The condition of stability of friction is that
the tangential component CQ of the resistance required shall not exceed the friction due to
the normal component; that is, that
Page 493
CQ ≯ ƒ · CP,
where ƒ denotes the coefficient of friction for the surfaces in question. The angle whose
tangent is the coefficient of friction is called the angle of repose, and is expressed
symbolically by—
φ = tan −1 ƒ.
Now CQ = CP · tan PCR;
consequently the condition of stability of friction is fulfilled if the angle PCR is not greater
than φ; that is to say, if the obliquity of the resistance required at the joint does not exceed
the angle of repose; and this condition ought to be fulfilled under all possible variations of
the load.
It is chiefly in masonry and earthwork that stability of friction is relied on.
§ 15. Stability of Friction in Earth.—The grains of a mass of loose earth are to be
regarded as so many separate pieces abutting against each other at joints in all possible
positions, and depending for their stability on friction. To determine whether a mass of
earth is stable at a given point, conceive that point to be traversed by planes in all possible
positions, and determine which position gives the greatest obliquity to the total pressure
exerted between the portions of the mass which abut against each other at the plane. The
condition of stability is that this obliquity shall not exceed the angle of repose of the earth.
The consequences of this principle are developed in a paper, “On the Stability of Loose
Earth,” already cited in § 2.
§ 16. Parallel Projections of Figures.—If any figure be referred to a system of co-
ordinates, rectangular or oblique, and if a second figure be constructed by means of a
second system of co-ordinates, rectangular or oblique, and either agreeing with or differing
from the first system in rectangularity or obliquity, but so related to the co-ordinates of the
first figure that for each point in the first figure there shall be a corresponding point in the
second figure, the lengths of whose co-ordinates shall bear respectively to the three
corresponding co-ordinates of the corresponding point in the first figure three ratios which
are the same for every pair of corresponding points in the two figures, these corresponding
figures are called parallel projections of each other. The properties of parallel projections
of most importance to the subject of the present article are the following:—
(1) A parallel projection of a straight line is a straight line.
(2) A parallel projection of a plane is a plane.
where ƒ denotes the coefficient of friction for the surfaces in question. The angle whose
tangent is the coefficient of friction is called the angle of repose, and is expressed
symbolically by—
φ = tan −1 ƒ.
Now CQ = CP · tan PCR;
consequently the condition of stability of friction is fulfilled if the angle PCR is not greater
than φ; that is to say, if the obliquity of the resistance required at the joint does not exceed
the angle of repose; and this condition ought to be fulfilled under all possible variations of
the load.
It is chiefly in masonry and earthwork that stability of friction is relied on.
§ 15. Stability of Friction in Earth.—The grains of a mass of loose earth are to be
regarded as so many separate pieces abutting against each other at joints in all possible
positions, and depending for their stability on friction. To determine whether a mass of
earth is stable at a given point, conceive that point to be traversed by planes in all possible
positions, and determine which position gives the greatest obliquity to the total pressure
exerted between the portions of the mass which abut against each other at the plane. The
condition of stability is that this obliquity shall not exceed the angle of repose of the earth.
The consequences of this principle are developed in a paper, “On the Stability of Loose
Earth,” already cited in § 2.
§ 16. Parallel Projections of Figures.—If any figure be referred to a system of co-
ordinates, rectangular or oblique, and if a second figure be constructed by means of a
second system of co-ordinates, rectangular or oblique, and either agreeing with or differing
from the first system in rectangularity or obliquity, but so related to the co-ordinates of the
first figure that for each point in the first figure there shall be a corresponding point in the
second figure, the lengths of whose co-ordinates shall bear respectively to the three
corresponding co-ordinates of the corresponding point in the first figure three ratios which
are the same for every pair of corresponding points in the two figures, these corresponding
figures are called parallel projections of each other. The properties of parallel projections
of most importance to the subject of the present article are the following:—
(1) A parallel projection of a straight line is a straight line.
(2) A parallel projection of a plane is a plane.
Page 494
(3) A parallel projection of a straight line or a plane surface divided in a given ratio is a
straight line or a plane surface divided in the same ratio.
(4) A parallel projection of a pair of equal and parallel straight lines, or plain surfaces, is
a pair of equal and parallel straight lines, or plane surfaces; whence it follows
(5) That a parallel projection of a parallelogram is a parallelogram, and
(6) That a parallel projection of a parallelepiped is a parallelepiped.
(7) A parallel projection of a pair of solids having a given ratio is a pair of solids having
the same ratio.
Though not essential for the purposes of the present article, the following consequence
will serve to illustrate the principle of parallel projections:—
(8) A parallel projection of a curve, or of a surface of a given algebraical order, is a curve
or a surface of the same order.
For example, all ellipsoids referred to co-ordinates parallel to any three conjugate
diameters are parallel projections of each other and of a sphere referred to rectangular co-
ordinates.
§ 17. Parallel Projections of Systems of Forces.—If a balanced system of forces be
represented by a system of lines, then will every parallel projection of that system of lines
represent a balanced system of forces.
For the condition of equilibrium of forces not parallel is that they shall be represented in
direction and magnitude by the sides and diagonals of certain parallelograms, and of
parallel forces that they shall divide certain straight lines in certain ratios; and the parallel
projection of a parallelogram is a parallelogram, and that of a straight line divided in a
given ratio is a straight line divided in the same ratio.
The resultant of a parallel projection of any system of forces is the projection of their
resultant; and the centre of gravity of a parallel projection of a solid is the projection of the
centre of gravity of the first solid.
§ 18. Principle of the Transformation of Structures.—Here we have the following
theorem: If a structure of a given figure have stability of position under a system of forces
represented by a given system of lines, then will any structure whose figure is a parallel
projection of that of the first structure have stability of position under a system of forces
represented by the corresponding projection of the first system of lines.
straight line or a plane surface divided in the same ratio.
(4) A parallel projection of a pair of equal and parallel straight lines, or plain surfaces, is
a pair of equal and parallel straight lines, or plane surfaces; whence it follows
(5) That a parallel projection of a parallelogram is a parallelogram, and
(6) That a parallel projection of a parallelepiped is a parallelepiped.
(7) A parallel projection of a pair of solids having a given ratio is a pair of solids having
the same ratio.
Though not essential for the purposes of the present article, the following consequence
will serve to illustrate the principle of parallel projections:—
(8) A parallel projection of a curve, or of a surface of a given algebraical order, is a curve
or a surface of the same order.
For example, all ellipsoids referred to co-ordinates parallel to any three conjugate
diameters are parallel projections of each other and of a sphere referred to rectangular co-
ordinates.
§ 17. Parallel Projections of Systems of Forces.—If a balanced system of forces be
represented by a system of lines, then will every parallel projection of that system of lines
represent a balanced system of forces.
For the condition of equilibrium of forces not parallel is that they shall be represented in
direction and magnitude by the sides and diagonals of certain parallelograms, and of
parallel forces that they shall divide certain straight lines in certain ratios; and the parallel
projection of a parallelogram is a parallelogram, and that of a straight line divided in a
given ratio is a straight line divided in the same ratio.
The resultant of a parallel projection of any system of forces is the projection of their
resultant; and the centre of gravity of a parallel projection of a solid is the projection of the
centre of gravity of the first solid.
§ 18. Principle of the Transformation of Structures.—Here we have the following
theorem: If a structure of a given figure have stability of position under a system of forces
represented by a given system of lines, then will any structure whose figure is a parallel
projection of that of the first structure have stability of position under a system of forces
represented by the corresponding projection of the first system of lines.
Page 495
For in the second structure the weights, external pressures, and resistances will balance
each other as in the first structure; the weights of the pieces and all other parallel systems
of forces will have the same ratios as in the first structure; and the several centres of
resistance will divide the depths of the joints in the same proportions as in the first
structure.
If the first structure have stability of friction, the second structure will have stability of
friction also, so long as the effect of the projection is not to increase the obliquity of the
resistance at any joint beyond the angle of repose.
The lines representing the forces in the second figure show their relative directions and
magnitudes. To find their absolute directions and magnitudes, a vertical line is to be drawn
in the first figure, of such a length as to represent the weight of a particular portion of the
structure. Then will the projection of that line in the projected figure indicate the vertical
direction, and represent the weight of the part of the second structure corresponding to the
before-mentioned portion of the first structure.
The foregoing “principle of the transformation of structures” was first announced,
though in a somewhat less comprehensive form, to the Royal Society on the 6th of March
1856. It is useful in practice, by enabling the engineer easily to deduce the conditions of
equilibrium and stability of structures of complex and unsymmetrical figures from those of
structures of simple and symmetrical figures. By its aid, for example, the whole of the
properties of elliptical arches, whether square or skew, whether level or sloping in their
span, are at once deduced by projection from those of symmetrical circular arches, and the
properties of ellipsoidal and elliptic-conoidal domes from those of hemispherical and
circular-conoidal domes; and the figures of arches fitted to resist the thrust of earth, which
is less horizontally than vertically in a certain given ratio, can be deduced by a projection
from those of arches fitted to resist the thrust of a liquid, which is of equal intensity,
horizontally and vertically.
§ 19. Conditions of Stiffness and Strength.—After the arrangement of the pieces of a
structure and the size and figure of their joints or surfaces of contact have been determined
so as to fulfil the conditions of stability,—conditions which depend mainly on the position
and direction of the resultant or total load on each piece, and the relative magnitude of the
loads on the different pieces—the dimensions of each piece singly have to be adjusted so as
to fulfil the conditions of stiffness and strength—conditions which depend not only on the
absolute magnitude of the load on each piece, and of the resistances by which it is
balanced, but also on the mode of distribution of the load over the piece, and of the
resistances over the joints.
each other as in the first structure; the weights of the pieces and all other parallel systems
of forces will have the same ratios as in the first structure; and the several centres of
resistance will divide the depths of the joints in the same proportions as in the first
structure.
If the first structure have stability of friction, the second structure will have stability of
friction also, so long as the effect of the projection is not to increase the obliquity of the
resistance at any joint beyond the angle of repose.
The lines representing the forces in the second figure show their relative directions and
magnitudes. To find their absolute directions and magnitudes, a vertical line is to be drawn
in the first figure, of such a length as to represent the weight of a particular portion of the
structure. Then will the projection of that line in the projected figure indicate the vertical
direction, and represent the weight of the part of the second structure corresponding to the
before-mentioned portion of the first structure.
The foregoing “principle of the transformation of structures” was first announced,
though in a somewhat less comprehensive form, to the Royal Society on the 6th of March
1856. It is useful in practice, by enabling the engineer easily to deduce the conditions of
equilibrium and stability of structures of complex and unsymmetrical figures from those of
structures of simple and symmetrical figures. By its aid, for example, the whole of the
properties of elliptical arches, whether square or skew, whether level or sloping in their
span, are at once deduced by projection from those of symmetrical circular arches, and the
properties of ellipsoidal and elliptic-conoidal domes from those of hemispherical and
circular-conoidal domes; and the figures of arches fitted to resist the thrust of earth, which
is less horizontally than vertically in a certain given ratio, can be deduced by a projection
from those of arches fitted to resist the thrust of a liquid, which is of equal intensity,
horizontally and vertically.
§ 19. Conditions of Stiffness and Strength.—After the arrangement of the pieces of a
structure and the size and figure of their joints or surfaces of contact have been determined
so as to fulfil the conditions of stability,—conditions which depend mainly on the position
and direction of the resultant or total load on each piece, and the relative magnitude of the
loads on the different pieces—the dimensions of each piece singly have to be adjusted so as
to fulfil the conditions of stiffness and strength—conditions which depend not only on the
absolute magnitude of the load on each piece, and of the resistances by which it is
balanced, but also on the mode of distribution of the load over the piece, and of the
resistances over the joints.
Page 496
The effect of the pressures applied to a piece, consisting of the load and the supporting
resistances, is to force the piece into a state of strain or disfigurement, which increases
until the elasticity, or resistance to strain, of the material causes it to exert a stress, or effort
to recover its figure, equal and opposite to the system of applied pressures. The condition
of stiffness is that the strain or disfigurement shall not be greater than is consistent with the
purposes of the structure; and the condition of strength is that the stress shall be within the
limits of that which the material can bear with safety against breaking. The ratio in which
the utmost stress before breaking exceeds the safe working stress is called the factor of
safety, and is determined empirically. It varies from three to twelve for various materials
and structures. (See Strength of Materials.)
PART II. THEORY OF MACHINES
§ 20. Parts of a Machine: Frame and Mechanism.—The parts of a machine may be
distinguished into two principal divisions,—the frame, or fixed parts, and the mechanism,
or moving parts. The frame is a structure which supports the pieces of the mechanism, and
to a certain extent determines the nature of their motions.
The form and arrangement of the pieces of the frame depend upon the arrangement and
the motions of the mechanism; the dimensions of the pieces of the frame required in order
to give it stability and strength are determined from the pressures applied to it by means of
the mechanism. It appears therefore that in general the mechanism is to be designed first
and the frame afterwards, and that the designing of the frame is regulated by the principles
of the stability of structures and of the strength and stiffness of materials,—care being
taken to adapt the frame to the most severe load which can be thrown upon it at any period
of the action of the mechanism.
Each independent piece of the mechanism also is a structure, and its dimensions are to
be adapted, according to the principles of the strength and stiffness of materials, to the most
severe load to which it can be subjected during the action of the machine.
§ 21. Definition and Division of the Theory of Machines.—From what has been said in
the last section it appears that the department of the art of designing machines which has
reference to the stability of the frame and to the stiffness and strength of the frame and
mechanism is a branch of the art of construction. It is therefore to be separated from the
theory of machines, properly speaking, which has reference to the action of machines
considered as moving. In the action of a machine the following three things take place:—
Firstly, Some natural source of energy communicates motion and force to a piece or
pieces of the mechanism, called the receiver of power or prime mover.
resistances, is to force the piece into a state of strain or disfigurement, which increases
until the elasticity, or resistance to strain, of the material causes it to exert a stress, or effort
to recover its figure, equal and opposite to the system of applied pressures. The condition
of stiffness is that the strain or disfigurement shall not be greater than is consistent with the
purposes of the structure; and the condition of strength is that the stress shall be within the
limits of that which the material can bear with safety against breaking. The ratio in which
the utmost stress before breaking exceeds the safe working stress is called the factor of
safety, and is determined empirically. It varies from three to twelve for various materials
and structures. (See Strength of Materials.)
PART II. THEORY OF MACHINES
§ 20. Parts of a Machine: Frame and Mechanism.—The parts of a machine may be
distinguished into two principal divisions,—the frame, or fixed parts, and the mechanism,
or moving parts. The frame is a structure which supports the pieces of the mechanism, and
to a certain extent determines the nature of their motions.
The form and arrangement of the pieces of the frame depend upon the arrangement and
the motions of the mechanism; the dimensions of the pieces of the frame required in order
to give it stability and strength are determined from the pressures applied to it by means of
the mechanism. It appears therefore that in general the mechanism is to be designed first
and the frame afterwards, and that the designing of the frame is regulated by the principles
of the stability of structures and of the strength and stiffness of materials,—care being
taken to adapt the frame to the most severe load which can be thrown upon it at any period
of the action of the mechanism.
Each independent piece of the mechanism also is a structure, and its dimensions are to
be adapted, according to the principles of the strength and stiffness of materials, to the most
severe load to which it can be subjected during the action of the machine.
§ 21. Definition and Division of the Theory of Machines.—From what has been said in
the last section it appears that the department of the art of designing machines which has
reference to the stability of the frame and to the stiffness and strength of the frame and
mechanism is a branch of the art of construction. It is therefore to be separated from the
theory of machines, properly speaking, which has reference to the action of machines
considered as moving. In the action of a machine the following three things take place:—
Firstly, Some natural source of energy communicates motion and force to a piece or
pieces of the mechanism, called the receiver of power or prime mover.
Page 497
Secondly, The motion and force are transmitted from the prime mover through the train
of mechanism to the working piece or pieces, and during that transmission the motion and
force are modified in amount and direction, so as to be rendered suitable for the purpose to
which they are to be applied.
Thirdly, The working piece or pieces by their motion, or by their motion and force
combined, produce some useful effect.
Such are the phenomena of the action of a machine, arranged in the order of causation.
But in studying or treating of the theory of machines, the order of simplicity is the best; and
in this order the first branch of the subject is the modification of motion and force by the
train of mechanism; the next is the effect or purpose of the machine; and the last, or most
complex, is the action of the prime mover.
The modification of motion and the modification of force take place together, and are
connected by certain laws; but in the study of the theory of machines, as well as in that of
pure mechanics, much advantage has been gained in point of clearness and simplicity by
first considering alone the principles of the modification of motion, which are founded
upon what is now known as Kinematics, and afterwards considering the principles of the
combined modification of motion and force, which are founded both on geometry and on
the laws of dynamics. The separation of kinematics from dynamics is due mainly to G.
Monge, Ampère and R. Willis.
The theory of machines in the present article will be considered under the following
heads:—
I. Pure Mechanism, or Applied Kinematics; being the theory of machines considered
simply as modifying motion.
II. Applied Dynamics; being the theory of machines considered as modifying both
motion and force.
Chap. I. On Pure Mechanism
§ 22. Division of the Subject.—Proceeding in the order of simplicity, the subject of Pure
Mechanism, or Applied Kinematics, may be thus divided:—
Division 1.—Motion of a point.
Division 2.—Motion of the surface of a fluid.
Division 3.—Motion of a rigid solid.
of mechanism to the working piece or pieces, and during that transmission the motion and
force are modified in amount and direction, so as to be rendered suitable for the purpose to
which they are to be applied.
Thirdly, The working piece or pieces by their motion, or by their motion and force
combined, produce some useful effect.
Such are the phenomena of the action of a machine, arranged in the order of causation.
But in studying or treating of the theory of machines, the order of simplicity is the best; and
in this order the first branch of the subject is the modification of motion and force by the
train of mechanism; the next is the effect or purpose of the machine; and the last, or most
complex, is the action of the prime mover.
The modification of motion and the modification of force take place together, and are
connected by certain laws; but in the study of the theory of machines, as well as in that of
pure mechanics, much advantage has been gained in point of clearness and simplicity by
first considering alone the principles of the modification of motion, which are founded
upon what is now known as Kinematics, and afterwards considering the principles of the
combined modification of motion and force, which are founded both on geometry and on
the laws of dynamics. The separation of kinematics from dynamics is due mainly to G.
Monge, Ampère and R. Willis.
The theory of machines in the present article will be considered under the following
heads:—
I. Pure Mechanism, or Applied Kinematics; being the theory of machines considered
simply as modifying motion.
II. Applied Dynamics; being the theory of machines considered as modifying both
motion and force.
Chap. I. On Pure Mechanism
§ 22. Division of the Subject.—Proceeding in the order of simplicity, the subject of Pure
Mechanism, or Applied Kinematics, may be thus divided:—
Division 1.—Motion of a point.
Division 2.—Motion of the surface of a fluid.
Division 3.—Motion of a rigid solid.
Page 498
Division 4.—Motions of a pair of connected pieces, or of an “elementary
combination” in mechanism.
Division 5.—Motions of trains of pieces of mechanism.
Division 6.—Motions of sets of more than two connected pieces, or of “aggregate
combinations.”
A point is the boundary of a line, which is the boundary of a surface, which is the
boundary of a volume. Points, lines and surfaces have no independent existence, and
consequently those divisions of this chapter which relate to their motions are only
preliminary to the subsequent divisions, which relate to the motions of bodies.
Division 1. Motion of a Point.
§ 23. Comparative Motion.—The comparative motion of two points is the relation which
exists between their motions, without having regard to their absolute amounts. It consists of
two elements,—the velocity ratio, which is the ratio of any two magnitudes bearing to each
other the proportions of the respective velocities of the two points at a given instant, and
the directional relation, which is the relation borne to each other by the respective
directions of the motions of the two points at the same given instant.
It is obvious that the motions of a pair of points may be varied in any manner, whether
by direct or by lateral deviation, and yet that their comparative motion may remain
constant, in consequence of the deviations taking place in the same proportions, in the
same directions and at the same instants for both points.
Robert Willis (1800-1875) has the merit of having been the first to simplify considerably
the theory of pure mechanism, by pointing out that that branch of mechanics relates wholly
to comparative motions.
The comparative motion of two points at a given instant is capable of being completely
expressed by one of Sir William Hamilton’s Quaternions,—the “tensor” expressing the
velocity ratio, and the “versor” the directional relation.
Graphical methods of analysis founded on this way of representing velocity and
acceleration were developed by R. H. Smith in a paper communicated to the Royal Society
of Edinburgh in 1885, and illustrations of the method will be found below.
Division 2. Motion of the Surface of a Fluid Mass.
combination” in mechanism.
Division 5.—Motions of trains of pieces of mechanism.
Division 6.—Motions of sets of more than two connected pieces, or of “aggregate
combinations.”
A point is the boundary of a line, which is the boundary of a surface, which is the
boundary of a volume. Points, lines and surfaces have no independent existence, and
consequently those divisions of this chapter which relate to their motions are only
preliminary to the subsequent divisions, which relate to the motions of bodies.
Division 1. Motion of a Point.
§ 23. Comparative Motion.—The comparative motion of two points is the relation which
exists between their motions, without having regard to their absolute amounts. It consists of
two elements,—the velocity ratio, which is the ratio of any two magnitudes bearing to each
other the proportions of the respective velocities of the two points at a given instant, and
the directional relation, which is the relation borne to each other by the respective
directions of the motions of the two points at the same given instant.
It is obvious that the motions of a pair of points may be varied in any manner, whether
by direct or by lateral deviation, and yet that their comparative motion may remain
constant, in consequence of the deviations taking place in the same proportions, in the
same directions and at the same instants for both points.
Robert Willis (1800-1875) has the merit of having been the first to simplify considerably
the theory of pure mechanism, by pointing out that that branch of mechanics relates wholly
to comparative motions.
The comparative motion of two points at a given instant is capable of being completely
expressed by one of Sir William Hamilton’s Quaternions,—the “tensor” expressing the
velocity ratio, and the “versor” the directional relation.
Graphical methods of analysis founded on this way of representing velocity and
acceleration were developed by R. H. Smith in a paper communicated to the Royal Society
of Edinburgh in 1885, and illustrations of the method will be found below.
Division 2. Motion of the Surface of a Fluid Mass.
Page 499
§ 24. General Principle.—A mass of fluid is used in mechanism to transmit motion and
force between two or more movable portions (called pistons or plungers) of the solid
envelope or vessel in which the fluid is contained; and, when such transmission is the sole
action, or the only appreciable action of the fluid mass, its volume is either absolutely
constant, by reason of its temperature and pressure being maintained constant, or not
sensibly varied.
Let a represent the area of the section of a piston made by a plane perpendicular to its
direction of motion, and v its velocity, which is to be considered as positive when outward,
and negative when inward. Then the variation of the cubic contents of the vessel in a unit
of time by reason of the motion of one piston is va. The condition that the volume of the
fluid mass shall remain unchanged requires that there shall be more than one piston, and
that the velocities and areas of the pistons shall be connected by the equation—
Σ · va = 0.
(1)
§ 25. Comparative Motion of Two Pistons.—If there be but two pistons, whose areas are
a1 and a2, and their velocities v1 and v2, their comparative motion is expressed by the
equation—
v2/v1 = −a1/a2;
(2)
that is to say, their velocities are opposite as to inwardness and outwardness and inversely
proportional to their areas.
§ 26. Applications: Hydraulic Press: Pneumatic Power-Transmitter.—In the hydraulic
press the vessel consists of two cylinders, viz. the pump-barrel and the press-barrel, each
having its piston, and of a passage connecting them having a valve opening towards the
press-barrel. The action of the enclosed water in transmitting motion takes place during the
inward stroke of the pump-plunger, when the above-mentioned valve is open; and at that
time the press-plunger moves outwards with a velocity which is less than the inward
velocity of the pump-plunger, in the same ratio that the area of the pump-plunger is less
than the area of the press-plunger. (See Hydraulics.)
In the pneumatic power-transmitter the motion of one piston is transmitted to another at
a distance by means of a mass of air contained in two cylinders and an intervening tube.
When the pressure and temperature of the air can be maintained constant, this machine
fulfils equation (2), like the hydraulic press. The amount and effect of the variations of
pressure and temperature undergone by the air depend on the principles of the mechanical
action of heat, or Thermodynamics (q.v.), and are foreign to the subject of pure mechanism.
force between two or more movable portions (called pistons or plungers) of the solid
envelope or vessel in which the fluid is contained; and, when such transmission is the sole
action, or the only appreciable action of the fluid mass, its volume is either absolutely
constant, by reason of its temperature and pressure being maintained constant, or not
sensibly varied.
Let a represent the area of the section of a piston made by a plane perpendicular to its
direction of motion, and v its velocity, which is to be considered as positive when outward,
and negative when inward. Then the variation of the cubic contents of the vessel in a unit
of time by reason of the motion of one piston is va. The condition that the volume of the
fluid mass shall remain unchanged requires that there shall be more than one piston, and
that the velocities and areas of the pistons shall be connected by the equation—
Σ · va = 0.
(1)
§ 25. Comparative Motion of Two Pistons.—If there be but two pistons, whose areas are
a1 and a2, and their velocities v1 and v2, their comparative motion is expressed by the
equation—
v2/v1 = −a1/a2;
(2)
that is to say, their velocities are opposite as to inwardness and outwardness and inversely
proportional to their areas.
§ 26. Applications: Hydraulic Press: Pneumatic Power-Transmitter.—In the hydraulic
press the vessel consists of two cylinders, viz. the pump-barrel and the press-barrel, each
having its piston, and of a passage connecting them having a valve opening towards the
press-barrel. The action of the enclosed water in transmitting motion takes place during the
inward stroke of the pump-plunger, when the above-mentioned valve is open; and at that
time the press-plunger moves outwards with a velocity which is less than the inward
velocity of the pump-plunger, in the same ratio that the area of the pump-plunger is less
than the area of the press-plunger. (See Hydraulics.)
In the pneumatic power-transmitter the motion of one piston is transmitted to another at
a distance by means of a mass of air contained in two cylinders and an intervening tube.
When the pressure and temperature of the air can be maintained constant, this machine
fulfils equation (2), like the hydraulic press. The amount and effect of the variations of
pressure and temperature undergone by the air depend on the principles of the mechanical
action of heat, or Thermodynamics (q.v.), and are foreign to the subject of pure mechanism.
Page 500
Division 3. Motion of a Rigid Solid.
§ 27. Motions Classed.—In problems of mechanism, each solid piece of the machine is
supposed to be so stiff and strong as not to undergo any sensible change of figure or
dimensions by the forces applied to it—a supposition which is realized in practice if the
machine is skilfully designed.
This being the case, the various possible motions of a rigid solid body may all be classed
under the following heads: (1) Shifting or Translation; (2) Turning or Rotation; (3) Motions
compounded of Shifting and Turning.
The most common forms for the paths of the points of a piece of mechanism, whose
motion is simple shifting, are the straight line and the circle.
Shifting in a straight line is regulated either by straight fixed guides, in contact with
which the moving piece slides, or by combinations of link-work, called parallel motions,
which will be described in the sequel. Shifting in a straight line is usually reciprocating;
that is to say, the piece, after shifting through a certain distance, returns to its original
position by reversing its motion.
Circular shifting is regulated by attaching two or more points of the shifting piece to
ends of equal and parallel rotating cranks, or by combinations of wheel-work to be
afterwards described. As an example of circular shifting may be cited the motion of the
coupling rod, by which the parallel and equal cranks upon two or more axles of a
locomotive engine are connected and made to rotate simultaneously. The coupling rod
remains always parallel to itself, and all its points describe equal and similar circles
relatively to the frame of the engine, and move in parallel directions with equal velocities
at the same instant.
§ 28. Rotation about a Fixed Axis: Lever, Wheel and Axle.—The fixed axis of a turning
body is a line fixed relatively to the body and relatively to the fixed space in which the
body turns. In mechanism it is usually the central line either of a rotating shaft or axle
having journals, gudgeons, or pivots turning in fixed bearings, or of a fixed spindle or dead
centre round which a rotating bush turns; but it may sometimes be entirely beyond the
limits of the turning body. For example, if a sliding piece moves in circular fixed guides,
that piece rotates about an ideal fixed axis traversing the centre of those guides.
Let the angular velocity of the rotation be denoted by α = dθ/dt, then the linear velocity
of any point A at the distance r from the axis is αr; and the path of that point is a circle of
the radius r described about the axis.
§ 27. Motions Classed.—In problems of mechanism, each solid piece of the machine is
supposed to be so stiff and strong as not to undergo any sensible change of figure or
dimensions by the forces applied to it—a supposition which is realized in practice if the
machine is skilfully designed.
This being the case, the various possible motions of a rigid solid body may all be classed
under the following heads: (1) Shifting or Translation; (2) Turning or Rotation; (3) Motions
compounded of Shifting and Turning.
The most common forms for the paths of the points of a piece of mechanism, whose
motion is simple shifting, are the straight line and the circle.
Shifting in a straight line is regulated either by straight fixed guides, in contact with
which the moving piece slides, or by combinations of link-work, called parallel motions,
which will be described in the sequel. Shifting in a straight line is usually reciprocating;
that is to say, the piece, after shifting through a certain distance, returns to its original
position by reversing its motion.
Circular shifting is regulated by attaching two or more points of the shifting piece to
ends of equal and parallel rotating cranks, or by combinations of wheel-work to be
afterwards described. As an example of circular shifting may be cited the motion of the
coupling rod, by which the parallel and equal cranks upon two or more axles of a
locomotive engine are connected and made to rotate simultaneously. The coupling rod
remains always parallel to itself, and all its points describe equal and similar circles
relatively to the frame of the engine, and move in parallel directions with equal velocities
at the same instant.
§ 28. Rotation about a Fixed Axis: Lever, Wheel and Axle.—The fixed axis of a turning
body is a line fixed relatively to the body and relatively to the fixed space in which the
body turns. In mechanism it is usually the central line either of a rotating shaft or axle
having journals, gudgeons, or pivots turning in fixed bearings, or of a fixed spindle or dead
centre round which a rotating bush turns; but it may sometimes be entirely beyond the
limits of the turning body. For example, if a sliding piece moves in circular fixed guides,
that piece rotates about an ideal fixed axis traversing the centre of those guides.
Let the angular velocity of the rotation be denoted by α = dθ/dt, then the linear velocity
of any point A at the distance r from the axis is αr; and the path of that point is a circle of
the radius r described about the axis.
Page 501
This is the principle of the modification of motion by the lever, which consists of a rigid
body turning about a fixed axis called a fulcrum, and having two points at the same or
different distances from that axis, and in the same or different directions, one of which
receives motion and the other transmits motion, modified in direction and velocity
according to the above law.
In the wheel and axle, motion is received and transmitted by two cylindrical surfaces of
different radii described about their common fixed axis of turning, their velocity-ratio
being that of their radii.
§ 29. Velocity Ratio of Components of Motion.—As the
distance between any two points in a rigid body is
invariable, the projections of their velocities upon the line
joining them must be equal. Hence it follows that, if A in
fig. 90 be a point in a rigid body CD, rotating round the
fixed axis F, the component of the velocity of A in any
direction AP parallel to the plane of rotation is equal to
the total velocity of the point m, found by letting fall Fm
perpendicular to AP; that is to say, is equal to Fig. 90.
α · Fm.
Hence also the ratio of the components of the velocities of two points A and B in the
directions AP and BW respectively, both in the plane of rotation, is equal to the ratio of the
perpendiculars Fm and Fn.
§ 30. Instantaneous Axis of a Cylinder rolling on a Cylinder.—Let a cylinder bbb, whose
axis of figure is B and angular velocity γ, roll on a fixed cylinder ααα, whose axis of figure
is A, either outside (as in fig. 91), when the rolling will be towards the same hand as the
rotation, or inside (as in fig. 92), when the rolling will be towards the opposite hand; and at
a given instant let T be the line of contact of the two cylindrical surfaces, which is at their
common intersection with the plane AB traversing the two axes of figure.
The line T on the surface bbb has for the instant no velocity in a direction perpendicular
to AB; because for the instant it touches, without sliding, the line T on the fixed surface
aaa.
The line T on the surface bbb has also for the instant no velocity in the plane AB; for it
has just ceased to move towards the fixed surface aaa, and is just about to begin to move
away from that surface.
body turning about a fixed axis called a fulcrum, and having two points at the same or
different distances from that axis, and in the same or different directions, one of which
receives motion and the other transmits motion, modified in direction and velocity
according to the above law.
In the wheel and axle, motion is received and transmitted by two cylindrical surfaces of
different radii described about their common fixed axis of turning, their velocity-ratio
being that of their radii.
§ 29. Velocity Ratio of Components of Motion.—As the
distance between any two points in a rigid body is
invariable, the projections of their velocities upon the line
joining them must be equal. Hence it follows that, if A in
fig. 90 be a point in a rigid body CD, rotating round the
fixed axis F, the component of the velocity of A in any
direction AP parallel to the plane of rotation is equal to
the total velocity of the point m, found by letting fall Fm
perpendicular to AP; that is to say, is equal to Fig. 90.
α · Fm.
Hence also the ratio of the components of the velocities of two points A and B in the
directions AP and BW respectively, both in the plane of rotation, is equal to the ratio of the
perpendiculars Fm and Fn.
§ 30. Instantaneous Axis of a Cylinder rolling on a Cylinder.—Let a cylinder bbb, whose
axis of figure is B and angular velocity γ, roll on a fixed cylinder ααα, whose axis of figure
is A, either outside (as in fig. 91), when the rolling will be towards the same hand as the
rotation, or inside (as in fig. 92), when the rolling will be towards the opposite hand; and at
a given instant let T be the line of contact of the two cylindrical surfaces, which is at their
common intersection with the plane AB traversing the two axes of figure.
The line T on the surface bbb has for the instant no velocity in a direction perpendicular
to AB; because for the instant it touches, without sliding, the line T on the fixed surface
aaa.
The line T on the surface bbb has also for the instant no velocity in the plane AB; for it
has just ceased to move towards the fixed surface aaa, and is just about to begin to move
away from that surface.
Page 502
The line of contact T, therefore, on the surface of the cylinder bbb, is for the instant at
rest, and is the “instantaneous axis” about which the cylinder bbb turns, together with any
body rigidly attached to that cylinder.
Fig. 91. Fig. 92.
To find, then, the direction and velocity at the given instant of any point P, either in or
rigidly attached to the rolling cylinder T, draw the plane PT; the direction of motion of P
will be perpendicular to that plane, and towards the right or left hand according to the
direction of the rotation of bbb; and the velocity of P will be
vP = γ·PT,
(3)
PT denoting the perpendicular distance of P from T. The path of P is a curve of the kind
called epitrochoids. If P is in the circumference of bbb, that path becomes an epicycloid.
The velocity of any point in the axis of figure B is
vB = γ·TB;
(4)
and the path of such a point is a circle described about A with the radius AB, being for
outside rolling the sum, and for inside rolling the difference, of the radii of the cylinders.
Let α denote the angular velocity with which the plane of axes AB rotates about the fixed
axis A. Then it is evident that
vB = α·AB,
(5)
and consequently that
α = γ·TB/AB.
(6)
rest, and is the “instantaneous axis” about which the cylinder bbb turns, together with any
body rigidly attached to that cylinder.
Fig. 91. Fig. 92.
To find, then, the direction and velocity at the given instant of any point P, either in or
rigidly attached to the rolling cylinder T, draw the plane PT; the direction of motion of P
will be perpendicular to that plane, and towards the right or left hand according to the
direction of the rotation of bbb; and the velocity of P will be
vP = γ·PT,
(3)
PT denoting the perpendicular distance of P from T. The path of P is a curve of the kind
called epitrochoids. If P is in the circumference of bbb, that path becomes an epicycloid.
The velocity of any point in the axis of figure B is
vB = γ·TB;
(4)
and the path of such a point is a circle described about A with the radius AB, being for
outside rolling the sum, and for inside rolling the difference, of the radii of the cylinders.
Let α denote the angular velocity with which the plane of axes AB rotates about the fixed
axis A. Then it is evident that
vB = α·AB,
(5)
and consequently that
α = γ·TB/AB.
(6)
Page 503
For internal rolling, as in fig. 92, AB is to be treated as negative, which will give a negative
value to α, indicating that in this case the rotation of AB round A is contrary to that of the
cylinder bbb.
The angular velocity of the rolling cylinder, relatively to the plane of axes AB, is
obviously given by the equation—
β=γ−α
whence β = γ · TA/AB
(7)
care being taken to attend to the sign of α, so that when that is negative the arithmetical
values of γ and α are to be added in order to give that of β.
The whole of the foregoing reasonings are applicable, not merely when aaa and bbb are
actual cylinders, but also when they are the osculating cylinders of a pair of cylindroidal
surfaces of varying curvature, A and B being the axes of curvature of the parts of those
surfaces which are in contact for the instant under consideration.
Fig. 93.
§ 31. Instantaneous Axis of a Cone rolling on a Cone.—Let Oaa (fig. 93) be a fixed
cone, OA its axis, Obb a cone rolling on it, OB the axis of the rolling cone, OT the line of
contact of the two cones at the instant under consideration. By reasoning similar to that of §
30, it appears that OT is the instantaneous axis of rotation of the rolling cone.
Let γ denote the total angular velocity of the rotation of the cone B about the
instantaneous axis, β its angular velocity about the axis OB relatively to the plane AOB,
and α the angular velocity with which the plane AOB turns round the axis OA. It is
required to find the ratios of those angular velocities.
Solution.—In OT take any point E, from which draw EC parallel to OA, and ED parallel
to OB, so as to construct the parallelogram OCED. Then
OD : OC : OE :: α : β : γ.
value to α, indicating that in this case the rotation of AB round A is contrary to that of the
cylinder bbb.
The angular velocity of the rolling cylinder, relatively to the plane of axes AB, is
obviously given by the equation—
β=γ−α
whence β = γ · TA/AB
(7)
care being taken to attend to the sign of α, so that when that is negative the arithmetical
values of γ and α are to be added in order to give that of β.
The whole of the foregoing reasonings are applicable, not merely when aaa and bbb are
actual cylinders, but also when they are the osculating cylinders of a pair of cylindroidal
surfaces of varying curvature, A and B being the axes of curvature of the parts of those
surfaces which are in contact for the instant under consideration.
Fig. 93.
§ 31. Instantaneous Axis of a Cone rolling on a Cone.—Let Oaa (fig. 93) be a fixed
cone, OA its axis, Obb a cone rolling on it, OB the axis of the rolling cone, OT the line of
contact of the two cones at the instant under consideration. By reasoning similar to that of §
30, it appears that OT is the instantaneous axis of rotation of the rolling cone.
Let γ denote the total angular velocity of the rotation of the cone B about the
instantaneous axis, β its angular velocity about the axis OB relatively to the plane AOB,
and α the angular velocity with which the plane AOB turns round the axis OA. It is
required to find the ratios of those angular velocities.
Solution.—In OT take any point E, from which draw EC parallel to OA, and ED parallel
to OB, so as to construct the parallelogram OCED. Then
OD : OC : OE :: α : β : γ.
Page 504
(8)
Or because of the proportionality of the sides of triangles to the sines of the opposite
angles,
sin TOB : sin TOA : sin AOB :: α : β : γ,
(8 a)
that is to say, the angular velocity about each axis is proportional to the sine of the angle
between the other two.
Demonstration.—From C draw CF perpendicular to OA, and CG perpendicular to OE
area EC
Then CF = 2 × ,
CE
area ECO
and CG = 2 × ;
OE
∴ CG : CF :: CE = OD : OE.
Let vc denote the linear velocity of the point C. Then
vc = α · CF = γ · CG
∴ γ : α :: CF : CG :: OE : OD,
which is one part of the solution above stated. From E draw EH perpendicular to OB, and
EK to OA. Then it can be shown as before that
EK : EH :: OC : OD.
Let vE be the linear velocity of the point E fixed in the plane of axes AOB. Then
vK = α · EK.
Now, as the line of contact OT is for the instant at rest on the rolling cone as well as on the
fixed cone, the linear velocity of the point E fixed to the plane AOB relatively to the rolling
cone is the same with its velocity relatively to the fixed cone. That is to say,
β · EH = vE = α · EK;
therefore
α : β :: EH : EK :: OD : OC,
which is the remainder of the solution.
Or because of the proportionality of the sides of triangles to the sines of the opposite
angles,
sin TOB : sin TOA : sin AOB :: α : β : γ,
(8 a)
that is to say, the angular velocity about each axis is proportional to the sine of the angle
between the other two.
Demonstration.—From C draw CF perpendicular to OA, and CG perpendicular to OE
area EC
Then CF = 2 × ,
CE
area ECO
and CG = 2 × ;
OE
∴ CG : CF :: CE = OD : OE.
Let vc denote the linear velocity of the point C. Then
vc = α · CF = γ · CG
∴ γ : α :: CF : CG :: OE : OD,
which is one part of the solution above stated. From E draw EH perpendicular to OB, and
EK to OA. Then it can be shown as before that
EK : EH :: OC : OD.
Let vE be the linear velocity of the point E fixed in the plane of axes AOB. Then
vK = α · EK.
Now, as the line of contact OT is for the instant at rest on the rolling cone as well as on the
fixed cone, the linear velocity of the point E fixed to the plane AOB relatively to the rolling
cone is the same with its velocity relatively to the fixed cone. That is to say,
β · EH = vE = α · EK;
therefore
α : β :: EH : EK :: OD : OC,
which is the remainder of the solution.
Page 505
The path of a point P in or attached to the rolling cone is a spherical epitrochoid traced
on the surface of a sphere of the radius OP. From P draw PQ perpendicular to the
instantaneous axis. Then the motion of P is perpendicular to the plane OPQ, and its velocity
is
vP = γ · PQ.
(9)
The whole of the foregoing reasonings are applicable, not merely when A and B are
actual regular cones, but also when they are the osculating regular cones of a pair of
irregular conical surfaces, having a common apex at O.
§ 32. Screw-like or Helical Motion.—Since any displacement in a plane can be
represented in general by a rotation, it follows that the only combination of translation and
rotation, in which a complex movement which is not a mere rotation is produced, occurs
when there is a translation perpendicular to the plane and parallel to the axis of rotation.
Such a complex motion is called screw-like or helical motion; for
each point in the body describes a helix or screw round the axis of
rotation, fixed or instantaneous as the case may be. To cause a body to
move in this manner it is usually made of a helical or screw-like
figure, and moves in a guide of a corresponding figure. Helical motion
and screws adapted to it are said to be right- or left-handed according
to the appearance presented by the rotation to an observer looking Fig. 94.
towards the direction of the translation. Thus the screw G in fig. 94 is
right-handed.
The translation of a body in helical motion is called its advance. Let vx denote the
velocity of advance at a given instant, which of course is common to all the particles of the
body; α the angular velocity of the rotation at the same instant; 2π = 6.2832 nearly, the
circumference of a circle of the radius unity. Then
T = 2π/α
(10)
is the time of one turn at the rate α; and
p = vxT = 2πvx/α
(11)
is the pitch or advance per turn—a length which expresses the comparative motion of the
translation and the rotation.
on the surface of a sphere of the radius OP. From P draw PQ perpendicular to the
instantaneous axis. Then the motion of P is perpendicular to the plane OPQ, and its velocity
is
vP = γ · PQ.
(9)
The whole of the foregoing reasonings are applicable, not merely when A and B are
actual regular cones, but also when they are the osculating regular cones of a pair of
irregular conical surfaces, having a common apex at O.
§ 32. Screw-like or Helical Motion.—Since any displacement in a plane can be
represented in general by a rotation, it follows that the only combination of translation and
rotation, in which a complex movement which is not a mere rotation is produced, occurs
when there is a translation perpendicular to the plane and parallel to the axis of rotation.
Such a complex motion is called screw-like or helical motion; for
each point in the body describes a helix or screw round the axis of
rotation, fixed or instantaneous as the case may be. To cause a body to
move in this manner it is usually made of a helical or screw-like
figure, and moves in a guide of a corresponding figure. Helical motion
and screws adapted to it are said to be right- or left-handed according
to the appearance presented by the rotation to an observer looking Fig. 94.
towards the direction of the translation. Thus the screw G in fig. 94 is
right-handed.
The translation of a body in helical motion is called its advance. Let vx denote the
velocity of advance at a given instant, which of course is common to all the particles of the
body; α the angular velocity of the rotation at the same instant; 2π = 6.2832 nearly, the
circumference of a circle of the radius unity. Then
T = 2π/α
(10)
is the time of one turn at the rate α; and
p = vxT = 2πvx/α
(11)
is the pitch or advance per turn—a length which expresses the comparative motion of the
translation and the rotation.
Page 506
The pitch of a screw is the distance, measured parallel to its axis, between two
successive turns of the same thread or helical projection.
Let r denote the perpendicular distance of a point in a body moving helically from the
axis. Then
vr = αr
(12)
is the component of the velocity of that point in a plane perpendicular to the axis, and its
total velocity is
v = √ {vx2 + vr2}.
(13)
The ratio of the two components of that velocity is
vx/vr = p/2πr = tan θ.
(14)
where θ denotes the angle made by the helical path of the point with a plane perpendicular
to the axis.
Division 4. Elementary Combinations in Mechanism
§ 33. Definitions.—An elementary combination in mechanism consists of two pieces
whose kinds of motion are determined by their connexion with the frame, and their
comparative motion by their connexion with each other—that connexion being effected
either by direct contact of the pieces, or by a connecting piece, which is not connected with
the frame, and whose motion depends entirely on the motions of the pieces which it
connects.
The piece whose motion is the cause is called the driver; the piece whose motion is the
effect, the follower.
The connexion of each of those two pieces with the frame is in general such as to
determine the path of every point in it. In the investigation, therefore, of the comparative
motion of the driver and follower, in an elementary combination, it is unnecessary to
consider relations of angular direction, which are already fixed by the connexion of each
piece with the frame; so that the inquiry is confined to the determination of the velocity
ratio, and of the directional relation, so far only as it expresses the connexion between
forward and backward movements of the driver and follower. When a continuous motion
of the driver produces a continuous motion of the follower, forward or backward, and a
reciprocating motion a motion reciprocating at the same instant, the directional relation is
successive turns of the same thread or helical projection.
Let r denote the perpendicular distance of a point in a body moving helically from the
axis. Then
vr = αr
(12)
is the component of the velocity of that point in a plane perpendicular to the axis, and its
total velocity is
v = √ {vx2 + vr2}.
(13)
The ratio of the two components of that velocity is
vx/vr = p/2πr = tan θ.
(14)
where θ denotes the angle made by the helical path of the point with a plane perpendicular
to the axis.
Division 4. Elementary Combinations in Mechanism
§ 33. Definitions.—An elementary combination in mechanism consists of two pieces
whose kinds of motion are determined by their connexion with the frame, and their
comparative motion by their connexion with each other—that connexion being effected
either by direct contact of the pieces, or by a connecting piece, which is not connected with
the frame, and whose motion depends entirely on the motions of the pieces which it
connects.
The piece whose motion is the cause is called the driver; the piece whose motion is the
effect, the follower.
The connexion of each of those two pieces with the frame is in general such as to
determine the path of every point in it. In the investigation, therefore, of the comparative
motion of the driver and follower, in an elementary combination, it is unnecessary to
consider relations of angular direction, which are already fixed by the connexion of each
piece with the frame; so that the inquiry is confined to the determination of the velocity
ratio, and of the directional relation, so far only as it expresses the connexion between
forward and backward movements of the driver and follower. When a continuous motion
of the driver produces a continuous motion of the follower, forward or backward, and a
reciprocating motion a motion reciprocating at the same instant, the directional relation is
Page 507
said to be constant. When a continuous motion produces a reciprocating motion, or vice
versa, or when a reciprocating motion produces a motion not reciprocating at the same
instant, the directional relation is said to be variable.
The line of action or of connexion of the driver and follower is a line traversing a pair of
points in the driver and follower respectively, which are so connected that the component
of their velocity relatively to each other, resolved along the line of connexion, is null. There
may be several or an indefinite number of lines of connexion, or there may be but one; and
a line of connexion may connect either the same pair of points or a succession of different
pairs.
§ 34. General Principle.—From the definition of a line of connexion it follows that the
components of the velocities of a pair of connected points along their line of connexion are
equal. And from this, and from the property of a rigid body, already stated in § 29, it
follows, that the components along a line of connexion of all the points traversed by that
line, whether in the driver or in the follower, are equal; and consequently, that the
velocities of any pair of points traversed by a line of connexion are to each other inversely
as the cosines, or directly as the secants, of the angles made by the paths of those points
with the line of connexion.
The general principle stated above in different forms serves to solve every problem in
which—the mode of connexion of a pair of pieces being given—it is required to find their
comparative motion at a given instant, or vice versa.
§ 35. Application to a Pair of Shifting Pieces.—In fig.
95, let P1P2 be the line of connexion of a pair of pieces,
each of which has a motion of translation or shifting.
Through any point T in that line draw TV1, TV2,
respectively parallel to the simultaneous direction of
motion of the pieces; through any other point A in the
line of connexion draw a plane perpendicular to that line,
Fig. 95.
cutting TV1, TV2 in V1, V2; then, velocity of piece 1 :
velocity of piece 2 :: TV1 : TV2. Also TA represents the
equal components of the velocities of the pieces parallel to their line of connexion, and the
line V1V2 represents their velocity relatively to each other.
§ 36. Application to a Pair of Turning Pieces.—Let α1, α2 be the angular velocities of a
pair of turning pieces; θ1, θ2 the angles which their line of connexion makes with their
respective planes of rotation; r1, r2 the common perpendiculars let fall from the line of
connexion upon the respective axes of rotation of the pieces. Then the equal components,
versa, or when a reciprocating motion produces a motion not reciprocating at the same
instant, the directional relation is said to be variable.
The line of action or of connexion of the driver and follower is a line traversing a pair of
points in the driver and follower respectively, which are so connected that the component
of their velocity relatively to each other, resolved along the line of connexion, is null. There
may be several or an indefinite number of lines of connexion, or there may be but one; and
a line of connexion may connect either the same pair of points or a succession of different
pairs.
§ 34. General Principle.—From the definition of a line of connexion it follows that the
components of the velocities of a pair of connected points along their line of connexion are
equal. And from this, and from the property of a rigid body, already stated in § 29, it
follows, that the components along a line of connexion of all the points traversed by that
line, whether in the driver or in the follower, are equal; and consequently, that the
velocities of any pair of points traversed by a line of connexion are to each other inversely
as the cosines, or directly as the secants, of the angles made by the paths of those points
with the line of connexion.
The general principle stated above in different forms serves to solve every problem in
which—the mode of connexion of a pair of pieces being given—it is required to find their
comparative motion at a given instant, or vice versa.
§ 35. Application to a Pair of Shifting Pieces.—In fig.
95, let P1P2 be the line of connexion of a pair of pieces,
each of which has a motion of translation or shifting.
Through any point T in that line draw TV1, TV2,
respectively parallel to the simultaneous direction of
motion of the pieces; through any other point A in the
line of connexion draw a plane perpendicular to that line,
Fig. 95.
cutting TV1, TV2 in V1, V2; then, velocity of piece 1 :
velocity of piece 2 :: TV1 : TV2. Also TA represents the
equal components of the velocities of the pieces parallel to their line of connexion, and the
line V1V2 represents their velocity relatively to each other.
§ 36. Application to a Pair of Turning Pieces.—Let α1, α2 be the angular velocities of a
pair of turning pieces; θ1, θ2 the angles which their line of connexion makes with their
respective planes of rotation; r1, r2 the common perpendiculars let fall from the line of
connexion upon the respective axes of rotation of the pieces. Then the equal components,
Page 508
along the line of connexion, of the velocities of the points where those perpendiculars meet
that line are—
α1r1 cos θ1 = α2r2 cos θ2;
consequently, the comparative motion of the pieces is given by the equation
α2 r1 cos θ1
= .
α1 r2 cos θ2 (15)
§ 37. Application to a Shifting Piece and a Turning Piece.—Let a shifting piece be
connected with a turning piece, and at a given instant let α1 be the angular velocity of the
turning piece, r1 the common perpendicular of its axis of rotation and the line of connexion,
θ1 the angle made by the line of connexion with the plane of rotation, θ2 the angle made by
the line of connexion with the direction of motion of the shifting piece, v2 the linear
velocity of that piece. Then
α1r1 cos θ1 = v2 cos θ2;
(16)
which equation expresses the comparative motion of the two pieces.
§ 38. Classification of Elementary Combinations in Mechanism.—The first systematic
classification of elementary combinations in mechanism was that founded by Monge, and
fully developed by Lanz and Bétancourt, which has been generally received, and has been
adopted in most treatises on applied mechanics. But that classification is founded on the
absolute instead of the comparative motions of the pieces, and is, for that reason, defective,
as Willis pointed out in his admirable treatise On the Principles of Mechanism.
Willis’s classification is founded, in the first place, on comparative motion, as expressed
by velocity ratio and directional relation, and in the second place, on the mode of
connexion of the driver and follower. He divides the elementary combinations in
mechanism into three classes, of which the characters are as follows:—
Class A: Directional relation constant; velocity ratio constant.
Class B: Directional relation constant; velocity ratio varying.
Class C: Directional relation changing periodically; velocity ratio constant or varying.
Each of those classes is subdivided by Willis into five divisions, of which the characters
are as follows:—
that line are—
α1r1 cos θ1 = α2r2 cos θ2;
consequently, the comparative motion of the pieces is given by the equation
α2 r1 cos θ1
= .
α1 r2 cos θ2 (15)
§ 37. Application to a Shifting Piece and a Turning Piece.—Let a shifting piece be
connected with a turning piece, and at a given instant let α1 be the angular velocity of the
turning piece, r1 the common perpendicular of its axis of rotation and the line of connexion,
θ1 the angle made by the line of connexion with the plane of rotation, θ2 the angle made by
the line of connexion with the direction of motion of the shifting piece, v2 the linear
velocity of that piece. Then
α1r1 cos θ1 = v2 cos θ2;
(16)
which equation expresses the comparative motion of the two pieces.
§ 38. Classification of Elementary Combinations in Mechanism.—The first systematic
classification of elementary combinations in mechanism was that founded by Monge, and
fully developed by Lanz and Bétancourt, which has been generally received, and has been
adopted in most treatises on applied mechanics. But that classification is founded on the
absolute instead of the comparative motions of the pieces, and is, for that reason, defective,
as Willis pointed out in his admirable treatise On the Principles of Mechanism.
Willis’s classification is founded, in the first place, on comparative motion, as expressed
by velocity ratio and directional relation, and in the second place, on the mode of
connexion of the driver and follower. He divides the elementary combinations in
mechanism into three classes, of which the characters are as follows:—
Class A: Directional relation constant; velocity ratio constant.
Class B: Directional relation constant; velocity ratio varying.
Class C: Directional relation changing periodically; velocity ratio constant or varying.
Each of those classes is subdivided by Willis into five divisions, of which the characters
are as follows:—
Page 509
Division A: Connexion by rolling contact.
” B: ” ” sliding contact.
” C: ” ” wrapping connectors.
” D: ” ” link-work.
” E: ” ” reduplication.
In the Reuleaux system of analysis of mechanisms the principle of comparative motion
is generalized, and mechanisms apparently very diverse in character are shown to be
founded on the same sequence of elementary combinations forming a kinematic chain. A
short description of this system is given in § 80, but in the present article the principle of
Willis’s classification is followed mainly. The arrangement is, however, modified by taking
the mode of connexion as the basis of the primary classification, and by removing the
subject of connexion by reduplication to the section of aggregate combinations. This
modified arrangement is adopted as being better suited than the original arrangement to the
limits of an article in an encyclopaedia; but it is not disputed that the original arrangement
may be the best for a separate treatise.
§ 39. Rolling Contact: Smooth Wheels and Racks.—In order that two pieces may move
in rolling contact, it is necessary that each pair of points in the two pieces which touch each
other should at the instant of contact be moving in the same direction with the same
velocity. In the case of two shifting pieces this would involve equal and parallel velocities
for all the points of each piece, so that there could be no rolling, and, in fact, the two pieces
would move like one; hence, in the case of rolling contact, either one or both of the pieces
must rotate.
The direction of motion of a point in a turning piece being perpendicular to a plane
passing through its axis, the condition that each pair of points in contact with each other
must move in the same direction leads to the following consequences:—
I. That, when both pieces rotate, their axes, and all their points of contact, lie in the same
plane.
II. That, when one piece rotates, and the other shifts, the axis of the rotating piece, and
all the points of contact, lie in a plane perpendicular to the direction of motion of the
shifting piece.
The condition that the velocity of each pair of points of contact must be equal leads to
the following consequences:—
III. That the angular velocities of a pair of turning pieces in rolling contact must be
inversely as the perpendicular distances of any pair of points of contact from the respective
” B: ” ” sliding contact.
” C: ” ” wrapping connectors.
” D: ” ” link-work.
” E: ” ” reduplication.
In the Reuleaux system of analysis of mechanisms the principle of comparative motion
is generalized, and mechanisms apparently very diverse in character are shown to be
founded on the same sequence of elementary combinations forming a kinematic chain. A
short description of this system is given in § 80, but in the present article the principle of
Willis’s classification is followed mainly. The arrangement is, however, modified by taking
the mode of connexion as the basis of the primary classification, and by removing the
subject of connexion by reduplication to the section of aggregate combinations. This
modified arrangement is adopted as being better suited than the original arrangement to the
limits of an article in an encyclopaedia; but it is not disputed that the original arrangement
may be the best for a separate treatise.
§ 39. Rolling Contact: Smooth Wheels and Racks.—In order that two pieces may move
in rolling contact, it is necessary that each pair of points in the two pieces which touch each
other should at the instant of contact be moving in the same direction with the same
velocity. In the case of two shifting pieces this would involve equal and parallel velocities
for all the points of each piece, so that there could be no rolling, and, in fact, the two pieces
would move like one; hence, in the case of rolling contact, either one or both of the pieces
must rotate.
The direction of motion of a point in a turning piece being perpendicular to a plane
passing through its axis, the condition that each pair of points in contact with each other
must move in the same direction leads to the following consequences:—
I. That, when both pieces rotate, their axes, and all their points of contact, lie in the same
plane.
II. That, when one piece rotates, and the other shifts, the axis of the rotating piece, and
all the points of contact, lie in a plane perpendicular to the direction of motion of the
shifting piece.
The condition that the velocity of each pair of points of contact must be equal leads to
the following consequences:—
III. That the angular velocities of a pair of turning pieces in rolling contact must be
inversely as the perpendicular distances of any pair of points of contact from the respective
Page 510
axes.
IV. That the linear velocity of a shifting piece in rolling contact with a turning piece is
equal to the product of the angular velocity of the turning piece by the perpendicular
distance from its axis to a pair of points of contact.
The line of contact is that line in which the points of contact are all situated. Respecting
this line, the above Principles III. and IV. lead to the following conclusions:—
V. That for a pair of turning pieces with parallel axes, and for a turning piece and a
shifting piece, the line of contact is straight, and parallel to the axes or axis; and hence that
the rolling surfaces are either plane or cylindrical (the term “cylindrical” including all
surfaces generated by the motion of a straight line parallel to itself).
VI. That for a pair of turning pieces with intersecting axes the line of contact is also
straight, and traverses the point of intersection of the axes; and hence that the rolling
surfaces are conical, with a common apex (the term “conical” including all surfaces
generated by the motion of a straight line which traverses a fixed point).
Turning pieces in rolling contact are called smooth or toothless wheels. Shifting pieces in
rolling contact with turning pieces may be called smooth or toothless racks.
VII. In a pair of pieces in rolling contact every straight line traversing the line of contact
is a line of connexion.
§ 40. Cylindrical Wheels and Smooth Racks.—In designing cylindrical wheels and
smooth racks, and determining their comparative motion, it is sufficient to consider a
section of the pair of pieces made by a plane perpendicular to the axis or axes.
The points where axes intersect the plane of section are called centres; the point where
the line of contact intersects it, the point of contact, or pitch-point; and the wheels are
described as circular, elliptical, &c., according to the forms of their sections made by that
plane.
When the point of contact of two wheels lies between their centres, they are said to be in
outside gearing; when beyond their centres, in inside gearing, because the rolling surface
of the larger wheel must in this case be turned inward or towards its centre.
From Principle III. of § 39 it appears that the angular velocity-ratio of a pair of wheels is
the inverse ratio of the distances of the point of contact from the centres respectively.
For outside gearing that ratio is negative, because the wheels turn contrary ways; for
inside gearing it is positive, because they turn the same way.
IV. That the linear velocity of a shifting piece in rolling contact with a turning piece is
equal to the product of the angular velocity of the turning piece by the perpendicular
distance from its axis to a pair of points of contact.
The line of contact is that line in which the points of contact are all situated. Respecting
this line, the above Principles III. and IV. lead to the following conclusions:—
V. That for a pair of turning pieces with parallel axes, and for a turning piece and a
shifting piece, the line of contact is straight, and parallel to the axes or axis; and hence that
the rolling surfaces are either plane or cylindrical (the term “cylindrical” including all
surfaces generated by the motion of a straight line parallel to itself).
VI. That for a pair of turning pieces with intersecting axes the line of contact is also
straight, and traverses the point of intersection of the axes; and hence that the rolling
surfaces are conical, with a common apex (the term “conical” including all surfaces
generated by the motion of a straight line which traverses a fixed point).
Turning pieces in rolling contact are called smooth or toothless wheels. Shifting pieces in
rolling contact with turning pieces may be called smooth or toothless racks.
VII. In a pair of pieces in rolling contact every straight line traversing the line of contact
is a line of connexion.
§ 40. Cylindrical Wheels and Smooth Racks.—In designing cylindrical wheels and
smooth racks, and determining their comparative motion, it is sufficient to consider a
section of the pair of pieces made by a plane perpendicular to the axis or axes.
The points where axes intersect the plane of section are called centres; the point where
the line of contact intersects it, the point of contact, or pitch-point; and the wheels are
described as circular, elliptical, &c., according to the forms of their sections made by that
plane.
When the point of contact of two wheels lies between their centres, they are said to be in
outside gearing; when beyond their centres, in inside gearing, because the rolling surface
of the larger wheel must in this case be turned inward or towards its centre.
From Principle III. of § 39 it appears that the angular velocity-ratio of a pair of wheels is
the inverse ratio of the distances of the point of contact from the centres respectively.
For outside gearing that ratio is negative, because the wheels turn contrary ways; for
inside gearing it is positive, because they turn the same way.
Page 511
If the velocity ratio is to be constant, as in Willis’s Class A, the
wheels must be circular; and this is the most common form for
wheels.
If the velocity ratio is to be variable, as in Willis’s Class B, the
figures of the wheels are a pair of rolling curves, subject to the
condition that the distance between their poles (which are the centres
of rotation) shall be constant.
The following is the geometrical relation which must exist
between such a pair of curves:—
Let C1, C2 (fig. 96) be the poles of a pair of rolling curves; T1, T2 Fig. 96.
any pair of points of contact; U1, U2 any other pair of points of
contact. Then, for every possible pair of points of contact, the two following equations
must be simultaneously fulfilled:—
Sum of radii, C1U1 + C2U2 = C1T1 + C2T2 = constant;
arc, T2U2 = T1U1.
(17)
A condition equivalent to the above, and necessarily connected with it, is, that at each
pair of points of contact the inclinations of the curves to their radii-vectores shall be equal
and contrary; or, denoting by r1, r2 the radii-vectores at any given pair of points of contact,
and s the length of the equal arcs measured from a certain fixed pair of points of contact—
dr2/ds = −dr1/ds;
(18)
which is the differential equation of a pair of rolling curves whose poles are at a constant
distance apart.
For full details as to rolling curves, see Willis’s work, already mentioned, and Clerk
Maxwell’s paper on Rolling Curves, Trans. Roy. Soc. Edin., 1849.
A rack, to work with a circular wheel, must be straight. To work with a wheel of any
other figure, its section must be a rolling curve, subject to the condition that the
perpendicular distance from the pole or centre of the wheel to a straight line parallel to the
direction of the motion of the rack shall be constant. Let r1 be the radius-vector of a point
of contact on the wheel, x2 the ordinate from the straight line before mentioned to the
corresponding point of contact on the rack. Then
dx2/ds = −dr1/ds
(19)
wheels must be circular; and this is the most common form for
wheels.
If the velocity ratio is to be variable, as in Willis’s Class B, the
figures of the wheels are a pair of rolling curves, subject to the
condition that the distance between their poles (which are the centres
of rotation) shall be constant.
The following is the geometrical relation which must exist
between such a pair of curves:—
Let C1, C2 (fig. 96) be the poles of a pair of rolling curves; T1, T2 Fig. 96.
any pair of points of contact; U1, U2 any other pair of points of
contact. Then, for every possible pair of points of contact, the two following equations
must be simultaneously fulfilled:—
Sum of radii, C1U1 + C2U2 = C1T1 + C2T2 = constant;
arc, T2U2 = T1U1.
(17)
A condition equivalent to the above, and necessarily connected with it, is, that at each
pair of points of contact the inclinations of the curves to their radii-vectores shall be equal
and contrary; or, denoting by r1, r2 the radii-vectores at any given pair of points of contact,
and s the length of the equal arcs measured from a certain fixed pair of points of contact—
dr2/ds = −dr1/ds;
(18)
which is the differential equation of a pair of rolling curves whose poles are at a constant
distance apart.
For full details as to rolling curves, see Willis’s work, already mentioned, and Clerk
Maxwell’s paper on Rolling Curves, Trans. Roy. Soc. Edin., 1849.
A rack, to work with a circular wheel, must be straight. To work with a wheel of any
other figure, its section must be a rolling curve, subject to the condition that the
perpendicular distance from the pole or centre of the wheel to a straight line parallel to the
direction of the motion of the rack shall be constant. Let r1 be the radius-vector of a point
of contact on the wheel, x2 the ordinate from the straight line before mentioned to the
corresponding point of contact on the rack. Then
dx2/ds = −dr1/ds
(19)
Page 512
is the differential equation of the pair of rolling curves.
To illustrate this subject, it may be mentioned that an ellipse rotating about one focus
rolls completely round in outside gearing with an equal and similar ellipse also rotating
about one focus, the distance between the axes of rotation being equal to the major axis of
the ellipses, and the velocity ratio varying from (1 + eccentricity)/(1 − eccentricity) to (1 −
eccentricity)/(1 + eccentricity); an hyperbola rotating about its further focus rolls in inside
gearing, through a limited arc, with an equal and similar hyperbola rotating about its nearer
focus, the distance between the axes of rotation being equal to the axis of the hyperbolas,
and the velocity ratio varying between (eccentricity + 1)/(eccentricity − 1) and unity; and a
parabola rotating about its focus rolls with an equal and similar parabola, shifting parallel
to its directrix.
§ 41. Conical or Bevel and Disk Wheels.—From
Principles III. and VI. of § 39 it appears that the angular
velocities of a pair of wheels whose axes meet in a point
are to each other inversely as the sines of the angles which
the axes of the wheels make with the line of contact.
Hence we have the following construction (figs. 97 and
98).—Let O be the apex or point of intersection of the two
axes OC1, OC2. The angular velocity ratio being given, it
is required to find the line of contact. On OC1, OC2 take
Fig. 97.
lengths OA1, OA2, respectively proportional to the angular
velocities of the pieces on whose axes they are taken.
Complete the parallelogram OA1EA2; the diagonal OET will be the line of contact
required.
When the velocity ratio is variable, the line of contact will shift its position in the plane
C1OC2, and the wheels will be cones, with eccentric or irregular bases. In every case which
occurs in practice, however, the velocity ratio is constant; the line of contact is constant in
position, and the rolling surfaces of the wheels are regular circular cones (when they are
called bevel wheels); or one of a pair of wheels may have a flat disk for its rolling surface,
as W2 in fig. 98, in which case it is a disk wheel. The rolling surfaces of actual wheels
consist of frusta or zones of the complete cones or disks, as shown by W1, W2 in figs. 97
and 98.
§ 42. Sliding Contact (lateral): Skew-Bevel Wheels.—An hyperboloid of revolution is a
surface resembling a sheaf or a dice box, generated by the rotation of a straight line round
an axis from which it is at a constant distance, and to which it is inclined at a constant
angle. If two such hyperboloids E, F, equal or unequal, be placed in the closest possible
To illustrate this subject, it may be mentioned that an ellipse rotating about one focus
rolls completely round in outside gearing with an equal and similar ellipse also rotating
about one focus, the distance between the axes of rotation being equal to the major axis of
the ellipses, and the velocity ratio varying from (1 + eccentricity)/(1 − eccentricity) to (1 −
eccentricity)/(1 + eccentricity); an hyperbola rotating about its further focus rolls in inside
gearing, through a limited arc, with an equal and similar hyperbola rotating about its nearer
focus, the distance between the axes of rotation being equal to the axis of the hyperbolas,
and the velocity ratio varying between (eccentricity + 1)/(eccentricity − 1) and unity; and a
parabola rotating about its focus rolls with an equal and similar parabola, shifting parallel
to its directrix.
§ 41. Conical or Bevel and Disk Wheels.—From
Principles III. and VI. of § 39 it appears that the angular
velocities of a pair of wheels whose axes meet in a point
are to each other inversely as the sines of the angles which
the axes of the wheels make with the line of contact.
Hence we have the following construction (figs. 97 and
98).—Let O be the apex or point of intersection of the two
axes OC1, OC2. The angular velocity ratio being given, it
is required to find the line of contact. On OC1, OC2 take
Fig. 97.
lengths OA1, OA2, respectively proportional to the angular
velocities of the pieces on whose axes they are taken.
Complete the parallelogram OA1EA2; the diagonal OET will be the line of contact
required.
When the velocity ratio is variable, the line of contact will shift its position in the plane
C1OC2, and the wheels will be cones, with eccentric or irregular bases. In every case which
occurs in practice, however, the velocity ratio is constant; the line of contact is constant in
position, and the rolling surfaces of the wheels are regular circular cones (when they are
called bevel wheels); or one of a pair of wheels may have a flat disk for its rolling surface,
as W2 in fig. 98, in which case it is a disk wheel. The rolling surfaces of actual wheels
consist of frusta or zones of the complete cones or disks, as shown by W1, W2 in figs. 97
and 98.
§ 42. Sliding Contact (lateral): Skew-Bevel Wheels.—An hyperboloid of revolution is a
surface resembling a sheaf or a dice box, generated by the rotation of a straight line round
an axis from which it is at a constant distance, and to which it is inclined at a constant
angle. If two such hyperboloids E, F, equal or unequal, be placed in the closest possible
Page 513
contact, as in fig. 99, they will touch each other
along one of the generating straight lines of each,
which will form their line of contact, and will be
inclined to the axes AG, BH in opposite
directions. The axes will not be parallel, nor will
they intersect each other.
Fig. 98.
The motion of two such hyperboloids, turning
in contact with each other, has hitherto been
classed amongst cases of rolling contact; but that
classification is not strictly correct, for, although
the component velocities of a pair of points of
contact in a direction at right angles to the line of
contact are equal, still, as the axes are parallel
neither to each other nor to the line of contact, Fig. 99.
the velocities of a pair of points of contact have
components along the line of contact which are
unequal, and their difference constitutes a lateral
sliding.
The directions and positions of the axes being
given, and the required angular velocity ratio, the
following construction serves to determine the
line of contact, by whose rotation round the two
axes respectively the hyperboloids are generated:
—
In fig. 100, let B1C1, B2C2 be the two axes; Fig. 100.
B1B2 their common perpendicular. Through any
point O in this common perpendicular draw OA1 parallel to B1C1 and OA2 parallel to
B2C2; make those lines proportional to the angular velocities about the axes to which they
are respectively parallel; complete the parallelogram OA1EA2, and draw the diagonal OE;
divide B1B2 in D into two parts, inversely proportional to the angular velocities about the
axes which they respectively adjoin; through D parallel to OE draw DT. This will be the
line of contact.
A pair of thin frusta of a pair of hyperboloids are used in practice to communicate
motion between a pair of axes neither parallel nor intersecting, and are called skew-bevel
wheels.
along one of the generating straight lines of each,
which will form their line of contact, and will be
inclined to the axes AG, BH in opposite
directions. The axes will not be parallel, nor will
they intersect each other.
Fig. 98.
The motion of two such hyperboloids, turning
in contact with each other, has hitherto been
classed amongst cases of rolling contact; but that
classification is not strictly correct, for, although
the component velocities of a pair of points of
contact in a direction at right angles to the line of
contact are equal, still, as the axes are parallel
neither to each other nor to the line of contact, Fig. 99.
the velocities of a pair of points of contact have
components along the line of contact which are
unequal, and their difference constitutes a lateral
sliding.
The directions and positions of the axes being
given, and the required angular velocity ratio, the
following construction serves to determine the
line of contact, by whose rotation round the two
axes respectively the hyperboloids are generated:
—
In fig. 100, let B1C1, B2C2 be the two axes; Fig. 100.
B1B2 their common perpendicular. Through any
point O in this common perpendicular draw OA1 parallel to B1C1 and OA2 parallel to
B2C2; make those lines proportional to the angular velocities about the axes to which they
are respectively parallel; complete the parallelogram OA1EA2, and draw the diagonal OE;
divide B1B2 in D into two parts, inversely proportional to the angular velocities about the
axes which they respectively adjoin; through D parallel to OE draw DT. This will be the
line of contact.
A pair of thin frusta of a pair of hyperboloids are used in practice to communicate
motion between a pair of axes neither parallel nor intersecting, and are called skew-bevel
wheels.
Page 514
In skew-bevel wheels the properties of a line of connexion are not possessed by every
line traversing the line of contact, but only by every line traversing the line of contact at
right angles.
If the velocity ratio to be communicated were variable, the point D would alter its
position, and the line DT its direction, at different periods of the motion, and the wheels
would be hyperboloids of an eccentric or irregular cross-section; but forms of this kind are
not used in practice.
§ 43. Sliding Contact (circular): Grooved Wheels.—As the adhesion or friction between
a pair of smooth wheels is seldom sufficient to prevent their slipping on each other,
contrivances are used to increase their mutual hold. One of those consists in forming the
rim of each wheel into a series of alternate ridges and grooves parallel to the plane of
rotation; it is applicable to cylindrical and bevel wheels, but not to skew-bevel wheels. The
comparative motion of a pair of wheels so ridged and grooved is the same as that of a pair
of smooth wheels in rolling contact, whose cylindrical or conical surfaces lie midway
between the tops of the ridges and bottoms of the grooves, and those ideal smooth surfaces
are called the pitch surfaces of the wheels.
The relative motion of the faces of contact of the ridges and grooves is a rotatory sliding
or grinding motion, about the line of contact of the pitch-surfaces as an instantaneous axis.
Grooved wheels have hitherto been but little used.
§ 44. Sliding Contact (direct): Teeth of Wheels, their Number and Pitch.—The ordinary
method of connecting a pair of wheels, or a wheel and a rack, and the only method which
ensures the exact maintenance of a given numerical velocity ratio, is by means of a series
of alternate ridges and hollows parallel or nearly parallel to the successive lines of contact
of the ideal smooth wheels whose velocity ratio would be the same with that of the toothed
wheels. The ridges are called teeth; the hollows, spaces. The teeth of the driver push those
of the follower before them, and in so doing sliding takes place between them in a direction
across their lines of contact.
The pitch-surfaces of a pair of toothed wheels are the ideal smooth surfaces which
would have the same comparative motion by rolling contact that the actual wheels have by
the sliding contact of their teeth. The pitch-circles of a pair of circular toothed wheels are
sections of their pitch-surfaces, made for spur-wheels (that is, for wheels whose axes are
parallel) by a plane at right angles to the axes, and for bevel wheels by a sphere described
about the common apex. For a pair of skew-bevel wheels the pitch-circles are a pair of
contiguous rectangular sections of the pitch-surfaces. The pitch-point is the point of contact
of the pitch-circles.
line traversing the line of contact, but only by every line traversing the line of contact at
right angles.
If the velocity ratio to be communicated were variable, the point D would alter its
position, and the line DT its direction, at different periods of the motion, and the wheels
would be hyperboloids of an eccentric or irregular cross-section; but forms of this kind are
not used in practice.
§ 43. Sliding Contact (circular): Grooved Wheels.—As the adhesion or friction between
a pair of smooth wheels is seldom sufficient to prevent their slipping on each other,
contrivances are used to increase their mutual hold. One of those consists in forming the
rim of each wheel into a series of alternate ridges and grooves parallel to the plane of
rotation; it is applicable to cylindrical and bevel wheels, but not to skew-bevel wheels. The
comparative motion of a pair of wheels so ridged and grooved is the same as that of a pair
of smooth wheels in rolling contact, whose cylindrical or conical surfaces lie midway
between the tops of the ridges and bottoms of the grooves, and those ideal smooth surfaces
are called the pitch surfaces of the wheels.
The relative motion of the faces of contact of the ridges and grooves is a rotatory sliding
or grinding motion, about the line of contact of the pitch-surfaces as an instantaneous axis.
Grooved wheels have hitherto been but little used.
§ 44. Sliding Contact (direct): Teeth of Wheels, their Number and Pitch.—The ordinary
method of connecting a pair of wheels, or a wheel and a rack, and the only method which
ensures the exact maintenance of a given numerical velocity ratio, is by means of a series
of alternate ridges and hollows parallel or nearly parallel to the successive lines of contact
of the ideal smooth wheels whose velocity ratio would be the same with that of the toothed
wheels. The ridges are called teeth; the hollows, spaces. The teeth of the driver push those
of the follower before them, and in so doing sliding takes place between them in a direction
across their lines of contact.
The pitch-surfaces of a pair of toothed wheels are the ideal smooth surfaces which
would have the same comparative motion by rolling contact that the actual wheels have by
the sliding contact of their teeth. The pitch-circles of a pair of circular toothed wheels are
sections of their pitch-surfaces, made for spur-wheels (that is, for wheels whose axes are
parallel) by a plane at right angles to the axes, and for bevel wheels by a sphere described
about the common apex. For a pair of skew-bevel wheels the pitch-circles are a pair of
contiguous rectangular sections of the pitch-surfaces. The pitch-point is the point of contact
of the pitch-circles.
Page 515
The pitch-surface of a wheel lies intermediate between the points of the teeth and the
bottoms of the hollows between them. That part of the acting surface of a tooth which
projects beyond the pitch-surface is called the face; that part which lies within the pitch-
surface, the flank.
Teeth, when not otherwise specified, are understood to be made in one piece with the
wheel, the material being generally cast-iron, brass or bronze. Separate teeth, fixed into
mortises in the rim of the wheel, are called cogs. A pinion is a small toothed wheel; a
trundle is a pinion with cylindrical staves for teeth.
The radius of the pitch-circle of a wheel is called the geometrical radius; a circle
touching the ends of the teeth is called the addendum circle, and its radius the real radius;
the difference between these radii, being the projection of the teeth beyond the pitch-
surface, is called the addendum.
The distance, measured along the pitch-circle, from the face of one tooth to the face of
the next, is called the pitch. The pitch and the number of teeth in wheels are regulated by
the following principles:—
I. In wheels which rotate continuously for one revolution or more, it is obviously
necessary that the pitch should be an aliquot part of the circumference.
In wheels which reciprocate without performing a complete revolution this condition is
not necessary. Such wheels are called sectors.
II. In order that a pair of wheels, or a wheel and a rack, may work correctly together, it is
in all cases essential that the pitch should be the same in each.
III. Hence, in any pair of circular wheels which work together, the numbers of teeth in a
complete circumference are directly as the radii and inversely as the angular velocities.
IV. Hence also, in any pair of circular wheels which rotate continuously for one
revolution or more, the ratio of the numbers of teeth and its reciprocal the angular velocity
ratio must be expressible in whole numbers.
From this principle arise problems of a kind which will be referred to in treating of
Trains of Mechanism.
V. Let n, N be the respective numbers of teeth in a pair of wheels, N being the greater.
Let t, T be a pair of teeth in the smaller and larger wheel respectively, which at a particular
instant work together. It is required to find, first, how many pairs of teeth must pass the line
of contact of the pitch-surfaces before t and T work together again (let this number be
called a); and, secondly, with how many different teeth of the larger wheel the tooth t will
bottoms of the hollows between them. That part of the acting surface of a tooth which
projects beyond the pitch-surface is called the face; that part which lies within the pitch-
surface, the flank.
Teeth, when not otherwise specified, are understood to be made in one piece with the
wheel, the material being generally cast-iron, brass or bronze. Separate teeth, fixed into
mortises in the rim of the wheel, are called cogs. A pinion is a small toothed wheel; a
trundle is a pinion with cylindrical staves for teeth.
The radius of the pitch-circle of a wheel is called the geometrical radius; a circle
touching the ends of the teeth is called the addendum circle, and its radius the real radius;
the difference between these radii, being the projection of the teeth beyond the pitch-
surface, is called the addendum.
The distance, measured along the pitch-circle, from the face of one tooth to the face of
the next, is called the pitch. The pitch and the number of teeth in wheels are regulated by
the following principles:—
I. In wheels which rotate continuously for one revolution or more, it is obviously
necessary that the pitch should be an aliquot part of the circumference.
In wheels which reciprocate without performing a complete revolution this condition is
not necessary. Such wheels are called sectors.
II. In order that a pair of wheels, or a wheel and a rack, may work correctly together, it is
in all cases essential that the pitch should be the same in each.
III. Hence, in any pair of circular wheels which work together, the numbers of teeth in a
complete circumference are directly as the radii and inversely as the angular velocities.
IV. Hence also, in any pair of circular wheels which rotate continuously for one
revolution or more, the ratio of the numbers of teeth and its reciprocal the angular velocity
ratio must be expressible in whole numbers.
From this principle arise problems of a kind which will be referred to in treating of
Trains of Mechanism.
V. Let n, N be the respective numbers of teeth in a pair of wheels, N being the greater.
Let t, T be a pair of teeth in the smaller and larger wheel respectively, which at a particular
instant work together. It is required to find, first, how many pairs of teeth must pass the line
of contact of the pitch-surfaces before t and T work together again (let this number be
called a); and, secondly, with how many different teeth of the larger wheel the tooth t will
Page 516
work at different times (let this number be called b); thirdly, with how many different teeth
of the smaller wheel the tooth T will work at different times (let this be called c).
Case 1. If n is a divisor of N,
a = N; b = N/n; c = 1.
(20)
Case 2. If the greatest common divisor of N and n be d, a number less than n, so that n =
md, N = Md; then
a = mN = Mn = Mmd; b = M; c = m.
(21)
Case 3. If N and n be prime to each other,
a = nN; b = N; c = n.
(22)
It is considered desirable by millwrights, with a view to the preservation of the
uniformity of shape of the teeth of a pair of wheels, that each given tooth in one wheel
should work with as many different teeth in the other wheel as possible. They therefore
study that the numbers of teeth in each pair of wheels which work together shall either be
prime to each other, or shall have their greatest common divisor as small as is consistent
with a velocity ratio suited for the purposes of the machine.
§ 45. Sliding Contact: Forms of the Teeth of Spur-wheels and Racks.—A line of
connexion of two pieces in sliding contact is a line perpendicular to their surfaces at a point
where they touch. Bearing this in mind, the principle of the comparative motion of a pair of
teeth belonging to a pair of spur-wheels, or to a spur-wheel and a rack, is found by
applying the principles stated generally in §§ 36 and 37 to the case of parallel axes for a
pair of spur-wheels, and to the case of an axis perpendicular to the direction of shifting for
a wheel and a rack.
In fig. 101, let C1, C2 be the centres of a pair of spur-wheels; B1IB1′, B2IB2′ portions of
their pitch-circles, touching at I, the pitch-point. Let the wheel 1 be the driver, and the
wheel 2 the follower.
Let D1TB1A1, D2TB2A2 be the positions, at a given instant, of the acting surfaces of a
pair of teeth in the driver and follower respectively, touching each other at T; the line of
connexion of those teeth is P1P2, perpendicular to their surfaces at T. Let C1P1, C2P2 be
perpendiculars let fall from the centres of the wheels on the line of contact. Then, by § 36,
the angular velocity-ratio is
of the smaller wheel the tooth T will work at different times (let this be called c).
Case 1. If n is a divisor of N,
a = N; b = N/n; c = 1.
(20)
Case 2. If the greatest common divisor of N and n be d, a number less than n, so that n =
md, N = Md; then
a = mN = Mn = Mmd; b = M; c = m.
(21)
Case 3. If N and n be prime to each other,
a = nN; b = N; c = n.
(22)
It is considered desirable by millwrights, with a view to the preservation of the
uniformity of shape of the teeth of a pair of wheels, that each given tooth in one wheel
should work with as many different teeth in the other wheel as possible. They therefore
study that the numbers of teeth in each pair of wheels which work together shall either be
prime to each other, or shall have their greatest common divisor as small as is consistent
with a velocity ratio suited for the purposes of the machine.
§ 45. Sliding Contact: Forms of the Teeth of Spur-wheels and Racks.—A line of
connexion of two pieces in sliding contact is a line perpendicular to their surfaces at a point
where they touch. Bearing this in mind, the principle of the comparative motion of a pair of
teeth belonging to a pair of spur-wheels, or to a spur-wheel and a rack, is found by
applying the principles stated generally in §§ 36 and 37 to the case of parallel axes for a
pair of spur-wheels, and to the case of an axis perpendicular to the direction of shifting for
a wheel and a rack.
In fig. 101, let C1, C2 be the centres of a pair of spur-wheels; B1IB1′, B2IB2′ portions of
their pitch-circles, touching at I, the pitch-point. Let the wheel 1 be the driver, and the
wheel 2 the follower.
Let D1TB1A1, D2TB2A2 be the positions, at a given instant, of the acting surfaces of a
pair of teeth in the driver and follower respectively, touching each other at T; the line of
connexion of those teeth is P1P2, perpendicular to their surfaces at T. Let C1P1, C2P2 be
perpendiculars let fall from the centres of the wheels on the line of contact. Then, by § 36,
the angular velocity-ratio is
Page 517
α2/α1 = C1P1/C2P2.
(23)
The following principles regulate the forms of the
teeth and their relative motions:—
I. The angular velocity ratio due to the sliding
contact of the teeth will be the same with that due to
the rolling contact of the pitch-circles, if the line of
connexion of the teeth cuts the line of centres at the
pitch-point.
For, let P1P2 cut the line of centres at I; then, by
similar triangles,
α1 : α2 :: C2P2 : C1P1 :: IC2 :: IC1; Fig. 101.
(24)
which is also the angular velocity ratio due to the rolling contact of the circles B1IB1′,
B2IB2′.
This principle determines the forms of all teeth of spur-wheels. It also determines the
forms of the teeth of straight racks, if one of the centres be removed, and a straight line
EIE′, parallel to the direction of motion of the rack, and perpendicular to C1IC2, be
substituted for a pitch-circle.
II. The component of the velocity of the point of contact of the teeth T along the line of
connexion is
α1 · C1P1 = α2 · C2P2.
(25)
III. The relative velocity perpendicular to P1P2 of the teeth at their point of contact—that
is, their velocity of sliding on each other—is found by supposing one of the wheels, such as
1, to be fixed, the line of centres C1C2 to rotate backwards round C1 with the angular
velocity α1, and the wheel 2 to rotate round C2 as before, with the angular velocity α2
relatively to the line of centres C1C2, so as to have the same motion as if its pitch-circle
rolled on the pitch-circle of the first wheel. Thus the relative motion of the wheels is
unchanged; but 1 is considered as fixed, and 2 has the total motion, that is, a rotation about
the instantaneous axis I, with the angular velocity α1 + α2. Hence the velocity of sliding is
that due to this rotation about I, with the radius IT; that is to say, its value is
(α1 + α2) · IT;
(23)
The following principles regulate the forms of the
teeth and their relative motions:—
I. The angular velocity ratio due to the sliding
contact of the teeth will be the same with that due to
the rolling contact of the pitch-circles, if the line of
connexion of the teeth cuts the line of centres at the
pitch-point.
For, let P1P2 cut the line of centres at I; then, by
similar triangles,
α1 : α2 :: C2P2 : C1P1 :: IC2 :: IC1; Fig. 101.
(24)
which is also the angular velocity ratio due to the rolling contact of the circles B1IB1′,
B2IB2′.
This principle determines the forms of all teeth of spur-wheels. It also determines the
forms of the teeth of straight racks, if one of the centres be removed, and a straight line
EIE′, parallel to the direction of motion of the rack, and perpendicular to C1IC2, be
substituted for a pitch-circle.
II. The component of the velocity of the point of contact of the teeth T along the line of
connexion is
α1 · C1P1 = α2 · C2P2.
(25)
III. The relative velocity perpendicular to P1P2 of the teeth at their point of contact—that
is, their velocity of sliding on each other—is found by supposing one of the wheels, such as
1, to be fixed, the line of centres C1C2 to rotate backwards round C1 with the angular
velocity α1, and the wheel 2 to rotate round C2 as before, with the angular velocity α2
relatively to the line of centres C1C2, so as to have the same motion as if its pitch-circle
rolled on the pitch-circle of the first wheel. Thus the relative motion of the wheels is
unchanged; but 1 is considered as fixed, and 2 has the total motion, that is, a rotation about
the instantaneous axis I, with the angular velocity α1 + α2. Hence the velocity of sliding is
that due to this rotation about I, with the radius IT; that is to say, its value is
(α1 + α2) · IT;
Page 518
(26)
so that it is greater the farther the point of contact is from the line of centres; and at the
instant when that point passes the line of centres, and coincides with the pitch-point, the
velocity of sliding is null, and the action of the teeth is, for the instant, that of rolling
contact.
IV. The path of contact is the line traversing the various positions of the point T. If the
line of connexion preserves always the same position, the path of contact coincides with it,
and is straight; in other cases the path of contact is curved.
It is divided by the pitch-point I into two parts—the arc or line of approach described by
T in approaching the line of centres, and the arc or line of recess described by T after
having passed the line of centres.
During the approach, the flank D1B1 of the driving tooth drives the face D2B2 of the
following tooth, and the teeth are sliding towards each other. During the recess (in which
the position of the teeth is exemplified in the figure by curves marked with accented
letters), the face B1′A1′ of the driving tooth drives the flank B2′A2′ of the following tooth,
and the teeth are sliding from each other.
The path of contact is bounded where the approach commences by the addendum-circle
of the follower, and where the recess terminates by the addendum-circle of the driver. The
length of the path of contact should be such that there shall always be at least one pair of
teeth in contact; and it is better still to make it so long that there shall always be at least two
pairs of teeth in contact.
V. The obliquity of the action of the teeth is the angle EIT = IC1, P1 = IC2P2.
In practice it is found desirable that the mean value of the obliquity of action during the
contact of teeth should not exceed 15°, nor the maximum value 30°.
It is unnecessary to give separate figures and demonstrations for inside gearing. The only
modification required in the formulae is, that in equation (26) the difference of the angular
velocities should be substituted for their sum.
§ 46. Involute Teeth.—The simplest form of tooth which fulfils the conditions of § 45 is
obtained in the following manner (see fig. 102). Let C1, C2 be the centres of two wheels,
B1IB1′, B2IB2′ their pitch-circles, I the pitch-point; let the obliquity of action of the teeth be
constant, so that the same straight line P1IP2 shall represent at once the constant line of
connexion of teeth and the path of contact. Draw C1P1, C2P2 perpendicular to P1IP2, and
with those lines as radii describe about the centres of the wheels the circles D1D1′, D2D2′,
so that it is greater the farther the point of contact is from the line of centres; and at the
instant when that point passes the line of centres, and coincides with the pitch-point, the
velocity of sliding is null, and the action of the teeth is, for the instant, that of rolling
contact.
IV. The path of contact is the line traversing the various positions of the point T. If the
line of connexion preserves always the same position, the path of contact coincides with it,
and is straight; in other cases the path of contact is curved.
It is divided by the pitch-point I into two parts—the arc or line of approach described by
T in approaching the line of centres, and the arc or line of recess described by T after
having passed the line of centres.
During the approach, the flank D1B1 of the driving tooth drives the face D2B2 of the
following tooth, and the teeth are sliding towards each other. During the recess (in which
the position of the teeth is exemplified in the figure by curves marked with accented
letters), the face B1′A1′ of the driving tooth drives the flank B2′A2′ of the following tooth,
and the teeth are sliding from each other.
The path of contact is bounded where the approach commences by the addendum-circle
of the follower, and where the recess terminates by the addendum-circle of the driver. The
length of the path of contact should be such that there shall always be at least one pair of
teeth in contact; and it is better still to make it so long that there shall always be at least two
pairs of teeth in contact.
V. The obliquity of the action of the teeth is the angle EIT = IC1, P1 = IC2P2.
In practice it is found desirable that the mean value of the obliquity of action during the
contact of teeth should not exceed 15°, nor the maximum value 30°.
It is unnecessary to give separate figures and demonstrations for inside gearing. The only
modification required in the formulae is, that in equation (26) the difference of the angular
velocities should be substituted for their sum.
§ 46. Involute Teeth.—The simplest form of tooth which fulfils the conditions of § 45 is
obtained in the following manner (see fig. 102). Let C1, C2 be the centres of two wheels,
B1IB1′, B2IB2′ their pitch-circles, I the pitch-point; let the obliquity of action of the teeth be
constant, so that the same straight line P1IP2 shall represent at once the constant line of
connexion of teeth and the path of contact. Draw C1P1, C2P2 perpendicular to P1IP2, and
with those lines as radii describe about the centres of the wheels the circles D1D1′, D2D2′,
Page 519
called base-circles. It is evident that the radii of the base-circles bear to each other the same
proportions as the radii of the pitch-circles, and also that
C1P1 = IC1 · cos obliquity
C2P2 = IC2 · cos obliquity.
(27)
(The obliquity which is found to answer best in practice is about 141⁄2°; its cosine is
about 31/22, and its sine about 1⁄4. These values though not absolutely exact, are near
enough to the truth for practical purposes.)
Suppose the base-circles to be a pair of circular
pulleys connected by means of a cord whose course
from pulley to pulley is P1IP2. As the line of
connexion of those pulleys is the same as that of the
proposed teeth, they will rotate with the required
velocity ratio. Now, suppose a tracing point T to be
fixed to the cord, so as to be carried along the path of
contact P1IP2, that point will trace on a plane rotating
along with the wheel 1 part of the involute of the
base-circle D1D1′, and on a plane rotating along with
the wheel 2 part of the involute of the base-circle
D2D2′; and the two curves so traced will always touch
each other in the required point of contact T, and will
Fig. 102.
therefore fulfil the condition required by Principle I.
of § 45.
Consequently, one of the forms suitable for the teeth of wheels is the involute of a circle;
and the obliquity of the action of such teeth is the angle whose cosine is the ratio of the
radius of their base-circle to that of the pitch-circle of the wheel.
All involute teeth of the same pitch work smoothly together.
To find the length of the path of contact on either side of the pitch-point I, it is to be
observed that the distance between the fronts of two successive teeth, as measured along
P1IP2, is less than the pitch in the ratio of cos obliquity : I; and consequently that, if
distances equal to the pitch be marked off either way from I towards P1 and P2 respectively,
as the extremities of the path of contact, and if, according to Principle IV. of § 45, the
addendum-circles be described through the points so found, there will always be at least
two pairs of teeth in action at once. In practice it is usual to make the path of contact
proportions as the radii of the pitch-circles, and also that
C1P1 = IC1 · cos obliquity
C2P2 = IC2 · cos obliquity.
(27)
(The obliquity which is found to answer best in practice is about 141⁄2°; its cosine is
about 31/22, and its sine about 1⁄4. These values though not absolutely exact, are near
enough to the truth for practical purposes.)
Suppose the base-circles to be a pair of circular
pulleys connected by means of a cord whose course
from pulley to pulley is P1IP2. As the line of
connexion of those pulleys is the same as that of the
proposed teeth, they will rotate with the required
velocity ratio. Now, suppose a tracing point T to be
fixed to the cord, so as to be carried along the path of
contact P1IP2, that point will trace on a plane rotating
along with the wheel 1 part of the involute of the
base-circle D1D1′, and on a plane rotating along with
the wheel 2 part of the involute of the base-circle
D2D2′; and the two curves so traced will always touch
each other in the required point of contact T, and will
Fig. 102.
therefore fulfil the condition required by Principle I.
of § 45.
Consequently, one of the forms suitable for the teeth of wheels is the involute of a circle;
and the obliquity of the action of such teeth is the angle whose cosine is the ratio of the
radius of their base-circle to that of the pitch-circle of the wheel.
All involute teeth of the same pitch work smoothly together.
To find the length of the path of contact on either side of the pitch-point I, it is to be
observed that the distance between the fronts of two successive teeth, as measured along
P1IP2, is less than the pitch in the ratio of cos obliquity : I; and consequently that, if
distances equal to the pitch be marked off either way from I towards P1 and P2 respectively,
as the extremities of the path of contact, and if, according to Principle IV. of § 45, the
addendum-circles be described through the points so found, there will always be at least
two pairs of teeth in action at once. In practice it is usual to make the path of contact
Page 520
somewhat longer, viz. about 2.4 times the pitch; and with this length of path, and the
obliquity already mentioned of 141⁄2°, the addendum is about 3.1 of the pitch.
The teeth of a rack, to work correctly with wheels having involute teeth, should have
plane surfaces perpendicular to the line of connexion, and consequently making with the
direction of motion of the rack angles equal to the complement of the obliquity of action.
§ 47. Teeth for a given Path of Contact: Sang’s Method.—In the preceding section the
form of the teeth is found by assuming a figure for the path of contact, viz. the straight line.
Any other convenient figure may be assumed for the path of contact, and the corresponding
forms of the teeth found by determining what curves a point T, moving along the assumed
path of contact, will trace on two disks rotating round the centres of the wheels with
angular velocities bearing that relation to the component velocity of T along TI, which is
given by Principle II. of § 45, and by equation (25). This method of finding the forms of
the teeth of wheels forms the subject of an elaborate and most interesting treatise by
Edward Sang.
All wheels having teeth of the same pitch, traced from the same path of contact, work
correctly together, and are said to belong to the same set.
§ 48. Teeth traced by Rolling Curves.—If
any curve R (fig. 103) be rolled on the inside
of the pitch-circle BB of a wheel, it appears,
from § 30, that the instantaneous axis of the
rolling curve at any instant will be at the point
I, where it touches the pitch-circle for the
moment, and that consequently the line AT,
traced by a tracing-point T, fixed to the
Fig. 103.
rolling curve upon the plane of the wheel,
will be everywhere perpendicular to the
straight line TI; so that the traced curve AT will be suitable for the flank of a tooth, in
which T is the point of contact corresponding to the position I of the pitch-point. If the
same rolling curve R, with the same tracing-point T, be rolled on the outside of any other
pitch-circle, it will have the face of a tooth suitable to work with the flank AT.
In like manner, if either the same or any other rolling curve R′ be rolled the opposite
way, on the outside of the pitch-circle BB, so that the tracing point T′ shall start from A, it
will trace the face AT′ of a tooth suitable to work with a flank traced by rolling the same
curve R′ with the same tracing-point T′ inside any other pitch-circle.
obliquity already mentioned of 141⁄2°, the addendum is about 3.1 of the pitch.
The teeth of a rack, to work correctly with wheels having involute teeth, should have
plane surfaces perpendicular to the line of connexion, and consequently making with the
direction of motion of the rack angles equal to the complement of the obliquity of action.
§ 47. Teeth for a given Path of Contact: Sang’s Method.—In the preceding section the
form of the teeth is found by assuming a figure for the path of contact, viz. the straight line.
Any other convenient figure may be assumed for the path of contact, and the corresponding
forms of the teeth found by determining what curves a point T, moving along the assumed
path of contact, will trace on two disks rotating round the centres of the wheels with
angular velocities bearing that relation to the component velocity of T along TI, which is
given by Principle II. of § 45, and by equation (25). This method of finding the forms of
the teeth of wheels forms the subject of an elaborate and most interesting treatise by
Edward Sang.
All wheels having teeth of the same pitch, traced from the same path of contact, work
correctly together, and are said to belong to the same set.
§ 48. Teeth traced by Rolling Curves.—If
any curve R (fig. 103) be rolled on the inside
of the pitch-circle BB of a wheel, it appears,
from § 30, that the instantaneous axis of the
rolling curve at any instant will be at the point
I, where it touches the pitch-circle for the
moment, and that consequently the line AT,
traced by a tracing-point T, fixed to the
Fig. 103.
rolling curve upon the plane of the wheel,
will be everywhere perpendicular to the
straight line TI; so that the traced curve AT will be suitable for the flank of a tooth, in
which T is the point of contact corresponding to the position I of the pitch-point. If the
same rolling curve R, with the same tracing-point T, be rolled on the outside of any other
pitch-circle, it will have the face of a tooth suitable to work with the flank AT.
In like manner, if either the same or any other rolling curve R′ be rolled the opposite
way, on the outside of the pitch-circle BB, so that the tracing point T′ shall start from A, it
will trace the face AT′ of a tooth suitable to work with a flank traced by rolling the same
curve R′ with the same tracing-point T′ inside any other pitch-circle.
Page 521
The figure of the path of contact is that traced on a fixed plane by the tracing-point,
when the rolling curve is rotated in such a manner as always to touch a fixed straight line
EIE (or E′I′E′, as the case may be) at a fixed point I (or I′).
If the same rolling curve and tracing-point be used to trace both the faces and the flanks
of the teeth of a number of wheels of different sizes but of the same pitch, all those wheels
will work correctly together, and will form a set. The teeth of a rack, of the same set, are
traced by rolling the rolling curve on both sides of a straight line.
The teeth of wheels of any figure, as well as of circular wheels, may be traced by rolling
curves on their pitch-surfaces; and all teeth of the same pitch, traced by the same rolling
curve with the same tracing-point, will work together correctly if their pitch-surfaces are in
rolling contact.
§ 49. Epicycloidal Teeth.—The most convenient
rolling curve is the circle. The path of contact which
it traces is identical with itself; and the flanks of the
teeth are internal and their faces external epicycloids
for wheels, and both flanks and faces are cycloids for
a rack.
For a pitch-circle of twice the radius of the rolling
or describing circle (as it is called) the internal
epicycloid is a straight line, being, in fact, a diameter
of the pitch-circle, so that the flanks of the teeth for
such a pitch-circle are planes radiating from the axis.
Fig. 104.
For a smaller pitch-circle the flanks would be convex
and in-curved or under-cut, which would be
inconvenient; therefore the smallest wheel of a set should have its pitch-circle of twice the
radius of the describing circle, so that the flanks may be either straight or concave.
In fig. 104 let BB′ be part of the pitch-circle of a wheel with epicycloidal teeth; CIC′ the
line of centres; I the pitch-point; EIE′ a straight tangent to the pitch-circle at that point; R
the internal and R′ the equal external describing circles, so placed as to touch the pitch-
circle and each other at I. Let DID′ be the path of contact, consisting of the arc of approach
DI and the arc of recess ID′. In order that there may always be at least two pairs of teeth in
action, each of those arcs should be equal to the pitch.
The obliquity of the action in passing the line of centres is nothing; the maximum
obliquity is the angle EID = E′ID; and the mean obliquity is one-half of that angle.
when the rolling curve is rotated in such a manner as always to touch a fixed straight line
EIE (or E′I′E′, as the case may be) at a fixed point I (or I′).
If the same rolling curve and tracing-point be used to trace both the faces and the flanks
of the teeth of a number of wheels of different sizes but of the same pitch, all those wheels
will work correctly together, and will form a set. The teeth of a rack, of the same set, are
traced by rolling the rolling curve on both sides of a straight line.
The teeth of wheels of any figure, as well as of circular wheels, may be traced by rolling
curves on their pitch-surfaces; and all teeth of the same pitch, traced by the same rolling
curve with the same tracing-point, will work together correctly if their pitch-surfaces are in
rolling contact.
§ 49. Epicycloidal Teeth.—The most convenient
rolling curve is the circle. The path of contact which
it traces is identical with itself; and the flanks of the
teeth are internal and their faces external epicycloids
for wheels, and both flanks and faces are cycloids for
a rack.
For a pitch-circle of twice the radius of the rolling
or describing circle (as it is called) the internal
epicycloid is a straight line, being, in fact, a diameter
of the pitch-circle, so that the flanks of the teeth for
such a pitch-circle are planes radiating from the axis.
Fig. 104.
For a smaller pitch-circle the flanks would be convex
and in-curved or under-cut, which would be
inconvenient; therefore the smallest wheel of a set should have its pitch-circle of twice the
radius of the describing circle, so that the flanks may be either straight or concave.
In fig. 104 let BB′ be part of the pitch-circle of a wheel with epicycloidal teeth; CIC′ the
line of centres; I the pitch-point; EIE′ a straight tangent to the pitch-circle at that point; R
the internal and R′ the equal external describing circles, so placed as to touch the pitch-
circle and each other at I. Let DID′ be the path of contact, consisting of the arc of approach
DI and the arc of recess ID′. In order that there may always be at least two pairs of teeth in
action, each of those arcs should be equal to the pitch.
The obliquity of the action in passing the line of centres is nothing; the maximum
obliquity is the angle EID = E′ID; and the mean obliquity is one-half of that angle.
Page 522
It appears from experience that the mean obliquity should not exceed 15°; therefore the
maximum obliquity should be about 30°; therefore the equal arcs DI and ID′ should each
be one-sixth of a circumference; therefore the circumference of the describing circle should
be six times the pitch.
It follows that the smallest pinion of a set in which pinion the flanks are straight should
have twelve teeth.
§ 50. Nearly Epicycloidal Teeth: Willis’s Method.—To facilitate the drawing of
epicycloidal teeth in practice, Willis showed how to approximate to their figure by means
of two circular arcs—one concave, for the flank, and the other convex, for the face—and
each having for its radius the mean radius of curvature of the epicycloidal arc. Willis’s
formulae are founded on the following properties of epicycloids:—
Let R be the radius of the pitch-circle; r that of the describing circle; θ the angle made by
the normal TI to the epicycloid at a given point T, with a tangent to the circle at I—that is,
the obliquity of the action at T.
Then the radius of curvature of the epicycloid at T is—
R−r
For an internal epicycloid, ρ = 4r sin θ
R − 2r
R+r
For an external epicycloid, ρ′ = 4r sin θ
R + 2r (28)
Also, to find the position of the centres of curvature relatively to the pitch-circle, we have,
denoting the chord of the describing circle TI by c, c = 2r sin θ; and therefore
R
For the flank, ρ − c = 2r sin θ
R − 2r
R
For the face, ρ′ − c = 2r sin θ
R + 2r (29)
For the proportions approved of by Willis, sin θ = 1⁄4 nearly; r = p (the pitch) nearly; c = 1⁄2p
nearly; and, if N be the number of teeth in the wheel, r/R = 6/N nearly; therefore,
approximately,
p N
ρ−c= ·
2 N − 12
p N
ρ−c= ·
2 N + 12 (30)
maximum obliquity should be about 30°; therefore the equal arcs DI and ID′ should each
be one-sixth of a circumference; therefore the circumference of the describing circle should
be six times the pitch.
It follows that the smallest pinion of a set in which pinion the flanks are straight should
have twelve teeth.
§ 50. Nearly Epicycloidal Teeth: Willis’s Method.—To facilitate the drawing of
epicycloidal teeth in practice, Willis showed how to approximate to their figure by means
of two circular arcs—one concave, for the flank, and the other convex, for the face—and
each having for its radius the mean radius of curvature of the epicycloidal arc. Willis’s
formulae are founded on the following properties of epicycloids:—
Let R be the radius of the pitch-circle; r that of the describing circle; θ the angle made by
the normal TI to the epicycloid at a given point T, with a tangent to the circle at I—that is,
the obliquity of the action at T.
Then the radius of curvature of the epicycloid at T is—
R−r
For an internal epicycloid, ρ = 4r sin θ
R − 2r
R+r
For an external epicycloid, ρ′ = 4r sin θ
R + 2r (28)
Also, to find the position of the centres of curvature relatively to the pitch-circle, we have,
denoting the chord of the describing circle TI by c, c = 2r sin θ; and therefore
R
For the flank, ρ − c = 2r sin θ
R − 2r
R
For the face, ρ′ − c = 2r sin θ
R + 2r (29)
For the proportions approved of by Willis, sin θ = 1⁄4 nearly; r = p (the pitch) nearly; c = 1⁄2p
nearly; and, if N be the number of teeth in the wheel, r/R = 6/N nearly; therefore,
approximately,
p N
ρ−c= ·
2 N − 12
p N
ρ−c= ·
2 N + 12 (30)
Page 523
Hence the following construction (fig. 105). Let
BB be part of the pitch-circle, and a the point where
a tooth is to cross it. Set off ab = ac − 1⁄2p. Draw
radii bd, ce; draw fb, cg, making angles of 751⁄2°
with those radii. Make bf = p′ − c, cg = p − c. From
f, with the radius fa, draw the circular arc ah; from
g, with the radius ga, draw the circular arc ak. Then
Fig. 105.
ah is the face and ak the flank of the tooth required.
To facilitate the application of this rule, Willis published tables of ρ − c and ρ′ − c, and
invented an instrument called the “odontograph.”
§ 51. Trundles and Pin-Wheels.—If a wheel or trundle have cylindrical pins or staves for
teeth, the faces of the teeth of a wheel suitable for driving it are described by first tracing
external epicycloids, by rolling the pitch-circle of the pin-wheel or trundle on the pitch-
circle of the driving-wheel, with the centre of a stave for a tracing-point, and then drawing
curves parallel to, and within the epicycloids, at a distance from them equal to the radius of
a stave. Trundles having only six staves will work with large wheels.
§ 52. Backs of Teeth and Spaces.—Toothed wheels being in general intended to rotate
either way, the backs of the teeth are made similar to the fronts. The space between two
teeth, measured on the pitch-circle, is made about 1⁄6th part wider than the thickness of the
tooth on the pitch-circle—that is to say,
Thickness of tooth = 5⁄11 pitch;
Width of space = 6⁄11 pitch.
The difference of 1⁄11 of the pitch is called the back-lash. The clearance allowed between the
points of teeth and the bottoms of the spaces between the teeth of the other wheel is about
one-tenth of the pitch.
§ 53. Stepped and Helical Teeth.—R. J. Hooke invented the making of the fronts of teeth
in a series of steps with a view to increase the smoothness of action. A wheel thus formed
resembles in shape a series of equal and similar toothed disks placed side by side, with the
teeth of each a little behind those of the preceding disk. He also invented, with the same
object, teeth whose fronts, instead of being parallel to the line of contact of the pitch-
circles, cross it obliquely, so as to be of a screw-like or helical form. In wheel-work of this
kind the contact of each pair of teeth commences at the foremost end of the helical front,
BB be part of the pitch-circle, and a the point where
a tooth is to cross it. Set off ab = ac − 1⁄2p. Draw
radii bd, ce; draw fb, cg, making angles of 751⁄2°
with those radii. Make bf = p′ − c, cg = p − c. From
f, with the radius fa, draw the circular arc ah; from
g, with the radius ga, draw the circular arc ak. Then
Fig. 105.
ah is the face and ak the flank of the tooth required.
To facilitate the application of this rule, Willis published tables of ρ − c and ρ′ − c, and
invented an instrument called the “odontograph.”
§ 51. Trundles and Pin-Wheels.—If a wheel or trundle have cylindrical pins or staves for
teeth, the faces of the teeth of a wheel suitable for driving it are described by first tracing
external epicycloids, by rolling the pitch-circle of the pin-wheel or trundle on the pitch-
circle of the driving-wheel, with the centre of a stave for a tracing-point, and then drawing
curves parallel to, and within the epicycloids, at a distance from them equal to the radius of
a stave. Trundles having only six staves will work with large wheels.
§ 52. Backs of Teeth and Spaces.—Toothed wheels being in general intended to rotate
either way, the backs of the teeth are made similar to the fronts. The space between two
teeth, measured on the pitch-circle, is made about 1⁄6th part wider than the thickness of the
tooth on the pitch-circle—that is to say,
Thickness of tooth = 5⁄11 pitch;
Width of space = 6⁄11 pitch.
The difference of 1⁄11 of the pitch is called the back-lash. The clearance allowed between the
points of teeth and the bottoms of the spaces between the teeth of the other wheel is about
one-tenth of the pitch.
§ 53. Stepped and Helical Teeth.—R. J. Hooke invented the making of the fronts of teeth
in a series of steps with a view to increase the smoothness of action. A wheel thus formed
resembles in shape a series of equal and similar toothed disks placed side by side, with the
teeth of each a little behind those of the preceding disk. He also invented, with the same
object, teeth whose fronts, instead of being parallel to the line of contact of the pitch-
circles, cross it obliquely, so as to be of a screw-like or helical form. In wheel-work of this
kind the contact of each pair of teeth commences at the foremost end of the helical front,
Page 524
and terminates at the aftermost end; and the helix is of such a pitch that the contact of one
pair of teeth shall not terminate until that of the next pair has commenced.
Stepped and helical teeth have the desired effect of increasing the smoothness of motion,
but they require more difficult and expensive workmanship than common teeth; and helical
teeth are, besides, open to the objection that they exert a laterally oblique pressure, which
tends to increase resistance, and unduly strain the machinery.
§ 54. Teeth of Bevel-Wheels.—The acting surfaces of the teeth of bevel-wheels are of the
conical kind, generated by the motion of a line passing through the common apex of the
pitch-cones, while its extremity is carried round the outlines of the cross section of the
teeth made by a sphere described about that apex.
The operations of describing the exact figures
of the teeth of bevel-wheels, whether by involutes
or by rolling curves, are in every respect
analogous to those for describing the figures of
the teeth of spur-wheels, except that in the case of
bevel-wheels all those operations are to be
performed on the surface of a sphere described
about the apex instead of on a plane, substituting
poles for centres, and great circles for straight
lines. Fig. 106.
In consideration of the practical difficulty, especially in the case of large wheels, of
obtaining an accurate spherical surface, and of drawing upon it when obtained, the
following approximate method, proposed originally by Tredgold, is generally used:—
Let O (fig. 106) be the common apex of a pair of bevel-wheels; OB1I, OB2I their pitch
cones; OC1, OC2 their axes; OI their line of contact. Perpendicular to OI draw A1IA2,
cutting the axes in A1, A2; make the outer rims of the patterns and of the wheels portions of
the cones A1B1I, A2B2I, of which the narrow zones occupied by the teeth will be
sufficiently near to a spherical surface described about O for practical purposes. To find the
figures of the teeth, draw on a flat surface circular arcs ID1, ID2, with the radii A1I, A2I;
those arcs will be the developments of arcs of the pitch-circles B1I, B2I, when the conical
surfaces A1B1I, A2B2I are spread out flat. Describe the figures of teeth for the developed
arcs as for a pair of spur-wheels; then wrap the developed arcs on the cones, so as to make
them coincide with the pitch-circles, and trace the teeth on the conical surfaces.
§ 55. Teeth of Skew-Bevel Wheels.—The crests of the teeth of a skew-bevel wheel are
parallel to the generating straight line of the hyperboloidal pitch-surface; and the transverse
pair of teeth shall not terminate until that of the next pair has commenced.
Stepped and helical teeth have the desired effect of increasing the smoothness of motion,
but they require more difficult and expensive workmanship than common teeth; and helical
teeth are, besides, open to the objection that they exert a laterally oblique pressure, which
tends to increase resistance, and unduly strain the machinery.
§ 54. Teeth of Bevel-Wheels.—The acting surfaces of the teeth of bevel-wheels are of the
conical kind, generated by the motion of a line passing through the common apex of the
pitch-cones, while its extremity is carried round the outlines of the cross section of the
teeth made by a sphere described about that apex.
The operations of describing the exact figures
of the teeth of bevel-wheels, whether by involutes
or by rolling curves, are in every respect
analogous to those for describing the figures of
the teeth of spur-wheels, except that in the case of
bevel-wheels all those operations are to be
performed on the surface of a sphere described
about the apex instead of on a plane, substituting
poles for centres, and great circles for straight
lines. Fig. 106.
In consideration of the practical difficulty, especially in the case of large wheels, of
obtaining an accurate spherical surface, and of drawing upon it when obtained, the
following approximate method, proposed originally by Tredgold, is generally used:—
Let O (fig. 106) be the common apex of a pair of bevel-wheels; OB1I, OB2I their pitch
cones; OC1, OC2 their axes; OI their line of contact. Perpendicular to OI draw A1IA2,
cutting the axes in A1, A2; make the outer rims of the patterns and of the wheels portions of
the cones A1B1I, A2B2I, of which the narrow zones occupied by the teeth will be
sufficiently near to a spherical surface described about O for practical purposes. To find the
figures of the teeth, draw on a flat surface circular arcs ID1, ID2, with the radii A1I, A2I;
those arcs will be the developments of arcs of the pitch-circles B1I, B2I, when the conical
surfaces A1B1I, A2B2I are spread out flat. Describe the figures of teeth for the developed
arcs as for a pair of spur-wheels; then wrap the developed arcs on the cones, so as to make
them coincide with the pitch-circles, and trace the teeth on the conical surfaces.
§ 55. Teeth of Skew-Bevel Wheels.—The crests of the teeth of a skew-bevel wheel are
parallel to the generating straight line of the hyperboloidal pitch-surface; and the transverse
Page 525
sections of the teeth at a given pitch-circle are similar to those of the teeth of a bevel-wheel
whose pitch surface is a cone touching the hyperboloidal surface at the given circle.
§ 56. Cams.—A cam is a single tooth, either rotating continuously or oscillating, and
driving a sliding or turning piece either constantly or at intervals. All the principles which
have been stated in § 45 as being applicable to teeth are applicable to cams; but in
designing cams it is not usual to determine or take into consideration the form of the ideal
pitch-surface, which would give the same comparative motion by rolling contact that the
cam gives by sliding contact.
§ 57. Screws.—The figure of a screw is that of a convex or concave cylinder, with one or
more helical projections, called threads, winding round it. Convex and concave screws are
distinguished technically by the respective names of male and female; a short concave
screw is called a nut; and when a screw is spoken of without qualification a convex screw is
usually understood.
The relation between the advance and the rotation, which compose the motion of a
screw working in contact with a fixed screw or helical guide, has already been
demonstrated in § 32; and the same relation exists between the magnitudes of the rotation
of a screw about a fixed axis and the advance of a shifting nut in which it rotates. The
advance of the nut takes place in the opposite direction to that of the advance of the screw
in the case in which the nut is fixed. The pitch or axial pitch of a screw has the meaning
assigned to it in that section, viz. the distance, measured parallel to the axis, between the
corresponding points in two successive turns of the same thread. If, therefore, the screw
has several equidistant threads, the true pitch is equal to the divided axial pitch, as
measured between two adjacent threads, multiplied by the number of threads.
If a helix be described round the screw, crossing each turn of the thread at right angles,
the distance between two corresponding points on two successive turns of the same thread,
measured along this normal helix, may be called the normal pitch; and when the screw has
more than one thread the normal pitch from thread to thread may be called the normal
divided pitch.
The distance from thread to thread, measured on a circle described about the axis of the
screw, called the pitch-circle, may be called the circumferential pitch; for a screw of one
thread it is one circumference; for a screw of n threads, (one circumference)/n.
Let r denote the radius of the pitch circle;
n the number of threads;
θ the obliquity of the threads to the pitch circle, and of the normal helix to the axis;
whose pitch surface is a cone touching the hyperboloidal surface at the given circle.
§ 56. Cams.—A cam is a single tooth, either rotating continuously or oscillating, and
driving a sliding or turning piece either constantly or at intervals. All the principles which
have been stated in § 45 as being applicable to teeth are applicable to cams; but in
designing cams it is not usual to determine or take into consideration the form of the ideal
pitch-surface, which would give the same comparative motion by rolling contact that the
cam gives by sliding contact.
§ 57. Screws.—The figure of a screw is that of a convex or concave cylinder, with one or
more helical projections, called threads, winding round it. Convex and concave screws are
distinguished technically by the respective names of male and female; a short concave
screw is called a nut; and when a screw is spoken of without qualification a convex screw is
usually understood.
The relation between the advance and the rotation, which compose the motion of a
screw working in contact with a fixed screw or helical guide, has already been
demonstrated in § 32; and the same relation exists between the magnitudes of the rotation
of a screw about a fixed axis and the advance of a shifting nut in which it rotates. The
advance of the nut takes place in the opposite direction to that of the advance of the screw
in the case in which the nut is fixed. The pitch or axial pitch of a screw has the meaning
assigned to it in that section, viz. the distance, measured parallel to the axis, between the
corresponding points in two successive turns of the same thread. If, therefore, the screw
has several equidistant threads, the true pitch is equal to the divided axial pitch, as
measured between two adjacent threads, multiplied by the number of threads.
If a helix be described round the screw, crossing each turn of the thread at right angles,
the distance between two corresponding points on two successive turns of the same thread,
measured along this normal helix, may be called the normal pitch; and when the screw has
more than one thread the normal pitch from thread to thread may be called the normal
divided pitch.
The distance from thread to thread, measured on a circle described about the axis of the
screw, called the pitch-circle, may be called the circumferential pitch; for a screw of one
thread it is one circumference; for a screw of n threads, (one circumference)/n.
Let r denote the radius of the pitch circle;
n the number of threads;
θ the obliquity of the threads to the pitch circle, and of the normal helix to the axis;
Page 526
Pa pitch
the axial
Pa/n = pa divided pitch;
Pn pitch
the normal
Pn/n = pn divided pitch;
Pc the circumferential pitch;
then
2πr
pc = pa cot θ = pn cos θ = ,
n
2πr tan θ
pa = pn sec θ = pc tan θ = ,
n
2πr sin θ
pn = pc sin θ = pa cos θ = .
n (31)
If a screw rotates, the number of threads which pass a fixed point in one revolution is the
number of threads in the screw.
A pair of convex screws, each rotating about its axis, are used as an elementary
combination to transmit motion by the sliding contact of their threads. Such screws are
commonly called endless screws. At the point of contact of the screws their threads must be
parallel; and their line of connexion is the common perpendicular to the acting surfaces of
the threads at their point of contact. Hence the following principles:—
I. If the screws are both right-handed or both left-handed, the angle between the
directions of their axes is the sum of their obliquities; if one is right-handed and the other
left-handed, that angle is the difference of their obliquities.
II. The normal pitch for a screw of one thread, and the normal divided pitch for a screw
of more than one thread, must be the same in each screw.
III. The angular velocities of the screws are inversely as their numbers of threads.
Hooke’s wheels with oblique or helical teeth are in fact screws of many threads, and of
large diameters as compared with their lengths.
The ordinary position of a pair of endless screws is with their axes at right angles to each
other. When one is of considerably greater diameter than the other, the larger is commonly
the axial
Pa/n = pa divided pitch;
Pn pitch
the normal
Pn/n = pn divided pitch;
Pc the circumferential pitch;
then
2πr
pc = pa cot θ = pn cos θ = ,
n
2πr tan θ
pa = pn sec θ = pc tan θ = ,
n
2πr sin θ
pn = pc sin θ = pa cos θ = .
n (31)
If a screw rotates, the number of threads which pass a fixed point in one revolution is the
number of threads in the screw.
A pair of convex screws, each rotating about its axis, are used as an elementary
combination to transmit motion by the sliding contact of their threads. Such screws are
commonly called endless screws. At the point of contact of the screws their threads must be
parallel; and their line of connexion is the common perpendicular to the acting surfaces of
the threads at their point of contact. Hence the following principles:—
I. If the screws are both right-handed or both left-handed, the angle between the
directions of their axes is the sum of their obliquities; if one is right-handed and the other
left-handed, that angle is the difference of their obliquities.
II. The normal pitch for a screw of one thread, and the normal divided pitch for a screw
of more than one thread, must be the same in each screw.
III. The angular velocities of the screws are inversely as their numbers of threads.
Hooke’s wheels with oblique or helical teeth are in fact screws of many threads, and of
large diameters as compared with their lengths.
The ordinary position of a pair of endless screws is with their axes at right angles to each
other. When one is of considerably greater diameter than the other, the larger is commonly
Page 527
called in practice a wheel, the name screw being applied to the smaller only; but they are
nevertheless both screws in fact.
To make the teeth of a pair of endless screws fit correctly and work smoothly, a hardened
steel screw is made of the figure of the smaller screw, with its thread or threads notched so
as to form a cutting tool; the larger screw, or “wheel,” is cast approximately of the required
figure; the larger screw and the steel screw are fitted up in their proper relative position,
and made to rotate in contact with each other by turning the steel screw, which cuts the
threads of the larger screw to their true figure.
§ 58. Coupling of Parallel Axes—Oldham’s Coupling.—
A coupling is a mode of connecting a pair of shafts so that
they shall rotate in the same direction with the same mean
angular velocity. If the axes of the shafts are in the same
straight line, the coupling consists in so connecting their
contiguous ends that they shall rotate as one piece; but if
the axes are not in the same straight line combinations of
mechanism are required. A coupling for parallel shafts
which acts by sliding contact was invented by Oldham, and
Fig. 107.
is represented in fig. 107. C1, C2 are the axes of the two
parallel shafts; D1, D2 two disks facing each other, fixed on the ends of the two shafts
respectively; E1E1 a bar sliding in a diametral groove in the face of D1; E2E2 a bar sliding
in a diametral groove in the face of D2: those bars are fixed together at A, so as to form a
rigid cross. The angular velocities of the two disks and of the cross are all equal at every
instant; the middle point of the cross, at A, revolves in the dotted circle described upon the
line of centres C1C2 as a diameter twice for each turn of the disks and cross; the
instantaneous axis of rotation of the cross at any instant is at I, the point in the circle C1C2
diametrically opposite to A.
Oldham’s coupling may be used with advantage where the axes of the shafts are
intended to be as nearly in the same straight line as is possible, but where there is some
doubt as to the practibility or permanency of their exact continuity.
§ 59. Wrapping Connectors—Belts, Cords and Chains.—Flat belts of leather or of gutta
percha, round cords of catgut, hemp or other material, and metal chains are used as
wrapping connectors to transmit rotatory motion between pairs of pulleys and drums.
Belts (the most frequently used of all wrapping connectors) require nearly cylindrical
pulleys. A belt tends to move towards that part of a pulley whose radius is greatest; pulleys
for belts, therefore, are slightly swelled in the middle, in order that the belt may remain on
the pulley, unless forcibly shifted. A belt when in motion is shifted off a pulley, or from one
nevertheless both screws in fact.
To make the teeth of a pair of endless screws fit correctly and work smoothly, a hardened
steel screw is made of the figure of the smaller screw, with its thread or threads notched so
as to form a cutting tool; the larger screw, or “wheel,” is cast approximately of the required
figure; the larger screw and the steel screw are fitted up in their proper relative position,
and made to rotate in contact with each other by turning the steel screw, which cuts the
threads of the larger screw to their true figure.
§ 58. Coupling of Parallel Axes—Oldham’s Coupling.—
A coupling is a mode of connecting a pair of shafts so that
they shall rotate in the same direction with the same mean
angular velocity. If the axes of the shafts are in the same
straight line, the coupling consists in so connecting their
contiguous ends that they shall rotate as one piece; but if
the axes are not in the same straight line combinations of
mechanism are required. A coupling for parallel shafts
which acts by sliding contact was invented by Oldham, and
Fig. 107.
is represented in fig. 107. C1, C2 are the axes of the two
parallel shafts; D1, D2 two disks facing each other, fixed on the ends of the two shafts
respectively; E1E1 a bar sliding in a diametral groove in the face of D1; E2E2 a bar sliding
in a diametral groove in the face of D2: those bars are fixed together at A, so as to form a
rigid cross. The angular velocities of the two disks and of the cross are all equal at every
instant; the middle point of the cross, at A, revolves in the dotted circle described upon the
line of centres C1C2 as a diameter twice for each turn of the disks and cross; the
instantaneous axis of rotation of the cross at any instant is at I, the point in the circle C1C2
diametrically opposite to A.
Oldham’s coupling may be used with advantage where the axes of the shafts are
intended to be as nearly in the same straight line as is possible, but where there is some
doubt as to the practibility or permanency of their exact continuity.
§ 59. Wrapping Connectors—Belts, Cords and Chains.—Flat belts of leather or of gutta
percha, round cords of catgut, hemp or other material, and metal chains are used as
wrapping connectors to transmit rotatory motion between pairs of pulleys and drums.
Belts (the most frequently used of all wrapping connectors) require nearly cylindrical
pulleys. A belt tends to move towards that part of a pulley whose radius is greatest; pulleys
for belts, therefore, are slightly swelled in the middle, in order that the belt may remain on
the pulley, unless forcibly shifted. A belt when in motion is shifted off a pulley, or from one
Page 528
pulley on to another of equal size alongside of it, by pressing against that part of the belt
which is moving towards the pulley.
Cords require either cylindrical drums with ledges or grooved pulleys.
Chains require pulleys or drums, grooved, notched and toothed, so as to fit the links of
the chain.
Wrapping connectors for communicating continuous motion are endless.
Wrapping connectors for communicating reciprocating motion have usually their ends
made fast to the pulleys or drums which they connect, and which in this case may be
sectors.
The line of connexion of two pieces connected by a
wrapping connector is the centre line of the belt, cord
or chain; and the comparative motions of the pieces are
determined by the principles of § 36 if both pieces turn,
and of § 37 if one turns and the other shifts, in which
latter case the motion must be reciprocating.
The pitch-line of a pulley or drum is a curve to
which the line of connexion is always a tangent—that
is to say, it is a curve parallel to the acting surface of
the pulley or drum, and distant from it by half the Fig. 108.
thickness of the wrapping connector.
Pulleys and drums for communicating a constant velocity ratio are circular. The effective
radius, or radius of the pitch-circle of a circular pulley or drum, is equal to the real radius
added to half the thickness of the connector. The angular velocities of a pair of connected
circular pulleys or drums are inversely as the effective radii.
A crossed belt, as in fig. 108, A, reverses the direction of the rotation communicated; an
uncrossed belt, as in fig. 108, B, preserves that direction.
The length L of an endless belt connecting a pair of pulleys whose effective radii are r1,
r2, with parallel axes whose distance apart is c, is given by the following formulae, in each
of which the first term, containing the radical, expresses the length of the straight parts of
the belt, and the remainder of the formula the length of the curved parts.
For a crossed belt:—
r1 + r2
L = 2 √ {c2 − (r1 + r2)2} + (r1 + r2) ( π − 2 sin−1 c
);
which is moving towards the pulley.
Cords require either cylindrical drums with ledges or grooved pulleys.
Chains require pulleys or drums, grooved, notched and toothed, so as to fit the links of
the chain.
Wrapping connectors for communicating continuous motion are endless.
Wrapping connectors for communicating reciprocating motion have usually their ends
made fast to the pulleys or drums which they connect, and which in this case may be
sectors.
The line of connexion of two pieces connected by a
wrapping connector is the centre line of the belt, cord
or chain; and the comparative motions of the pieces are
determined by the principles of § 36 if both pieces turn,
and of § 37 if one turns and the other shifts, in which
latter case the motion must be reciprocating.
The pitch-line of a pulley or drum is a curve to
which the line of connexion is always a tangent—that
is to say, it is a curve parallel to the acting surface of
the pulley or drum, and distant from it by half the Fig. 108.
thickness of the wrapping connector.
Pulleys and drums for communicating a constant velocity ratio are circular. The effective
radius, or radius of the pitch-circle of a circular pulley or drum, is equal to the real radius
added to half the thickness of the connector. The angular velocities of a pair of connected
circular pulleys or drums are inversely as the effective radii.
A crossed belt, as in fig. 108, A, reverses the direction of the rotation communicated; an
uncrossed belt, as in fig. 108, B, preserves that direction.
The length L of an endless belt connecting a pair of pulleys whose effective radii are r1,
r2, with parallel axes whose distance apart is c, is given by the following formulae, in each
of which the first term, containing the radical, expresses the length of the straight parts of
the belt, and the remainder of the formula the length of the curved parts.
For a crossed belt:—
r1 + r2
L = 2 √ {c2 − (r1 + r2)2} + (r1 + r2) ( π − 2 sin−1 c
);
Page 529
(32 A)
and for an uncrossed belt:—
r1 − r2
L = 2 √ {c2 − (r1 − r2)2 } + π (r1 + r2 + 2 (r1 − r2) sin−1 ;
c (32 B)
in which r1 is the greater radius, and r2 the less.
When the axes of a pair of pulleys are not parallel, the pulleys should be so placed that
the part of the belt which is approaching each pulley shall be in the plane of the pulley.
§ 60. Speed-Cones.—A pair of speed-cones (fig. 109) is a contrivance for varying and
adjusting the velocity ratio communicated between a pair of parallel shafts by means of a
belt. The speed-cones are either continuous cones or conoids, as A, B, whose velocity ratio
can be varied gradually while they are in motion by shifting the belt, or sets of pulleys
whose radii vary by steps, as C, D, in which case the velocity ratio can be changed by
shifting the belt from one pair of pulleys to another.
In order that the belt may fit accurately in every
possible position on a pair of speed-cones, the
quantity L must be constant, in equations (32 A) or
(32 B), according as the belt is crossed or
uncrossed.
For a crossed belt, as in A and C, fig. 109, L
depends solely on c and on r1 + r2. Now c is
constant because the axes are parallel; therefore
the sum of the radii of the pitch-circles connected
in every position of the belt is to be constant. That
condition is fulfilled by a pair of continuous cones
Fig. 109.
generated by the revolution of two straight lines
inclined opposite ways to their respective axes at equal angles.
For an uncrossed belt, the quantity L in equation (32 B) is to be made constant. The
exact fulfilment of this condition requires the solution of a transcendental equation; but it
may be fulfilled with accuracy sufficient for practical purposes by using, instead of (32 B)
the following approximate equation:—
L nearly = 2c + π (r1 + r2) + (r1 − r2)2 / c.
(33)
and for an uncrossed belt:—
r1 − r2
L = 2 √ {c2 − (r1 − r2)2 } + π (r1 + r2 + 2 (r1 − r2) sin−1 ;
c (32 B)
in which r1 is the greater radius, and r2 the less.
When the axes of a pair of pulleys are not parallel, the pulleys should be so placed that
the part of the belt which is approaching each pulley shall be in the plane of the pulley.
§ 60. Speed-Cones.—A pair of speed-cones (fig. 109) is a contrivance for varying and
adjusting the velocity ratio communicated between a pair of parallel shafts by means of a
belt. The speed-cones are either continuous cones or conoids, as A, B, whose velocity ratio
can be varied gradually while they are in motion by shifting the belt, or sets of pulleys
whose radii vary by steps, as C, D, in which case the velocity ratio can be changed by
shifting the belt from one pair of pulleys to another.
In order that the belt may fit accurately in every
possible position on a pair of speed-cones, the
quantity L must be constant, in equations (32 A) or
(32 B), according as the belt is crossed or
uncrossed.
For a crossed belt, as in A and C, fig. 109, L
depends solely on c and on r1 + r2. Now c is
constant because the axes are parallel; therefore
the sum of the radii of the pitch-circles connected
in every position of the belt is to be constant. That
condition is fulfilled by a pair of continuous cones
Fig. 109.
generated by the revolution of two straight lines
inclined opposite ways to their respective axes at equal angles.
For an uncrossed belt, the quantity L in equation (32 B) is to be made constant. The
exact fulfilment of this condition requires the solution of a transcendental equation; but it
may be fulfilled with accuracy sufficient for practical purposes by using, instead of (32 B)
the following approximate equation:—
L nearly = 2c + π (r1 + r2) + (r1 − r2)2 / c.
(33)
Page 530
The following is the most convenient practical rule for the application of this equation:
—
Let the speed-cones be equal and similar conoids, as in B, fig. 109, but with their large
and small ends turned opposite ways. Let r1 be the radius of the large end of each, r2 that of
the small end, r0 that of the middle; and let v be the sagitta, measured perpendicular to the
axes, of the arc by whose revolution each of the conoids is generated, or, in other words,
the bulging of the conoids in the middle of their length. Then
v = r0 − (r1 + r2) / 2 = (r1 − r2)2 / 2πc.
(34)
2π = 6.2832; but 6 may be used in most practical cases without sensible error.
The radii at the middle and end being thus determined, make the generating curve an arc
either of a circle or of a parabola.
§ 61. Linkwork in General.—The pieces which are connected by linkwork, if they rotate
or oscillate, are usually called cranks, beams and levers. The link by which they are
connected is a rigid rod or bar, which may be straight or of any other figure; the straight
figure being the most favourable to strength, is always used when there is no special reason
to the contrary. The link is known by various names in various circumstances, such as
coupling-rod, connecting-rod, crank-rod, eccentric-rod, &c. It is attached to the pieces
which it connects by two pins, about which it is free to turn. The effect of the link is to
maintain the distance between the axes of those pins invariable; hence the common
perpendicular of the axes of the pins is the line of connexion, and its extremities may be
called the connected points. In a turning piece, the perpendicular let fall from its connected
point upon its axis of rotation is the arm or crank-arm.
The axes of rotation of a pair of turning pieces connected by a link are almost always
parallel, and perpendicular to the line of connexion in which case the angular velocity ratio
at any instant is the reciprocal of the ratio of the common perpendiculars let fall from the
line of connexion upon the respective axes of rotation.
If at any instant the direction of one of the crank-arms coincides with the line of
connexion, the common perpendicular of the line of connexion and the axis of that crank-
arm vanishes, and the directional relation of the motions becomes indeterminate. The
position of the connected point of the crank-arm in question at such an instant is called a
dead-point. The velocity of the other connected point at such an instant is null, unless it
also reaches a dead-point at the same instant, so that the line of connexion is in the plane of
the two axes of rotation, in which case the velocity ratio is indeterminate. Examples of
—
Let the speed-cones be equal and similar conoids, as in B, fig. 109, but with their large
and small ends turned opposite ways. Let r1 be the radius of the large end of each, r2 that of
the small end, r0 that of the middle; and let v be the sagitta, measured perpendicular to the
axes, of the arc by whose revolution each of the conoids is generated, or, in other words,
the bulging of the conoids in the middle of their length. Then
v = r0 − (r1 + r2) / 2 = (r1 − r2)2 / 2πc.
(34)
2π = 6.2832; but 6 may be used in most practical cases without sensible error.
The radii at the middle and end being thus determined, make the generating curve an arc
either of a circle or of a parabola.
§ 61. Linkwork in General.—The pieces which are connected by linkwork, if they rotate
or oscillate, are usually called cranks, beams and levers. The link by which they are
connected is a rigid rod or bar, which may be straight or of any other figure; the straight
figure being the most favourable to strength, is always used when there is no special reason
to the contrary. The link is known by various names in various circumstances, such as
coupling-rod, connecting-rod, crank-rod, eccentric-rod, &c. It is attached to the pieces
which it connects by two pins, about which it is free to turn. The effect of the link is to
maintain the distance between the axes of those pins invariable; hence the common
perpendicular of the axes of the pins is the line of connexion, and its extremities may be
called the connected points. In a turning piece, the perpendicular let fall from its connected
point upon its axis of rotation is the arm or crank-arm.
The axes of rotation of a pair of turning pieces connected by a link are almost always
parallel, and perpendicular to the line of connexion in which case the angular velocity ratio
at any instant is the reciprocal of the ratio of the common perpendiculars let fall from the
line of connexion upon the respective axes of rotation.
If at any instant the direction of one of the crank-arms coincides with the line of
connexion, the common perpendicular of the line of connexion and the axis of that crank-
arm vanishes, and the directional relation of the motions becomes indeterminate. The
position of the connected point of the crank-arm in question at such an instant is called a
dead-point. The velocity of the other connected point at such an instant is null, unless it
also reaches a dead-point at the same instant, so that the line of connexion is in the plane of
the two axes of rotation, in which case the velocity ratio is indeterminate. Examples of
Page 531
dead-points, and of the means of preventing the inconvenience which they tend to
occasion, will appear in the sequel.
§ 62. Coupling of Parallel Axes.—Two or more parallel shafts (such as those of a
locomotive engine, with two or more pairs of driving wheels) are made to rotate with
constantly equal angular velocities by having equal cranks, which are maintained parallel
by a coupling-rod of such a length that the line of connexion is equal to the distance
between the axes. The cranks pass their dead-points simultaneously. To obviate the
unsteadiness of motion which this tends to cause, the shafts are provided with a second set
of cranks at right angles to the first, connected by means of a similar coupling-rod, so that
one set of cranks pass their dead points at the instant when the other set are farthest from
theirs.
§ 63. Comparative Motion of Connected Points.—As the link is a rigid body, it is
obvious that its action in communicating motion may be determined by finding the
comparative motion of the connected points, and this is often the most convenient method
of proceeding.
If a connected point belongs to a turning piece, the direction of its motion at a given
instant is perpendicular to the plane containing the axis and crank-arm of the piece. If a
connected point belongs to a shifting piece, the direction of its motion at any instant is
given, and a plane can be drawn perpendicular to that direction.
The line of intersection of the planes perpendicular to the paths of the two connected
points at a given instant is the instantaneous axis of the link at that instant; and the
velocities of the connected points are directly as their distances from that axis.
In drawing on a plane surface, the two planes
perpendicular to the paths of the connected points
are represented by two lines (being their sections by
a plane normal to them), and the instantaneous axis
by a point (fig. 110); and, should the length of the
two lines render it impracticable to produce them
until they actually intersect, the velocity ratio of the
connected points may be found by the principle that
it is equal to the ratio of the segments which a line
parallel to the line of connexion cuts off from any
Fig. 110.
two lines drawn from a given point, perpendicular
respectively to the paths of the connected points.
occasion, will appear in the sequel.
§ 62. Coupling of Parallel Axes.—Two or more parallel shafts (such as those of a
locomotive engine, with two or more pairs of driving wheels) are made to rotate with
constantly equal angular velocities by having equal cranks, which are maintained parallel
by a coupling-rod of such a length that the line of connexion is equal to the distance
between the axes. The cranks pass their dead-points simultaneously. To obviate the
unsteadiness of motion which this tends to cause, the shafts are provided with a second set
of cranks at right angles to the first, connected by means of a similar coupling-rod, so that
one set of cranks pass their dead points at the instant when the other set are farthest from
theirs.
§ 63. Comparative Motion of Connected Points.—As the link is a rigid body, it is
obvious that its action in communicating motion may be determined by finding the
comparative motion of the connected points, and this is often the most convenient method
of proceeding.
If a connected point belongs to a turning piece, the direction of its motion at a given
instant is perpendicular to the plane containing the axis and crank-arm of the piece. If a
connected point belongs to a shifting piece, the direction of its motion at any instant is
given, and a plane can be drawn perpendicular to that direction.
The line of intersection of the planes perpendicular to the paths of the two connected
points at a given instant is the instantaneous axis of the link at that instant; and the
velocities of the connected points are directly as their distances from that axis.
In drawing on a plane surface, the two planes
perpendicular to the paths of the connected points
are represented by two lines (being their sections by
a plane normal to them), and the instantaneous axis
by a point (fig. 110); and, should the length of the
two lines render it impracticable to produce them
until they actually intersect, the velocity ratio of the
connected points may be found by the principle that
it is equal to the ratio of the segments which a line
parallel to the line of connexion cuts off from any
Fig. 110.
two lines drawn from a given point, perpendicular
respectively to the paths of the connected points.
Page 532
To illustrate this by one example. Let C1 be the axis, and T1 the connected point of the
beam of a steam-engine; T1T2 the connecting or crank-rod; T2 the other connected point,
and the centre of the crank-pin; C2 the axis of the crank and its shaft. Let v1 denote the
velocity of T1 at any given instant; v2 that of T2. To find the ratio of these velocities,
produce C1T1, C2T2 till they intersect in K; K is the instantaneous axis of the connecting
rod, and the velocity ratio is
v1 : v2 :: KT1 : KT2.
(35)
Should K be inconveniently far off, draw any triangle with its sides respectively parallel to
C1T1, C2T2 and T1T2; the ratio of the two sides first mentioned will be the velocity ratio
required. For example, draw C2A parallel to C1T1, cutting T1T2 in A; then
v1 : v2 :: C2A : C2T2.
(36)
§ 64. Eccentric.—An eccentric circular disk fixed on a shaft, and used to give a
reciprocating motion to a rod, is in effect a crank-pin of sufficiently large diameter to
surround the shaft, and so to avoid the weakening of the shaft which would arise from
bending it so as to form an ordinary crank. The centre of the eccentric is its connected
point; and its eccentricity, or the distance from that centre to the axis of the shaft, is its
crank-arm.
An eccentric may be made capable of having its eccentricity altered by means of an
adjusting screw, so as to vary the extent of the reciprocating motion which it
communicates.
§ 65. Reciprocating Pieces—Stroke—Dead-Points.—The distance between the
extremities of the path of the connected point in a reciprocating piece (such as the piston of
a steam-engine) is called the stroke or length of stroke of that piece. When it is connected
with a continuously turning piece (such as the crank of a steam-engine) the ends of the
stroke of the reciprocating piece correspond to the dead-points of the path of the connected
point of the turning piece, where the line of connexion is continuous with or coincides with
the crank-arm.
Let S be the length of stroke of the reciprocating piece, L the length of the line of
connexion, and R the crank-arm of the continuously turning piece. Then, if the two ends of
the stroke be in one straight line with the axis of the crank,
S = 2R;
(37)
beam of a steam-engine; T1T2 the connecting or crank-rod; T2 the other connected point,
and the centre of the crank-pin; C2 the axis of the crank and its shaft. Let v1 denote the
velocity of T1 at any given instant; v2 that of T2. To find the ratio of these velocities,
produce C1T1, C2T2 till they intersect in K; K is the instantaneous axis of the connecting
rod, and the velocity ratio is
v1 : v2 :: KT1 : KT2.
(35)
Should K be inconveniently far off, draw any triangle with its sides respectively parallel to
C1T1, C2T2 and T1T2; the ratio of the two sides first mentioned will be the velocity ratio
required. For example, draw C2A parallel to C1T1, cutting T1T2 in A; then
v1 : v2 :: C2A : C2T2.
(36)
§ 64. Eccentric.—An eccentric circular disk fixed on a shaft, and used to give a
reciprocating motion to a rod, is in effect a crank-pin of sufficiently large diameter to
surround the shaft, and so to avoid the weakening of the shaft which would arise from
bending it so as to form an ordinary crank. The centre of the eccentric is its connected
point; and its eccentricity, or the distance from that centre to the axis of the shaft, is its
crank-arm.
An eccentric may be made capable of having its eccentricity altered by means of an
adjusting screw, so as to vary the extent of the reciprocating motion which it
communicates.
§ 65. Reciprocating Pieces—Stroke—Dead-Points.—The distance between the
extremities of the path of the connected point in a reciprocating piece (such as the piston of
a steam-engine) is called the stroke or length of stroke of that piece. When it is connected
with a continuously turning piece (such as the crank of a steam-engine) the ends of the
stroke of the reciprocating piece correspond to the dead-points of the path of the connected
point of the turning piece, where the line of connexion is continuous with or coincides with
the crank-arm.
Let S be the length of stroke of the reciprocating piece, L the length of the line of
connexion, and R the crank-arm of the continuously turning piece. Then, if the two ends of
the stroke be in one straight line with the axis of the crank,
S = 2R;
(37)
Page 533
and if these ends be not in one straight line with that axis, then S, L − R, and L + R, are the
three sides of a triangle, having the angle opposite S at that axis; so that, if θ be the
supplement of the arc between the dead-points,
S2 = 2 (L2 + R2) − 2 (L2 − R2) cos θ,
2L2 + 2R2 − S2
cos θ = .
2 (L2 − R2) (38)
§ 66. Coupling of Intersecting Axes—Hooke’s
Universal Joint.—Intersecting axes are coupled by
a contrivance of Hooke’s, known as the “universal
joint,” which belongs to the class of linkwork (see
fig. 111). Let O be the point of intersection of the
axes OC1, OC2, and θ their angle of inclination to
each other. The pair of shafts C1, C2 terminate in a
pair of forks F1, F2 in bearings at the extremities of Fig. 111.
which turn the gudgeons at the ends of the arms of
a rectangular cross, having its centre at O. This cross is the link; the connected points are
the centres of the bearings F1, F2. At each instant each of those points moves at right angles
to the central plane of its shaft and fork, therefore the line of intersection of the central
planes of the two forks at any instant is the instantaneous axis of the cross, and the velocity
ratio of the points F1, F2 (which, as the forks are equal, is also the angular velocity ratio of
the shafts) is equal to the ratio of the distances of those points from that instantaneous axis.
The mean value of that velocity ratio is that of equality, for each successive quarter-turn is
made by both shafts in the same time; but its actual value fluctuates between the limits:—
α2 1
= when F1 is the plane of OC1C2
α1 cos θ
α2
and = cos θ when F2 is in that plane.
α1 (39)
Its value at intermediate instants is given by the following equations: let φ1, φ2 be the
angles respectively made by the central planes of the forks and shafts with the plane OC1C2
at a given instant; then
cos θ = tan φ1 tan φ2,
α2 dφ2 tan φ1 + cot φ1
=− = .
α1 dφ1 tan φ2 + cot φ2 (40)
three sides of a triangle, having the angle opposite S at that axis; so that, if θ be the
supplement of the arc between the dead-points,
S2 = 2 (L2 + R2) − 2 (L2 − R2) cos θ,
2L2 + 2R2 − S2
cos θ = .
2 (L2 − R2) (38)
§ 66. Coupling of Intersecting Axes—Hooke’s
Universal Joint.—Intersecting axes are coupled by
a contrivance of Hooke’s, known as the “universal
joint,” which belongs to the class of linkwork (see
fig. 111). Let O be the point of intersection of the
axes OC1, OC2, and θ their angle of inclination to
each other. The pair of shafts C1, C2 terminate in a
pair of forks F1, F2 in bearings at the extremities of Fig. 111.
which turn the gudgeons at the ends of the arms of
a rectangular cross, having its centre at O. This cross is the link; the connected points are
the centres of the bearings F1, F2. At each instant each of those points moves at right angles
to the central plane of its shaft and fork, therefore the line of intersection of the central
planes of the two forks at any instant is the instantaneous axis of the cross, and the velocity
ratio of the points F1, F2 (which, as the forks are equal, is also the angular velocity ratio of
the shafts) is equal to the ratio of the distances of those points from that instantaneous axis.
The mean value of that velocity ratio is that of equality, for each successive quarter-turn is
made by both shafts in the same time; but its actual value fluctuates between the limits:—
α2 1
= when F1 is the plane of OC1C2
α1 cos θ
α2
and = cos θ when F2 is in that plane.
α1 (39)
Its value at intermediate instants is given by the following equations: let φ1, φ2 be the
angles respectively made by the central planes of the forks and shafts with the plane OC1C2
at a given instant; then
cos θ = tan φ1 tan φ2,
α2 dφ2 tan φ1 + cot φ1
=− = .
α1 dφ1 tan φ2 + cot φ2 (40)
Page 534
§ 67. Intermittent Linkwork—Click and Ratchet.—A click acting upon a ratchet-wheel or
rack, which it pushes or pulls through a certain arc at each forward stroke and leaves at rest
at each backward stroke, is an example of intermittent linkwork. During the forward stroke
the action of the click is governed by the principles of linkwork; during the backward
stroke that action ceases. A catch or pall, turning on a fixed axis, prevents the ratchet-wheel
or rack from reversing its motion.
Division 5.—Trains of Mechanism.
§ 68. General Principles..—A train of mechanism consists of a series of pieces each of
which is follower to that which drives it and driver to that which follows it.
The comparative motion of the first driver and last follower is obtained by combining
the proportions expressing by their terms the velocity ratios and by their signs the
directional relations of the several elementary combinations of which the train consists.
§ 69. Trains of Wheelwork.—Let A1, A2, A3, &c., Am−1, Am denote a series of axes, and
α1, α2, α3, &c., αm−1, αm their angular velocities. Let the axis A1 carry a wheel of N1 teeth,
driving a wheel of n2 teeth on the axis A2, which carries also a wheel of N2 teeth, driving a
wheel of n3 teeth on the axis A3, and so on; the numbers of teeth in drivers being denoted
by N′s, and in followers by n’s, and the axes to which the wheels are fixed being denoted
by numbers. Then the resulting velocity ratio is denoted by
αm α2 α3 αm N1 · N2 ... &c. ... Nm−1
= · · &c. ... = ;
α1 α1 α2 αm−1 n2 · n3 ... &c. ... nm (41)
that is to say, the velocity ratio of the last and first axes is the ratio of the product of the
numbers of teeth in the drivers to the product of the numbers of teeth in the followers.
Supposing all the wheels to be in outside gearing, then, as each elementary combination
reverses the direction of rotation, and as the number of elementary combinations m − 1 is
one less than the number of axes m, it is evident that if m is odd the direction of rotation is
preserved, and if even reversed.
It is often a question of importance to determine the number of teeth in a train of wheels
best suited for giving a determinate velocity ratio to two axes. It was shown by Young that,
to do this with the least total number of teeth, the velocity ratio of each elementary
combination should approximate as nearly as possible to 3.59. This would in many cases
give too many axes; and, as a useful practical rule, it may be laid down that from 3 to 6
ought to be the limit of the velocity ratio of an elementary combination in wheel-work. The
smallest number of teeth in a pinion for epicycloidal teeth ought to be twelve (see § 49)—
rack, which it pushes or pulls through a certain arc at each forward stroke and leaves at rest
at each backward stroke, is an example of intermittent linkwork. During the forward stroke
the action of the click is governed by the principles of linkwork; during the backward
stroke that action ceases. A catch or pall, turning on a fixed axis, prevents the ratchet-wheel
or rack from reversing its motion.
Division 5.—Trains of Mechanism.
§ 68. General Principles..—A train of mechanism consists of a series of pieces each of
which is follower to that which drives it and driver to that which follows it.
The comparative motion of the first driver and last follower is obtained by combining
the proportions expressing by their terms the velocity ratios and by their signs the
directional relations of the several elementary combinations of which the train consists.
§ 69. Trains of Wheelwork.—Let A1, A2, A3, &c., Am−1, Am denote a series of axes, and
α1, α2, α3, &c., αm−1, αm their angular velocities. Let the axis A1 carry a wheel of N1 teeth,
driving a wheel of n2 teeth on the axis A2, which carries also a wheel of N2 teeth, driving a
wheel of n3 teeth on the axis A3, and so on; the numbers of teeth in drivers being denoted
by N′s, and in followers by n’s, and the axes to which the wheels are fixed being denoted
by numbers. Then the resulting velocity ratio is denoted by
αm α2 α3 αm N1 · N2 ... &c. ... Nm−1
= · · &c. ... = ;
α1 α1 α2 αm−1 n2 · n3 ... &c. ... nm (41)
that is to say, the velocity ratio of the last and first axes is the ratio of the product of the
numbers of teeth in the drivers to the product of the numbers of teeth in the followers.
Supposing all the wheels to be in outside gearing, then, as each elementary combination
reverses the direction of rotation, and as the number of elementary combinations m − 1 is
one less than the number of axes m, it is evident that if m is odd the direction of rotation is
preserved, and if even reversed.
It is often a question of importance to determine the number of teeth in a train of wheels
best suited for giving a determinate velocity ratio to two axes. It was shown by Young that,
to do this with the least total number of teeth, the velocity ratio of each elementary
combination should approximate as nearly as possible to 3.59. This would in many cases
give too many axes; and, as a useful practical rule, it may be laid down that from 3 to 6
ought to be the limit of the velocity ratio of an elementary combination in wheel-work. The
smallest number of teeth in a pinion for epicycloidal teeth ought to be twelve (see § 49)—
Page 535
but it is better, for smoothness of motion, not to go below fifteen; and for involute teeth the
smallest number is about twenty-four.
Let B/C be the velocity ratio required, reduced to its least terms, and let B be greater
than C. If B/C is not greater than 6, and C lies between the prescribed minimum number of
teeth (which may be called t) and its double 2t, then one pair of wheels will answer the
purpose, and B and C will themselves be the numbers required. Should B and C be
inconveniently large, they are, if possible, to be resolved into factors, and those factors (or
if they are too small, multiples of them) used for the number of teeth. Should B or C, or
both, be at once inconveniently large and prime, then, instead of the exact ratio B/C some
ratio approximating to that ratio, and capable of resolution into convenient factors, is to be
found by the method of continued fractions.
Should B/C be greater than 6, the best number of elementary combinations m − 1 will lie
between
log B − log C log B − log C
and .
log 6 log 3
Then, if possible, B and C themselves are to be resolved each into m − 1 factors
(counting 1 as a factor), which factors, or multiples of them, shall be not less than t nor
greater than 6t; or if B and C contain inconveniently large prime factors, an approximate
velocity ratio, found by the method of continued fractions, is to be substituted for B/C as
before.
So far as the resultant velocity ratio is concerned, the order of the drivers N and of the
followers n is immaterial: but to secure equable wear of the teeth, as explained in § 44, the
wheels ought to be so arranged that, for each elementary combination, the greatest common
divisor of N and n shall be either 1, or as small as possible.
§ 70. Double Hooke’s Coupling.—It has been shown in § 66 that the velocity ratio of a
pair of shafts coupled by a universal joint fluctuates between the limits cos θ and 1/cos θ.
Hence one or both of the shafts must have a vibratory and unsteady motion, injurious to the
mechanism and framework. To obviate this evil a short intermediate shaft is introduced,
making equal angles with the first and last shaft, coupled with each of them by a Hooke’s
joint, and having its own two forks in the same plane. Let α1, α2, α3 be the angular
velocities of the first, intermediate, and last shaft in this train of two Hooke’s couplings.
Then, from the principles of § 60 it is evident that at each instant α2/α1 = α2/α3, and
consequently that α3 = α1; so that the fluctuations of angular velocity ratio caused by the
first coupling are exactly neutralized by the second, and the first and last shafts have equal
angular velocities at each instant.
smallest number is about twenty-four.
Let B/C be the velocity ratio required, reduced to its least terms, and let B be greater
than C. If B/C is not greater than 6, and C lies between the prescribed minimum number of
teeth (which may be called t) and its double 2t, then one pair of wheels will answer the
purpose, and B and C will themselves be the numbers required. Should B and C be
inconveniently large, they are, if possible, to be resolved into factors, and those factors (or
if they are too small, multiples of them) used for the number of teeth. Should B or C, or
both, be at once inconveniently large and prime, then, instead of the exact ratio B/C some
ratio approximating to that ratio, and capable of resolution into convenient factors, is to be
found by the method of continued fractions.
Should B/C be greater than 6, the best number of elementary combinations m − 1 will lie
between
log B − log C log B − log C
and .
log 6 log 3
Then, if possible, B and C themselves are to be resolved each into m − 1 factors
(counting 1 as a factor), which factors, or multiples of them, shall be not less than t nor
greater than 6t; or if B and C contain inconveniently large prime factors, an approximate
velocity ratio, found by the method of continued fractions, is to be substituted for B/C as
before.
So far as the resultant velocity ratio is concerned, the order of the drivers N and of the
followers n is immaterial: but to secure equable wear of the teeth, as explained in § 44, the
wheels ought to be so arranged that, for each elementary combination, the greatest common
divisor of N and n shall be either 1, or as small as possible.
§ 70. Double Hooke’s Coupling.—It has been shown in § 66 that the velocity ratio of a
pair of shafts coupled by a universal joint fluctuates between the limits cos θ and 1/cos θ.
Hence one or both of the shafts must have a vibratory and unsteady motion, injurious to the
mechanism and framework. To obviate this evil a short intermediate shaft is introduced,
making equal angles with the first and last shaft, coupled with each of them by a Hooke’s
joint, and having its own two forks in the same plane. Let α1, α2, α3 be the angular
velocities of the first, intermediate, and last shaft in this train of two Hooke’s couplings.
Then, from the principles of § 60 it is evident that at each instant α2/α1 = α2/α3, and
consequently that α3 = α1; so that the fluctuations of angular velocity ratio caused by the
first coupling are exactly neutralized by the second, and the first and last shafts have equal
angular velocities at each instant.
Page 536
§ 71. Converging and Diverging Trains of Mechanism.—Two or more trains of
mechanism may converge into one—as when the two pistons of a pair of steam-engines,
each through its own connecting-rod, act upon one crank-shaft. One train of mechanism
may diverge into two or more—as when a single shaft, driven by a prime mover, carries
several pulleys, each of which drives a different machine. The principles of comparative
motion in such converging and diverging trains are the same as in simple trains.
Division 6.—Aggregate Combinations.
§ 72. General Principles.—Willis designated as “aggregate combinations” those
assemblages of pieces of mechanism in which the motion of one follower is the resultant of
component motions impressed on it by more than one driver. Two classes of aggregate
combinations may be distinguished which, though not different in their actual nature, differ
in the data which they present to the designer, and in the method of solution to be followed
in questions respecting them.
Class I. comprises those cases in which a piece A is not carried directly by the frame C,
but by another piece B, relatively to which the motion of A is given—the motion of the
piece B relatively to the frame C being also given. Then the motion of A relatively to the
frame C is the resultant of the motion of A relatively to B and of B relatively to C; and that
resultant is to be found by the principles already explained in Division 3 of this Chapter §§
27-32.
Class II. comprises those cases in which the motions of three points in one follower are
determined by their connexions with two or with three different drivers.
This classification is founded on the kinds of problems arising from the combinations.
Willis adopts another classification founded on the objects of the combinations, which
objects he divides into two classes, viz. (1) to produce aggregate velocity, or a velocity
which is the resultant of two or more components in the same path, and (2) to produce an
aggregate path—that is, to make a given point in a rigid body move in an assigned path by
communicating certain motions to other points in that body.
It is seldom that one of these effects is produced without at the same time producing the
other; but the classification of Willis depends upon which of those two effects, even
supposing them to occur together, is the practical object of the mechanism.
§ 73. Differential Windlass.—The axis C (fig. 112) carries a larger barrel AE and a
smaller barrel DB, rotating as one piece with the angular velocity α1 in the direction AE.
The pulley or sheave FG has a weight W hung to its centre. A cord has one end made fast
to and wrapped round the barrel AE; it passes from A under the sheave FG, and has the
mechanism may converge into one—as when the two pistons of a pair of steam-engines,
each through its own connecting-rod, act upon one crank-shaft. One train of mechanism
may diverge into two or more—as when a single shaft, driven by a prime mover, carries
several pulleys, each of which drives a different machine. The principles of comparative
motion in such converging and diverging trains are the same as in simple trains.
Division 6.—Aggregate Combinations.
§ 72. General Principles.—Willis designated as “aggregate combinations” those
assemblages of pieces of mechanism in which the motion of one follower is the resultant of
component motions impressed on it by more than one driver. Two classes of aggregate
combinations may be distinguished which, though not different in their actual nature, differ
in the data which they present to the designer, and in the method of solution to be followed
in questions respecting them.
Class I. comprises those cases in which a piece A is not carried directly by the frame C,
but by another piece B, relatively to which the motion of A is given—the motion of the
piece B relatively to the frame C being also given. Then the motion of A relatively to the
frame C is the resultant of the motion of A relatively to B and of B relatively to C; and that
resultant is to be found by the principles already explained in Division 3 of this Chapter §§
27-32.
Class II. comprises those cases in which the motions of three points in one follower are
determined by their connexions with two or with three different drivers.
This classification is founded on the kinds of problems arising from the combinations.
Willis adopts another classification founded on the objects of the combinations, which
objects he divides into two classes, viz. (1) to produce aggregate velocity, or a velocity
which is the resultant of two or more components in the same path, and (2) to produce an
aggregate path—that is, to make a given point in a rigid body move in an assigned path by
communicating certain motions to other points in that body.
It is seldom that one of these effects is produced without at the same time producing the
other; but the classification of Willis depends upon which of those two effects, even
supposing them to occur together, is the practical object of the mechanism.
§ 73. Differential Windlass.—The axis C (fig. 112) carries a larger barrel AE and a
smaller barrel DB, rotating as one piece with the angular velocity α1 in the direction AE.
The pulley or sheave FG has a weight W hung to its centre. A cord has one end made fast
to and wrapped round the barrel AE; it passes from A under the sheave FG, and has the
Page 537
other end wrapped round and made fast to the barrel BD. Required the
relation between the velocity of translation v2 of W and the angular
velocity α1 of the differential barrel.
In this case v2 is an aggregate velocity, produced by the joint action
of the two drivers AE and BD, transmitted by wrapping connectors to
FG, and combined by that sheave so as to act on the follower W, whose
motion is the same with that of the centre of FG.
The velocity of the point F is α1·AC, upward motion being
considered positive. The velocity of the point G is −α1·CB, downward
Fig. 112.
motion being negative. Hence the instantaneous axis of the sheave FG
is in the diameter FG, at the distance
FG AC − BC
·
2 AC + BC
from the centre towards G; the angular velocity of the sheave is
AC + BC
α2 = α 1 · ;
FG
and, consequently, the velocity of its centre is
FG AC − BC α1 (AC − BC)
v2 = α2 · · = ,
2 AC + BC 2 (42)
or the mean between the velocities of the two vertical parts of the cord.
If the cord be fixed to the framework at the point B, instead of being wound on a barrel,
the velocity of W is half that of AF.
A case containing several sheaves is called a block. A fall-block is attached to a fixed
point; a running-block is movable to and from a fall-block, with which it is connected by
two or more plies of a rope. The whole combination constitutes a tackle or purchase. (See
Pulleys for practical applications of these principles.)
§ 74. Differential Screw.—On the same axis let there be two screws of the respective
pitches p1 and p2, made in one piece, and rotating with the angular velocity α. Let this
piece be called B. Let the first screw turn in a fixed nut C, and the second in a sliding nut
A. The velocity of advance of B relatively to C is (according to § 32) αp1, and of A
relatively to B (according to § 57) −αp2; hence the velocity of A relatively to C is
relation between the velocity of translation v2 of W and the angular
velocity α1 of the differential barrel.
In this case v2 is an aggregate velocity, produced by the joint action
of the two drivers AE and BD, transmitted by wrapping connectors to
FG, and combined by that sheave so as to act on the follower W, whose
motion is the same with that of the centre of FG.
The velocity of the point F is α1·AC, upward motion being
considered positive. The velocity of the point G is −α1·CB, downward
Fig. 112.
motion being negative. Hence the instantaneous axis of the sheave FG
is in the diameter FG, at the distance
FG AC − BC
·
2 AC + BC
from the centre towards G; the angular velocity of the sheave is
AC + BC
α2 = α 1 · ;
FG
and, consequently, the velocity of its centre is
FG AC − BC α1 (AC − BC)
v2 = α2 · · = ,
2 AC + BC 2 (42)
or the mean between the velocities of the two vertical parts of the cord.
If the cord be fixed to the framework at the point B, instead of being wound on a barrel,
the velocity of W is half that of AF.
A case containing several sheaves is called a block. A fall-block is attached to a fixed
point; a running-block is movable to and from a fall-block, with which it is connected by
two or more plies of a rope. The whole combination constitutes a tackle or purchase. (See
Pulleys for practical applications of these principles.)
§ 74. Differential Screw.—On the same axis let there be two screws of the respective
pitches p1 and p2, made in one piece, and rotating with the angular velocity α. Let this
piece be called B. Let the first screw turn in a fixed nut C, and the second in a sliding nut
A. The velocity of advance of B relatively to C is (according to § 32) αp1, and of A
relatively to B (according to § 57) −αp2; hence the velocity of A relatively to C is
Page 538
α (p1 − p2),
(46)
being the same with the velocity of advance of a screw of the pitch p1 − p2. This
combination, called Hunter’s or the differential screw, combines the strength of a large
thread with the slowness of motion due to a small one.
§ 75. Epicyclic Trains.—The term epicyclic train is used by Willis to denote a train of
wheels carried by an arm, and having certain rotations relatively to that arm, which itself
rotates. The arm may either be driven by the wheels or assist in driving them. The
comparative motions of the wheels and of the arm, and the aggregate paths traced by
points in the wheels, are determined by the principles of the composition of rotations, and
of the description of rolling curves, explained in §§ 30, 31.
§ 76. Link Motion.—A slide valve operated by a link motion receives an aggregate
motion from the mechanism driving it. (See Steam-engine for a description of this and
other types of mechanism of this class.)
§ 77. Parallel Motions.—A parallel motion is a
combination of turning pieces in mechanism designed
to guide the motion of a reciprocating piece either
exactly or approximately in a straight line, so as to
avoid the friction which arises from the use of straight
guides for that purpose.
Fig. 113 represents an exact parallel motion, first
proposed, it is believed, by Scott Russell. The arm CD
turns on the axis C, and is jointed at D to the middle of Fig. 113.
the bar ADB, whose length is double of that of CD,
and one of whose ends B is jointed to a slider, sliding in straight guides along the line CB.
Draw BE perpendicular to CB, cutting CD produced in E, then E is the instantaneous axis
of the bar ADB; and the direction of motion of A is at every instant perpendicular to EA—
that is, along the straight line ACa. While the stroke of A is ACa, extending to equal
distances on either side of C, and equal to twice the chord of the arc Dd, the stroke of B is
only equal to twice the sagitta; and thus A is guided through a comparatively long stroke by
the sliding of B through a comparatively short stroke, and by rotatory motions at the joints
C, D, B.
(46)
being the same with the velocity of advance of a screw of the pitch p1 − p2. This
combination, called Hunter’s or the differential screw, combines the strength of a large
thread with the slowness of motion due to a small one.
§ 75. Epicyclic Trains.—The term epicyclic train is used by Willis to denote a train of
wheels carried by an arm, and having certain rotations relatively to that arm, which itself
rotates. The arm may either be driven by the wheels or assist in driving them. The
comparative motions of the wheels and of the arm, and the aggregate paths traced by
points in the wheels, are determined by the principles of the composition of rotations, and
of the description of rolling curves, explained in §§ 30, 31.
§ 76. Link Motion.—A slide valve operated by a link motion receives an aggregate
motion from the mechanism driving it. (See Steam-engine for a description of this and
other types of mechanism of this class.)
§ 77. Parallel Motions.—A parallel motion is a
combination of turning pieces in mechanism designed
to guide the motion of a reciprocating piece either
exactly or approximately in a straight line, so as to
avoid the friction which arises from the use of straight
guides for that purpose.
Fig. 113 represents an exact parallel motion, first
proposed, it is believed, by Scott Russell. The arm CD
turns on the axis C, and is jointed at D to the middle of Fig. 113.
the bar ADB, whose length is double of that of CD,
and one of whose ends B is jointed to a slider, sliding in straight guides along the line CB.
Draw BE perpendicular to CB, cutting CD produced in E, then E is the instantaneous axis
of the bar ADB; and the direction of motion of A is at every instant perpendicular to EA—
that is, along the straight line ACa. While the stroke of A is ACa, extending to equal
distances on either side of C, and equal to twice the chord of the arc Dd, the stroke of B is
only equal to twice the sagitta; and thus A is guided through a comparatively long stroke by
the sliding of B through a comparatively short stroke, and by rotatory motions at the joints
C, D, B.
Page 539
Fig. 114. Fig. 115.
§ 78.* An example of an approximate straight-line motion composed of three bars fixed
to a frame is shown in fig. 114. It is due to P. L. Tchebichev of St Petersburg. The links AB
and CD are equal in length and are centred respectively at A and C. The ends D and B are
joined by a link DB. If the respective lengths are made in the proportions AC : CD : DB =
1 : 1.3 : 0.4 the middle point P of DB will describe an approximately straight line parallel
to AC within limits of length about equal to AC. C. N. Peaucellier, a French engineer
officer, was the first, in 1864, to invent a linkwork with which an exact straight line could
be drawn. The linkwork is shown in fig. 115, from which it will be seen that it consists of a
rhombus of four equal bars ABCD, jointed at opposite corners with two equal bars BE and
DE. The seventh link AF is equal in length to halt the distance EA when the mechanism is
in its central position. The points E and F are fixed. It can be proved that the point C
always moves in a straight line at right angles to the line EF. The more general property of
the mechanism corresponding to proportions between the lengths FA and EF other than that
of equality is that the curve described by the point C is the inverse of the curve described
by A. There are other arrangements of bars giving straight-line motions, and these
arrangements together with the general properties of mechanisms of this kind are discussed
in How to Draw a Straight Line by A. B. Kempe (London, 1877).
§ 79.* The Pantograph.—If a parallelogram
of links (fig. 116), be fixed at any one point a in
any one of the links produced in either direction,
and if any straight line be drawn from this point
to cut the links in the points b and c, then the
points a, b, c will be in a straight line for all Fig. 116.
positions of the mechanism, and if the point b be
guided in any curve whatever, the point c will
trace a similar curve to a scale enlarged in the
ratio ab : ac. This property of the parallelogram
§ 78.* An example of an approximate straight-line motion composed of three bars fixed
to a frame is shown in fig. 114. It is due to P. L. Tchebichev of St Petersburg. The links AB
and CD are equal in length and are centred respectively at A and C. The ends D and B are
joined by a link DB. If the respective lengths are made in the proportions AC : CD : DB =
1 : 1.3 : 0.4 the middle point P of DB will describe an approximately straight line parallel
to AC within limits of length about equal to AC. C. N. Peaucellier, a French engineer
officer, was the first, in 1864, to invent a linkwork with which an exact straight line could
be drawn. The linkwork is shown in fig. 115, from which it will be seen that it consists of a
rhombus of four equal bars ABCD, jointed at opposite corners with two equal bars BE and
DE. The seventh link AF is equal in length to halt the distance EA when the mechanism is
in its central position. The points E and F are fixed. It can be proved that the point C
always moves in a straight line at right angles to the line EF. The more general property of
the mechanism corresponding to proportions between the lengths FA and EF other than that
of equality is that the curve described by the point C is the inverse of the curve described
by A. There are other arrangements of bars giving straight-line motions, and these
arrangements together with the general properties of mechanisms of this kind are discussed
in How to Draw a Straight Line by A. B. Kempe (London, 1877).
§ 79.* The Pantograph.—If a parallelogram
of links (fig. 116), be fixed at any one point a in
any one of the links produced in either direction,
and if any straight line be drawn from this point
to cut the links in the points b and c, then the
points a, b, c will be in a straight line for all Fig. 116.
positions of the mechanism, and if the point b be
guided in any curve whatever, the point c will
trace a similar curve to a scale enlarged in the
ratio ab : ac. This property of the parallelogram
Page 540
is utilized in the construction of the pantograph,
an instrument used for obtaining a copy of a map
or drawing on a different scale. Professor J. J.
Sylvester discovered that this property of the
parallelogram is not confined to points lying in
one line with the fixed point. Thus if b (fig. 117)
Fig. 117.
be any point on the link CD, and if a point c be
taken on the link DE such that the triangles CbD
and DcE are similar and similarly situated with regard to their respective links, then the
ratio of the distances ab and ac is constant, and the angle bac is constant for all positions of
the mechanism; so that, if b is guided in any curve, the point c will describe a similar curve
turned through an angle bac, the scales of the curves being in the ratio ab to ac. Sylvester
called an instrument based on this property a plagiograph or a skew pantograph.
The combination of the parallelogram with a straight-line motion, for guiding one of the
points in a straight line, is illustrated in Watt’s parallel motion for steam-engines. (See
Steam-engine.)
§ 80.* The Reuleaux System of Analysis.—If two pieces, A and B, (fig. 118) are jointed
together by a pin, the pin being fixed, say, to A, the only relative motion possible between
the pieces is one of turning about the axis of the pin. Whatever motion the pair of pieces
may have as a whole each separate piece shares in common, and this common motion in no
way affects the relative motion of A and B. The motion of one piece is said to be
completely constrained relatively to the other piece. Again, the pieces A and B (fig. 119)
are paired together as a slide, and the only relative motion possible between them now is
that of sliding, and therefore the motion of one relatively to the other is completely
constrained. The pieces may be paired together as a screw and nut, in which case the
relative motion is compounded of turning with sliding.
Fig. 118. Fig. 119.
These combinations of pieces are known individually as kinematic pairs of elements, or
briefly kinematic pairs. The three pairs mentioned above have each the peculiarity that
an instrument used for obtaining a copy of a map
or drawing on a different scale. Professor J. J.
Sylvester discovered that this property of the
parallelogram is not confined to points lying in
one line with the fixed point. Thus if b (fig. 117)
Fig. 117.
be any point on the link CD, and if a point c be
taken on the link DE such that the triangles CbD
and DcE are similar and similarly situated with regard to their respective links, then the
ratio of the distances ab and ac is constant, and the angle bac is constant for all positions of
the mechanism; so that, if b is guided in any curve, the point c will describe a similar curve
turned through an angle bac, the scales of the curves being in the ratio ab to ac. Sylvester
called an instrument based on this property a plagiograph or a skew pantograph.
The combination of the parallelogram with a straight-line motion, for guiding one of the
points in a straight line, is illustrated in Watt’s parallel motion for steam-engines. (See
Steam-engine.)
§ 80.* The Reuleaux System of Analysis.—If two pieces, A and B, (fig. 118) are jointed
together by a pin, the pin being fixed, say, to A, the only relative motion possible between
the pieces is one of turning about the axis of the pin. Whatever motion the pair of pieces
may have as a whole each separate piece shares in common, and this common motion in no
way affects the relative motion of A and B. The motion of one piece is said to be
completely constrained relatively to the other piece. Again, the pieces A and B (fig. 119)
are paired together as a slide, and the only relative motion possible between them now is
that of sliding, and therefore the motion of one relatively to the other is completely
constrained. The pieces may be paired together as a screw and nut, in which case the
relative motion is compounded of turning with sliding.
Fig. 118. Fig. 119.
These combinations of pieces are known individually as kinematic pairs of elements, or
briefly kinematic pairs. The three pairs mentioned above have each the peculiarity that
Page 541
contact between the two pieces forming the pair is distributed over a surface. Kinematic
pairs which have surface contact are classified as lower pairs. Kinematic pairs in which
contact takes place along a line only are classified as higher pairs. A pair of spur wheels in
gear is an example of a higher pair, because the wheels have contact between their teeth
along lines only.
A kinematic link of the simplest form is made by joining up the halves of two kinematic
pairs by means of a rigid link. Thus if A1B1 represent a turning pair, and A2B2 a second
turning pair, the rigid link formed by joining B1 to B2 is a kinematic link. Four links of this
kind are shown in fig. 120 joined up to form a closed kinematic chain.
In order that a kinematic chain may be
made the basis of a mechanism, every point
in any link of it must be completely
constrained with regard to every other link.
Thus in fig. 120 the motion of a point a in
the link A1A2 is completely constrained with
regard to the link B1B4 by the turning pair
Fig. 120.
A1B1, and it can be proved that the motion
of a relatively to the non-adjacent link A3A4 is completely constrained, and therefore the
four-bar chain, as it is called, can be and is used as the basis of many mechanisms. Another
way of considering the question of constraint is to imagine any one link of the chain fixed;
then, however the chain be moved, the path of a point, as a, will always remain the same.
In a five-bar chain, if a is a point in a link non-adjacent to a fixed link, its path is
indeterminate. Still another way of stating the matter is to say that, if any one link in the
chain be fixed, any point in the chain must have only one degree of freedom. In a five-bar
chain a point, as a, in a link non-adjacent to the fixed link has two degrees of freedom and
the chain cannot therefore be used for a mechanism. These principles may be applied to
examine any possible combination of links forming a kinematic chain in order to test its
suitability for use as a mechanism. Compound chains are formed by the superposition of
two or more simple chains, and in these more complex chains links will be found carrying
three, or even more, halves of kinematic pairs. The Joy valve gear mechanism is a good
example of a compound kinematic chain.
A chain built up of three turning pairs
and one sliding pair, and known as the
slider crank chain, is shown in fig. 121.
It will be seen that the piece A1 can only
slide relatively to the piece B1, and these
two pieces therefore form the sliding
pairs which have surface contact are classified as lower pairs. Kinematic pairs in which
contact takes place along a line only are classified as higher pairs. A pair of spur wheels in
gear is an example of a higher pair, because the wheels have contact between their teeth
along lines only.
A kinematic link of the simplest form is made by joining up the halves of two kinematic
pairs by means of a rigid link. Thus if A1B1 represent a turning pair, and A2B2 a second
turning pair, the rigid link formed by joining B1 to B2 is a kinematic link. Four links of this
kind are shown in fig. 120 joined up to form a closed kinematic chain.
In order that a kinematic chain may be
made the basis of a mechanism, every point
in any link of it must be completely
constrained with regard to every other link.
Thus in fig. 120 the motion of a point a in
the link A1A2 is completely constrained with
regard to the link B1B4 by the turning pair
Fig. 120.
A1B1, and it can be proved that the motion
of a relatively to the non-adjacent link A3A4 is completely constrained, and therefore the
four-bar chain, as it is called, can be and is used as the basis of many mechanisms. Another
way of considering the question of constraint is to imagine any one link of the chain fixed;
then, however the chain be moved, the path of a point, as a, will always remain the same.
In a five-bar chain, if a is a point in a link non-adjacent to a fixed link, its path is
indeterminate. Still another way of stating the matter is to say that, if any one link in the
chain be fixed, any point in the chain must have only one degree of freedom. In a five-bar
chain a point, as a, in a link non-adjacent to the fixed link has two degrees of freedom and
the chain cannot therefore be used for a mechanism. These principles may be applied to
examine any possible combination of links forming a kinematic chain in order to test its
suitability for use as a mechanism. Compound chains are formed by the superposition of
two or more simple chains, and in these more complex chains links will be found carrying
three, or even more, halves of kinematic pairs. The Joy valve gear mechanism is a good
example of a compound kinematic chain.
A chain built up of three turning pairs
and one sliding pair, and known as the
slider crank chain, is shown in fig. 121.
It will be seen that the piece A1 can only
slide relatively to the piece B1, and these
two pieces therefore form the sliding
Page 542
pair. The piece A1 carries the pin B4, Fig. 121.
which is one half of the turning pair A4
B4. The piece A1 together with the pin B4 therefore form a kinematic link A1B4. The other
links of the chain are, B1A2, B2B3, A3A4. In order to convert a chain into a mechanism it is
necessary to fix one link in it. Any one of the links may be fixed. It follows therefore that
there are as many possible mechanisms as there are links in the chain. For example, there is
a well-known mechanism corresponding to the fixing of three of the four links of the slider
crank chain (fig. 121). If the link d is fixed the chain at once becomes the mechanism of the
ordinary steam engine; if the link e is fixed the mechanism obtained is that of the
oscillating cylinder steam engine; if the link c is fixed the mechanism becomes either the
Whitworth quick-return motion or the slot-bar motion, depending upon the proportion
between the lengths of the links c and e. These different mechanisms are called inversions
of the slider crank chain. What was the fixed framework of the mechanism in one case
becomes a moving link in an inversion.
The Reuleaux system, therefore, consists essentially of the analysis of every mechanism
into a kinematic chain, and since each link of the chain may be the fixed frame of a
mechanism quite diverse mechanisms are found to be merely inversions of the same
kinematic chain. Franz Reuleaux’s Kinematics of Machinery, translated by Sir A. B. W.
Kennedy (London, 1876), is the book in which the system is set forth in all its
completeness. In Mechanics of Machinery, by Sir A. B. W. Kennedy (London, 1886), the
system was used for the first time in an English textbook, and now it has found its way into
most modern textbooks relating to the subject of mechanism.
§ 81.* Centrodes, Instantaneous Centres, Velocity Image, Velocity Diagram.—Problems
concerning the relative motion of the several parts of a kinematic chain may be considered
in two ways, in addition to the way hitherto used in this article and based on the principle
of § 34. The first is by the method of instantaneous centres, already exemplified in § 63,
and rolling centroids, developed by Reuleaux in connexion with his method of analysis.
The second is by means of Professor R. H. Smith’s method already referred to in § 23.
Method 1.—By reference to § 30 it will be seen that the motion of a cylinder rolling on a
fixed cylinder is one of rotation about an instantaneous axis T, and that the velocity both as
regards direction and magnitude is the same as if the rolling piece B were for the instant
turning about a fixed axis coincident with the instantaneous axis. If the rolling cylinder B
and its path A now be assumed to receive a common plane motion, what was before the
velocity of the point P becomes the velocity of P relatively to the cylinder A, since the
motion of B relatively to A still takes place about the instantaneous axis T. If B stops
rolling, then the two cylinders continue to move as though they were parts of a rigid body.
Notice that the shape of either rolling curve (fig. 91 or 92) may be found by considering
which is one half of the turning pair A4
B4. The piece A1 together with the pin B4 therefore form a kinematic link A1B4. The other
links of the chain are, B1A2, B2B3, A3A4. In order to convert a chain into a mechanism it is
necessary to fix one link in it. Any one of the links may be fixed. It follows therefore that
there are as many possible mechanisms as there are links in the chain. For example, there is
a well-known mechanism corresponding to the fixing of three of the four links of the slider
crank chain (fig. 121). If the link d is fixed the chain at once becomes the mechanism of the
ordinary steam engine; if the link e is fixed the mechanism obtained is that of the
oscillating cylinder steam engine; if the link c is fixed the mechanism becomes either the
Whitworth quick-return motion or the slot-bar motion, depending upon the proportion
between the lengths of the links c and e. These different mechanisms are called inversions
of the slider crank chain. What was the fixed framework of the mechanism in one case
becomes a moving link in an inversion.
The Reuleaux system, therefore, consists essentially of the analysis of every mechanism
into a kinematic chain, and since each link of the chain may be the fixed frame of a
mechanism quite diverse mechanisms are found to be merely inversions of the same
kinematic chain. Franz Reuleaux’s Kinematics of Machinery, translated by Sir A. B. W.
Kennedy (London, 1876), is the book in which the system is set forth in all its
completeness. In Mechanics of Machinery, by Sir A. B. W. Kennedy (London, 1886), the
system was used for the first time in an English textbook, and now it has found its way into
most modern textbooks relating to the subject of mechanism.
§ 81.* Centrodes, Instantaneous Centres, Velocity Image, Velocity Diagram.—Problems
concerning the relative motion of the several parts of a kinematic chain may be considered
in two ways, in addition to the way hitherto used in this article and based on the principle
of § 34. The first is by the method of instantaneous centres, already exemplified in § 63,
and rolling centroids, developed by Reuleaux in connexion with his method of analysis.
The second is by means of Professor R. H. Smith’s method already referred to in § 23.
Method 1.—By reference to § 30 it will be seen that the motion of a cylinder rolling on a
fixed cylinder is one of rotation about an instantaneous axis T, and that the velocity both as
regards direction and magnitude is the same as if the rolling piece B were for the instant
turning about a fixed axis coincident with the instantaneous axis. If the rolling cylinder B
and its path A now be assumed to receive a common plane motion, what was before the
velocity of the point P becomes the velocity of P relatively to the cylinder A, since the
motion of B relatively to A still takes place about the instantaneous axis T. If B stops
rolling, then the two cylinders continue to move as though they were parts of a rigid body.
Notice that the shape of either rolling curve (fig. 91 or 92) may be found by considering
Page 543
each fixed in turn and then tracing out the locus of the instantaneous axis. These rolling
cylinders are sometimes called axodes, and a section of an axode in a plane parallel to the
plane of motion is called a centrode. The axode is hence the locus of the instantaneous axis,
whilst the centrode is the locus of the instantaneous centre in any plane parallel to the plane
of motion. There is no restriction on the shape of these rolling axodes; they may have any
shape consistent with rolling (that is, no slipping is permitted), and the relative velocity of a
point P is still found by considering it with regard to the instantaneous centre.
Reuleaux has shown that the relative motion of any pair of non-adjacent links of a
kinematic chain is determined by the rolling together of two ideal cylindrical surfaces
(cylindrical being used here in the general sense), each of which may be assumed to be
formed by the extension of the material of the link to which it corresponds. These surfaces
have contact at the instantaneous axis, which is now called the instantaneous axis of the
two links concerned. To find the form of these surfaces corresponding to a particular pair of
non-adjacent links, consider each link of the pair fixed in turn, then the locus of the
instantaneous axis is the axode corresponding to the fixed link, or, considering a plane of
motion only, the locus of the instantaneous centre is the centrode corresponding to the fixed
link.
To find the instantaneous centre for a particular link corresponding to any given
configuration of the kinematic chain, it is only necessary to know the direction of motion
of any two points in the link, since lines through these points respectively at right angles to
their directions of motion intersect in the instantaneous centre.
To illustrate this principle, consider the
four-bar chain shown in fig. 122 made up
of the four links, a, b, c, d. Let a be the
fixed link, and consider the link c. Its
extremities are moving respectively in
directions at right angles to the links b and
d; hence produce the links b and d to meet
in the point Oac. This point is the
instantaneous centre of the motion of the
link c relatively to the fixed link a, a fact Fig. 122.
indicated by the suffix ac placed after the
letter O. The process being repeated for different values of the angle θ the curve through
the several points Oac is the centroid which may be imagined as formed by an extension of
the material of the link a. To find the corresponding centroid for the link c, fix c and repeat
the process. Again, imagine d fixed, then the instantaneous centre Obd of b with regard to d
is found by producing the links c and a to intersect in Obd, and the shapes of the centroids
cylinders are sometimes called axodes, and a section of an axode in a plane parallel to the
plane of motion is called a centrode. The axode is hence the locus of the instantaneous axis,
whilst the centrode is the locus of the instantaneous centre in any plane parallel to the plane
of motion. There is no restriction on the shape of these rolling axodes; they may have any
shape consistent with rolling (that is, no slipping is permitted), and the relative velocity of a
point P is still found by considering it with regard to the instantaneous centre.
Reuleaux has shown that the relative motion of any pair of non-adjacent links of a
kinematic chain is determined by the rolling together of two ideal cylindrical surfaces
(cylindrical being used here in the general sense), each of which may be assumed to be
formed by the extension of the material of the link to which it corresponds. These surfaces
have contact at the instantaneous axis, which is now called the instantaneous axis of the
two links concerned. To find the form of these surfaces corresponding to a particular pair of
non-adjacent links, consider each link of the pair fixed in turn, then the locus of the
instantaneous axis is the axode corresponding to the fixed link, or, considering a plane of
motion only, the locus of the instantaneous centre is the centrode corresponding to the fixed
link.
To find the instantaneous centre for a particular link corresponding to any given
configuration of the kinematic chain, it is only necessary to know the direction of motion
of any two points in the link, since lines through these points respectively at right angles to
their directions of motion intersect in the instantaneous centre.
To illustrate this principle, consider the
four-bar chain shown in fig. 122 made up
of the four links, a, b, c, d. Let a be the
fixed link, and consider the link c. Its
extremities are moving respectively in
directions at right angles to the links b and
d; hence produce the links b and d to meet
in the point Oac. This point is the
instantaneous centre of the motion of the
link c relatively to the fixed link a, a fact Fig. 122.
indicated by the suffix ac placed after the
letter O. The process being repeated for different values of the angle θ the curve through
the several points Oac is the centroid which may be imagined as formed by an extension of
the material of the link a. To find the corresponding centroid for the link c, fix c and repeat
the process. Again, imagine d fixed, then the instantaneous centre Obd of b with regard to d
is found by producing the links c and a to intersect in Obd, and the shapes of the centroids
Page 544
belonging respectively to the links b and d can be found as before. The axis about which a
pair of adjacent links turn is a permanent axis, and is of course the axis of the pin which
forms the point. Adding the centres corresponding to these several axes to the figure, it will
be seen that there are six centres in connexion with the four-bar chain of which four are
permanent and two are instantaneous or virtual centres; and, further, that whatever be the
configuration of the chain these centres group themselves into three sets of three, each set
lying on a straight line. This peculiarity is not an accident or a special property of the four-
bar chain, but is an illustration of a general law regarding the subject discovered by
Aronhold and Sir A. B. W. Kennedy independently, which may be thus stated: If any three
bodies, a, b, c, have plane motion their three virtual centres, Oab, Obc, Oac, are three points
on one straight line. A proof of this will be found in The Mechanics of Machinery quoted
above. Having obtained the set of instantaneous centres for a chain, suppose a is the fixed
link of the chain and c any other link; then Oac is the instantaneous centre of the two links
and may be considered for the instant as the trace of an axis fixed to an extension of the
link a about which c is turning, and thus problems of instantaneous velocity concerning the
link c are solved as though the link c were merely rotating for the instant about a fixed axis
coincident with the instantaneous axis.
Fig. 123. Fig. 124.
Method 2.—The second method is based upon the vector representation of velocity, and
may be illustrated by applying it to the four-bar chain. Let AD (fig. 123) be the fixed link.
Consider the link BC, and let it be required to find the velocity of the point B having given
the velocity of the point C. The principle upon which the solution is based is that the only
motion which B can have relatively to an axis through C fixed to the link CD is one of
turning about C. Choose any pole O (fig. 124). From this pole set out Oc to represent the
velocity of the point C. The direction of this must be at right angles to the line CD, because
this is the only direction possible to the point C. If the link BC moves without turning, Oc
will also represent the velocity of the point B; but, if the link is turning, B can only move
about the axis C, and its direction of motion is therefore at right angles to the line CB.
pair of adjacent links turn is a permanent axis, and is of course the axis of the pin which
forms the point. Adding the centres corresponding to these several axes to the figure, it will
be seen that there are six centres in connexion with the four-bar chain of which four are
permanent and two are instantaneous or virtual centres; and, further, that whatever be the
configuration of the chain these centres group themselves into three sets of three, each set
lying on a straight line. This peculiarity is not an accident or a special property of the four-
bar chain, but is an illustration of a general law regarding the subject discovered by
Aronhold and Sir A. B. W. Kennedy independently, which may be thus stated: If any three
bodies, a, b, c, have plane motion their three virtual centres, Oab, Obc, Oac, are three points
on one straight line. A proof of this will be found in The Mechanics of Machinery quoted
above. Having obtained the set of instantaneous centres for a chain, suppose a is the fixed
link of the chain and c any other link; then Oac is the instantaneous centre of the two links
and may be considered for the instant as the trace of an axis fixed to an extension of the
link a about which c is turning, and thus problems of instantaneous velocity concerning the
link c are solved as though the link c were merely rotating for the instant about a fixed axis
coincident with the instantaneous axis.
Fig. 123. Fig. 124.
Method 2.—The second method is based upon the vector representation of velocity, and
may be illustrated by applying it to the four-bar chain. Let AD (fig. 123) be the fixed link.
Consider the link BC, and let it be required to find the velocity of the point B having given
the velocity of the point C. The principle upon which the solution is based is that the only
motion which B can have relatively to an axis through C fixed to the link CD is one of
turning about C. Choose any pole O (fig. 124). From this pole set out Oc to represent the
velocity of the point C. The direction of this must be at right angles to the line CD, because
this is the only direction possible to the point C. If the link BC moves without turning, Oc
will also represent the velocity of the point B; but, if the link is turning, B can only move
about the axis C, and its direction of motion is therefore at right angles to the line CB.
Page 545
Hence set out the possible direction of B′s motion in the velocity diagram, namely cb1, at
right angles to CB. But the point B must also move at right angles to AB in the case under
consideration. Hence draw a line through O in the velocity diagram at right angles to AB to
cut cb1 in b. Then Ob is the velocity of the point b in magnitude and direction, and cb is the
tangential velocity of B relatively to C. Moreover, whatever be the actual magnitudes of the
velocities, the instantaneous velocity ratio of the points C and B is given by the ratio
Oc/Ob.
A most important property of the diagram (figs. 123 and 124) is the following: If points
X and x are taken dividing the link BC and the tangential velocity cb, so that cx:xb =
CX:XB, then Ox represents the velocity of the point X in magnitude and direction. The line
cb has been called the velocity image of the rod, since it may be looked upon as a scale
drawing of the rod turned through 90° from the actual rod. Or, put in another way, if the
link CB is drawn to scale on the new length cb in the velocity diagram (fig. 124), then a
vector drawn from O to any point on the new drawing of the rod will represent the velocity
of that point of the actual rod in magnitude and direction. It will be understood that there is
a new velocity diagram for every new configuration of the mechanism, and that in each
new diagram the image of the rod will be different in scale. Following the method indicated
above for a kinematic chain in general, there will be obtained a velocity diagram similar to
that of fig. 124 for each configuration of the mechanism, a diagram in which the velocity of
the several points in the chain utilized for drawing the diagram will appear to the same
scale, all radiating from the pole O. The lines joining the ends of these several velocities
are the several tangential velocities, each being the velocity image of a link in the chain.
These several images are not to the same scale, so that although the images may be
considered to form collectively an image of the chain itself, the several members of this
chain-image are to different scales in any one velocity diagram, and thus the chain-image is
distorted from the actual proportions of the mechanism which it represents.
§ 82.* Acceleration Diagram. Acceleration Image.—
Although it is possible to obtain the acceleration of
points in a kinematic chain with one link fixed by
methods which utilize the instantaneous centres of the
chain, the vector method more readily lends itself to this
purpose. It should be understood that the instantaneous
centre considered in the preceding paragraphs is
available only for estimating relative velocities; it
Fig. 125.
cannot be used in a similar manner for questions
regarding acceleration. That is to say, although the
instantaneous centre is a centre of no velocity for the instant, it is not a centre of no
acceleration, and in fact the centre of no acceleration is in general a quite different point.
right angles to CB. But the point B must also move at right angles to AB in the case under
consideration. Hence draw a line through O in the velocity diagram at right angles to AB to
cut cb1 in b. Then Ob is the velocity of the point b in magnitude and direction, and cb is the
tangential velocity of B relatively to C. Moreover, whatever be the actual magnitudes of the
velocities, the instantaneous velocity ratio of the points C and B is given by the ratio
Oc/Ob.
A most important property of the diagram (figs. 123 and 124) is the following: If points
X and x are taken dividing the link BC and the tangential velocity cb, so that cx:xb =
CX:XB, then Ox represents the velocity of the point X in magnitude and direction. The line
cb has been called the velocity image of the rod, since it may be looked upon as a scale
drawing of the rod turned through 90° from the actual rod. Or, put in another way, if the
link CB is drawn to scale on the new length cb in the velocity diagram (fig. 124), then a
vector drawn from O to any point on the new drawing of the rod will represent the velocity
of that point of the actual rod in magnitude and direction. It will be understood that there is
a new velocity diagram for every new configuration of the mechanism, and that in each
new diagram the image of the rod will be different in scale. Following the method indicated
above for a kinematic chain in general, there will be obtained a velocity diagram similar to
that of fig. 124 for each configuration of the mechanism, a diagram in which the velocity of
the several points in the chain utilized for drawing the diagram will appear to the same
scale, all radiating from the pole O. The lines joining the ends of these several velocities
are the several tangential velocities, each being the velocity image of a link in the chain.
These several images are not to the same scale, so that although the images may be
considered to form collectively an image of the chain itself, the several members of this
chain-image are to different scales in any one velocity diagram, and thus the chain-image is
distorted from the actual proportions of the mechanism which it represents.
§ 82.* Acceleration Diagram. Acceleration Image.—
Although it is possible to obtain the acceleration of
points in a kinematic chain with one link fixed by
methods which utilize the instantaneous centres of the
chain, the vector method more readily lends itself to this
purpose. It should be understood that the instantaneous
centre considered in the preceding paragraphs is
available only for estimating relative velocities; it
Fig. 125.
cannot be used in a similar manner for questions
regarding acceleration. That is to say, although the
instantaneous centre is a centre of no velocity for the instant, it is not a centre of no
acceleration, and in fact the centre of no acceleration is in general a quite different point.
Page 546
The general principle on which the method of drawing an acceleration diagram depends is
that if a link CB (fig. 125) have plane motion and the acceleration of any point C be given
in magnitude and direction, the acceleration of any other point B is the vector sum of the
acceleration of C, the radial acceleration of B about C and the tangential acceleration of B
about C. Let A be any origin, and let Ac represent the acceleration of the point C, ct the
radial acceleration of B about C which must be in a direction parallel to BC, and tb the
tangential acceleration of B about C, which must of course be at right angles to ct; then the
vector sum of these three magnitudes is Ab, and this vector represents the acceleration of
the point B. The directions of the radial and tangential accelerations of the point B are
always known when the position of the link is assigned, since these are to be drawn
respectively parallel to and at right angles to the link itself. The magnitude of the radial
acceleration is given by the expression v2/BC, v being the velocity of the point B about the
point C. This velocity can always be found from the velocity diagram of the chain of which
the link forms a part. If dw/dt is the angular acceleration of the link, dw/dt × CB is the
tangential acceleration of the point B about the point C. Generally this tangential
acceleration is unknown in magnitude, and it becomes part of the problem to find it. An
important property of the diagram is that if points X and x are taken dividing the link CB
and the whole acceleration of B about C, namely, cb in the same ratio, then Ax represents
the acceleration of the point X in magnitude and direction; cb is called the acceleration
image of the rod. In applying this principle to the drawing of an acceleration diagram for a
mechanism, the velocity diagram of the mechanism must be first drawn in order to afford
the means of calculating the several radial accelerations of the links. Then assuming that
the acceleration of one point of a particular link of the mechanism is known together with
the corresponding configuration of the mechanism, the two vectors Ac and ct can be drawn.
The direction of tb, the third vector in the diagram, is also known, so that the problem is
reduced to the condition that b is somewhere on the line tb. Then other conditions
consequent upon the fact that the link forms part of a kinematic chain operate to enable b to
be fixed. These methods are set forth and exemplified in Graphics, by R. H. Smith
(London, 1889). Examples, completely worked out, of velocity and acceleration diagrams
for the slider crank chain, the four-bar chain, and the mechanism of the Joy valve gear will
be found in ch. ix. of Valves and Valve Gear Mechanism, by W. E. Dalby (London, 1906).
Chapter II. On Applied Dynamics.
§ 83. Laws of Motion.—The action of a machine in transmitting force and motion
simultaneously, or performing work, is governed, in common with the phenomena of
moving bodies in general, by two “laws of motion.”
that if a link CB (fig. 125) have plane motion and the acceleration of any point C be given
in magnitude and direction, the acceleration of any other point B is the vector sum of the
acceleration of C, the radial acceleration of B about C and the tangential acceleration of B
about C. Let A be any origin, and let Ac represent the acceleration of the point C, ct the
radial acceleration of B about C which must be in a direction parallel to BC, and tb the
tangential acceleration of B about C, which must of course be at right angles to ct; then the
vector sum of these three magnitudes is Ab, and this vector represents the acceleration of
the point B. The directions of the radial and tangential accelerations of the point B are
always known when the position of the link is assigned, since these are to be drawn
respectively parallel to and at right angles to the link itself. The magnitude of the radial
acceleration is given by the expression v2/BC, v being the velocity of the point B about the
point C. This velocity can always be found from the velocity diagram of the chain of which
the link forms a part. If dw/dt is the angular acceleration of the link, dw/dt × CB is the
tangential acceleration of the point B about the point C. Generally this tangential
acceleration is unknown in magnitude, and it becomes part of the problem to find it. An
important property of the diagram is that if points X and x are taken dividing the link CB
and the whole acceleration of B about C, namely, cb in the same ratio, then Ax represents
the acceleration of the point X in magnitude and direction; cb is called the acceleration
image of the rod. In applying this principle to the drawing of an acceleration diagram for a
mechanism, the velocity diagram of the mechanism must be first drawn in order to afford
the means of calculating the several radial accelerations of the links. Then assuming that
the acceleration of one point of a particular link of the mechanism is known together with
the corresponding configuration of the mechanism, the two vectors Ac and ct can be drawn.
The direction of tb, the third vector in the diagram, is also known, so that the problem is
reduced to the condition that b is somewhere on the line tb. Then other conditions
consequent upon the fact that the link forms part of a kinematic chain operate to enable b to
be fixed. These methods are set forth and exemplified in Graphics, by R. H. Smith
(London, 1889). Examples, completely worked out, of velocity and acceleration diagrams
for the slider crank chain, the four-bar chain, and the mechanism of the Joy valve gear will
be found in ch. ix. of Valves and Valve Gear Mechanism, by W. E. Dalby (London, 1906).
Chapter II. On Applied Dynamics.
§ 83. Laws of Motion.—The action of a machine in transmitting force and motion
simultaneously, or performing work, is governed, in common with the phenomena of
moving bodies in general, by two “laws of motion.”
Page 547
Division 1. Balanced Forces in Machines of Uniform Velocity.
§ 84. Application of Force to Mechanism.—Forces are applied in units of weight; and
the unit most commonly employed in Britain is the pound avoirdupois. The action of a
force applied to a body is always in reality distributed over some definite space, either a
volume of three dimensions or a surface of two. An example of a force distributed
throughout a volume is the weight of the body itself, which acts on every particle, however
small. The pressure exerted between two bodies at their surface of contact, or between the
two parts of one body on either side of an ideal surface of separation, is an example of a
force distributed over a surface. The mode of distribution of a force applied to a solid body
requires to be considered when its stiffness and strength are treated of; but, in questions
respecting the action of a force upon a rigid body considered as a whole, the resultant of
the distributed force, determined according to the principles of statics, and considered as
acting in a single line and applied at a single point, may, for the occasion, be substituted for
the force as really distributed. Thus, the weight of each separate piece in a machine is
treated as acting wholly at its centre of gravity, and each pressure applied to it as acting at a
point called the centre of pressure of the surface to which the pressure is really applied.
§ 85. Forces applied to Mechanism Classed.—If θ be the obliquity of a force F applied
to a piece of a machine—that is, the angle made by the direction of the force with the
direction of motion of its point of application—then by the principles of statics, F may be
resolved into two rectangular components, viz.:—
Along the direction of motion, P = F cos θ
Across the direction of motion, Q = F sin θ
(49)
If the component along the direction of motion acts with the motion, it is called an effort;
if against the motion, a resistance. The component across the direction of motion is a
lateral pressure; the unbalanced lateral pressure on any piece, or part of a piece, is
deflecting force. A lateral pressure may increase resistance by causing friction; the friction
so caused acts against the motion, and is a resistance, but the lateral pressure causing it is
not a resistance. Resistances are distinguished into useful and prejudicial, according as they
arise from the useful effect produced by the machine or from other causes.
§ 86. Work.—Work consists in moving against resistance. The work is said to be
performed, and the resistance overcome. Work is measured by the product of the resistance
into the distance through which its point of application is moved. The unit of work
commonly used in Britain is a resistance of one pound overcome through a distance of one
foot, and is called a foot-pound.
§ 84. Application of Force to Mechanism.—Forces are applied in units of weight; and
the unit most commonly employed in Britain is the pound avoirdupois. The action of a
force applied to a body is always in reality distributed over some definite space, either a
volume of three dimensions or a surface of two. An example of a force distributed
throughout a volume is the weight of the body itself, which acts on every particle, however
small. The pressure exerted between two bodies at their surface of contact, or between the
two parts of one body on either side of an ideal surface of separation, is an example of a
force distributed over a surface. The mode of distribution of a force applied to a solid body
requires to be considered when its stiffness and strength are treated of; but, in questions
respecting the action of a force upon a rigid body considered as a whole, the resultant of
the distributed force, determined according to the principles of statics, and considered as
acting in a single line and applied at a single point, may, for the occasion, be substituted for
the force as really distributed. Thus, the weight of each separate piece in a machine is
treated as acting wholly at its centre of gravity, and each pressure applied to it as acting at a
point called the centre of pressure of the surface to which the pressure is really applied.
§ 85. Forces applied to Mechanism Classed.—If θ be the obliquity of a force F applied
to a piece of a machine—that is, the angle made by the direction of the force with the
direction of motion of its point of application—then by the principles of statics, F may be
resolved into two rectangular components, viz.:—
Along the direction of motion, P = F cos θ
Across the direction of motion, Q = F sin θ
(49)
If the component along the direction of motion acts with the motion, it is called an effort;
if against the motion, a resistance. The component across the direction of motion is a
lateral pressure; the unbalanced lateral pressure on any piece, or part of a piece, is
deflecting force. A lateral pressure may increase resistance by causing friction; the friction
so caused acts against the motion, and is a resistance, but the lateral pressure causing it is
not a resistance. Resistances are distinguished into useful and prejudicial, according as they
arise from the useful effect produced by the machine or from other causes.
§ 86. Work.—Work consists in moving against resistance. The work is said to be
performed, and the resistance overcome. Work is measured by the product of the resistance
into the distance through which its point of application is moved. The unit of work
commonly used in Britain is a resistance of one pound overcome through a distance of one
foot, and is called a foot-pound.
Page 548
Work is distinguished into useful work and prejudicial or lost work, according as it is
performed in producing the useful effect of the machine, or in overcoming prejudicial
resistance.
§ 87. Energy: Potential Energy.—Energy means capacity for performing work. The
energy of an effort, or potential energy, is measured by the product of the effort into the
distance through which its point of application is capable of being moved. The unit of
energy is the same with the unit of work.
When the point of application of an effort has been moved through a given distance,
energy is said to have been exerted to an amount expressed by the product of the effort into
the distance through which its point of application has been moved.
§ 88. Variable Effort and Resistance.—If an effort has different magnitudes during
different portions of the motion of its point of application through a given distance, let each
different magnitude of the effort P be multiplied by the length Δs of the corresponding
portion of the path of the point of application; the sum
Σ · PΔs
(50)
is the whole energy exerted. If the effort varies by insensible gradations, the energy exerted
is the integral or limit towards which that sum approaches continually as the divisions of
the path are made smaller and more numerous, and is expressed by
∫ P ds.
(51)
Similar processes are applicable to the finding of the work performed in overcoming a
varying resistance.
The work done by a machine can be actually measured by means of a dynamometer
(q.v.).
§ 89. Principle of the Equality of Energy and Work.—From the first law of motion it
follows that in a machine whose pieces move with uniform velocities the efforts and
resistances must balance each other. Now from the laws of statics it is known that, in order
that a system of forces applied to a system of connected points may be in equilibrium, it is
necessary that the sum formed by putting together the products of the forces by the
respective distances through which their points of application are capable of moving
simultaneously, each along the direction of the force applied to it, shall be zero,—products
being considered positive or negative according as the direction of the forces and the
possible motions of their points of application are the same or opposite.
performed in producing the useful effect of the machine, or in overcoming prejudicial
resistance.
§ 87. Energy: Potential Energy.—Energy means capacity for performing work. The
energy of an effort, or potential energy, is measured by the product of the effort into the
distance through which its point of application is capable of being moved. The unit of
energy is the same with the unit of work.
When the point of application of an effort has been moved through a given distance,
energy is said to have been exerted to an amount expressed by the product of the effort into
the distance through which its point of application has been moved.
§ 88. Variable Effort and Resistance.—If an effort has different magnitudes during
different portions of the motion of its point of application through a given distance, let each
different magnitude of the effort P be multiplied by the length Δs of the corresponding
portion of the path of the point of application; the sum
Σ · PΔs
(50)
is the whole energy exerted. If the effort varies by insensible gradations, the energy exerted
is the integral or limit towards which that sum approaches continually as the divisions of
the path are made smaller and more numerous, and is expressed by
∫ P ds.
(51)
Similar processes are applicable to the finding of the work performed in overcoming a
varying resistance.
The work done by a machine can be actually measured by means of a dynamometer
(q.v.).
§ 89. Principle of the Equality of Energy and Work.—From the first law of motion it
follows that in a machine whose pieces move with uniform velocities the efforts and
resistances must balance each other. Now from the laws of statics it is known that, in order
that a system of forces applied to a system of connected points may be in equilibrium, it is
necessary that the sum formed by putting together the products of the forces by the
respective distances through which their points of application are capable of moving
simultaneously, each along the direction of the force applied to it, shall be zero,—products
being considered positive or negative according as the direction of the forces and the
possible motions of their points of application are the same or opposite.
Page 549
In other words, the sum of the negative products is equal to the sum of the positive
products. This principle, applied to a machine whose parts move with uniform velocities, is
equivalent to saying that in any given interval of time the energy exerted is equal to the
work performed.
The symbolical expression of this law is as follows: let efforts be applied to one or any
number of points of a machine; let any one of these efforts be represented by P, and the
distance traversed by its point of application in a given interval of time by ds; let
resistances be overcome at one or any number of points of the same machine; let any one
of these resistances be denoted by R, and the distance traversed by its point of application
in the given interval of time by ds′; then
Σ · P ds = Σ · R ds′.
(52)
The lengths ds, ds′ are proportional to the velocities of the points to whose paths they
belong, and the proportions of those velocities to each other are deducible from the
construction of the machine by the principles of pure mechanism explained in Chapter I.
§ 90. Static Equilibrium of Mechanisms.—The principle stated in the preceding section,
namely, that the energy exerted is equal to the work performed, enables the ratio of the
components of the forces acting in the respective directions of motion at two points of a
mechanism, one being the point of application of the effort, and the other the point of
application of the resistance, to be readily found. Removing the summation signs in
equation (52) in order to restrict its application to two points and dividing by the common
time interval during which the respective small displacements ds and ds′ were made, it
becomes P ds/dt = R ds′/dt, that is, Pv = Rv′, which shows that the force ratio is the inverse
of the velocity ratio. It follows at once that any method which may be available for the
determination of the velocity ratio is equally available for the determination of the force
ratio, it being clearly understood that the forces involved are the components of the actual
forces resolved in the direction of motion of the points. The relation between the effort and
the resistance may be found by means of this principle for all kinds of mechanisms, when
the friction produced by the components of the forces across the direction of motion of the
two points is neglected. Consider the following example:—
A four-bar chain having the configuration shown in fig. 126 supports a load P at the
point x. What load is required at the point y to maintain the configuration shown, both
loads being supposed to act vertically? Find the instantaneous centre Obd, and resolve each
load in the respective directions of motion of the points x and y; thus there are obtained the
components P cos θ and R cos φ. Let the mechanism have a small motion; then, for the
instant, the link b is turning about its instantaneous centre Obd, and, if ω is its instantaneous
products. This principle, applied to a machine whose parts move with uniform velocities, is
equivalent to saying that in any given interval of time the energy exerted is equal to the
work performed.
The symbolical expression of this law is as follows: let efforts be applied to one or any
number of points of a machine; let any one of these efforts be represented by P, and the
distance traversed by its point of application in a given interval of time by ds; let
resistances be overcome at one or any number of points of the same machine; let any one
of these resistances be denoted by R, and the distance traversed by its point of application
in the given interval of time by ds′; then
Σ · P ds = Σ · R ds′.
(52)
The lengths ds, ds′ are proportional to the velocities of the points to whose paths they
belong, and the proportions of those velocities to each other are deducible from the
construction of the machine by the principles of pure mechanism explained in Chapter I.
§ 90. Static Equilibrium of Mechanisms.—The principle stated in the preceding section,
namely, that the energy exerted is equal to the work performed, enables the ratio of the
components of the forces acting in the respective directions of motion at two points of a
mechanism, one being the point of application of the effort, and the other the point of
application of the resistance, to be readily found. Removing the summation signs in
equation (52) in order to restrict its application to two points and dividing by the common
time interval during which the respective small displacements ds and ds′ were made, it
becomes P ds/dt = R ds′/dt, that is, Pv = Rv′, which shows that the force ratio is the inverse
of the velocity ratio. It follows at once that any method which may be available for the
determination of the velocity ratio is equally available for the determination of the force
ratio, it being clearly understood that the forces involved are the components of the actual
forces resolved in the direction of motion of the points. The relation between the effort and
the resistance may be found by means of this principle for all kinds of mechanisms, when
the friction produced by the components of the forces across the direction of motion of the
two points is neglected. Consider the following example:—
A four-bar chain having the configuration shown in fig. 126 supports a load P at the
point x. What load is required at the point y to maintain the configuration shown, both
loads being supposed to act vertically? Find the instantaneous centre Obd, and resolve each
load in the respective directions of motion of the points x and y; thus there are obtained the
components P cos θ and R cos φ. Let the mechanism have a small motion; then, for the
instant, the link b is turning about its instantaneous centre Obd, and, if ω is its instantaneous
Page 550
angular velocity, the velocity of the point x
is ωr, and the velocity of the point y is ωs.
Hence, by the principle just stated, P cos θ
× ωr = R cos φ × ωs. But, p and q being
respectively the perpendiculars to the lines
of action of the forces, this equation
reduces to Pp = Rq, which shows that the
ratio of the two forces may be found by
taking moments about the instantaneous
centre of the link on which they act.
The forces P and R may, however, act on
different links. The general problem may
then be thus stated: Given a mechanism of
which r is the fixed link, and s and t any
other two links, given also a force ƒs,
acting on the link s, to find the force ƒt
acting in a given direction on the link t,
which will keep the mechanism in static
equilibrium. The graphic solution of this Fig. 126.
problem may be effected thus:—
(1) Find the three virtual centres Ors, Ort, Ost, which must be three points in a line.
(2) Resolve ƒs into two components, one of which, namely, ƒq, passes through Ors and
may be neglected, and the other ƒp passes through Ost.
(3) Find the point M, where ƒp joins the given direction of ƒt, and resolve ƒp into two
components, of which one is in the direction MOrt, and may be neglected
because it passes through Ort, and the other is in the given direction of ƒt and is
therefore the force required.
This statement of the problem and the solution is due to Sir A. B. W. Kennedy, and is
given in ch. 8 of his Mechanics of Machinery. Another general solution of the problem is
given in the Proc. Lond. Math. Soc. (1878-1879), by the same author. An example of the
method of solution stated above, and taken from the Mechanics of Machinery, is illustrated
by the mechanism fig. 127, which is an epicyclic train of three wheels with the first wheel r
fixed. Let it be required to find the vertical force which must act at the pitch radius of the
last wheel t to balance exactly a force ƒs acting vertically downwards on the arm at the
point indicated in the figure. The two links concerned are the last wheel t and the arm s, the
is ωr, and the velocity of the point y is ωs.
Hence, by the principle just stated, P cos θ
× ωr = R cos φ × ωs. But, p and q being
respectively the perpendiculars to the lines
of action of the forces, this equation
reduces to Pp = Rq, which shows that the
ratio of the two forces may be found by
taking moments about the instantaneous
centre of the link on which they act.
The forces P and R may, however, act on
different links. The general problem may
then be thus stated: Given a mechanism of
which r is the fixed link, and s and t any
other two links, given also a force ƒs,
acting on the link s, to find the force ƒt
acting in a given direction on the link t,
which will keep the mechanism in static
equilibrium. The graphic solution of this Fig. 126.
problem may be effected thus:—
(1) Find the three virtual centres Ors, Ort, Ost, which must be three points in a line.
(2) Resolve ƒs into two components, one of which, namely, ƒq, passes through Ors and
may be neglected, and the other ƒp passes through Ost.
(3) Find the point M, where ƒp joins the given direction of ƒt, and resolve ƒp into two
components, of which one is in the direction MOrt, and may be neglected
because it passes through Ort, and the other is in the given direction of ƒt and is
therefore the force required.
This statement of the problem and the solution is due to Sir A. B. W. Kennedy, and is
given in ch. 8 of his Mechanics of Machinery. Another general solution of the problem is
given in the Proc. Lond. Math. Soc. (1878-1879), by the same author. An example of the
method of solution stated above, and taken from the Mechanics of Machinery, is illustrated
by the mechanism fig. 127, which is an epicyclic train of three wheels with the first wheel r
fixed. Let it be required to find the vertical force which must act at the pitch radius of the
last wheel t to balance exactly a force ƒs acting vertically downwards on the arm at the
point indicated in the figure. The two links concerned are the last wheel t and the arm s, the
Page 551
wheel r being the fixed link of the mechanism.
The virtual centres Ors, Ost are at the respective
axes of the wheels r and t, and the centre Ort
divides the line through these two points
externally in the ratio of the train of wheels. The
figure sufficiently indicates the various steps of
the solution.
The relation between the effort and the
resistance in a machine to include the effect of
friction at the joints has been investigated in a
paper by Professor Fleeming Jenkin, “On the
Fig. 127.
application of graphic methods to the
determination of the efficiency of machinery”
(Trans. Roy. Soc. Ed., vol. 28). It is shown that a machine may at any instant be represented
by a frame of links the stresses in which are identical with the pressures at the joints of the
mechanism. This self-strained frame is called the dynamic frame of the machine. The
driving and resisting efforts are represented by elastic links in the dynamic frame, and
when the frame with its elastic links is drawn the stresses in the several members of it may
be determined by means of reciprocal figures. Incidentally the method gives the pressures
at every joint of the mechanism.
§ 91. Efficiency.—The efficiency of a machine is the ratio of the useful work to the total
work—that is, to the energy exerted—and is represented by
Σ · Ruds′ Σ · Ru ds′ Σ · Ru ds′ U
= = = .
Σ · R ds′ Σ · Ru ds′ + Σ · Rp ds′ Σ · P ds E (53)
Ru being taken to represent useful and Rp prejudicial resistances. The more nearly the
efficiency of a machine approaches to unity the better is the machine.
§ 92. Power and Effect.—The power of a machine is the energy exerted, and the effect
the useful work performed, in some interval of time of definite length, such as a second, an
hour, or a day.
The unit of power, called conventionally a horse-power, is 550 foot-pounds per second,
or 33,000 foot-pounds per minute, or 1,980,000 foot-pounds per hour.
§ 93. Modulus of a Machine.—In the investigation of the properties of a machine, the
useful resistances to be overcome and the useful work to be performed are usually given.
The prejudicial resistances arc generally functions of the useful resistances of the weights
The virtual centres Ors, Ost are at the respective
axes of the wheels r and t, and the centre Ort
divides the line through these two points
externally in the ratio of the train of wheels. The
figure sufficiently indicates the various steps of
the solution.
The relation between the effort and the
resistance in a machine to include the effect of
friction at the joints has been investigated in a
paper by Professor Fleeming Jenkin, “On the
Fig. 127.
application of graphic methods to the
determination of the efficiency of machinery”
(Trans. Roy. Soc. Ed., vol. 28). It is shown that a machine may at any instant be represented
by a frame of links the stresses in which are identical with the pressures at the joints of the
mechanism. This self-strained frame is called the dynamic frame of the machine. The
driving and resisting efforts are represented by elastic links in the dynamic frame, and
when the frame with its elastic links is drawn the stresses in the several members of it may
be determined by means of reciprocal figures. Incidentally the method gives the pressures
at every joint of the mechanism.
§ 91. Efficiency.—The efficiency of a machine is the ratio of the useful work to the total
work—that is, to the energy exerted—and is represented by
Σ · Ruds′ Σ · Ru ds′ Σ · Ru ds′ U
= = = .
Σ · R ds′ Σ · Ru ds′ + Σ · Rp ds′ Σ · P ds E (53)
Ru being taken to represent useful and Rp prejudicial resistances. The more nearly the
efficiency of a machine approaches to unity the better is the machine.
§ 92. Power and Effect.—The power of a machine is the energy exerted, and the effect
the useful work performed, in some interval of time of definite length, such as a second, an
hour, or a day.
The unit of power, called conventionally a horse-power, is 550 foot-pounds per second,
or 33,000 foot-pounds per minute, or 1,980,000 foot-pounds per hour.
§ 93. Modulus of a Machine.—In the investigation of the properties of a machine, the
useful resistances to be overcome and the useful work to be performed are usually given.
The prejudicial resistances arc generally functions of the useful resistances of the weights
Page 552
of the pieces of the mechanism, and of their form and arrangement; and, having been
determined, they serve for the computation of the lost work, which, being added to the
useful work, gives the expenditure of energy required. The result of this investigation,
expressed in the form of an equation between this energy and the useful work, is called by
Moseley the modulus of the machine. The general form of the modulus may be expressed
thus—
E = U + φ (U, A) + ψ (A),
(54)
where A denotes some quantity or set of quantities depending on the form, arrangement,
weight and other properties of the mechanism. Moseley, however, has pointed out that in
most cases this equation takes the much more simple form of
E = (1 + A) U + B,
(55)
where A and B are constants, depending on the form, arrangement and weight of the
mechanism. The efficiency corresponding to the last equation is
U 1
= .
E 1 + A + B/U (56)
§ 94. Trains of Mechanism.—In applying the preceding principles to a train of
mechanism, it may either be treated as a whole, or it may be considered in sections
consisting of single pieces, or of any convenient portion of the train—each section being
treated as a machine, driven by the effort applied to it and energy exerted upon it through
its line of connexion with the preceding section, performing useful work by driving the
following section, and losing work by overcoming its own prejudicial resistances. It is
evident that the efficiency of the whole train is the product of the efficiencies of its sections.
§ 95. Rotating Pieces: Couples of Forces.—It is often convenient to express the energy
exerted upon and the work performed by a turning piece in a machine in terms of the
moment of the couples of forces acting on it, and of the angular velocity. The ordinary
British unit of moment is a foot-pound; but it is to be remembered that this is a foot-pound
of a different sort from the unit of energy and work.
If a force be applied to a turning piece in a line not passing through its axis, the axis will
press against its bearings with an equal and parallel force, and the equal and opposite
reaction of the bearings will constitute, together with the first-mentioned force, a couple
whose arm is the perpendicular distance from the axis to the line of action of the first force.
determined, they serve for the computation of the lost work, which, being added to the
useful work, gives the expenditure of energy required. The result of this investigation,
expressed in the form of an equation between this energy and the useful work, is called by
Moseley the modulus of the machine. The general form of the modulus may be expressed
thus—
E = U + φ (U, A) + ψ (A),
(54)
where A denotes some quantity or set of quantities depending on the form, arrangement,
weight and other properties of the mechanism. Moseley, however, has pointed out that in
most cases this equation takes the much more simple form of
E = (1 + A) U + B,
(55)
where A and B are constants, depending on the form, arrangement and weight of the
mechanism. The efficiency corresponding to the last equation is
U 1
= .
E 1 + A + B/U (56)
§ 94. Trains of Mechanism.—In applying the preceding principles to a train of
mechanism, it may either be treated as a whole, or it may be considered in sections
consisting of single pieces, or of any convenient portion of the train—each section being
treated as a machine, driven by the effort applied to it and energy exerted upon it through
its line of connexion with the preceding section, performing useful work by driving the
following section, and losing work by overcoming its own prejudicial resistances. It is
evident that the efficiency of the whole train is the product of the efficiencies of its sections.
§ 95. Rotating Pieces: Couples of Forces.—It is often convenient to express the energy
exerted upon and the work performed by a turning piece in a machine in terms of the
moment of the couples of forces acting on it, and of the angular velocity. The ordinary
British unit of moment is a foot-pound; but it is to be remembered that this is a foot-pound
of a different sort from the unit of energy and work.
If a force be applied to a turning piece in a line not passing through its axis, the axis will
press against its bearings with an equal and parallel force, and the equal and opposite
reaction of the bearings will constitute, together with the first-mentioned force, a couple
whose arm is the perpendicular distance from the axis to the line of action of the first force.
Page 553
A couple is said to be right or left handed with reference to the observer, according to the
direction in which it tends to turn the body, and is a driving couple or a resisting couple
according as its tendency is with or against that of the actual rotation.
Let dt be an interval of time, α the angular velocity of the piece; then αdt is the angle
through which it turns in the interval dt, and ds = v dt = rα dt is the distance through which
the point of application of the force moves. Let P represent an effort, so that Pr is a driving
couple, then
P ds = Pv dt = Prα dt = Mα dt
(57)
is the energy exerted by the couple M in the interval dt; and a similar equation gives the
work performed in overcoming a resisting couple. When several couples act on one piece,
the resultant of their moments is to be multiplied by the common angular velocity of the
whole piece.
§ 96. Reduction of Forces to a given Point, and of Couples to the Axis of a given Piece.
—In computations respecting machines it is often convenient to substitute for a force
applied to a given point, or a couple applied to a given piece, the equivalent force or couple
applied to some other point or piece; that is to say, the force or couple, which, if applied to
the other point or piece, would exert equal energy or employ equal work. The principles of
this reduction are that the ratio of the given to the equivalent force is the reciprocal of the
ratio of the velocities of their points of application, and the ratio of the given to the
equivalent couple is the reciprocal of the ratio of the angular velocities of the pieces to
which they are applied.
These velocity ratios are known by the construction of the mechanism, and are
independent of the absolute speed.
§ 97. Balanced Lateral Pressure of Guides and Bearings.—The most important part of
the lateral pressure on a piece of mechanism is the reaction of its guides, if it is a sliding
piece, or of the bearings of its axis, if it is a turning piece; and the balanced portion of this
reaction is equal and opposite to the resultant of all the other forces applied to the piece, its
own weight included. There may be or may not be an unbalanced component in this
pressure, due to the deviated motion. Its laws will be considered in the sequel.
§ 98. Friction. Unguents.—The most important kind of resistance in machines is the
friction or rubbing resistance of surfaces which slide over each other. The direction of the
resistance of friction is opposite to that in which the sliding takes place. Its magnitude is
the product of the normal pressure or force which presses the rubbing surfaces together in a
direction perpendicular to themselves into a specific constant already mentioned in § 14, as
direction in which it tends to turn the body, and is a driving couple or a resisting couple
according as its tendency is with or against that of the actual rotation.
Let dt be an interval of time, α the angular velocity of the piece; then αdt is the angle
through which it turns in the interval dt, and ds = v dt = rα dt is the distance through which
the point of application of the force moves. Let P represent an effort, so that Pr is a driving
couple, then
P ds = Pv dt = Prα dt = Mα dt
(57)
is the energy exerted by the couple M in the interval dt; and a similar equation gives the
work performed in overcoming a resisting couple. When several couples act on one piece,
the resultant of their moments is to be multiplied by the common angular velocity of the
whole piece.
§ 96. Reduction of Forces to a given Point, and of Couples to the Axis of a given Piece.
—In computations respecting machines it is often convenient to substitute for a force
applied to a given point, or a couple applied to a given piece, the equivalent force or couple
applied to some other point or piece; that is to say, the force or couple, which, if applied to
the other point or piece, would exert equal energy or employ equal work. The principles of
this reduction are that the ratio of the given to the equivalent force is the reciprocal of the
ratio of the velocities of their points of application, and the ratio of the given to the
equivalent couple is the reciprocal of the ratio of the angular velocities of the pieces to
which they are applied.
These velocity ratios are known by the construction of the mechanism, and are
independent of the absolute speed.
§ 97. Balanced Lateral Pressure of Guides and Bearings.—The most important part of
the lateral pressure on a piece of mechanism is the reaction of its guides, if it is a sliding
piece, or of the bearings of its axis, if it is a turning piece; and the balanced portion of this
reaction is equal and opposite to the resultant of all the other forces applied to the piece, its
own weight included. There may be or may not be an unbalanced component in this
pressure, due to the deviated motion. Its laws will be considered in the sequel.
§ 98. Friction. Unguents.—The most important kind of resistance in machines is the
friction or rubbing resistance of surfaces which slide over each other. The direction of the
resistance of friction is opposite to that in which the sliding takes place. Its magnitude is
the product of the normal pressure or force which presses the rubbing surfaces together in a
direction perpendicular to themselves into a specific constant already mentioned in § 14, as
Page 554
the coefficient of friction, which depends on the nature and condition of the surfaces of the
unguent, if any, with which they are covered. The total pressure exerted between the
rubbing surfaces is the resultant of the normal pressure and of the friction, and its obliquity,
or inclination to the common perpendicular of the surfaces, is the angle of repose formerly
mentioned in § 14, whose tangent is the coefficient of friction. Thus, let N be the normal
pressure, R the friction, T the total pressure, ƒ the coefficient of friction, and φ the angle of
repose; then
ƒ = tan φ
R = ƒN = N tan φ = T sin φ
(58)
Experiments on friction have been made by Coulomb, Samuel Vince, John Rennie,
James Wood, D. Rankine and others. The most complete and elaborate experiments are
those of Morin, published in his Notions fondamentales de mécanique, and republished in
Britain in the works of Moseley and Gordon.
The experiments of Beauchamp Tower (“Report of Friction Experiments,” Proc. Inst.
Mech. Eng., 1883) showed that when oil is supplied to a journal by means of an oil bath the
coefficient of friction varies nearly inversely as the load on the bearing, thus making the
product of the load on the bearing and the coefficient of friction a constant. Mr Tower’s
experiments were carried out at nearly constant temperature. The more recent experiments
of Lasche (Zeitsch, Verein Deutsche Ingen., 1902, 46, 1881) show that the product of the
coefficient of friction, the load on the bearing, and the temperature is approximately
constant. For further information on this point and on Osborne Reynolds’s theory of
lubrication see Bearings and Lubrication.
§ 99. Work of Friction. Moment of Friction.—The work performed in a unit of time in
overcoming the friction of a pair of surfaces is the product of the friction by the velocity of
sliding of the surfaces over each other, if that is the same throughout the whole extent of
the rubbing surfaces. If that velocity is different for different portions of the rubbing
surfaces, the velocity of each portion is to be multiplied by the friction of that portion, and
the results summed or integrated.
When the relative motion of the rubbing surfaces is one of rotation, the work of friction
in a unit of time, for a portion of the rubbing surfaces at a given distance from the axis of
rotation, may be found by multiplying together the friction of that portion, its distance from
the axis, and the angular velocity. The product of the force of friction by the distance at
which it acts from the axis of rotation is called the moment of friction. The total moment of
friction of a pair of rotating rubbing surfaces is the sum or integral of the moments of
friction of their several portions.
unguent, if any, with which they are covered. The total pressure exerted between the
rubbing surfaces is the resultant of the normal pressure and of the friction, and its obliquity,
or inclination to the common perpendicular of the surfaces, is the angle of repose formerly
mentioned in § 14, whose tangent is the coefficient of friction. Thus, let N be the normal
pressure, R the friction, T the total pressure, ƒ the coefficient of friction, and φ the angle of
repose; then
ƒ = tan φ
R = ƒN = N tan φ = T sin φ
(58)
Experiments on friction have been made by Coulomb, Samuel Vince, John Rennie,
James Wood, D. Rankine and others. The most complete and elaborate experiments are
those of Morin, published in his Notions fondamentales de mécanique, and republished in
Britain in the works of Moseley and Gordon.
The experiments of Beauchamp Tower (“Report of Friction Experiments,” Proc. Inst.
Mech. Eng., 1883) showed that when oil is supplied to a journal by means of an oil bath the
coefficient of friction varies nearly inversely as the load on the bearing, thus making the
product of the load on the bearing and the coefficient of friction a constant. Mr Tower’s
experiments were carried out at nearly constant temperature. The more recent experiments
of Lasche (Zeitsch, Verein Deutsche Ingen., 1902, 46, 1881) show that the product of the
coefficient of friction, the load on the bearing, and the temperature is approximately
constant. For further information on this point and on Osborne Reynolds’s theory of
lubrication see Bearings and Lubrication.
§ 99. Work of Friction. Moment of Friction.—The work performed in a unit of time in
overcoming the friction of a pair of surfaces is the product of the friction by the velocity of
sliding of the surfaces over each other, if that is the same throughout the whole extent of
the rubbing surfaces. If that velocity is different for different portions of the rubbing
surfaces, the velocity of each portion is to be multiplied by the friction of that portion, and
the results summed or integrated.
When the relative motion of the rubbing surfaces is one of rotation, the work of friction
in a unit of time, for a portion of the rubbing surfaces at a given distance from the axis of
rotation, may be found by multiplying together the friction of that portion, its distance from
the axis, and the angular velocity. The product of the force of friction by the distance at
which it acts from the axis of rotation is called the moment of friction. The total moment of
friction of a pair of rotating rubbing surfaces is the sum or integral of the moments of
friction of their several portions.
Page 555
To express this symbolically, let du represent the area of a portion of a pair of rubbing
surfaces at a distance r from the axis of their relative rotation; p the intensity of the normal
pressure at du per unit of area; and ƒ the coefficient of friction. Then the moment of friction
of du is ƒpr du;
the total moment of friction is ƒ ∫ pr·du;
and the work performed in a unit cf time in overcoming friction, when the angular
velocity is α, is αƒ ∫ pr·du.
(59)
It is evident that the moment of friction, and the work lost by being performed in
overcoming friction, are less in a rotating piece as the bearings are of smaller radius. But a
limit is put to the diminution of the radii of journals and pivots by the conditions of
durability and of proper lubrication, and also by conditions of strength and stiffness.
§ 100. Total Pressure between Journal and Bearing.—A single piece rotating with a
uniform velocity has four mutually balanced forces applied to it: (l) the effort exerted on it
by the piece which drives it; (2) the resistance of the piece which follows it—which may be
considered for the purposes of the present question as useful resistance; (3) its weight; and
(4) the reaction of its own cylindrical bearings. There are given the following data:—
The direction of the effort.
The direction of the useful resistance.
The weight of the piece and the direction in which it acts.
The magnitude of the useful resistance.
The radius of the bearing r.
The angle of repose φ, corresponding to the friction of the journal on the bearing.
And there are required the following:—
The direction of the reaction of the bearing.
The magnitude of that reaction.
The magnitude of the effort.
Let the useful resistance and the weight of the piece be compounded by the principles of
statics into one force, and let this be called the given force.
surfaces at a distance r from the axis of their relative rotation; p the intensity of the normal
pressure at du per unit of area; and ƒ the coefficient of friction. Then the moment of friction
of du is ƒpr du;
the total moment of friction is ƒ ∫ pr·du;
and the work performed in a unit cf time in overcoming friction, when the angular
velocity is α, is αƒ ∫ pr·du.
(59)
It is evident that the moment of friction, and the work lost by being performed in
overcoming friction, are less in a rotating piece as the bearings are of smaller radius. But a
limit is put to the diminution of the radii of journals and pivots by the conditions of
durability and of proper lubrication, and also by conditions of strength and stiffness.
§ 100. Total Pressure between Journal and Bearing.—A single piece rotating with a
uniform velocity has four mutually balanced forces applied to it: (l) the effort exerted on it
by the piece which drives it; (2) the resistance of the piece which follows it—which may be
considered for the purposes of the present question as useful resistance; (3) its weight; and
(4) the reaction of its own cylindrical bearings. There are given the following data:—
The direction of the effort.
The direction of the useful resistance.
The weight of the piece and the direction in which it acts.
The magnitude of the useful resistance.
The radius of the bearing r.
The angle of repose φ, corresponding to the friction of the journal on the bearing.
And there are required the following:—
The direction of the reaction of the bearing.
The magnitude of that reaction.
The magnitude of the effort.
Let the useful resistance and the weight of the piece be compounded by the principles of
statics into one force, and let this be called the given force.
Page 556
The directions of the effort and of the given force are either
parallel or meet in a point. If they are parallel, the direction of
the reaction of the bearing is also parallel to them; if they meet
in a point, the direction of the reaction traverses the same point.
Also, let AAA, fig. 128, be a section of the bearing, and C its
axis; then the direction of the reaction, at the point where it
intersects the circle AAA, must make the angle φ with the Fig. 128.
radius of that circle; that is to say, it must be a line such as PT
touching the smaller circle BB, whose radius is r · sin φ. The side on which it touches that
circle is determined by the fact that the obliquity of the reaction is such as to oppose the
rotation.
Thus is determined the direction of the reaction of the bearing; and the magnitude of that
reaction and of the effort are then found by the principles of the equilibrium of three forces
already stated in § 7.
The work lost in overcoming the friction of the bearing is the same as that which would
be performed in overcoming at the circumference of the small circle BB a resistance equal
to the whole pressure between the journal and bearing.
In order to diminish that pressure to the smallest possible amount, the effort, and the
resultant of the useful resistance, and the weight of the piece (called above the “given
force”) ought to be opposed to each other as directly as is practicable consistently with the
purposes of the machine.
An investigation of the forces acting on a bearing and journal lubricated by an oil bath
will be found in a paper by Osborne Reynolds in the Phil. Trans. pt. i. (1886). (See also
Bearings.)
§ 101. Friction of Pivots and Collars.—When a shaft is acted upon by a force tending to
shift it lengthways, that force must be balanced by the reaction of a bearing against a pivot
at the end of the shaft; or, if that be impossible, against one or more collars, or rings
projecting from the body of the shaft. The bearing of the pivot is called a step or footstep.
Pivots require great hardness, and are usually made of steel. The flat pivot is a cylinder of
steel having a plane circular end as a rubbing surface. Let N be the total pressure sustained
by a flat pivot of the radius r; if that pressure be uniformly distributed, which is the case
when the rubbing surfaces of the pivot and its step are both true planes, the intensity of the
pressure is
p = N / πr2;
(60)
parallel or meet in a point. If they are parallel, the direction of
the reaction of the bearing is also parallel to them; if they meet
in a point, the direction of the reaction traverses the same point.
Also, let AAA, fig. 128, be a section of the bearing, and C its
axis; then the direction of the reaction, at the point where it
intersects the circle AAA, must make the angle φ with the Fig. 128.
radius of that circle; that is to say, it must be a line such as PT
touching the smaller circle BB, whose radius is r · sin φ. The side on which it touches that
circle is determined by the fact that the obliquity of the reaction is such as to oppose the
rotation.
Thus is determined the direction of the reaction of the bearing; and the magnitude of that
reaction and of the effort are then found by the principles of the equilibrium of three forces
already stated in § 7.
The work lost in overcoming the friction of the bearing is the same as that which would
be performed in overcoming at the circumference of the small circle BB a resistance equal
to the whole pressure between the journal and bearing.
In order to diminish that pressure to the smallest possible amount, the effort, and the
resultant of the useful resistance, and the weight of the piece (called above the “given
force”) ought to be opposed to each other as directly as is practicable consistently with the
purposes of the machine.
An investigation of the forces acting on a bearing and journal lubricated by an oil bath
will be found in a paper by Osborne Reynolds in the Phil. Trans. pt. i. (1886). (See also
Bearings.)
§ 101. Friction of Pivots and Collars.—When a shaft is acted upon by a force tending to
shift it lengthways, that force must be balanced by the reaction of a bearing against a pivot
at the end of the shaft; or, if that be impossible, against one or more collars, or rings
projecting from the body of the shaft. The bearing of the pivot is called a step or footstep.
Pivots require great hardness, and are usually made of steel. The flat pivot is a cylinder of
steel having a plane circular end as a rubbing surface. Let N be the total pressure sustained
by a flat pivot of the radius r; if that pressure be uniformly distributed, which is the case
when the rubbing surfaces of the pivot and its step are both true planes, the intensity of the
pressure is
p = N / πr2;
(60)
Page 557
and, introducing this value into equation 59, the moment of friction of the flat pivot is found
to be
23 ⁄ ƒNr
(61)
or two-thirds of that of a cylindrical journal of the same radius under the same normal
pressure.
The friction of a conical pivot exceeds that of a flat pivot of the same radius, and under
the same pressure, in the proportion of the side of the cone to the radius of its base.
The moment of friction of a collar is given by the formula—
r3 − r′3
⁄ ƒN
23 ,
r2 − r′2 (62)
where r is the external and r′ the internal radius.
In the cup and ball pivot the end of the shaft and the step
present two recesses facing each other, into which art fitted two
shallow cups of steel or hard bronze. Between the concave
spherical surfaces of those cups is placed a steel ball, being either
a complete sphere or a lens having convex surfaces of a somewhat
less radius than the concave surfaces of the cups. The moment of
friction of this pivot is at first almost inappreciable from the
extreme smallness of the radius of the circles of contact of the ball
and cups, but, as they wear, that radius and the moment of friction
increase. Fig. 129.
It appears that the rapidity with which a rubbing surface wears
away is proportional to the friction and to the velocity jointly, or nearly so. Hence the
pivots already mentioned wear unequally at different points, and tend to alter their figures.
Schiele has invented a pivot which preserves its original figure by wearing equally at all
points in a direction parallel to its axis. The following are the principles on which this
equality of wear depends:—
The rapidity of wear of a surface measured in an oblique direction is to the rapidity of
wear measured normally as the secant of the obliquity is to unity. Let OX (fig. 129) be the
axis of a pivot, and let RPC be a portion of a curve such that at any point P the secant of the
obliquity to the normal of the curve of a line parallel to the axis is inversely proportional to
the ordinate PY, to which the velocity of P is proportional. The rotation of that curve round
OX will generate the form of pivot required. Now let PT be a tangent to the curve at P,
to be
23 ⁄ ƒNr
(61)
or two-thirds of that of a cylindrical journal of the same radius under the same normal
pressure.
The friction of a conical pivot exceeds that of a flat pivot of the same radius, and under
the same pressure, in the proportion of the side of the cone to the radius of its base.
The moment of friction of a collar is given by the formula—
r3 − r′3
⁄ ƒN
23 ,
r2 − r′2 (62)
where r is the external and r′ the internal radius.
In the cup and ball pivot the end of the shaft and the step
present two recesses facing each other, into which art fitted two
shallow cups of steel or hard bronze. Between the concave
spherical surfaces of those cups is placed a steel ball, being either
a complete sphere or a lens having convex surfaces of a somewhat
less radius than the concave surfaces of the cups. The moment of
friction of this pivot is at first almost inappreciable from the
extreme smallness of the radius of the circles of contact of the ball
and cups, but, as they wear, that radius and the moment of friction
increase. Fig. 129.
It appears that the rapidity with which a rubbing surface wears
away is proportional to the friction and to the velocity jointly, or nearly so. Hence the
pivots already mentioned wear unequally at different points, and tend to alter their figures.
Schiele has invented a pivot which preserves its original figure by wearing equally at all
points in a direction parallel to its axis. The following are the principles on which this
equality of wear depends:—
The rapidity of wear of a surface measured in an oblique direction is to the rapidity of
wear measured normally as the secant of the obliquity is to unity. Let OX (fig. 129) be the
axis of a pivot, and let RPC be a portion of a curve such that at any point P the secant of the
obliquity to the normal of the curve of a line parallel to the axis is inversely proportional to
the ordinate PY, to which the velocity of P is proportional. The rotation of that curve round
OX will generate the form of pivot required. Now let PT be a tangent to the curve at P,
Page 558
cutting OX in T; PT = PY × secant obliquity, and this is to be a constant quantity; hence the
curve is that known as the tractory of the straight line OX, in which PT = OR = constant.
This curve is described by having a fixed straight edge parallel to OX, along which slides a
slider carrying a pin whose centre is T. On that pin turns an arm, carrying at a point P a
tracing-point, pencil or pen. Should the pen have a nib of two jaws, like those of an
ordinary drawing-pen, the plane of the jaws must pass through PT. Then, while T is slid
along the axis from O towards X, P will be drawn after it from R towards C along the
tractory. This curve, being an asymptote to its axis, is capable of being indefinitely
prolonged towards X; but in designing pivots it should stop before the angle PTY becomes
less than the angle of repose of the rubbing surfaces, otherwise the pivot will be liable to
stick in its bearing. The moment of friction of “Schiele’s anti-friction pivot,” as it is called,
is equal to that of a cylindrical journal of the radius OR = PT the constant tangent, under
the same pressure.
Records of experiments on the friction of a pivot bearing will be found in the Proc. Inst.
Mech. Eng. (1891), and on the friction of a collar bearing ib. May 1888.
§ 102. Friction of Teeth.—Let N be the normal pressure exerted between a pair of teeth
of a pair of wheels; s the total distance through which they slide upon each other; n the
number of pairs of teeth which pass the plane of axis in a unit of time; then
nƒNs
(63)
is the work lost in unity of time by the friction of the teeth. The sliding s is composed of
two parts, which take place during the approach and recess respectively. Let those be
denoted by s1 and s2, so that s = s1 + s2. In § 45 the velocity of sliding at any instant has
been given, viz. u = c (α1 + α2), where u is that velocity, c the distance T1 at any instant
from the point of contact of the teeth to the pitch-point, and α1, α2 the respective angular
velocities of the wheels.
Let v be the common velocity of the two pitch-circles, r1, r2, their radii; then the above
equation becomes
1 1
u = cv ( r1 + r2 ).
To apply this to involute teeth, let c1 be the length of the approach, c2 that of the recess,
u1, the mean volocity of sliding during the approach, u2 that during the recess; then
c1v 1 1 c2v 1 1
u1 =
2
( r1 + r2 ); u2 =
2
( r1 + r2 )
curve is that known as the tractory of the straight line OX, in which PT = OR = constant.
This curve is described by having a fixed straight edge parallel to OX, along which slides a
slider carrying a pin whose centre is T. On that pin turns an arm, carrying at a point P a
tracing-point, pencil or pen. Should the pen have a nib of two jaws, like those of an
ordinary drawing-pen, the plane of the jaws must pass through PT. Then, while T is slid
along the axis from O towards X, P will be drawn after it from R towards C along the
tractory. This curve, being an asymptote to its axis, is capable of being indefinitely
prolonged towards X; but in designing pivots it should stop before the angle PTY becomes
less than the angle of repose of the rubbing surfaces, otherwise the pivot will be liable to
stick in its bearing. The moment of friction of “Schiele’s anti-friction pivot,” as it is called,
is equal to that of a cylindrical journal of the radius OR = PT the constant tangent, under
the same pressure.
Records of experiments on the friction of a pivot bearing will be found in the Proc. Inst.
Mech. Eng. (1891), and on the friction of a collar bearing ib. May 1888.
§ 102. Friction of Teeth.—Let N be the normal pressure exerted between a pair of teeth
of a pair of wheels; s the total distance through which they slide upon each other; n the
number of pairs of teeth which pass the plane of axis in a unit of time; then
nƒNs
(63)
is the work lost in unity of time by the friction of the teeth. The sliding s is composed of
two parts, which take place during the approach and recess respectively. Let those be
denoted by s1 and s2, so that s = s1 + s2. In § 45 the velocity of sliding at any instant has
been given, viz. u = c (α1 + α2), where u is that velocity, c the distance T1 at any instant
from the point of contact of the teeth to the pitch-point, and α1, α2 the respective angular
velocities of the wheels.
Let v be the common velocity of the two pitch-circles, r1, r2, their radii; then the above
equation becomes
1 1
u = cv ( r1 + r2 ).
To apply this to involute teeth, let c1 be the length of the approach, c2 that of the recess,
u1, the mean volocity of sliding during the approach, u2 that during the recess; then
c1v 1 1 c2v 1 1
u1 =
2
( r1 + r2 ); u2 =
2
( r1 + r2 )
Page 559
also, let θ be the obliquity of the action; then the times occupied by the approach and recess
are respectively
c1 c2
, ;
v cos θ v cos θ
giving, finally, for the length of sliding between each pair of teeth,
c12 + c22 1 1
s = s 1 + s2 =
2 cos θ
( r1 +
r2 ) (64)
which, substituted in equation (63), gives the work lost in a unit of time by the friction of
involute teeth. This result, which is exact for involute teeth, is approximately true for teeth
of any figure.
For inside gearing, if r1 be the less radius and r2 the greater, 1/r1 − 1/r2 is to be
substituted for 1/r1 + 1/r2.
§ 103. Friction of Cords and Belts.—A flexible band, such as a cord, rope, belt or strap,
may be used either to exert an effort or a resistance upon a pulley round which it wraps. In
either case the tangential force, whether effort or resistance, exerted between the band and
the pulley is their mutual friction, caused by and proportional to the normal pressure
between them.
Let T1 be the tension of the free part of the band at that side towards which it tends to
draw the pulley, or from which the pulley tends to draw it; T2 the tension of the free part at
the other side; T the tension of the band at any intermediate point of its arc of contact with
the pulley; θ the ratio of the length of that arc to the radius of the pulley; dθ the ratio of an
indefinitely small element of that arc to the radius; F = T1 − T2 the total friction between
the band and the pulley; dF the elementary portion of that friction due to the elementary arc
dθ; ƒ the coefficient of friction between the materials of the band and pulley.
Then, according to a well-known principle in statics, the normal pressure at the
elementary arc dθ is T dθ, T being the mean tension of the band at that elementary arc;
consequently the friction on that arc is dF = ƒT dθ. Now that friction is also the difference
between the tensions of the band at the two ends of the elementary arc, or dT = dF = ƒT dθ;
which equation, being integrated throughout the entire arc of contact, gives the following
formulae:—
T1
hyp log. = ƒθ
T2
T1 = eƒθ
are respectively
c1 c2
, ;
v cos θ v cos θ
giving, finally, for the length of sliding between each pair of teeth,
c12 + c22 1 1
s = s 1 + s2 =
2 cos θ
( r1 +
r2 ) (64)
which, substituted in equation (63), gives the work lost in a unit of time by the friction of
involute teeth. This result, which is exact for involute teeth, is approximately true for teeth
of any figure.
For inside gearing, if r1 be the less radius and r2 the greater, 1/r1 − 1/r2 is to be
substituted for 1/r1 + 1/r2.
§ 103. Friction of Cords and Belts.—A flexible band, such as a cord, rope, belt or strap,
may be used either to exert an effort or a resistance upon a pulley round which it wraps. In
either case the tangential force, whether effort or resistance, exerted between the band and
the pulley is their mutual friction, caused by and proportional to the normal pressure
between them.
Let T1 be the tension of the free part of the band at that side towards which it tends to
draw the pulley, or from which the pulley tends to draw it; T2 the tension of the free part at
the other side; T the tension of the band at any intermediate point of its arc of contact with
the pulley; θ the ratio of the length of that arc to the radius of the pulley; dθ the ratio of an
indefinitely small element of that arc to the radius; F = T1 − T2 the total friction between
the band and the pulley; dF the elementary portion of that friction due to the elementary arc
dθ; ƒ the coefficient of friction between the materials of the band and pulley.
Then, according to a well-known principle in statics, the normal pressure at the
elementary arc dθ is T dθ, T being the mean tension of the band at that elementary arc;
consequently the friction on that arc is dF = ƒT dθ. Now that friction is also the difference
between the tensions of the band at the two ends of the elementary arc, or dT = dF = ƒT dθ;
which equation, being integrated throughout the entire arc of contact, gives the following
formulae:—
T1
hyp log. = ƒθ
T2
T1 = eƒθ
Page 560
T2
F = T1 − T2 = T1 (1 − e − ƒθ) = T2 (eƒθ − 1)
(65)
When a belt connecting a pair of pulleys has the tensions of its two sides originally
equal, the pulleys being at rest, and when the pulleys are next set in motion, so that one of
them drives the other by means of the belt, it is found that the advancing side of the belt is
exactly as much tightened as the returning side is slackened, so that the mean tension
remains unchanged. Its value is given by this formula—
T1 + T 2 eƒθ + 1
=
2 2 (eƒθ − 1) (66)
which is useful in determining the original tension required to enable a belt to transmit a
given force between two pulleys.
The equations 65 and 66 are applicable to a kind of brake called a friction-strap, used to
stop or moderate the velocity of machines by being tightened round a pulley. The strap is
usually of iron, and the pulley of hard wood.
Let α denote the arc of contact expressed in turns and fractions of a turn; then
θ = 6.2832a
eƒθ = number whose common logarithm is 2.7288ƒa
(67)
See also Dynamometer for illustrations of the use of what are essentially friction-straps
of different forms for the measurement of the brake horse-power of an engine or motor.
§ 104. Stiffness of Ropes.—Ropes offer a resistance to being bent, and, when bent, to
being straightened again, which arises from the mutual friction of their fibres. It increases
with the sectional area of the rope, and is inversely proportional to the radius of the curve
into which it is bent.
The work lost in pulling a given length of rope over a pulley is found by multiplying the
length of the rope in feet by its stiffness in pounds, that stiffness being the excess of the
tension at the leading side of the rope above that at the following side, which is necessary
to bend it into a curve fitting the pulley, and then to straighten it again.
The following empirical formulae for the stiffness of hempen ropes have been deduced
by Morin from the experiments of Coulomb:—
F = T1 − T2 = T1 (1 − e − ƒθ) = T2 (eƒθ − 1)
(65)
When a belt connecting a pair of pulleys has the tensions of its two sides originally
equal, the pulleys being at rest, and when the pulleys are next set in motion, so that one of
them drives the other by means of the belt, it is found that the advancing side of the belt is
exactly as much tightened as the returning side is slackened, so that the mean tension
remains unchanged. Its value is given by this formula—
T1 + T 2 eƒθ + 1
=
2 2 (eƒθ − 1) (66)
which is useful in determining the original tension required to enable a belt to transmit a
given force between two pulleys.
The equations 65 and 66 are applicable to a kind of brake called a friction-strap, used to
stop or moderate the velocity of machines by being tightened round a pulley. The strap is
usually of iron, and the pulley of hard wood.
Let α denote the arc of contact expressed in turns and fractions of a turn; then
θ = 6.2832a
eƒθ = number whose common logarithm is 2.7288ƒa
(67)
See also Dynamometer for illustrations of the use of what are essentially friction-straps
of different forms for the measurement of the brake horse-power of an engine or motor.
§ 104. Stiffness of Ropes.—Ropes offer a resistance to being bent, and, when bent, to
being straightened again, which arises from the mutual friction of their fibres. It increases
with the sectional area of the rope, and is inversely proportional to the radius of the curve
into which it is bent.
The work lost in pulling a given length of rope over a pulley is found by multiplying the
length of the rope in feet by its stiffness in pounds, that stiffness being the excess of the
tension at the leading side of the rope above that at the following side, which is necessary
to bend it into a curve fitting the pulley, and then to straighten it again.
The following empirical formulae for the stiffness of hempen ropes have been deduced
by Morin from the experiments of Coulomb:—
Page 561
Let F be the stiffness in pounds avoirdupois; d the diameter of the rope in inches, n =
48d2 for white ropes and 35d2 for tarred ropes; r the effective radius of the pulley in inches;
T the tension in pounds. Then
n
For white ropes, F = (0.0012 + 0.001026n + 0.0012T).
r
n
For tarred ropes, F = (0.006 + 0.001392n + 0.00168T).
r (68)
§ 105. Friction-Couplings.—Friction is useful as a means of communicating motion
where sudden changes either of force or velocity take place, because, being limited in
amount, it may be so adjusted as to limit the forces which strain the pieces of the
mechanism within the bounds of safety. Amongst contrivances for effecting this object are
friction-cones. A rotating shaft carries upon a cylindrical portion of its figure a wheel or
pulley turning loosely on it, and consequently capable of remaining at rest when the shaft is
in motion. This pulley has fixed to one side, and concentric with it, a short frustum of a
hollow cone. At a small distance from the pulley the shaft carries a short frustum of a solid
cone accurately turned to fit the hollow cone. This frustum is made always to turn along
with the shaft by being fitted on a square portion of it, or by means of a rib and groove, or
otherwise, but is capable of a slight longitudinal motion, so as to be pressed into, or
withdrawn from, the hollow cone by means of a lever. When the cones are pressed together
or engaged, their friction causes the pulley to rotate along with the shaft; when they are
disengaged, the pulley is free to stand still. The angle made by the sides of the cones with
the axis should not be less than the angle of repose. In the friction-clutch, a pulley loose on
a shaft has a hoop or gland made to embrace it more or less tightly by means of a screw;
this hoop has short projecting arms or ears. A fork or clutch rotates along with the shaft,
and is capable of being moved longitudinally by a handle. When the clutch is moved
towards the hoop, its arms catch those of the hoop, and cause the hoop to rotate and to
communicate its rotation to the pulley by friction. There are many other contrivances of the
same class, but the two just mentioned may serve for examples.
§ 106. Heat of Friction: Unguents.—The work lost in friction is employed in producing
heat. This fact is very obvious, and has been known from a remote period; but the exact
determination of the proportion of the work lost to the heat produced, and the experimental
proof that that proportion is the same under all circumstances and with all materials, solid,
liquid and gaseous, are comparatively recent achievements of J. P. Joule. The quantity of
work which produces a British unit of heat (or so much heat as elevates the temperature of
one pound of pure water, at or near ordinary atmospheric temperatures, by 1° F.) is 772
foot-pounds. This constant, now designated as “Joule’s equivalent,” is the principal
experimental datum of the science of thermodynamics.
48d2 for white ropes and 35d2 for tarred ropes; r the effective radius of the pulley in inches;
T the tension in pounds. Then
n
For white ropes, F = (0.0012 + 0.001026n + 0.0012T).
r
n
For tarred ropes, F = (0.006 + 0.001392n + 0.00168T).
r (68)
§ 105. Friction-Couplings.—Friction is useful as a means of communicating motion
where sudden changes either of force or velocity take place, because, being limited in
amount, it may be so adjusted as to limit the forces which strain the pieces of the
mechanism within the bounds of safety. Amongst contrivances for effecting this object are
friction-cones. A rotating shaft carries upon a cylindrical portion of its figure a wheel or
pulley turning loosely on it, and consequently capable of remaining at rest when the shaft is
in motion. This pulley has fixed to one side, and concentric with it, a short frustum of a
hollow cone. At a small distance from the pulley the shaft carries a short frustum of a solid
cone accurately turned to fit the hollow cone. This frustum is made always to turn along
with the shaft by being fitted on a square portion of it, or by means of a rib and groove, or
otherwise, but is capable of a slight longitudinal motion, so as to be pressed into, or
withdrawn from, the hollow cone by means of a lever. When the cones are pressed together
or engaged, their friction causes the pulley to rotate along with the shaft; when they are
disengaged, the pulley is free to stand still. The angle made by the sides of the cones with
the axis should not be less than the angle of repose. In the friction-clutch, a pulley loose on
a shaft has a hoop or gland made to embrace it more or less tightly by means of a screw;
this hoop has short projecting arms or ears. A fork or clutch rotates along with the shaft,
and is capable of being moved longitudinally by a handle. When the clutch is moved
towards the hoop, its arms catch those of the hoop, and cause the hoop to rotate and to
communicate its rotation to the pulley by friction. There are many other contrivances of the
same class, but the two just mentioned may serve for examples.
§ 106. Heat of Friction: Unguents.—The work lost in friction is employed in producing
heat. This fact is very obvious, and has been known from a remote period; but the exact
determination of the proportion of the work lost to the heat produced, and the experimental
proof that that proportion is the same under all circumstances and with all materials, solid,
liquid and gaseous, are comparatively recent achievements of J. P. Joule. The quantity of
work which produces a British unit of heat (or so much heat as elevates the temperature of
one pound of pure water, at or near ordinary atmospheric temperatures, by 1° F.) is 772
foot-pounds. This constant, now designated as “Joule’s equivalent,” is the principal
experimental datum of the science of thermodynamics.
Page 562
A more recent determination (Phil. Trans., 1897), by Osborne Reynolds and W. M.
Moorby, gives 778 as the mean value of Joule’s equivalent through the range of 32° to 212°
F. See also the papers of Rowland in the Proc. Amer. Acad. (1879), and Griffiths, Phil.
Trans. (1893).
The heat produced by friction, when moderate in amount, is useful in softening and
liquefying thick unguents; but when excessive it is prejudicial, by decomposing the
unguents, and sometimes even by softening the metal of the bearings, and raising their
temperature so high as to set fire to neighbouring combustible matters.
Excessive heating is prevented by a constant and copious supply of a good unguent. The
elevation of temperature produced by the friction of a journal is sometimes used as an
experimental test of the quality of unguents. For modern methods of forced lubrication see
Bearings.
§ 107. Rolling Resistance.—By the rolling of two surfaces over each other without
sliding a resistance is caused which is called sometimes “rolling friction,” but more
correctly rolling resistance. It is of the nature of a couple, resisting rotation. Its moment is
found by multiplying the normal pressure between the rolling surfaces by an arm, whose
length depends on the nature of the rolling surfaces, and the work lost in a unit of time in
overcoming it is the product of its moment by the angular velocity of the rolling surfaces
relatively to each other. The following are approximate values of the arm in decimals of a
foot:—
Oak upon oak 0.006 (Coulomb).
Lignum vitae on oak 0.004 ”
Cast iron on cast iron 0.002 (Tredgold).
§ 108. Reciprocating Forces: Stored and Restored Energy.—When a force acts on a
machine alternately as an effort and as a resistance, it may be called a reciprocating force.
Of this kind is the weight of any piece in the mechanism whose centre of gravity
alternately rises and falls; for during the rise of the centre of gravity that weight acts as a
resistance, and energy is employed in lifting it to an amount expressed by the product of
the weight into the vertical height of its rise; and during the fall of the centre of gravity the
weight acts as an effort, and exerts in assisting to perform the work of the machine an
amount of energy exactly equal to that which had previously been employed in lifting it.
Thus that amount of energy is not lost, but has its operation deferred; and it is said to be
stored when the weight is lifted, and restored when it falls.
Moorby, gives 778 as the mean value of Joule’s equivalent through the range of 32° to 212°
F. See also the papers of Rowland in the Proc. Amer. Acad. (1879), and Griffiths, Phil.
Trans. (1893).
The heat produced by friction, when moderate in amount, is useful in softening and
liquefying thick unguents; but when excessive it is prejudicial, by decomposing the
unguents, and sometimes even by softening the metal of the bearings, and raising their
temperature so high as to set fire to neighbouring combustible matters.
Excessive heating is prevented by a constant and copious supply of a good unguent. The
elevation of temperature produced by the friction of a journal is sometimes used as an
experimental test of the quality of unguents. For modern methods of forced lubrication see
Bearings.
§ 107. Rolling Resistance.—By the rolling of two surfaces over each other without
sliding a resistance is caused which is called sometimes “rolling friction,” but more
correctly rolling resistance. It is of the nature of a couple, resisting rotation. Its moment is
found by multiplying the normal pressure between the rolling surfaces by an arm, whose
length depends on the nature of the rolling surfaces, and the work lost in a unit of time in
overcoming it is the product of its moment by the angular velocity of the rolling surfaces
relatively to each other. The following are approximate values of the arm in decimals of a
foot:—
Oak upon oak 0.006 (Coulomb).
Lignum vitae on oak 0.004 ”
Cast iron on cast iron 0.002 (Tredgold).
§ 108. Reciprocating Forces: Stored and Restored Energy.—When a force acts on a
machine alternately as an effort and as a resistance, it may be called a reciprocating force.
Of this kind is the weight of any piece in the mechanism whose centre of gravity
alternately rises and falls; for during the rise of the centre of gravity that weight acts as a
resistance, and energy is employed in lifting it to an amount expressed by the product of
the weight into the vertical height of its rise; and during the fall of the centre of gravity the
weight acts as an effort, and exerts in assisting to perform the work of the machine an
amount of energy exactly equal to that which had previously been employed in lifting it.
Thus that amount of energy is not lost, but has its operation deferred; and it is said to be
stored when the weight is lifted, and restored when it falls.
Page 563
In a machine of which each piece is to move with a uniform velocity, if the effort and the
resistance be constant, the weight of each piece must be balanced on its axis, so that it may
produce lateral pressure only, and not act as a reciprocating force. But if the effort and the
resistance be alternately in excess, the uniformity of speed may still be preserved by so
adjusting some moving weight in the mechanism that when the effort is in excess it may be
lifted, and so balance and employ the excess of effort, and that when the resistance is in
excess it may fall, and so balance and overcome the excess of resistance—thus storing the
periodical excess of energy and restoring that energy to perform the periodical excess of
work.
Other forces besides gravity may be used as reciprocating forces for storing and
restoring energy—for example, the elasticity of a spring or of a mass of air.
In most of the delusive machines commonly called “perpetual motions,” of which so
many are patented in each year, and which are expected by their inventors to perform work
without receiving energy, the fundamental fallacy consists in an expectation that some
reciprocating force shall restore more energy than it has been the means of storing.
Division 2. Deflecting Forces.
§ 109. Deflecting Force for Translation in a Curved Path.—In machinery, deflecting
force is supplied by the tenacity of some piece, such as a crank, which guides the deflected
body in its curved path, and is unbalanced, being employed in producing deflexion, and not
in balancing another force.
§ 110. Centrifugal Force of a Rotating Body.—The centrifugal force exerted by a
rotating body on its axis of rotation is the same in magnitude as if the mass of the body
were concentrated at its centre of gravity, and acts in a plane passing through the axis of
rotation and the centre of gravity of the body.
The particles of a rotating body exert centrifugal forces on each other, which strain the
body, and tend to tear it asunder, but these forces balance each other, and do not affect the
resultant centrifugal force exerted on the axis of rotation.3
If the axis of rotation traverses the centre of gravity of the body, the centrifugal force
exerted on that axis is nothing.
Hence, unless there be some reason to the contrary, each piece of a machine should be
balanced on its axis of rotation; otherwise the centrifugal force will cause strains, vibration
and increased friction, and a tendency of the shafts to jump out of their bearings.
resistance be constant, the weight of each piece must be balanced on its axis, so that it may
produce lateral pressure only, and not act as a reciprocating force. But if the effort and the
resistance be alternately in excess, the uniformity of speed may still be preserved by so
adjusting some moving weight in the mechanism that when the effort is in excess it may be
lifted, and so balance and employ the excess of effort, and that when the resistance is in
excess it may fall, and so balance and overcome the excess of resistance—thus storing the
periodical excess of energy and restoring that energy to perform the periodical excess of
work.
Other forces besides gravity may be used as reciprocating forces for storing and
restoring energy—for example, the elasticity of a spring or of a mass of air.
In most of the delusive machines commonly called “perpetual motions,” of which so
many are patented in each year, and which are expected by their inventors to perform work
without receiving energy, the fundamental fallacy consists in an expectation that some
reciprocating force shall restore more energy than it has been the means of storing.
Division 2. Deflecting Forces.
§ 109. Deflecting Force for Translation in a Curved Path.—In machinery, deflecting
force is supplied by the tenacity of some piece, such as a crank, which guides the deflected
body in its curved path, and is unbalanced, being employed in producing deflexion, and not
in balancing another force.
§ 110. Centrifugal Force of a Rotating Body.—The centrifugal force exerted by a
rotating body on its axis of rotation is the same in magnitude as if the mass of the body
were concentrated at its centre of gravity, and acts in a plane passing through the axis of
rotation and the centre of gravity of the body.
The particles of a rotating body exert centrifugal forces on each other, which strain the
body, and tend to tear it asunder, but these forces balance each other, and do not affect the
resultant centrifugal force exerted on the axis of rotation.3
If the axis of rotation traverses the centre of gravity of the body, the centrifugal force
exerted on that axis is nothing.
Hence, unless there be some reason to the contrary, each piece of a machine should be
balanced on its axis of rotation; otherwise the centrifugal force will cause strains, vibration
and increased friction, and a tendency of the shafts to jump out of their bearings.
Page 564
§ 111. Centrifugal Couples of a Rotating Body.—Besides the tendency (if any) of the
combined centrifugal forces of the particles of a rotating body to shift the axis of rotation,
they may also tend to turn it out of its original direction. The latter tendency is called a
centrifugal couple, and vanishes for rotation about a principal axis.
It is essential to the steady motion of every rapidly rotating piece in a machine that its
axis of rotation should not merely traverse its centre of gravity, but should be a permanent
axis; for otherwise the centrifugal couples will increase friction, produce oscillation of the
shaft and tend to make it leave its bearings.
The principles of this and the preceding section are those which regulate the adjustment
of the weight and position of the counterpoises which are placed between the spokes of the
driving-wheels of locomotive engines.
§ 112.* Method of computing the position and
magnitudes of balance weights which must be added to a
given system of arbitrarily chosen rotating masses in
order to make the common axis of rotation a permanent
axis.—The method here briefly explained is taken from a
paper by W. E. Dalby, “The Balancing of Engines with
special reference to Marine Work,” Trans. Inst. Nav.
(From Balancing of Engines, by
Arch. (1899). Let the weight (fig. 130), attached to a truly
permission of Edward Arnold.)
turned disk, be rotated by the shaft OX, and conceive that
the shaft is held in a bearing at one point, O. The force Fig. 130.
required to constrain the weight to move in a circle, that
is the deviating force, produces an equal and opposite reaction on the shaft, whose amount
F is equal to the centrifugal force Wa2r/g ℔, where r is the radius of the mass centre of the
weight, and a is its angular velocity in radians per second. Transferring this force to the
point O, it is equivalent to, (1) a force at O equal and parallel to F, and, (2) a centrifugal
couple of Fa foot-pounds. In order that OX may be a permanent axis it is necessary that
there should be a sufficient number of weights attached to the shaft and so distributed that
when each is referred to the point O
(1) ΣF = 0
(2) ΣFa = 0
(a)
The plane through O to which the shaft is perpendicular is called the reference plane,
because all the transferred forces act in that plane at the point O. The plane through the
radius of the weight containing the axis OX is called the axial plane because it contains the
combined centrifugal forces of the particles of a rotating body to shift the axis of rotation,
they may also tend to turn it out of its original direction. The latter tendency is called a
centrifugal couple, and vanishes for rotation about a principal axis.
It is essential to the steady motion of every rapidly rotating piece in a machine that its
axis of rotation should not merely traverse its centre of gravity, but should be a permanent
axis; for otherwise the centrifugal couples will increase friction, produce oscillation of the
shaft and tend to make it leave its bearings.
The principles of this and the preceding section are those which regulate the adjustment
of the weight and position of the counterpoises which are placed between the spokes of the
driving-wheels of locomotive engines.
§ 112.* Method of computing the position and
magnitudes of balance weights which must be added to a
given system of arbitrarily chosen rotating masses in
order to make the common axis of rotation a permanent
axis.—The method here briefly explained is taken from a
paper by W. E. Dalby, “The Balancing of Engines with
special reference to Marine Work,” Trans. Inst. Nav.
(From Balancing of Engines, by
Arch. (1899). Let the weight (fig. 130), attached to a truly
permission of Edward Arnold.)
turned disk, be rotated by the shaft OX, and conceive that
the shaft is held in a bearing at one point, O. The force Fig. 130.
required to constrain the weight to move in a circle, that
is the deviating force, produces an equal and opposite reaction on the shaft, whose amount
F is equal to the centrifugal force Wa2r/g ℔, where r is the radius of the mass centre of the
weight, and a is its angular velocity in radians per second. Transferring this force to the
point O, it is equivalent to, (1) a force at O equal and parallel to F, and, (2) a centrifugal
couple of Fa foot-pounds. In order that OX may be a permanent axis it is necessary that
there should be a sufficient number of weights attached to the shaft and so distributed that
when each is referred to the point O
(1) ΣF = 0
(2) ΣFa = 0
(a)
The plane through O to which the shaft is perpendicular is called the reference plane,
because all the transferred forces act in that plane at the point O. The plane through the
radius of the weight containing the axis OX is called the axial plane because it contains the
Page 565
forces forming the couple due to the transference of F to the reference plane. Substituting
the values of F in (a) the two conditions become
α2
(1) (W1r1 + W2r2 + W3r3 + ...) =0
g
α2
(2) (W1a1r1 + W2a2r2 + ... ) =0
g (b)
In order that these conditions may obtain, the quantities in the brackets must be zero, since
the factor α2/g is not zero. Hence finally the conditions which must be satisfied by the
system of weights in order that the axis of rotation may be a permanent axis is
(1) (W1r1 + W2r2 + W3r3) = 0
(2) (W1a1r1 + W2a2r2 + W3a3r3) = 0
(c)
It must be remembered that these are all directed quantities, and that their respective sums
are to be taken by drawing vector polygons. In drawing these polygons the magnitude of
the vector of the type Wr is the product Wr, and the direction of the vector is from the shaft
outwards towards the weight W, parallel to the radius r. For the vector representing a
couple of the type War, if the masses are all on the same side of the reference plane, the
direction of drawing is from the axis outwards; if the masses are some on one side of the
reference plane and some on the other side, the direction of drawing is from the axis
outwards towards the weight for all masses on the one side, and from the mass inwards
towards the axis for all weights on the other side, drawing always parallel to the direction
defined by the radius r. The magnitude of the vector is the product War. The conditions (c)
may thus be expressed: first, that the sum of the vectors Wr must form a closed polygon,
and, second, that the sum of the vectors War must form a closed polygon. The general
problem in practice is, given a system of weights attached to a shaft, to find the respective
weights and positions of two balance weights or counterpoises which must be added to the
system in order to make the shaft a permanent axis, the planes in which the balance weights
are to revolve also being given. To solve this the reference plane must be chosen so that it
coincides with the plane of revolution of one of the as yet unknown balance weights. The
balance weight in this plane has therefore no couple corresponding to it. Hence by drawing
a couple polygon for the given weights the vector which is required to close the polygon is
at once found and from it the magnitude and position of the balance weight which must be
added to the system to balance the couples follow at once. Then, transferring the product
Wr corresponding with this balance weight to the reference plane, proceed to draw the
the values of F in (a) the two conditions become
α2
(1) (W1r1 + W2r2 + W3r3 + ...) =0
g
α2
(2) (W1a1r1 + W2a2r2 + ... ) =0
g (b)
In order that these conditions may obtain, the quantities in the brackets must be zero, since
the factor α2/g is not zero. Hence finally the conditions which must be satisfied by the
system of weights in order that the axis of rotation may be a permanent axis is
(1) (W1r1 + W2r2 + W3r3) = 0
(2) (W1a1r1 + W2a2r2 + W3a3r3) = 0
(c)
It must be remembered that these are all directed quantities, and that their respective sums
are to be taken by drawing vector polygons. In drawing these polygons the magnitude of
the vector of the type Wr is the product Wr, and the direction of the vector is from the shaft
outwards towards the weight W, parallel to the radius r. For the vector representing a
couple of the type War, if the masses are all on the same side of the reference plane, the
direction of drawing is from the axis outwards; if the masses are some on one side of the
reference plane and some on the other side, the direction of drawing is from the axis
outwards towards the weight for all masses on the one side, and from the mass inwards
towards the axis for all weights on the other side, drawing always parallel to the direction
defined by the radius r. The magnitude of the vector is the product War. The conditions (c)
may thus be expressed: first, that the sum of the vectors Wr must form a closed polygon,
and, second, that the sum of the vectors War must form a closed polygon. The general
problem in practice is, given a system of weights attached to a shaft, to find the respective
weights and positions of two balance weights or counterpoises which must be added to the
system in order to make the shaft a permanent axis, the planes in which the balance weights
are to revolve also being given. To solve this the reference plane must be chosen so that it
coincides with the plane of revolution of one of the as yet unknown balance weights. The
balance weight in this plane has therefore no couple corresponding to it. Hence by drawing
a couple polygon for the given weights the vector which is required to close the polygon is
at once found and from it the magnitude and position of the balance weight which must be
added to the system to balance the couples follow at once. Then, transferring the product
Wr corresponding with this balance weight to the reference plane, proceed to draw the
Page 566
force polygon. The vector required to close it will determine the second balance weight, the
work may be checked by taking the reference plane to coincide with the plane of revolution
of the second balance weight and then re-determining them, or by taking a reference plane
anywhere and including the two balance weights trying if condition (c) is satisfied.
When a weight is reciprocated, the equal and opposite force required for its acceleration
at any instant appears as an unbalanced force on the frame of the machine to which the
weight belongs. In the particular case, where the motion is of the kind known as “simple
harmonic” the disturbing force on the frame due to the reciprocation of the weight is equal
to the component of the centrifugal force in the line of stroke due to a weight equal to the
reciprocated weight supposed concentrated at the crank pin. Using this principle the
method of finding the balance weights to be added to a given system of reciprocating
weights in order to produce a system of forces on the frame continuously in equilibrium is
exactly the same as that just explained for a system of revolving weights, because for the
purpose of finding the balance weights each reciprocating weight may be supposed
attached to the crank pin which operates it, thus forming an equivalent revolving system.
The balance weights found as part of the equivalent revolving system when reciprocated by
their respective crank pins form the balance weights for the given reciprocating system.
These conditions may be exactly realized by a system of weights reciprocated by slotted
bars, the crank shaft driving the slotted bars rotating uniformly. In practice reciprocation is
usually effected through a connecting rod, as in the case of steam engines. In balancing the
mechanism of a steam engine it is often sufficiently accurate to consider the motion of the
pistons as simple harmonic, and the effect on the framework of the acceleration of the
connecting rod may be approximately allowed for by distributing the weight of the rod
between the crank pin and the piston inversely as the centre of gravity of the rod divides
the distance between the centre of the cross head pin and the centre of the crank pin. The
moving parts of the engine are then divided into two complete and independent systems,
namely, one system of revolving weights consisting of crank pins, crank arms, &c.,
attached to and revolving with the crank shaft, and a second system of reciprocating
weights consisting of the pistons, cross-heads, &c., supposed to be moving each in its line
of stroke with simple harmonic motion. The balance weights are to be separately calculated
for each system, the one set being added to the crank shaft as revolving weights, and the
second set being included with the reciprocating weights and operated by a properly placed
crank on the crank shaft. Balance weights added in this way to a set of reciprocating
weights are sometimes called bob-weights. In the case of locomotives the balance weights
required to balance the pistons are added as revolving weights to the crank shaft system,
and in fact are generally combined with the weights required to balance the revolving
system so as to form one weight, the counterpoise referred to in the preceding section,
which is seen between the spokes of the wheels of a locomotive. Although this method
work may be checked by taking the reference plane to coincide with the plane of revolution
of the second balance weight and then re-determining them, or by taking a reference plane
anywhere and including the two balance weights trying if condition (c) is satisfied.
When a weight is reciprocated, the equal and opposite force required for its acceleration
at any instant appears as an unbalanced force on the frame of the machine to which the
weight belongs. In the particular case, where the motion is of the kind known as “simple
harmonic” the disturbing force on the frame due to the reciprocation of the weight is equal
to the component of the centrifugal force in the line of stroke due to a weight equal to the
reciprocated weight supposed concentrated at the crank pin. Using this principle the
method of finding the balance weights to be added to a given system of reciprocating
weights in order to produce a system of forces on the frame continuously in equilibrium is
exactly the same as that just explained for a system of revolving weights, because for the
purpose of finding the balance weights each reciprocating weight may be supposed
attached to the crank pin which operates it, thus forming an equivalent revolving system.
The balance weights found as part of the equivalent revolving system when reciprocated by
their respective crank pins form the balance weights for the given reciprocating system.
These conditions may be exactly realized by a system of weights reciprocated by slotted
bars, the crank shaft driving the slotted bars rotating uniformly. In practice reciprocation is
usually effected through a connecting rod, as in the case of steam engines. In balancing the
mechanism of a steam engine it is often sufficiently accurate to consider the motion of the
pistons as simple harmonic, and the effect on the framework of the acceleration of the
connecting rod may be approximately allowed for by distributing the weight of the rod
between the crank pin and the piston inversely as the centre of gravity of the rod divides
the distance between the centre of the cross head pin and the centre of the crank pin. The
moving parts of the engine are then divided into two complete and independent systems,
namely, one system of revolving weights consisting of crank pins, crank arms, &c.,
attached to and revolving with the crank shaft, and a second system of reciprocating
weights consisting of the pistons, cross-heads, &c., supposed to be moving each in its line
of stroke with simple harmonic motion. The balance weights are to be separately calculated
for each system, the one set being added to the crank shaft as revolving weights, and the
second set being included with the reciprocating weights and operated by a properly placed
crank on the crank shaft. Balance weights added in this way to a set of reciprocating
weights are sometimes called bob-weights. In the case of locomotives the balance weights
required to balance the pistons are added as revolving weights to the crank shaft system,
and in fact are generally combined with the weights required to balance the revolving
system so as to form one weight, the counterpoise referred to in the preceding section,
which is seen between the spokes of the wheels of a locomotive. Although this method
Page 567
balances the pistons in the horizontal plane, and thus allows the pull of the engine on the
train to be exerted without the variation due to the reciprocation of the pistons, yet the force
balanced horizontally is introduced vertically and appears as a variation of pressure on the
rail. In practice about two-thirds of the reciprocating weight is balanced in order to keep
this variation of rail pressure within safe limits. The assumption that the pistons of an
engine move with simple harmonic motion is increasingly erroneous as the ratio of the
length of the crank r, to the length of the connecting rod l increases. A more accurate
though still approximate expression for the force on the frame due to the acceleration of the
piston whose weight is W is given by
W r
g ω2r { cos θ + l cos 2θ }
The conditions regulating the balancing of a system of weights reciprocating under the
action of accelerating forces given by the above expression are investigated in a paper by
Otto Schlick, “On Balancing of Steam Engines,” Trans, Inst. Nav. Arch. (1900), and in a
paper by W. E. Dalby, “On the Balancing of the Reciprocating Parts of Engines, including
the Effect of the Connecting Rod” (ibid., 1901). A still more accurate expression than the
above is obtained by expansion in a Fourier series, regarding which and its bearing on
balancing engines see a paper by J. H. Macalpine, “A Solution of the Vibration Problem”
(ibid., 1901). The whole subject is dealt with in a treatise, The Balancing of Engines, by W.
E. Dalby (London, 1906). Most of the original papers on this subject of engine balancing
are to be found in the Transactions of the Institution of Naval Architects.
§ 113.* Centrifugal Whirling of Shafts.—When a system of revolving masses is balanced
so that the conditions of the preceding section are fulfilled, the centre of gravity of the
system lies on the axis of revolution. If there is the slightest displacement of the centre of
gravity of the system from the axis of revolution a force acts on the shaft tending to deflect
it, and varies as the deflexion and as the square of the speed. If the shaft is therefore to
revolve stably, this force must be balanced at any instant by the elastic resistance of the
shaft to deflexion. To take a simple case, suppose a shaft, supported on two bearings to
carry a disk of weight W at its centre, and let the centre of gravity of the disk be at a
distance e from the axis of rotation, this small distance being due to imperfections of
material or faulty construction. Neglecting the mass of the shaft itself, when the shaft
rotates with an angular velocity a, the centrifugal force Wa2e/g will act upon the shaft and
cause its axis to deflect from the axis of rotation a distance, y say. The elastic resistance
evoked by this deflexion is proportional to the deflexion, so that if c is a constant
depending upon the form, material and method of support of the shaft, the following
equality must hold if the shaft is to rotate stably at the stated speed—
train to be exerted without the variation due to the reciprocation of the pistons, yet the force
balanced horizontally is introduced vertically and appears as a variation of pressure on the
rail. In practice about two-thirds of the reciprocating weight is balanced in order to keep
this variation of rail pressure within safe limits. The assumption that the pistons of an
engine move with simple harmonic motion is increasingly erroneous as the ratio of the
length of the crank r, to the length of the connecting rod l increases. A more accurate
though still approximate expression for the force on the frame due to the acceleration of the
piston whose weight is W is given by
W r
g ω2r { cos θ + l cos 2θ }
The conditions regulating the balancing of a system of weights reciprocating under the
action of accelerating forces given by the above expression are investigated in a paper by
Otto Schlick, “On Balancing of Steam Engines,” Trans, Inst. Nav. Arch. (1900), and in a
paper by W. E. Dalby, “On the Balancing of the Reciprocating Parts of Engines, including
the Effect of the Connecting Rod” (ibid., 1901). A still more accurate expression than the
above is obtained by expansion in a Fourier series, regarding which and its bearing on
balancing engines see a paper by J. H. Macalpine, “A Solution of the Vibration Problem”
(ibid., 1901). The whole subject is dealt with in a treatise, The Balancing of Engines, by W.
E. Dalby (London, 1906). Most of the original papers on this subject of engine balancing
are to be found in the Transactions of the Institution of Naval Architects.
§ 113.* Centrifugal Whirling of Shafts.—When a system of revolving masses is balanced
so that the conditions of the preceding section are fulfilled, the centre of gravity of the
system lies on the axis of revolution. If there is the slightest displacement of the centre of
gravity of the system from the axis of revolution a force acts on the shaft tending to deflect
it, and varies as the deflexion and as the square of the speed. If the shaft is therefore to
revolve stably, this force must be balanced at any instant by the elastic resistance of the
shaft to deflexion. To take a simple case, suppose a shaft, supported on two bearings to
carry a disk of weight W at its centre, and let the centre of gravity of the disk be at a
distance e from the axis of rotation, this small distance being due to imperfections of
material or faulty construction. Neglecting the mass of the shaft itself, when the shaft
rotates with an angular velocity a, the centrifugal force Wa2e/g will act upon the shaft and
cause its axis to deflect from the axis of rotation a distance, y say. The elastic resistance
evoked by this deflexion is proportional to the deflexion, so that if c is a constant
depending upon the form, material and method of support of the shaft, the following
equality must hold if the shaft is to rotate stably at the stated speed—
Page 568
W
(y + e) a2 = cy,
g
from which y = Wa2e / (gc − Wa2).
This expression shows that as a increases y increases until when Wa2 = gc, y becomes
infinitely large. The corresponding value of a, namely √(gc/W), is called the critical
velocity of the shaft, and is the speed at which the shaft ceases to rotate stably and at which
centrifugal whirling begins. The general problem is to find the value of a corresponding to
all kinds of loadings on shafts supported in any manner. The question was investigated by
Rankine in an article in the Engineer (April 9, 1869). Professor A. G. Greenhill treated the
problem of the centrifugal whirling of an unloaded shaft with different supporting
conditions in a paper “On the Strength of Shafting exposed both to torsion and to end
thrust,” Proc. Inst. Mech. Eng. (1883). Professor S. Dunkerley (“On the Whirling and
Vibration of Shafts,” Phil. Trans., 1894) investigated the question for the cases of loaded
and unloaded shafts, and, owing to the complication arising from the application of the
general theory to the cases of loaded shafts, devised empirical formulae for the critical
speeds of shafts loaded with heavy pulleys, based generally upon the following
assumption, which is stated for the case of a shaft carrying one pulley: If N1, N2 be the
separate speeds of whirl of the shaft and pulley on the assumption that the effect of one is
neglected when that of the other is under consideration, then the resulting speed of whirl
due to both causes combined may be taken to be of the form N1N2 √(N21 + N12) where N
means revolutions per minute. This form is extended to include the cases of several pulleys
on the same shaft. The interesting and important part of the investigation is that a number
of experiments were made on small shafts arranged in different ways and loaded in
different ways, and the speed at which whirling actually occurred was compared with the
speed calculated from formulae of the general type indicated above. The agreement
between the observed and calculated values of the critical speeds was in most cases quite
remarkable. In a paper by Dr C. Chree, “The Whirling and Transverse Vibrations of
Rotating Shafts,” Proc. Phys. Soc. Lon., vol. 19 (1904); also Phil. Mag., vol. 7 (1904), the
question is investigated from a new mathematical point of view, and expressions for the
whirling of loaded shafts are obtained without the necessity of any assumption of the kind
stated above. An elementary presentation of the problem from a practical point of view will
be found in Steam Turbines, by Dr A. Stodola (London, 1905).
§ 114. Revolving Pendulum. Governors.—In fig. 131 AO represents an upright axis or
spindle; B a weight called a bob, suspended by rod OB from a horizontal axis at O, carried
by the vertical axis. When the spindle is at rest the bob hangs close to it; when the spindle
rotates, the bob, being made to revolve round it, diverges until the resultant of the
centrifugal force and the weight of the bob is a force acting at O in the direction OB, and
(y + e) a2 = cy,
g
from which y = Wa2e / (gc − Wa2).
This expression shows that as a increases y increases until when Wa2 = gc, y becomes
infinitely large. The corresponding value of a, namely √(gc/W), is called the critical
velocity of the shaft, and is the speed at which the shaft ceases to rotate stably and at which
centrifugal whirling begins. The general problem is to find the value of a corresponding to
all kinds of loadings on shafts supported in any manner. The question was investigated by
Rankine in an article in the Engineer (April 9, 1869). Professor A. G. Greenhill treated the
problem of the centrifugal whirling of an unloaded shaft with different supporting
conditions in a paper “On the Strength of Shafting exposed both to torsion and to end
thrust,” Proc. Inst. Mech. Eng. (1883). Professor S. Dunkerley (“On the Whirling and
Vibration of Shafts,” Phil. Trans., 1894) investigated the question for the cases of loaded
and unloaded shafts, and, owing to the complication arising from the application of the
general theory to the cases of loaded shafts, devised empirical formulae for the critical
speeds of shafts loaded with heavy pulleys, based generally upon the following
assumption, which is stated for the case of a shaft carrying one pulley: If N1, N2 be the
separate speeds of whirl of the shaft and pulley on the assumption that the effect of one is
neglected when that of the other is under consideration, then the resulting speed of whirl
due to both causes combined may be taken to be of the form N1N2 √(N21 + N12) where N
means revolutions per minute. This form is extended to include the cases of several pulleys
on the same shaft. The interesting and important part of the investigation is that a number
of experiments were made on small shafts arranged in different ways and loaded in
different ways, and the speed at which whirling actually occurred was compared with the
speed calculated from formulae of the general type indicated above. The agreement
between the observed and calculated values of the critical speeds was in most cases quite
remarkable. In a paper by Dr C. Chree, “The Whirling and Transverse Vibrations of
Rotating Shafts,” Proc. Phys. Soc. Lon., vol. 19 (1904); also Phil. Mag., vol. 7 (1904), the
question is investigated from a new mathematical point of view, and expressions for the
whirling of loaded shafts are obtained without the necessity of any assumption of the kind
stated above. An elementary presentation of the problem from a practical point of view will
be found in Steam Turbines, by Dr A. Stodola (London, 1905).
§ 114. Revolving Pendulum. Governors.—In fig. 131 AO represents an upright axis or
spindle; B a weight called a bob, suspended by rod OB from a horizontal axis at O, carried
by the vertical axis. When the spindle is at rest the bob hangs close to it; when the spindle
rotates, the bob, being made to revolve round it, diverges until the resultant of the
centrifugal force and the weight of the bob is a force acting at O in the direction OB, and
Page 569
then it revolves steadily in a circle. This combination is
called a revolving, centrifugal, or conical pendulum.
Revolving pendulums are usually constructed with pairs of
rods and bobs, as OB, Ob, hung at opposite sides of the
spindle, that the centrifugal forces exerted at the point O
may balance each other.
In finding the position in which the bob will revolve with
a given angular velocity, a, for most practical cases
connected with machinery the mass of the rod may be
considered as insensible compared with that of the bob. Let
Fig. 131.
the bob be a sphere, and from the centre of that sphere draw
BH = y perpendicular to OA. Let OH = z; let W be the
weight of the bob, F its centrifugal force. Then the condition of its steady revolution is W :
F :: z : y; that is to say, y/z = F/W = yα2/g; consequently
z = g/α2
(69)
Or, if n = α 2π = α/6.2832 be the number of turns or fractions of a turn in a second,
g 0.8165 ft. 9.79771 in.
z= = =
4π2n2 n2 n2 (70)
z is called the altitude of the pendulum.
If the rod of a revolving pendulum be jointed, as in fig. 132,
not to a point in the vertical axis, but to the end of a projecting
arm C, the position in which the bob will revolve will be the same
as if the rod were jointed to the point O, where its prolongation
cuts the vertical axis.
A revolving pendulum is an essential part of most of the Fig. 132.
contrivances called governors, for regulating the speed of prime
movers, for further particulars of which see Steam Engine.
Division 3. Working of Machines of Varying Velocity.
§ 115. General Principles.—In order that the velocity of every piece of a machine may
be uniform, it is necessary that the forces acting on each piece should be always exactly
balanced. Also, in order that the forces acting on each piece of a machine may be always
exactly balanced, it is necessary that the velocity of that piece should be uniform.
called a revolving, centrifugal, or conical pendulum.
Revolving pendulums are usually constructed with pairs of
rods and bobs, as OB, Ob, hung at opposite sides of the
spindle, that the centrifugal forces exerted at the point O
may balance each other.
In finding the position in which the bob will revolve with
a given angular velocity, a, for most practical cases
connected with machinery the mass of the rod may be
considered as insensible compared with that of the bob. Let
Fig. 131.
the bob be a sphere, and from the centre of that sphere draw
BH = y perpendicular to OA. Let OH = z; let W be the
weight of the bob, F its centrifugal force. Then the condition of its steady revolution is W :
F :: z : y; that is to say, y/z = F/W = yα2/g; consequently
z = g/α2
(69)
Or, if n = α 2π = α/6.2832 be the number of turns or fractions of a turn in a second,
g 0.8165 ft. 9.79771 in.
z= = =
4π2n2 n2 n2 (70)
z is called the altitude of the pendulum.
If the rod of a revolving pendulum be jointed, as in fig. 132,
not to a point in the vertical axis, but to the end of a projecting
arm C, the position in which the bob will revolve will be the same
as if the rod were jointed to the point O, where its prolongation
cuts the vertical axis.
A revolving pendulum is an essential part of most of the Fig. 132.
contrivances called governors, for regulating the speed of prime
movers, for further particulars of which see Steam Engine.
Division 3. Working of Machines of Varying Velocity.
§ 115. General Principles.—In order that the velocity of every piece of a machine may
be uniform, it is necessary that the forces acting on each piece should be always exactly
balanced. Also, in order that the forces acting on each piece of a machine may be always
exactly balanced, it is necessary that the velocity of that piece should be uniform.
Page 570
An excess of the effort exerted on any piece, above that which is necessary to balance
the resistance, is accompanied with acceleration; a deficiency of the effort, with
retardation.
When a machine is being started from a state of rest, and brought by degrees up to its
proper speed, the effort must be in excess; when it is being retarded for the purpose of
stopping it, the resistance must be in excess.
An excess of effort above resistance involves an excess of energy exerted above work
performed; that excess of energy is employed in producing acceleration.
An excess of resistance above effort involves an excess of work performed above energy
expended; that excess of work is performed by means of the retardation of the machinery.
When a machine undergoes alternate acceleration and retardation, so that at certain
instants of time, occurring at the end of intervals called periods or cycles, it returns to its
original speed, then in each of those periods or cycles the alternate excesses of energy and
of work neutralize each other; and at the end of each cycle the principle of the equality of
energy and work stated in § 87, with all its consequences, is verified exactly as in the case
of machines of uniform speed.
At intermediate instants, however, other principles have also to be taken into account,
which are deduced from the second law of motion, as applied to direct deviation, or
acceleration and retardation.
§ 116. Energy of Acceleration and Work of Retardation for a Shifting Body.—Let w be
the weight of a body which has a motion of translation in any path, and in the course of the
interval of time Δt let its velocity be increased at a uniform rate of acceleration from v1 to
v2. The rate of acceleration will be
dv/dt = const. = (v2 − v1) Δt;
and to produce this acceleration a uniform effort will be required, expressed by
P = w (v2 − v1) gΔt
(71)
(The product wv/g of the mass of a body by its velocity is called its momentum; so that
the effort required is found by dividing the increase of momentum by the time in which it is
produced.)
To find the energy which has to be exerted to produce the acceleration from v1 to v2, it is
to be observed that the distance through which the effort P acts during the acceleration is
the resistance, is accompanied with acceleration; a deficiency of the effort, with
retardation.
When a machine is being started from a state of rest, and brought by degrees up to its
proper speed, the effort must be in excess; when it is being retarded for the purpose of
stopping it, the resistance must be in excess.
An excess of effort above resistance involves an excess of energy exerted above work
performed; that excess of energy is employed in producing acceleration.
An excess of resistance above effort involves an excess of work performed above energy
expended; that excess of work is performed by means of the retardation of the machinery.
When a machine undergoes alternate acceleration and retardation, so that at certain
instants of time, occurring at the end of intervals called periods or cycles, it returns to its
original speed, then in each of those periods or cycles the alternate excesses of energy and
of work neutralize each other; and at the end of each cycle the principle of the equality of
energy and work stated in § 87, with all its consequences, is verified exactly as in the case
of machines of uniform speed.
At intermediate instants, however, other principles have also to be taken into account,
which are deduced from the second law of motion, as applied to direct deviation, or
acceleration and retardation.
§ 116. Energy of Acceleration and Work of Retardation for a Shifting Body.—Let w be
the weight of a body which has a motion of translation in any path, and in the course of the
interval of time Δt let its velocity be increased at a uniform rate of acceleration from v1 to
v2. The rate of acceleration will be
dv/dt = const. = (v2 − v1) Δt;
and to produce this acceleration a uniform effort will be required, expressed by
P = w (v2 − v1) gΔt
(71)
(The product wv/g of the mass of a body by its velocity is called its momentum; so that
the effort required is found by dividing the increase of momentum by the time in which it is
produced.)
To find the energy which has to be exerted to produce the acceleration from v1 to v2, it is
to be observed that the distance through which the effort P acts during the acceleration is
Page 571
Δs = (v2 + v1) Δt/2;
consequently, the energy of acceleration is
PΔs = w (v2 − v1) (v2 + v1) / 2g = w (v22 − v12) 2g,
(72)
being proportional to the increase in the square of the velocity, and independent of the time.
In order to produce a retardation from the greater velocity v2 to the less velocity v1, it is
necessary to apply to the body a resistance connected with the retardation and the time by
an equation identical in every respect with equation (71), except by the substitution of a
resistance for an effort; and in overcoming that resistance the body performs work to an
amount determined by equation (72), putting Rds for Pas.
§ 117. Energy Stored and Restored by Deviations of Velocity.—Thus a body alternately
accelerated and retarded, so as to be brought back to its original speed, performs work
during its retardation exactly equal in amount to the energy exerted upon it during its
acceleration; so that that energy may be considered as stored during the acceleration, and
restored during the retardation, in a manner analogous to the operation of a reciprocating
force (§ 108).
Let there be given the mean velocity V = 1⁄2 (v2 + v1) of a body whose weight is w, and
let it be required to determine the fluctuation of velocity v2 − v1, and the extreme velocities
v1, v2, which that body must have, in order alternately to store and restore an amount of
energy E. By equation (72) we have
E = w (v22 − v12) / 2g
which, being divided by V = 1⁄2(v2 + v1), gives
E/V = w (v2 − v1) / g;
and consequently
v2 − v1 = gE / Vw
(73)
The ratio of this fluctuation to the mean velocity, sometimes called the unsteadiness of the
motion of the body, is
(v2 − v1) V = gE / V2w.
(74)
consequently, the energy of acceleration is
PΔs = w (v2 − v1) (v2 + v1) / 2g = w (v22 − v12) 2g,
(72)
being proportional to the increase in the square of the velocity, and independent of the time.
In order to produce a retardation from the greater velocity v2 to the less velocity v1, it is
necessary to apply to the body a resistance connected with the retardation and the time by
an equation identical in every respect with equation (71), except by the substitution of a
resistance for an effort; and in overcoming that resistance the body performs work to an
amount determined by equation (72), putting Rds for Pas.
§ 117. Energy Stored and Restored by Deviations of Velocity.—Thus a body alternately
accelerated and retarded, so as to be brought back to its original speed, performs work
during its retardation exactly equal in amount to the energy exerted upon it during its
acceleration; so that that energy may be considered as stored during the acceleration, and
restored during the retardation, in a manner analogous to the operation of a reciprocating
force (§ 108).
Let there be given the mean velocity V = 1⁄2 (v2 + v1) of a body whose weight is w, and
let it be required to determine the fluctuation of velocity v2 − v1, and the extreme velocities
v1, v2, which that body must have, in order alternately to store and restore an amount of
energy E. By equation (72) we have
E = w (v22 − v12) / 2g
which, being divided by V = 1⁄2(v2 + v1), gives
E/V = w (v2 − v1) / g;
and consequently
v2 − v1 = gE / Vw
(73)
The ratio of this fluctuation to the mean velocity, sometimes called the unsteadiness of the
motion of the body, is
(v2 − v1) V = gE / V2w.
(74)
Page 572
§ 118. Actual Energy of a Shifting Body.—The energy which must be exerted on a body
of the weight w, to accelerate it from a state of rest up to a given velocity of translation v,
and the equal amount of work which that body is capable of performing by overcoming
resistance while being retarded from the same velocity of translation v to a state of rest, is
wv2 / 2g.
(75)
This is called the actual energy of the motion of the body, and is half the quantity which
in some treatises is called vis viva.
The energy stored or restored, as the case may be, by the deviations of velocity of a body
or a system of bodies, is the amount by which the actual energy is increased or diminished.
§ 119. Principle of the Conservation of Energy in Machines.—The following principle,
expressing the general law of the action of machines with a velocity uniform or varying,
includes the law of the equality of energy and work stated in § 89 for machines of uniform
speed.
In any given interval during the working of a machine, the energy exerted added to the
energy restored is equal to the energy stored added to the work performed.
§ 120. Actual Energy of Circular Translation—Moment of Inertia.—Let a small body of
the weight w undergo translation in a circular path of the radius ρ, with the angular velocity
of deflexion α, so that the common linear velocity of all its particles is v = αρ. Then the
actual energy of that body is
wv2 / 2g = wa2ρ2 / 2g.
(76)
By comparing this with the expression for the centrifugal force (wa2ρ/g), it appears that
the actual energy of a revolving body is equal to the potential energy Fρ/2 due to the action
of the deflecting force along one-half of the radius of curvature of the path of the body.
The product wρ2/g, by which the half-square of the angular velocity is multiplied, is
called the moment of inertia of the revolving body.
§ 121. Flywheels.—A flywheel is a rotating piece in a machine, generally shaped like a
wheel (that is to say, consisting of a rim with spokes), and suited to store and restore energy
by the periodical variations in its angular velocity.
The principles according to which variations of angular velocity store and restore energy
are the same as those of § 117, only substituting moment of inertia for mass, and angular
for linear velocity.
of the weight w, to accelerate it from a state of rest up to a given velocity of translation v,
and the equal amount of work which that body is capable of performing by overcoming
resistance while being retarded from the same velocity of translation v to a state of rest, is
wv2 / 2g.
(75)
This is called the actual energy of the motion of the body, and is half the quantity which
in some treatises is called vis viva.
The energy stored or restored, as the case may be, by the deviations of velocity of a body
or a system of bodies, is the amount by which the actual energy is increased or diminished.
§ 119. Principle of the Conservation of Energy in Machines.—The following principle,
expressing the general law of the action of machines with a velocity uniform or varying,
includes the law of the equality of energy and work stated in § 89 for machines of uniform
speed.
In any given interval during the working of a machine, the energy exerted added to the
energy restored is equal to the energy stored added to the work performed.
§ 120. Actual Energy of Circular Translation—Moment of Inertia.—Let a small body of
the weight w undergo translation in a circular path of the radius ρ, with the angular velocity
of deflexion α, so that the common linear velocity of all its particles is v = αρ. Then the
actual energy of that body is
wv2 / 2g = wa2ρ2 / 2g.
(76)
By comparing this with the expression for the centrifugal force (wa2ρ/g), it appears that
the actual energy of a revolving body is equal to the potential energy Fρ/2 due to the action
of the deflecting force along one-half of the radius of curvature of the path of the body.
The product wρ2/g, by which the half-square of the angular velocity is multiplied, is
called the moment of inertia of the revolving body.
§ 121. Flywheels.—A flywheel is a rotating piece in a machine, generally shaped like a
wheel (that is to say, consisting of a rim with spokes), and suited to store and restore energy
by the periodical variations in its angular velocity.
The principles according to which variations of angular velocity store and restore energy
are the same as those of § 117, only substituting moment of inertia for mass, and angular
for linear velocity.
Page 573
Let W be the weight of a flywheel, R its radius of gyration, a2 its maximum, a1 its
minimum, and A = 1⁄2 (α2 + α1) its mean angular velocity. Let
I/S = (α2 − α2) / A
denote the unsteadiness of the motion of the flywheel; the denominator S of this fraction is
called the steadiness. Let e denote the quantity by which the energy exerted in each cycle
of the working of the machine alternately exceeds and falls short of the work performed,
and which has consequently to be alternately stored by acceleration and restored by
retardation of the flywheel. The value of this periodical excess is—
e = R2W (α22 − α12), 2g,
(77)
from which, dividing both sides by A2, we obtain the following equations:—
e / A2 = R2W / gS
R2WA2 / 2g = Se / 2.
(78)
The latter of these equations may be thus expressed in words: The actual energy due to the
rotation of the fly, with its mean angular velocity, is equal to one-half of the periodical
excess of energy multiplied by the steadiness.
In ordinary machinery S = about 32; in machinery for fine purposes S = from 50 to 60;
and when great steadiness is required S = from 100 to 150.
The periodical excess e may arise either from variations in the effort exerted by the
prime mover, or from variations in the resistance of the work, or from both these causes
combined. When but one flywheel is used, it should be placed in as direct connexion as
possible with that part of the mechanism where the greatest amount of the periodical excess
originates; but when it originates at two or more points, it is best to have a flywheel in
connexion with each of these points. For example, in a machine-work, the steam-engine,
which is the prime mover of the various tools, has a flywheel on the crank-shaft to store
and restore the periodical excess of energy arising from the variations in the effort exerted
by the connecting-rod upon the crank; and each of the slotting machines, punching
machines, riveting machines, and other tools has a flywheel of its own to store and restore
energy, so as to enable the very different resistances opposed to those tools at different
times to be overcome without too great unsteadiness of motion. For tools performing useful
work at intervals, and having only their own friction to overcome during the intermediate
intervals, e should be assumed equal to the whole work performed at each separate
operation.
minimum, and A = 1⁄2 (α2 + α1) its mean angular velocity. Let
I/S = (α2 − α2) / A
denote the unsteadiness of the motion of the flywheel; the denominator S of this fraction is
called the steadiness. Let e denote the quantity by which the energy exerted in each cycle
of the working of the machine alternately exceeds and falls short of the work performed,
and which has consequently to be alternately stored by acceleration and restored by
retardation of the flywheel. The value of this periodical excess is—
e = R2W (α22 − α12), 2g,
(77)
from which, dividing both sides by A2, we obtain the following equations:—
e / A2 = R2W / gS
R2WA2 / 2g = Se / 2.
(78)
The latter of these equations may be thus expressed in words: The actual energy due to the
rotation of the fly, with its mean angular velocity, is equal to one-half of the periodical
excess of energy multiplied by the steadiness.
In ordinary machinery S = about 32; in machinery for fine purposes S = from 50 to 60;
and when great steadiness is required S = from 100 to 150.
The periodical excess e may arise either from variations in the effort exerted by the
prime mover, or from variations in the resistance of the work, or from both these causes
combined. When but one flywheel is used, it should be placed in as direct connexion as
possible with that part of the mechanism where the greatest amount of the periodical excess
originates; but when it originates at two or more points, it is best to have a flywheel in
connexion with each of these points. For example, in a machine-work, the steam-engine,
which is the prime mover of the various tools, has a flywheel on the crank-shaft to store
and restore the periodical excess of energy arising from the variations in the effort exerted
by the connecting-rod upon the crank; and each of the slotting machines, punching
machines, riveting machines, and other tools has a flywheel of its own to store and restore
energy, so as to enable the very different resistances opposed to those tools at different
times to be overcome without too great unsteadiness of motion. For tools performing useful
work at intervals, and having only their own friction to overcome during the intermediate
intervals, e should be assumed equal to the whole work performed at each separate
operation.
Page 574
§ 122. Brakes.—A brake is an apparatus for stopping and diminishing the velocity of a
machine by friction, such as the friction-strap already referred to in § 103. To find the
distance s through which a brake, exerting the friction F, must rub in order to stop a
machine having the total actual energy E at the moment when the brake begins to act,
reduce, by the principles of § 96, the various efforts and other resistances of the machine
which act at the same time with the friction of the brake to the rubbing surface of the brake,
and let R be their resultant—positive if resistance, negative if effort preponderates. Then
s = E / (F + R).
(79)
§ 123. Energy distributed between two Bodies: Projection and Propulsion.—Hitherto the
effort by which a machine is moved has been treated as a force exerted between a movable
body and a fixed body, so that the whole energy exerted by it is employed upon the
movable body, and none upon the fixed body. This conception is sensibly realized in
practice when one of the two bodies between which the effort acts is either so heavy as
compared with the other, or has so great a resistance opposed to its motion, that it may,
without sensible error, be treated as fixed. But there are cases in which the motions of both
bodies are appreciable, and must be taken into account—such as the projection of
projectiles, where the velocity of the recoil or backward motion of the gun bears an
appreciable proportion to the forward motion of the projectile; and such as the propulsion
of vessels, where the velocity of the water thrown backward by the paddle, screw or other
propeller bears a very considerable proportion to the velocity of the water moved forwards
and sideways by the ship. In cases of this kind the energy exerted by the effort is
distributed between the two bodies between which the effort is exerted in shares
proportional to the velocities of the two bodies during the action of the effort; and those
velocities are to each other directly as the portions of the effort unbalanced by resistance on
the respective bodies, and inversely as the weights of the bodies.
To express this symbolically, let W1, W2 be the weights of the bodies; P the effort
exerted between them; S the distance through which it acts; R1, R2 the resistances opposed
to the effort overcome by W1, W2 respectively; E1, E2 the shares of the whole energy E
exerted upon W1, W2 respectively. Then
E : E1 : E2
W2 (P − R1) + W1 (P − R2) P − R1 P − R2
:: : : .
W1W2 W1 W2 (80)
If R1 = R2, which is the case when the resistance, as well as the effort, arises from the
mutual actions of the two bodies, the above becomes,
machine by friction, such as the friction-strap already referred to in § 103. To find the
distance s through which a brake, exerting the friction F, must rub in order to stop a
machine having the total actual energy E at the moment when the brake begins to act,
reduce, by the principles of § 96, the various efforts and other resistances of the machine
which act at the same time with the friction of the brake to the rubbing surface of the brake,
and let R be their resultant—positive if resistance, negative if effort preponderates. Then
s = E / (F + R).
(79)
§ 123. Energy distributed between two Bodies: Projection and Propulsion.—Hitherto the
effort by which a machine is moved has been treated as a force exerted between a movable
body and a fixed body, so that the whole energy exerted by it is employed upon the
movable body, and none upon the fixed body. This conception is sensibly realized in
practice when one of the two bodies between which the effort acts is either so heavy as
compared with the other, or has so great a resistance opposed to its motion, that it may,
without sensible error, be treated as fixed. But there are cases in which the motions of both
bodies are appreciable, and must be taken into account—such as the projection of
projectiles, where the velocity of the recoil or backward motion of the gun bears an
appreciable proportion to the forward motion of the projectile; and such as the propulsion
of vessels, where the velocity of the water thrown backward by the paddle, screw or other
propeller bears a very considerable proportion to the velocity of the water moved forwards
and sideways by the ship. In cases of this kind the energy exerted by the effort is
distributed between the two bodies between which the effort is exerted in shares
proportional to the velocities of the two bodies during the action of the effort; and those
velocities are to each other directly as the portions of the effort unbalanced by resistance on
the respective bodies, and inversely as the weights of the bodies.
To express this symbolically, let W1, W2 be the weights of the bodies; P the effort
exerted between them; S the distance through which it acts; R1, R2 the resistances opposed
to the effort overcome by W1, W2 respectively; E1, E2 the shares of the whole energy E
exerted upon W1, W2 respectively. Then
E : E1 : E2
W2 (P − R1) + W1 (P − R2) P − R1 P − R2
:: : : .
W1W2 W1 W2 (80)
If R1 = R2, which is the case when the resistance, as well as the effort, arises from the
mutual actions of the two bodies, the above becomes,
Page 575
E : E1 : E2
:: W1 + W2 : W2 : W1,
(81)
that is to say, the energy is exerted on the bodies in shares inversely proportional to their
weights; and they receive accelerations inversely proportional to their weights, according
to the principle of dynamics, already quoted in a note to § 110, that the mutual actions of a
system of bodies do not affect the motion of their common centre of gravity.
For example, if the weight of a gun be 160 times that of its ball 160⁄161 of the energy
exerted by the powder in exploding will be employed in propelling the ball, and 1⁄161 in
producing the recoil of the gun, provided the gun up to the instant of the ball’s quitting the
muzzle meets with no resistance to its recoil except the friction of the ball.
§ 124. Centre of Percussion.—It is obviously desirable that the deviations or changes of
motion of oscillating pieces in machinery should, as far as possible, be effected by forces
applied at their centres of percussion.
If the deviation be a translation—that is, an equal change of motion of all the particles of
the body—the centre of percussion is obviously the centre of gravity itself; and, according
to the second law of motion, if dv be the deviation of velocity to be produced in the interval
dt, and W the weight of the body, then
W dv
P= ·
g dt (82)
is the unbalanced effort required.
If the deviation be a rotation about an axis traversing the centre of gravity, there is no
centre of percussion; for such a deviation can only be produced by a couple of forces, and
not by any single force. Let dα be the deviation of angular velocity to be produced in the
interval dt, and I the moment of the inertia of the body about an axis through its centre of
gravity; then 1⁄2Id(α2) = Iα dα is the variation of the body’s actual energy. Let M be the
moment of the unbalanced couple required to produce the deviation; then by equation 57, §
104, the energy exerted by this couple in the interval dt is Mα dt, which, being equated to
the variation of energy, gives
dα R2W dα
M=I = · .
dt g dt (83)
R is called the radius of gyration of the body with regard to an axis through its centre of
gravity.
:: W1 + W2 : W2 : W1,
(81)
that is to say, the energy is exerted on the bodies in shares inversely proportional to their
weights; and they receive accelerations inversely proportional to their weights, according
to the principle of dynamics, already quoted in a note to § 110, that the mutual actions of a
system of bodies do not affect the motion of their common centre of gravity.
For example, if the weight of a gun be 160 times that of its ball 160⁄161 of the energy
exerted by the powder in exploding will be employed in propelling the ball, and 1⁄161 in
producing the recoil of the gun, provided the gun up to the instant of the ball’s quitting the
muzzle meets with no resistance to its recoil except the friction of the ball.
§ 124. Centre of Percussion.—It is obviously desirable that the deviations or changes of
motion of oscillating pieces in machinery should, as far as possible, be effected by forces
applied at their centres of percussion.
If the deviation be a translation—that is, an equal change of motion of all the particles of
the body—the centre of percussion is obviously the centre of gravity itself; and, according
to the second law of motion, if dv be the deviation of velocity to be produced in the interval
dt, and W the weight of the body, then
W dv
P= ·
g dt (82)
is the unbalanced effort required.
If the deviation be a rotation about an axis traversing the centre of gravity, there is no
centre of percussion; for such a deviation can only be produced by a couple of forces, and
not by any single force. Let dα be the deviation of angular velocity to be produced in the
interval dt, and I the moment of the inertia of the body about an axis through its centre of
gravity; then 1⁄2Id(α2) = Iα dα is the variation of the body’s actual energy. Let M be the
moment of the unbalanced couple required to produce the deviation; then by equation 57, §
104, the energy exerted by this couple in the interval dt is Mα dt, which, being equated to
the variation of energy, gives
dα R2W dα
M=I = · .
dt g dt (83)
R is called the radius of gyration of the body with regard to an axis through its centre of
gravity.
Page 576
Now (fig. 133) let the required deviation be a rotation of the
body BB about an axis O, not traversing the centre of gravity G,
dα being, as before, the deviation of angular velocity to be
produced in the interval dt. A rotation with the angular velocity
α about an axis O may be considered as compounded of a
rotation with the same angular velocity about an axis drawn
through G parallel to O and a translation with the velocity α.
OG, OG being the perpendicular distance between the two axes.
Hence the required deviation may be regarded as compounded
of a deviation of translation dv = OG · dα, to produce which Fig. 133.
there would be required, according to equation (82), a force
applied at G perpendicular to the plane OG—
W dα
P= · OG ·
g dt (84)
and a deviation dα of rotation about an axis drawn through G parallel to O, to produce
which there would be required a couple of the moment M given by equation (83).
According to the principles of statics, the resultant of the force P, applied at G
perpendicular to the plane OG, and the couple M is a force equal and parallel to P, but
applied at a distance GC from G, in the prolongation of the perpendicular OG, whose value
is
GC = M / P = R2 / OG.
(85)
Thus is determined the position of the centre of percussion C, corresponding to the axis of
rotation O. It is obvious from this equation that, for an axis of rotation parallel to O
traversing C, the centre of percussion is at the point where the perpendicular OG meets O.
§ 125.* To find the moment of inertia of a body about an axis through its centre of
gravity experimentally.—Suspend the body from any conveniently selected axis O (fig. 48)
and hang near it a small plumb bob. Adjust the length of the plumb-line until it and the
body oscillate together in unison. The length of the plumb-line, measured from its point of
suspension to the centre of the bob, is for all practical purposes equal to the length OC, C
being therefore the centre of percussion corresponding to the selected axis O. From
equation (85)
R2 = CG × OG = (OC − OG) OG.
The position of G can be found experimentally; hence OG is known, and the quantity R2
can be calculated, from which and the ascertained weight W of the body the moment of
body BB about an axis O, not traversing the centre of gravity G,
dα being, as before, the deviation of angular velocity to be
produced in the interval dt. A rotation with the angular velocity
α about an axis O may be considered as compounded of a
rotation with the same angular velocity about an axis drawn
through G parallel to O and a translation with the velocity α.
OG, OG being the perpendicular distance between the two axes.
Hence the required deviation may be regarded as compounded
of a deviation of translation dv = OG · dα, to produce which Fig. 133.
there would be required, according to equation (82), a force
applied at G perpendicular to the plane OG—
W dα
P= · OG ·
g dt (84)
and a deviation dα of rotation about an axis drawn through G parallel to O, to produce
which there would be required a couple of the moment M given by equation (83).
According to the principles of statics, the resultant of the force P, applied at G
perpendicular to the plane OG, and the couple M is a force equal and parallel to P, but
applied at a distance GC from G, in the prolongation of the perpendicular OG, whose value
is
GC = M / P = R2 / OG.
(85)
Thus is determined the position of the centre of percussion C, corresponding to the axis of
rotation O. It is obvious from this equation that, for an axis of rotation parallel to O
traversing C, the centre of percussion is at the point where the perpendicular OG meets O.
§ 125.* To find the moment of inertia of a body about an axis through its centre of
gravity experimentally.—Suspend the body from any conveniently selected axis O (fig. 48)
and hang near it a small plumb bob. Adjust the length of the plumb-line until it and the
body oscillate together in unison. The length of the plumb-line, measured from its point of
suspension to the centre of the bob, is for all practical purposes equal to the length OC, C
being therefore the centre of percussion corresponding to the selected axis O. From
equation (85)
R2 = CG × OG = (OC − OG) OG.
The position of G can be found experimentally; hence OG is known, and the quantity R2
can be calculated, from which and the ascertained weight W of the body the moment of
Page 577
inertia about an axis through G, namely, W/g × R2, can be computed.
§ 126.* To find the force competent to produce
the instantaneous acceleration of any link of a
mechanism.—In many practical problems it is
necessary to know the magnitude and position of
the forces acting to produce the accelerations of
the several links of a mechanism. For a given
link, this force is the resultant of all the
accelerating forces distributed through the
substance of the material of the link required to
produce the requisite acceleration of each particle,
and the determination of this force depends upon
the principles of the two preceding sections. The
Fig. 134.
investigation of the distribution of the forces
through the material and the stress consequently
produced belongs to the subject of the Strength of Materials (q.v.). Let BK (fig. 134) be
any link moving in any manner in a plane, and let G be its centre of gravity. Then its
motion may be analysed into (1) a translation of its centre of gravity; and (2) a rotation
about an axis through its centre of gravity perpendicular to its plane of motion. Let α be the
acceleration of the centre of gravity and let A be the angular acceleration about the axis
through the centre of gravity; then the force required to produce the translation of the
centre of gravity is F = Wα/g, and the couple required to produce the angular acceleration
about the centre of gravity is M = IA/g, W and I being respectively the weight and the
moment of inertia of the link about the axis through the centre of gravity. The couple M
may be produced by shifting the force F parallel to itself through a distance x. such that Fx
= M. When the link forms part of a mechanism the respective accelerations of two points in
the link can be determined by means of the velocity and acceleration diagrams described in
§ 82, it being understood that the motion of one link in the mechanism is prescribed, for
instance, in the steam-engine’s mechanism that the crank shall revolve uniformly. Let the
acceleration of the two points B and K therefore be supposed known. The problem is now
to find the acceleration α and A. Take any pole O (fig. 49), and set out Ob equal to the
acceleration of B and Ok equal to the acceleration of K. Join bk and take the point g so that
KG: GB = kg : gb. Og is then the acceleration of the centre of gravity and the force F can
therefore be immediately calculated. To find the angular acceleration A, draw kt, bt
respectively parallel to and at right angles to the link KB. Then tb represents the angular
acceleration of the point B relatively to the point K and hence tb/KB is the value of A, the
angular acceleration of the link. Its moment of inertia about G can be found experimentally
by the method explained in § 125, and then the value of the couple M can be computed.
§ 126.* To find the force competent to produce
the instantaneous acceleration of any link of a
mechanism.—In many practical problems it is
necessary to know the magnitude and position of
the forces acting to produce the accelerations of
the several links of a mechanism. For a given
link, this force is the resultant of all the
accelerating forces distributed through the
substance of the material of the link required to
produce the requisite acceleration of each particle,
and the determination of this force depends upon
the principles of the two preceding sections. The
Fig. 134.
investigation of the distribution of the forces
through the material and the stress consequently
produced belongs to the subject of the Strength of Materials (q.v.). Let BK (fig. 134) be
any link moving in any manner in a plane, and let G be its centre of gravity. Then its
motion may be analysed into (1) a translation of its centre of gravity; and (2) a rotation
about an axis through its centre of gravity perpendicular to its plane of motion. Let α be the
acceleration of the centre of gravity and let A be the angular acceleration about the axis
through the centre of gravity; then the force required to produce the translation of the
centre of gravity is F = Wα/g, and the couple required to produce the angular acceleration
about the centre of gravity is M = IA/g, W and I being respectively the weight and the
moment of inertia of the link about the axis through the centre of gravity. The couple M
may be produced by shifting the force F parallel to itself through a distance x. such that Fx
= M. When the link forms part of a mechanism the respective accelerations of two points in
the link can be determined by means of the velocity and acceleration diagrams described in
§ 82, it being understood that the motion of one link in the mechanism is prescribed, for
instance, in the steam-engine’s mechanism that the crank shall revolve uniformly. Let the
acceleration of the two points B and K therefore be supposed known. The problem is now
to find the acceleration α and A. Take any pole O (fig. 49), and set out Ob equal to the
acceleration of B and Ok equal to the acceleration of K. Join bk and take the point g so that
KG: GB = kg : gb. Og is then the acceleration of the centre of gravity and the force F can
therefore be immediately calculated. To find the angular acceleration A, draw kt, bt
respectively parallel to and at right angles to the link KB. Then tb represents the angular
acceleration of the point B relatively to the point K and hence tb/KB is the value of A, the
angular acceleration of the link. Its moment of inertia about G can be found experimentally
by the method explained in § 125, and then the value of the couple M can be computed.
Page 578
The value of x is found immediately from the quotient M/F. Hence the magnitude F and the
position of F relatively to the centre of gravity of the link, necessary to give rise to the
couple M, are known, and this force is therefore the resultant force required.
§ 127.* Alternative construction for finding the position of
F relatively to the centre of gravity of the link.—Let B and K
be any two points in the link which for greater generality are
taken in fig. 135, so that the centre of gravity G is not in the
line joining them. First find the value of R experimentally.
Then produce the given directions of acceleration of B and K
to meet in O; draw a circle through the three points B, K and
O; produce the line joining O and G to cut the circle in Y; and
take a point Z on the line OY so that YG × GZ = R2. Then Z Fig. 135.
is a point in the line of action of the force F. This useful
theorem is due to G. T. Bennett, of Emmanuel College, Cambridge. A proof of it and three
corollaries are given in appendix 4 of the second edition of Dalby’s Balancing of Engines
(London, 1906). It is to be noticed that only the directions of the accelerations of two
points are required to find the point Z.
For an example of the application of the principles of the two preceding sections to a
practical problem see Valve and Valve Gear Mechanisms, by W. E. Dalby (London, 1906),
where the inertia stresses brought upon the several links of a Joy valve gear, belonging to
an express passenger engine of the Lancashire & Yorkshire railway, are investigated for an
engine-speed of 68 m. an hour.
Fig. 136.
§ 128.* The Connecting Rod Problem.—A particular problem of practical importance is
the determination of the force producing the motion of the connecting rod of a steam-
engine mechanism of the usual type. The methods of the two preceding sections may be
used when the acceleration of two points in the rod are known. In this problem it is usually
assumed that the crank pin K (fig. 136) moves with uniform velocity, so that if α is its
position of F relatively to the centre of gravity of the link, necessary to give rise to the
couple M, are known, and this force is therefore the resultant force required.
§ 127.* Alternative construction for finding the position of
F relatively to the centre of gravity of the link.—Let B and K
be any two points in the link which for greater generality are
taken in fig. 135, so that the centre of gravity G is not in the
line joining them. First find the value of R experimentally.
Then produce the given directions of acceleration of B and K
to meet in O; draw a circle through the three points B, K and
O; produce the line joining O and G to cut the circle in Y; and
take a point Z on the line OY so that YG × GZ = R2. Then Z Fig. 135.
is a point in the line of action of the force F. This useful
theorem is due to G. T. Bennett, of Emmanuel College, Cambridge. A proof of it and three
corollaries are given in appendix 4 of the second edition of Dalby’s Balancing of Engines
(London, 1906). It is to be noticed that only the directions of the accelerations of two
points are required to find the point Z.
For an example of the application of the principles of the two preceding sections to a
practical problem see Valve and Valve Gear Mechanisms, by W. E. Dalby (London, 1906),
where the inertia stresses brought upon the several links of a Joy valve gear, belonging to
an express passenger engine of the Lancashire & Yorkshire railway, are investigated for an
engine-speed of 68 m. an hour.
Fig. 136.
§ 128.* The Connecting Rod Problem.—A particular problem of practical importance is
the determination of the force producing the motion of the connecting rod of a steam-
engine mechanism of the usual type. The methods of the two preceding sections may be
used when the acceleration of two points in the rod are known. In this problem it is usually
assumed that the crank pin K (fig. 136) moves with uniform velocity, so that if α is its
Page 579
angular velocity and r its radius, the acceleration is α2r in a direction along the crank arm
from the crank pin to the centre of the shaft. Thus the acceleration of one point K is known
completely. The acceleration of a second point, usually taken at the centre of the crosshead
pin, can be found by the principles of § 82, but several special geometrical constructions
have been devised for this purpose, notably the construction of Klein,4 discovered also
independently by Kirsch.5 But probably the most convenient is the construction due to G.
T. Bennett6 which is as follows: Let OK be the crank and KB the connecting rod. On the
connecting rod take a point L such that KL × KB = KO2. Then, the crank standing at any
angle with the line of stroke, draw LP at right angles to the connecting rod, PN at right
angles to the line of stroke OB and NA at right angles to the connecting rod; then AO is the
acceleration of the point B to the scale on which KO represents the acceleration of the
point K. The proof of this construction is given in The Balancing of Engines.
The finding of F may be continued thus: join AK, then AK is the acceleration image of
the rod, OKA being the acceleration diagram. Through G, the centre of gravity of the rod,
draw Gg parallel to the line of stroke, thus dividing the image at g in the proportion that the
connecting rod is divided by G. Hence Og represents the acceleration of the centre of
gravity and, the weight of the connecting rod being ascertained, F can be immediately
calculated. To find a point in its line of action, take a point Q on the rod such that KG × GQ
= R2, R having been determined experimentally by the method of § 125; join G with O and
through Q draw a line parallel to BO to cut GO in Z. Z is a point in the line of action of the
resultant force F; hence through Z draw a line parallel to Og. The force F acts in this line,
and thus the problem is completely solved. The above construction for Z is a corollary of
the general theorem given in § 127.
§ 129. Impact. Impact or collision is a pressure of short duration exerted between two
bodies.
The effects of impact are sometimes an alteration of the distribution of actual energy
between the two bodies, and always a loss of a portion of that energy, depending on the
imperfection of the elasticity of the bodies, in permanently altering their figures, and
producing heat. The determination of the distribution of the actual energy after collision
and of the loss of energy is effected by means of the following principles:—
I. The motion of the common centre of gravity of the two bodies is unchanged by the
collision.
II. The loss of energy consists of a certain proportion of that part of the actual energy of
the bodies which is due to their motion relatively to their common centre of gravity.
from the crank pin to the centre of the shaft. Thus the acceleration of one point K is known
completely. The acceleration of a second point, usually taken at the centre of the crosshead
pin, can be found by the principles of § 82, but several special geometrical constructions
have been devised for this purpose, notably the construction of Klein,4 discovered also
independently by Kirsch.5 But probably the most convenient is the construction due to G.
T. Bennett6 which is as follows: Let OK be the crank and KB the connecting rod. On the
connecting rod take a point L such that KL × KB = KO2. Then, the crank standing at any
angle with the line of stroke, draw LP at right angles to the connecting rod, PN at right
angles to the line of stroke OB and NA at right angles to the connecting rod; then AO is the
acceleration of the point B to the scale on which KO represents the acceleration of the
point K. The proof of this construction is given in The Balancing of Engines.
The finding of F may be continued thus: join AK, then AK is the acceleration image of
the rod, OKA being the acceleration diagram. Through G, the centre of gravity of the rod,
draw Gg parallel to the line of stroke, thus dividing the image at g in the proportion that the
connecting rod is divided by G. Hence Og represents the acceleration of the centre of
gravity and, the weight of the connecting rod being ascertained, F can be immediately
calculated. To find a point in its line of action, take a point Q on the rod such that KG × GQ
= R2, R having been determined experimentally by the method of § 125; join G with O and
through Q draw a line parallel to BO to cut GO in Z. Z is a point in the line of action of the
resultant force F; hence through Z draw a line parallel to Og. The force F acts in this line,
and thus the problem is completely solved. The above construction for Z is a corollary of
the general theorem given in § 127.
§ 129. Impact. Impact or collision is a pressure of short duration exerted between two
bodies.
The effects of impact are sometimes an alteration of the distribution of actual energy
between the two bodies, and always a loss of a portion of that energy, depending on the
imperfection of the elasticity of the bodies, in permanently altering their figures, and
producing heat. The determination of the distribution of the actual energy after collision
and of the loss of energy is effected by means of the following principles:—
I. The motion of the common centre of gravity of the two bodies is unchanged by the
collision.
II. The loss of energy consists of a certain proportion of that part of the actual energy of
the bodies which is due to their motion relatively to their common centre of gravity.
Page 580
Unless there is some special reason for using impact in machines, it ought to be avoided,
on account not only of the waste of energy which it causes, but from the damage which it
occasions to the frame and mechanism.
on account not only of the waste of energy which it causes, but from the damage which it
occasions to the frame and mechanism.
Page 581
( ; )
1 In view of the great authority of the author, the late Professor Macquorn Rankine, it
has been thought desirable to retain the greater part of this article as it appeared in the 9th
edition of the Encyclopaedia Britannica. Considerable additions, however, have been
introduced in order to indicate subsequent developments of the subject; the new sections
are numbered continuously with the old, but are distinguished by an asterisk. Also, two
short chapters which concluded the original article have been omitted—ch. iii., “On
Purposes and Effects of Machines,” which was really a classification of machines, because
the classification of Franz Reuleaux is now usually followed, and ch. iv., “Applied
Energetics, or Theory of Prime Movers,” because its subject matter is now treated in
various special articles, e.g. Hydraulics, Steam Engine, Gas Engine, Oil Engine, and fully
developed in Rankine’s The Steam Engine and Other Prime Movers (London, 1902). (Ed.
E.B.)
2 Since the relation discussed in § 7 was enunciated by Rankine, an enormous
development has taken place in the subject of Graphic Statics, the first comprehensive
textbook on the subject being Die Graphische Statik by K. Culmann, published at Zürich in
1866. Many of the graphical methods therein given have now passed into the textbooks
usually studied by engineers. One of the most beautiful graphical constructions regularly
used by engineers and known as “the method of reciprocal figures” is that for finding the
loads supported by the several members of a braced structure, having given a system of
external loads. The method was discovered by Clerk Maxwell, and the complete theory is
discussed and exemplified in a paper “On Reciprocal Figures, Frames and Diagrams of
Forces,” Trans. Roy. Soc. Ed., vol. xxvi. (1870). Professor M. W. Crofton read a paper on
“Stress-Diagrams in Warren and Lattice Girders” at the meeting of the Mathematical
Society (April 13, 1871), and Professor O. Henrici illustrated the subject by a simple and
ingenious notation. The application of the method of reciprocal figures was facilitated by a
system of notation published in Economics of Construction in relation to framed
Structures, by Robert H. Bow (London, 1873). A notable work on the general subject is that
of Luigi Cremona, translated from the Italian by Professor T. H. Beare (Oxford, 1890), and
a discussion of the subject of reciprocal figures from the special point of view of the
engineering student is given in Vectors and Rotors by Henrici and Turner (London, 1903).
See also above under “Theoretical Mechanics,” Part 1. § 5.
3 This is a particular case of a more general principle, that the motion of the centre of
gravity of a body is not affected by the mutual actions of its parts.
4 J. F. Klein, “New Constructions of the Force of Inertia of Connecting Rods and
Couplers and Constructions of the Pressures on their Pins,” Journ. Franklin Inst., vol. 132
(Sept. and Oct., 1891).
1 In view of the great authority of the author, the late Professor Macquorn Rankine, it
has been thought desirable to retain the greater part of this article as it appeared in the 9th
edition of the Encyclopaedia Britannica. Considerable additions, however, have been
introduced in order to indicate subsequent developments of the subject; the new sections
are numbered continuously with the old, but are distinguished by an asterisk. Also, two
short chapters which concluded the original article have been omitted—ch. iii., “On
Purposes and Effects of Machines,” which was really a classification of machines, because
the classification of Franz Reuleaux is now usually followed, and ch. iv., “Applied
Energetics, or Theory of Prime Movers,” because its subject matter is now treated in
various special articles, e.g. Hydraulics, Steam Engine, Gas Engine, Oil Engine, and fully
developed in Rankine’s The Steam Engine and Other Prime Movers (London, 1902). (Ed.
E.B.)
2 Since the relation discussed in § 7 was enunciated by Rankine, an enormous
development has taken place in the subject of Graphic Statics, the first comprehensive
textbook on the subject being Die Graphische Statik by K. Culmann, published at Zürich in
1866. Many of the graphical methods therein given have now passed into the textbooks
usually studied by engineers. One of the most beautiful graphical constructions regularly
used by engineers and known as “the method of reciprocal figures” is that for finding the
loads supported by the several members of a braced structure, having given a system of
external loads. The method was discovered by Clerk Maxwell, and the complete theory is
discussed and exemplified in a paper “On Reciprocal Figures, Frames and Diagrams of
Forces,” Trans. Roy. Soc. Ed., vol. xxvi. (1870). Professor M. W. Crofton read a paper on
“Stress-Diagrams in Warren and Lattice Girders” at the meeting of the Mathematical
Society (April 13, 1871), and Professor O. Henrici illustrated the subject by a simple and
ingenious notation. The application of the method of reciprocal figures was facilitated by a
system of notation published in Economics of Construction in relation to framed
Structures, by Robert H. Bow (London, 1873). A notable work on the general subject is that
of Luigi Cremona, translated from the Italian by Professor T. H. Beare (Oxford, 1890), and
a discussion of the subject of reciprocal figures from the special point of view of the
engineering student is given in Vectors and Rotors by Henrici and Turner (London, 1903).
See also above under “Theoretical Mechanics,” Part 1. § 5.
3 This is a particular case of a more general principle, that the motion of the centre of
gravity of a body is not affected by the mutual actions of its parts.
4 J. F. Klein, “New Constructions of the Force of Inertia of Connecting Rods and
Couplers and Constructions of the Pressures on their Pins,” Journ. Franklin Inst., vol. 132
(Sept. and Oct., 1891).
Page 582
5 Prof. Kirsch, “Über die graphische Bestimmung der Kolbenbeschleunigung,” Zeitsch.
Verein deutsche Ingen. (1890), p. 1320.
6 Dalby, The Balancing of Engines (London, 1906), app. 1.
MECHANICVILLE, a village of Saratoga county, New York,
U.S.A., on the west bank of the Hudson River, about 20 m. N. of Albany;
on the Delaware & Hudson and Boston & Maine railways. Pop. (1900),
4695 (702 foreign-born); (1905, state census), 5877; (1910) 6,634. It lies
partly within Stillwater and partly within Half-Moon townships, in the
bottom-lands at the mouth of the Anthony Kill, about 1-1/2 m. S. of the
mouth of the Hoosick River. On the north and south are hills reaching a
maximum height of 200 ft. There is ample water power, and there are
manufactures of paper, sash and blinds, fibre, &c. From a dam here power
is derived for the General Electric Company at Schenectady. The first
settlement in this vicinity was made in what is now Half-Moon township
about 1680. Mechanicville (originally called Burrow) was chartered by the
county court in 1859, and incorporated as a village in 1870. It was the
birthplace of Colonel Ephraim Elmer Ellsworth (1837-1861), the first
Federal officer to lose his life in the Civil War.
Verein deutsche Ingen. (1890), p. 1320.
6 Dalby, The Balancing of Engines (London, 1906), app. 1.
MECHANICVILLE, a village of Saratoga county, New York,
U.S.A., on the west bank of the Hudson River, about 20 m. N. of Albany;
on the Delaware & Hudson and Boston & Maine railways. Pop. (1900),
4695 (702 foreign-born); (1905, state census), 5877; (1910) 6,634. It lies
partly within Stillwater and partly within Half-Moon townships, in the
bottom-lands at the mouth of the Anthony Kill, about 1-1/2 m. S. of the
mouth of the Hoosick River. On the north and south are hills reaching a
maximum height of 200 ft. There is ample water power, and there are
manufactures of paper, sash and blinds, fibre, &c. From a dam here power
is derived for the General Electric Company at Schenectady. The first
settlement in this vicinity was made in what is now Half-Moon township
about 1680. Mechanicville (originally called Burrow) was chartered by the
county court in 1859, and incorporated as a village in 1870. It was the
birthplace of Colonel Ephraim Elmer Ellsworth (1837-1861), the first
Federal officer to lose his life in the Civil War.
Page 583
MECHITHARISTS, a congregation of Armenian monks in
communion with the Church of Rome. The founder, Mechithar, was born at
Sebaste in Armenia, 1676. He entered a monastery, but under the influence
of Western missionaries he became possessed with the idea of propagating
Western ideas and culture in Armenia, and of converting the Armenian
Church from its monophysitism and uniting it to the Latin Church.
Mechithar set out for Rome in 1695 to make his ecclesiastical studies there,
but he was compelled by illness to abandon the journey and return to
Armenia. In 1696 he was ordained priest and for four years worked among
his people. In 1700 he went to Constantinople and began to gather disciples
around him. Mechithar formally joined the Latin Church, and in 1701, with
sixteen companions, he formed a definitely religious institute of which he
became the superior. Their Uniat propaganda encountered the opposition of
the Armenians and they were compelled to move to the Morea, at that time
Venetian territory, and there built a monastery, 1706. On the outbreak of
hostilities between the Turks and Venetians they migrated to Venice, and the
island of St Lazzaro was bestowed on them, 1717. This has since been the
headquarters of the congregation, and here Mechithar died in 1749, leaving
his institute firmly established. The rule followed at first was that attributed
to St Anthony; but when they settled in the West modifications from the
Benedictine rule were introduced, and the Mechitharists are numbered
among the lesser orders affiliated to the Benedictines. They have ever been
faithful to their founder’s programme. Their work has been fourfold: (1)
communion with the Church of Rome. The founder, Mechithar, was born at
Sebaste in Armenia, 1676. He entered a monastery, but under the influence
of Western missionaries he became possessed with the idea of propagating
Western ideas and culture in Armenia, and of converting the Armenian
Church from its monophysitism and uniting it to the Latin Church.
Mechithar set out for Rome in 1695 to make his ecclesiastical studies there,
but he was compelled by illness to abandon the journey and return to
Armenia. In 1696 he was ordained priest and for four years worked among
his people. In 1700 he went to Constantinople and began to gather disciples
around him. Mechithar formally joined the Latin Church, and in 1701, with
sixteen companions, he formed a definitely religious institute of which he
became the superior. Their Uniat propaganda encountered the opposition of
the Armenians and they were compelled to move to the Morea, at that time
Venetian territory, and there built a monastery, 1706. On the outbreak of
hostilities between the Turks and Venetians they migrated to Venice, and the
island of St Lazzaro was bestowed on them, 1717. This has since been the
headquarters of the congregation, and here Mechithar died in 1749, leaving
his institute firmly established. The rule followed at first was that attributed
to St Anthony; but when they settled in the West modifications from the
Benedictine rule were introduced, and the Mechitharists are numbered
among the lesser orders affiliated to the Benedictines. They have ever been
faithful to their founder’s programme. Their work has been fourfold: (1)
Page 584
they have brought out editions of important patristic works, some
Armenian, others translated into Armenian from Greek and Syriac originals
no longer extant; (2) they print and circulate Armenian literature among the
Armenians, and thereby exercise a powerful educational influence; (3) they
carry on schools both in Europe and Asia, in which Uniat Armenian boys
receive a good secondary education; (4) they work as Uniat missioners in
Armenia. The congregation is divided into two branches, the head houses
being at St Lazzaro and Vienna. They have fifteen establishments in various
places in Asia Minor and Europe. There are some 150 monks, all
Armenians; they use the Armenian language and rite in the liturgy.
See Vita del servo di Dio Mechitar (Venice, 1901); E. Boré, Saint-
Lazare (1835); Max Heimbucher, Orden u. Kongregationen (1907) I. §
37; and the articles in Wetzer u. Welte, Kirchenlexicon (ed. 2) and
Herzog, Realencyklopädie (ed. 3), also articles by Sargisean, a
Mechitharist, in Rivista storica benedettina (1906), “La Congregazione
Mechitarista.”
(E. C. B.)
MECKLENBURG, a territory in northern Germany, on the
Baltic Sea, extending from 53° 4′ to 54° 22′ N. and from 10° 35′ to 13° 57′
E., unequally divided into the two grand duchies of Mecklenburg-Schwerin
and Mecklenburg-Strelitz.
Mecklenburg-schwerin is bounded N. by the Baltic Sea, W. by the
principality of Ratzeburg and Schleswig-Holstein, S. by Brandenburg and
Armenian, others translated into Armenian from Greek and Syriac originals
no longer extant; (2) they print and circulate Armenian literature among the
Armenians, and thereby exercise a powerful educational influence; (3) they
carry on schools both in Europe and Asia, in which Uniat Armenian boys
receive a good secondary education; (4) they work as Uniat missioners in
Armenia. The congregation is divided into two branches, the head houses
being at St Lazzaro and Vienna. They have fifteen establishments in various
places in Asia Minor and Europe. There are some 150 monks, all
Armenians; they use the Armenian language and rite in the liturgy.
See Vita del servo di Dio Mechitar (Venice, 1901); E. Boré, Saint-
Lazare (1835); Max Heimbucher, Orden u. Kongregationen (1907) I. §
37; and the articles in Wetzer u. Welte, Kirchenlexicon (ed. 2) and
Herzog, Realencyklopädie (ed. 3), also articles by Sargisean, a
Mechitharist, in Rivista storica benedettina (1906), “La Congregazione
Mechitarista.”
(E. C. B.)
MECKLENBURG, a territory in northern Germany, on the
Baltic Sea, extending from 53° 4′ to 54° 22′ N. and from 10° 35′ to 13° 57′
E., unequally divided into the two grand duchies of Mecklenburg-Schwerin
and Mecklenburg-Strelitz.
Mecklenburg-schwerin is bounded N. by the Baltic Sea, W. by the
principality of Ratzeburg and Schleswig-Holstein, S. by Brandenburg and
Page 585
Hanover, and E. by Pomerania and Mecklenburg-Strelitz. It embraces the
duchies of Schwerin and Güstrow, the district of Rostock, the principality of
Schwerin, and the barony of Wismar, besides several small enclaves
(Ahrensberg, Rosson, Tretzeband, &c.) in the adjacent territories. Its area is
5080 sq. m. Pop. (1905), 625,045.
Mecklenburg-strelitz consists of two detached parts, the duchy of
Strelitz on the E. of Mecklenburg-Schwerin, and the principality of
Ratzeburg on the W. The first is bounded by Mecklenburg-Schwerin,
Pomerania and Brandenburg, the second by Mecklenburg-Schwerin,
Lauenburg, and the territory of the free town of Lübeck. Their joint area is
1130 sq. m. Pop. (1905), 103,451.
Mecklenburg lies wholly within the great North-European plain, and
its flat surface is interrupted only by one range of low hills, intersecting
the country from south-east to north-west, and forming the watershed
between the Baltic Sea and the Elbe. Its highest point, the Helpter
Berg, is 587 ft. above sea-level. The coast-line runs for 65 m. along the
Baltic (without including indentations), for the most part in flat sandy
stretches covered with dunes. The chief inlets are Wismar Bay, the
Salzhaff, and the roads of Warnemünde. The rivers are numerous
though small; most of them are affluents of the Elbe, which traverses a
small portion of Mecklenburg. Several are navigable, and the facilities
for inland water traffic are increased by canals. Lakes are numerous;
about four hundred, covering an area of 500 sq. m., are reckoned in the
two duchies. The largest is Lake Müritz, 52 sq. m. in extent. The
climate resembles that of Great Britain, but the winters are generally
more severe; the mean annual temperature is 48° F., and the annual
rainfall is about 28 in. Although there are long stretches of marshy
moorland along the coast, the soil is on the whole productive. About
57% of the total area of Mecklenburg-Schwerin consists of cultivated
duchies of Schwerin and Güstrow, the district of Rostock, the principality of
Schwerin, and the barony of Wismar, besides several small enclaves
(Ahrensberg, Rosson, Tretzeband, &c.) in the adjacent territories. Its area is
5080 sq. m. Pop. (1905), 625,045.
Mecklenburg-strelitz consists of two detached parts, the duchy of
Strelitz on the E. of Mecklenburg-Schwerin, and the principality of
Ratzeburg on the W. The first is bounded by Mecklenburg-Schwerin,
Pomerania and Brandenburg, the second by Mecklenburg-Schwerin,
Lauenburg, and the territory of the free town of Lübeck. Their joint area is
1130 sq. m. Pop. (1905), 103,451.
Mecklenburg lies wholly within the great North-European plain, and
its flat surface is interrupted only by one range of low hills, intersecting
the country from south-east to north-west, and forming the watershed
between the Baltic Sea and the Elbe. Its highest point, the Helpter
Berg, is 587 ft. above sea-level. The coast-line runs for 65 m. along the
Baltic (without including indentations), for the most part in flat sandy
stretches covered with dunes. The chief inlets are Wismar Bay, the
Salzhaff, and the roads of Warnemünde. The rivers are numerous
though small; most of them are affluents of the Elbe, which traverses a
small portion of Mecklenburg. Several are navigable, and the facilities
for inland water traffic are increased by canals. Lakes are numerous;
about four hundred, covering an area of 500 sq. m., are reckoned in the
two duchies. The largest is Lake Müritz, 52 sq. m. in extent. The
climate resembles that of Great Britain, but the winters are generally
more severe; the mean annual temperature is 48° F., and the annual
rainfall is about 28 in. Although there are long stretches of marshy
moorland along the coast, the soil is on the whole productive. About
57% of the total area of Mecklenburg-Schwerin consists of cultivated
Page 586
land, 18% of forest, and 13% of heath and pasture. In Mecklenburg-
Strelitz the corresponding figures are 47, 21 and 10%. Agriculture is by
far the most important industry in both duchies. The chief crops are
rye, oats, wheat, potatoes and hay. Smaller areas are devoted to maize,
buckwheat, pease, rape, hemp, flax, hops and tobacco. The extensive
pastures support large herds of sheep and cattle, including a noteworthy
breed of merino sheep. The horses of Mecklenburg are of a fine sturdy
quality and highly esteemed. Red deer, wild swine and various other
game are found in the forests. The industrial establishments include a
few iron-foundries, wool-spinning mills, carriage and machine
factories, dyeworks, tanneries, brick-fields, soap-works, breweries,
distilleries, numerous limekilns and tar-boiling works, tobacco and
cigar factories, and numerous mills of various kinds. Mining is
insignificant, though a fair variety of minerals is represented in the
district. Amber is found on and near the Baltic coast. Rostock,
Warnemünde and Wismar are the principal commercial centres. The
chief exports are grain and other agricultural produce, live stock,
spirits, wood and wool; the chief imports are colonial produce, iron,
coal, salt, wine, beer and tobacco. The horse and wool markets of
Mecklenburg are largely attended by buyers from various parts of
Germany. Fishing is carried on extensively in the numerous inland
lakes.
In 1907 the grand dukes of both duchies promised a constitution to
their subjects. The duchies had always been under a government of
feudal character, the grand dukes having the executive entirely in their
hands (though acting through ministers), while the duchies shared a
diet (Landtag), meeting for a short session each year, and at other times
represented by a committee, and consisting of the proprietors of
knights’ estates (Rittergüter), known as the Ritterschaft, and the
Landschaft or burgomasters of certain towns. Mecklenburg-Schwerin
Strelitz the corresponding figures are 47, 21 and 10%. Agriculture is by
far the most important industry in both duchies. The chief crops are
rye, oats, wheat, potatoes and hay. Smaller areas are devoted to maize,
buckwheat, pease, rape, hemp, flax, hops and tobacco. The extensive
pastures support large herds of sheep and cattle, including a noteworthy
breed of merino sheep. The horses of Mecklenburg are of a fine sturdy
quality and highly esteemed. Red deer, wild swine and various other
game are found in the forests. The industrial establishments include a
few iron-foundries, wool-spinning mills, carriage and machine
factories, dyeworks, tanneries, brick-fields, soap-works, breweries,
distilleries, numerous limekilns and tar-boiling works, tobacco and
cigar factories, and numerous mills of various kinds. Mining is
insignificant, though a fair variety of minerals is represented in the
district. Amber is found on and near the Baltic coast. Rostock,
Warnemünde and Wismar are the principal commercial centres. The
chief exports are grain and other agricultural produce, live stock,
spirits, wood and wool; the chief imports are colonial produce, iron,
coal, salt, wine, beer and tobacco. The horse and wool markets of
Mecklenburg are largely attended by buyers from various parts of
Germany. Fishing is carried on extensively in the numerous inland
lakes.
In 1907 the grand dukes of both duchies promised a constitution to
their subjects. The duchies had always been under a government of
feudal character, the grand dukes having the executive entirely in their
hands (though acting through ministers), while the duchies shared a
diet (Landtag), meeting for a short session each year, and at other times
represented by a committee, and consisting of the proprietors of
knights’ estates (Rittergüter), known as the Ritterschaft, and the
Landschaft or burgomasters of certain towns. Mecklenburg-Schwerin
Page 587
returns six members to the Reichstag and Mecklenburg-Strelitz one
member.
In Mecklenburg-Schwerin the chief towns are Rostock (with a
university), Schwerin, and Wismar the capital. The capital of
Mecklenburg-Strelitz is Neu-Strelitz. The peasantry of Mecklenburg
retain traces of their Slavonic origin, especially in speech, but their
peculiarities have been much modified by amalgamation with German
colonists. The townspeople and nobility are almost wholly of Saxon
strain. The slowness of the increase in population is chiefly accounted
for by emigration.
History.—The Teutonic peoples, who in the time of Tacitus occupied the
region now known as Mecklenburg, were succeeded in the 6th century by
some Slavonic tribes, one of these being the Obotrites, whose chief fortress
was Michilenburg, the modern Mecklenburg, near Wismar; hence the name
of the country. Though partly subdued by Charlemagne towards the close of
the 8th century, they soon regained their independence, and until the 10th
century no serious effort was made by their Christian neighbours to subject
them. Then the German king, Henry the Fowler, reduced the Slavs of
Mecklenburg to obedience and introduced Christianity among them. During
the period of weakness through which the German kingdom passed under
the later Ottos, however, they wrenched themselves free from this bondage;
the 11th and the early part of the 12th century saw the ebb and flow of the
tide of conquest, and then came the effective subjugation of Mecklenburg
by Henry the Lion, duke of Saxony. The Obotrite prince Niklot was killed
in battle in 1160 whilst resisting the Saxons, but his son Pribislaus (d. 1178)
submitted to Henry the Lion, married his daughter to the son of the duke,
embraced Christianity, and was permitted to retain his office. His
descendants and successors, the present grand dukes of Mecklenburg, are
the only ruling princes of Slavonic origin in Germany. Henry the Lion
member.
In Mecklenburg-Schwerin the chief towns are Rostock (with a
university), Schwerin, and Wismar the capital. The capital of
Mecklenburg-Strelitz is Neu-Strelitz. The peasantry of Mecklenburg
retain traces of their Slavonic origin, especially in speech, but their
peculiarities have been much modified by amalgamation with German
colonists. The townspeople and nobility are almost wholly of Saxon
strain. The slowness of the increase in population is chiefly accounted
for by emigration.
History.—The Teutonic peoples, who in the time of Tacitus occupied the
region now known as Mecklenburg, were succeeded in the 6th century by
some Slavonic tribes, one of these being the Obotrites, whose chief fortress
was Michilenburg, the modern Mecklenburg, near Wismar; hence the name
of the country. Though partly subdued by Charlemagne towards the close of
the 8th century, they soon regained their independence, and until the 10th
century no serious effort was made by their Christian neighbours to subject
them. Then the German king, Henry the Fowler, reduced the Slavs of
Mecklenburg to obedience and introduced Christianity among them. During
the period of weakness through which the German kingdom passed under
the later Ottos, however, they wrenched themselves free from this bondage;
the 11th and the early part of the 12th century saw the ebb and flow of the
tide of conquest, and then came the effective subjugation of Mecklenburg
by Henry the Lion, duke of Saxony. The Obotrite prince Niklot was killed
in battle in 1160 whilst resisting the Saxons, but his son Pribislaus (d. 1178)
submitted to Henry the Lion, married his daughter to the son of the duke,
embraced Christianity, and was permitted to retain his office. His
descendants and successors, the present grand dukes of Mecklenburg, are
the only ruling princes of Slavonic origin in Germany. Henry the Lion
Page 588
introduced German settlers and restored the bishoprics of Ratzeburg and
Schwerin; in 1170 the emperor Frederick I. made Pribislaus a prince of the
empire. From 1214 to 1227 Mecklenburg was under the supremacy of
Denmark; then, in 1229, after it had been regained by the Germans, there
took place the first of the many divisions of territory which with subsequent
reunions constitute much of its complicated history. At this time the country
was divided between four princes, grandsons of duke Henry Borwin, who
had died two years previously. But in less than a century the families of two
of these princes became extinct, and after dividing into three branches a
third family suffered the same fate in 1436. There then remained only the
line ruling in Mecklenburg proper, and the princes of this family, in addition
to inheriting the lands of their dead kinsmen, made many additions to their
territory, including the counties of Schwerin and of Strelitz. In 1352 the two
princes of this family made a division of their lands, Stargard being
separated from the rest of the country to form a principality for John (d.
1393), but on the extinction of his line in 1471 the whole of Mecklenburg
was again united under a single ruler. One member of this family, Albert (c.
1338-1412), was king of Sweden from 1364 to 1389. In 1348 the emperor
Charles IV. had raised Mecklenburg to the rank of a duchy, and in 1418 the
university of Rostock was founded.
The troubles which arose from the rivalry and jealousy of two or more
joint rulers incited the prelates, the nobles and the burghers to form a union
among themselves, and the results of this are still visible in the existence of
the Landesunion for the whole country which was established in 1523.
About the same time the teaching of Luther and the reformers was
welcomed in Mecklenburg, although Duke Albert (d. 1547) soon reverted to
the Catholic faith; in 1549 Lutheranism was recognized as the state religion;
a little later the churches and schools were reformed and most of the
monasteries were suppressed. A division of the land which took place in
1555 was of short duration, but a more important one was effected in 1611,
Schwerin; in 1170 the emperor Frederick I. made Pribislaus a prince of the
empire. From 1214 to 1227 Mecklenburg was under the supremacy of
Denmark; then, in 1229, after it had been regained by the Germans, there
took place the first of the many divisions of territory which with subsequent
reunions constitute much of its complicated history. At this time the country
was divided between four princes, grandsons of duke Henry Borwin, who
had died two years previously. But in less than a century the families of two
of these princes became extinct, and after dividing into three branches a
third family suffered the same fate in 1436. There then remained only the
line ruling in Mecklenburg proper, and the princes of this family, in addition
to inheriting the lands of their dead kinsmen, made many additions to their
territory, including the counties of Schwerin and of Strelitz. In 1352 the two
princes of this family made a division of their lands, Stargard being
separated from the rest of the country to form a principality for John (d.
1393), but on the extinction of his line in 1471 the whole of Mecklenburg
was again united under a single ruler. One member of this family, Albert (c.
1338-1412), was king of Sweden from 1364 to 1389. In 1348 the emperor
Charles IV. had raised Mecklenburg to the rank of a duchy, and in 1418 the
university of Rostock was founded.
The troubles which arose from the rivalry and jealousy of two or more
joint rulers incited the prelates, the nobles and the burghers to form a union
among themselves, and the results of this are still visible in the existence of
the Landesunion for the whole country which was established in 1523.
About the same time the teaching of Luther and the reformers was
welcomed in Mecklenburg, although Duke Albert (d. 1547) soon reverted to
the Catholic faith; in 1549 Lutheranism was recognized as the state religion;
a little later the churches and schools were reformed and most of the
monasteries were suppressed. A division of the land which took place in
1555 was of short duration, but a more important one was effected in 1611,
Page 589
although Duke John Albert I. (d. 1576) had introduced the principle of
primogeniture and had forbidden all further divisions of territory. By this
partition John Albert’s grandson Adolphus Frederick I. (d. 1658) received
Schwerin, and another grandson John Albert II. (d. 1636) received Güstrow.
The town of Rostock “with its university and high court of justice” was
declared to be common property, while the Diet or Landtag also retained its
joint character, its meetings being held alternately at Sternberg and at
Malchin.
During the early part of the Thirty Years’ War the dukes of Mecklenburg-
Schwerin and Mecklenburg-Güstrow were on the Protestant side, but about
1627 they submitted to the emperor Ferdinand II. This did not prevent
Ferdinand from promising their land to Wallenstein, who, having driven out
the dukes, was invested with the duchies in 1629 and ruled them until 1631.
In this year the former rulers were restored by Gustavus Adolphus of
Sweden, and in 1635 they came to terms with the emperor and signed the
peace of Prague, but their land continued to be ravaged by both sides until
the conclusion of the war. In 1648 by the Treaty of Westphalia, Wismar and
some other parts of Mecklenburg were surrendered to Sweden, the
recompense assigned to the duchies including the secularized bishoprics of
Schwerin and of Ratzeburg. The sufferings of the peasants in Mecklenburg
during the Thirty Years’ War were not exceeded by those of their class in
any other part of Germany; most of them were reduced to a state of serfdom
and in some cases whole villages vanished. Christian Louis who ruled
Mecklenburg-Schwerin from 1658 until his death in 1692 was, like his
father Adolphus Frederick, frequently at variance with the estates of the
land and with members of his family. He was a Roman Catholic and a
supporter of Louis XIV., and his country suffered severely during the wars
waged by France and her allies in Germany.
primogeniture and had forbidden all further divisions of territory. By this
partition John Albert’s grandson Adolphus Frederick I. (d. 1658) received
Schwerin, and another grandson John Albert II. (d. 1636) received Güstrow.
The town of Rostock “with its university and high court of justice” was
declared to be common property, while the Diet or Landtag also retained its
joint character, its meetings being held alternately at Sternberg and at
Malchin.
During the early part of the Thirty Years’ War the dukes of Mecklenburg-
Schwerin and Mecklenburg-Güstrow were on the Protestant side, but about
1627 they submitted to the emperor Ferdinand II. This did not prevent
Ferdinand from promising their land to Wallenstein, who, having driven out
the dukes, was invested with the duchies in 1629 and ruled them until 1631.
In this year the former rulers were restored by Gustavus Adolphus of
Sweden, and in 1635 they came to terms with the emperor and signed the
peace of Prague, but their land continued to be ravaged by both sides until
the conclusion of the war. In 1648 by the Treaty of Westphalia, Wismar and
some other parts of Mecklenburg were surrendered to Sweden, the
recompense assigned to the duchies including the secularized bishoprics of
Schwerin and of Ratzeburg. The sufferings of the peasants in Mecklenburg
during the Thirty Years’ War were not exceeded by those of their class in
any other part of Germany; most of them were reduced to a state of serfdom
and in some cases whole villages vanished. Christian Louis who ruled
Mecklenburg-Schwerin from 1658 until his death in 1692 was, like his
father Adolphus Frederick, frequently at variance with the estates of the
land and with members of his family. He was a Roman Catholic and a
supporter of Louis XIV., and his country suffered severely during the wars
waged by France and her allies in Germany.
Page 590
In June 1692 when Christian Louis died in exile and without sons, a
dispute arose about the succession to his duchy between his brother
Adolphus Frederick and his nephew Frederick William. The emperor and
the rulers of Sweden and of Brandenburg took part in this struggle which
was intensified when, three years later, on the death of Duke Gustavus
Adolphus, the family ruling over Mecklenburg-Güstrow became extinct. At
length the partition Treaty of Hamburg was signed on the 8th of March
1701, and a new division of the country was made. Mecklenburg was
divided between the two claimants, the shares given to each being
represented by the existing duchies of Mecklenburg-Schwerin, the part
which fell to Frederick William, and Mecklenburg-Strelitz, the share of
Adolphus Frederick. At the same time the principle of primogeniture was
again asserted, and the right of summoning the joint Landtag was reserved
to the ruler of Mecklenburg-Schwerin.
Mecklenburg-Schwerin began its existence by a series of constitutional
struggles between the duke and the nobles. The heavy debt incurred by
Duke Charles Leopold (d. 1747), who had joined Russia in a war against
Sweden, brought matters to a crisis; the emperor Charles VI. interfered and
in 1728 the imperial court of justice declared the duke incapable of
governing and his brother Christian Louis was appointed administrator of
the duchy. Under this prince, who became ruler de jure in 1747, there was
signed in April 1755 the convention of Rostock by which a new constitution
was framed for the duchy. By this instrument all power was in the hands of
the duke, the nobles and the upper classes generally, the lower classes being
entirely unrepresented. During the Seven Years’ War Duke Frederick (d.
1785) took up a hostile attitude towards Frederick the Great, and in
consequence Mecklenburg was occupied by Prussian troops, but in other
ways his rule was beneficial to the country. In the early years of the French
revolutionary wars Duke Frederick Francis I. (1756-1837) remained
neutral, and in 1803 he regained Wismar from Sweden, but in 1806 his land
dispute arose about the succession to his duchy between his brother
Adolphus Frederick and his nephew Frederick William. The emperor and
the rulers of Sweden and of Brandenburg took part in this struggle which
was intensified when, three years later, on the death of Duke Gustavus
Adolphus, the family ruling over Mecklenburg-Güstrow became extinct. At
length the partition Treaty of Hamburg was signed on the 8th of March
1701, and a new division of the country was made. Mecklenburg was
divided between the two claimants, the shares given to each being
represented by the existing duchies of Mecklenburg-Schwerin, the part
which fell to Frederick William, and Mecklenburg-Strelitz, the share of
Adolphus Frederick. At the same time the principle of primogeniture was
again asserted, and the right of summoning the joint Landtag was reserved
to the ruler of Mecklenburg-Schwerin.
Mecklenburg-Schwerin began its existence by a series of constitutional
struggles between the duke and the nobles. The heavy debt incurred by
Duke Charles Leopold (d. 1747), who had joined Russia in a war against
Sweden, brought matters to a crisis; the emperor Charles VI. interfered and
in 1728 the imperial court of justice declared the duke incapable of
governing and his brother Christian Louis was appointed administrator of
the duchy. Under this prince, who became ruler de jure in 1747, there was
signed in April 1755 the convention of Rostock by which a new constitution
was framed for the duchy. By this instrument all power was in the hands of
the duke, the nobles and the upper classes generally, the lower classes being
entirely unrepresented. During the Seven Years’ War Duke Frederick (d.
1785) took up a hostile attitude towards Frederick the Great, and in
consequence Mecklenburg was occupied by Prussian troops, but in other
ways his rule was beneficial to the country. In the early years of the French
revolutionary wars Duke Frederick Francis I. (1756-1837) remained
neutral, and in 1803 he regained Wismar from Sweden, but in 1806 his land
Page 591
was overrun by the French and in 1808 he joined the Confederation of the
Rhine. He was the first member of the confederation to abandon Napoleon,
to whose armies he had sent a contingent, and in 1813-1814 he fought
against France. In 1815 he joined the Germanic Confederation (Bund) and
took the title of grand duke. In 1819 serfdom was abolished in his
dominions. During the movement of 1848 the duchy witnessed a
considerable agitation in favour of a more liberal constitution, but in the
subsequent reaction all the concessions which had been made to the
democracy were withdrawn and further restrictive measures were
introduced in 1851 and 1852.
Mecklenburg-Strelitz adopted the constitution of the sister duchy by an
act of September 1755. In 1806 it was spared the infliction of a French
occupation through the good offices of the king of Bavaria; in 1808 its
duke, Charles (d. 1816), joined the confederation of the Rhine, but in 1813
he withdrew therefrom. Having been a member of the alliance against
Napoleon he joined the Germanic confederation in 1815 and assumed the
title of grand duke.
In 1866 both the grand dukes of Mecklenburg joined the North German
confederation and the Zollverein, and began to pass more and more under
the influence of Prussia, who in the war with Austria had been aided by the
soldiers of Mecklenburg-Schwerin. In the Franco-German War also Prussia
received valuable assistance from Mecklenburg, Duke Frederick Francis II.
(1823-1883), an ardent advocate of German unity, holding a high command
in her armies. In 1871 the two grand duchies became states of the German
Empire. There was now a renewal of the agitation for a more democratic
constitution, and the German Reichstag gave some countenance to this
movement. In 1897 Frederick Francis IV. (b. 1882) succeeded his father
Frederick Francis III. (1851-1897) as grand duke of Mecklenburg-
Schwerin, and in 1904 Adolphus Frederick (b. 1848) a son of the grand
Rhine. He was the first member of the confederation to abandon Napoleon,
to whose armies he had sent a contingent, and in 1813-1814 he fought
against France. In 1815 he joined the Germanic Confederation (Bund) and
took the title of grand duke. In 1819 serfdom was abolished in his
dominions. During the movement of 1848 the duchy witnessed a
considerable agitation in favour of a more liberal constitution, but in the
subsequent reaction all the concessions which had been made to the
democracy were withdrawn and further restrictive measures were
introduced in 1851 and 1852.
Mecklenburg-Strelitz adopted the constitution of the sister duchy by an
act of September 1755. In 1806 it was spared the infliction of a French
occupation through the good offices of the king of Bavaria; in 1808 its
duke, Charles (d. 1816), joined the confederation of the Rhine, but in 1813
he withdrew therefrom. Having been a member of the alliance against
Napoleon he joined the Germanic confederation in 1815 and assumed the
title of grand duke.
In 1866 both the grand dukes of Mecklenburg joined the North German
confederation and the Zollverein, and began to pass more and more under
the influence of Prussia, who in the war with Austria had been aided by the
soldiers of Mecklenburg-Schwerin. In the Franco-German War also Prussia
received valuable assistance from Mecklenburg, Duke Frederick Francis II.
(1823-1883), an ardent advocate of German unity, holding a high command
in her armies. In 1871 the two grand duchies became states of the German
Empire. There was now a renewal of the agitation for a more democratic
constitution, and the German Reichstag gave some countenance to this
movement. In 1897 Frederick Francis IV. (b. 1882) succeeded his father
Frederick Francis III. (1851-1897) as grand duke of Mecklenburg-
Schwerin, and in 1904 Adolphus Frederick (b. 1848) a son of the grand
Page 592
duke Frederick William (1819-1904) and his wife Augusta Carolina,
daughter of Adolphus Frederick, duke of Cambridge, became grand duke of
Mecklenburg-Strelitz. The grand dukes still style themselves princes of the
Wends.
See F. A. Rudloff, Pragmatisches Handbuch der mecklenburgischen
Geschichte (Schwerin, 1780-1822); C. C. F. von Lützow, Versuch einer
pragmatischen Geschichte von Mecklenburg (Berlin, 1827-1835);
Mecklenburgische Geschichte in Einzeldarstellungen, edited by R.
Beltz, C. Beyer, W. P. Graff and others; C. Hegel, Geschichte der
mecklenburgischen Landstände bis 1555 (Rostock, 1856); A. Mayer,
Geschichte des Grossherzogtums Mecklenburg-Strelitz 1816-1890
(New Strelitz, 1890); Tolzien, Die Grossherzöge von Mecklenburg-
Schwerin (Wismar, 1904); Lehsten, Der Adel Mecklenburgs seit dem
landesgrundgesetslichen Erbvergleich (Rostock, 1864); the
Mecklenburgisches Urkundenbuch in 21 vols. (Schwerin, 1873-1903);
the Jahrbücher des Vereins für mecklenburgische Geschichte und
Altertumskunde (Schwerin, 1836 fol.); and W. Raabe,
Mecklenburgische Vaterlandskunde (Wismar, 1894-1896); von
Hirschfeld, Friedrich Franz II., Grossherzog von Mecklenburg-
Schwerin und seine Vorgänger (Leipzig, 1891); Volz, Friedrich Franz
II. (Wismar, 1893); C. Schröder, Friedrich Franz III. (Schwerin, 1898);
Bartold, Friedrich Wilhelm, Grossherzog von Mecklenburg-Strelitz und
Augusta Carolina (New Strelitz, 1893); and H. Sachsse,
Mecklenburgische Urkunden und Daten (Rostock, 1900).
daughter of Adolphus Frederick, duke of Cambridge, became grand duke of
Mecklenburg-Strelitz. The grand dukes still style themselves princes of the
Wends.
See F. A. Rudloff, Pragmatisches Handbuch der mecklenburgischen
Geschichte (Schwerin, 1780-1822); C. C. F. von Lützow, Versuch einer
pragmatischen Geschichte von Mecklenburg (Berlin, 1827-1835);
Mecklenburgische Geschichte in Einzeldarstellungen, edited by R.
Beltz, C. Beyer, W. P. Graff and others; C. Hegel, Geschichte der
mecklenburgischen Landstände bis 1555 (Rostock, 1856); A. Mayer,
Geschichte des Grossherzogtums Mecklenburg-Strelitz 1816-1890
(New Strelitz, 1890); Tolzien, Die Grossherzöge von Mecklenburg-
Schwerin (Wismar, 1904); Lehsten, Der Adel Mecklenburgs seit dem
landesgrundgesetslichen Erbvergleich (Rostock, 1864); the
Mecklenburgisches Urkundenbuch in 21 vols. (Schwerin, 1873-1903);
the Jahrbücher des Vereins für mecklenburgische Geschichte und
Altertumskunde (Schwerin, 1836 fol.); and W. Raabe,
Mecklenburgische Vaterlandskunde (Wismar, 1894-1896); von
Hirschfeld, Friedrich Franz II., Grossherzog von Mecklenburg-
Schwerin und seine Vorgänger (Leipzig, 1891); Volz, Friedrich Franz
II. (Wismar, 1893); C. Schröder, Friedrich Franz III. (Schwerin, 1898);
Bartold, Friedrich Wilhelm, Grossherzog von Mecklenburg-Strelitz und
Augusta Carolina (New Strelitz, 1893); and H. Sachsse,
Mecklenburgische Urkunden und Daten (Rostock, 1900).
Page 593
Page 594
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BRITANNICA, 11TH EDITION, "MATTER" TO "MECKLENBURG" ***
Updated editions will replace the previous one—the old editions will
be renamed.
Creating the works from print editions not protected by U.S. copyright
law means that no one owns a United States copyright in these works,
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