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Title: Popular scientific lectures

Author: Ernst Mach

Translator: Thomas J. McCormack

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*** START OF THE PROJECT GUTENBERG EBOOK POPULAR
SCIENTIFIC LECTURES ***

POPULAR SCIENTIFIC
LECTURES.

BY THE SAME AUTHOR.
The Science of Mechanics. Translated from the Second German
Edition by T. J. McCormack. 250 Cuts and Illustrations. 534 Pages.
Half Morocco, Gilt Top. Price, $2.50.
Contributions to the Analysis of the Sensations. Translated by C. M.
Williams. With Notes and New Additions by the Author. 200 Pages. 36
Cuts. Price, $1.00.
Popular Scientific Lectures. Translated by T. J. McCormack. Third
Revised and Enlarged Edition. 411 Pages. 59 Cuts. Cloth, $1.50;
Paper, 50 cents.

THE OPEN COURT PUBLISHING CO.,
324 DEARBORN ST., CHICAGO.

POPULAR
SCIENTIFIC LECTURES
BY
ERNST MACH
FORMERLY PROFESSOR OF PHYSICS IN THE UNIVERSITY OF PRAGUE, NOW
PROFESSOR OF THE HISTORY AND THEORY OF INDUCTIVE SCIENCE IN THE
UNIVERSITY OF VIENNA

TRANSLATED
BY
THOMAS J. McCORMACK
THIRD EDITION, REVISED AND ENLARGED

WITH FIFTY-NINE CUTS AND DIAGRAMS

CHICAGO
THE OPEN COURT PUBLISHING COMPANY
FOR SALE BY
Kegan Paul, Trench, Truebner & Co., LONDON
1898

COPYRIGHT
By The Open Court Publishing Co.
Pages 1-258 in 1894.
Pages 338-374 in 1894.
Pages 259-281 in 1896.
Pages 282-308 in 1897.
Pages 309-337 in 1898.

AUTHOR'S PREFACE TO THE FIRST
EDITION.
Popular lectures, owing to the knowledge they presuppose, and the time
they occupy, can afford only a modicum of instruction. They must select for
this purpose easy subjects, and restrict themselves to the exposition of the
simplest and the most essential points. Nevertheless, by an appropriate
choice of the matter, the charm and the poetry of research can be conveyed
by them. It is only necessary to set forth the attractive and the alluring
features of a problem, and to show what broad domains of fact can be
illuminated by the light radiating from the solution of a single and ofttimes
unobtrusive point.
Furthermore, such lectures can exercise a favorable influence by showing
the substantial sameness of scientific and every-day thought. The public, in
this way, loses its shyness towards scientific questions, and acquires an
interest in scientific work which is a great help to the inquirer. The latter, in
his turn, is brought to understand that his work is a small part only of the
universal process of life, and that the results of his labors must redound to
the benefit not only of himself and a few of his associates, but to that of the
collective whole.
I sincerely hope that these lectures, in the present excellent translation, will
be productive of good in the direction indicated.
E. Mach.
Prague, December, 1894.

TRANSLATOR'S NOTE TO THE THIRD
EDITION.
The present third edition of this work has been enlarged by the addition of a
new lecture, "On Some Phenomena Attending the Flight of Projectiles." The
additions to the second consisted of the following four lectures and articles:
Professor Mach's Vienna Inaugural Lecture, "The Part Played by Accident
in Invention and Discovery," the lecture on "Sensations of Orientation,"
recently delivered and summing up the results of an important
psychological investigation, and two historical articles (see Appendix) on
Acoustics and Sight.
The lectures extend over a long period, from 1864 to 1898, and differ
greatly in style, contents, and purpose. They were first published in
collected form in English; afterwards two German editions were called for.
As the dates of the first five lectures are not given in the footnotes they are
here appended. The first lecture, "On the Forms of Liquids," was delivered
in 1868 and published with that "On Symmetry" in 1872 (Prague). The
second and third lectures, on acoustics, were first published in 1865 (Graz);
the fourth and fifth, on optics, in 1867 (Graz). They belong to the earliest
period of Professor Mach's scientific activity, and with the lectures on
electrostatics and education will more than realise the hope expressed in the
author's Preface.
The eighth, ninth, tenth, eleventh, and twelfth lectures are of a more
philosophical character and deal principally with the methods and nature of
scientific inquiry. In the ideas summarised in them will be found one of the
most important contributions to the theory of knowledge made in the last
quarter of a century. Significant hints in psychological method, and
exemplary specimen-researches in psychology and physics, are also
presented; while in physics many ideas find their first discussion that
afterwards, under other names and other authorship, became rallying-cries
in this department of inquiry.
All the proofs of this translation have been read by Professor Mach himself.

T. J. McCormack.
La Salle, Ill., May, 1898.

TABLE OF CONTENTS.
page
The Forms of Liquids 1
The Fibres of Corti 17
On the Causes of Harmony 32
The Velocity of Light 48
Why Has Man Two Eyes? 66
On Symmetry 89
On the Fundamental Concepts of Electrostatics 107
On the Principle of the Conservation of Energy 137
On the Economical Nature of Physical Inquiry 186
On Transformation and Adaptation in Scientific Thought 214
On the Principle of Comparison in Physics 236
On the Part Played by Accident in Invention and Discovery
259
On Sensations of Orientation 282
On Some Phenomena Attending the Flight of Projectiles 309
On Instruction in the Classics and the Mathematico-Physical
Sciences 338
Appendixes.
I. A Contribution to the History of Acoustics 375
II. Remarks on the Theory of Spatial Vision 386
Index 393

THE FORMS OF LIQUIDS.
What thinkest thou, dear Euthyphron, that the holy is, and the just, and the
good? Is the holy holy because the gods love it, or are the gods holy
because they love the holy? By such easy questions did the wise Socrates
make the market-place of Athens unsafe and relieve presumptuous young
statesmen of the burden of imaginary knowledge, by showing them how
confused, unclear, and self-contradictory their ideas were.
You know the fate of the importunate questioner. So called good society
avoided him on the promenade. Only the ignorant accompanied him. And
finally he drank the cup of hemlock—a lot which we ofttimes wish would
fall to modern critics of his stamp.
What we have learned from Socrates, however,—our inheritance from him,
—is scientific criticism. Every one who busies himself with science
recognises how unsettled and indefinite the notions are which he has
brought with him from common life, and how, on a minute examination of
things, old differences are effaced and new ones introduced. The history of
science is full of examples of this constant change, development, and
clarification of ideas.
But we will not linger by this general consideration of the fluctuating
character of ideas, which becomes a source of real uncomfortableness,
when we reflect that it applies to almost every notion of life. Rather shall
we observe by the study of a physical example how much a thing changes
when it is closely examined, and how it assumes, when thus considered,
increasing definiteness of form.
The majority of you think, perhaps, you know quite well the distinction
between a liquid and a solid. And precisely persons who have never busied
themselves with physics will consider this question one of the easiest that
can be put. But the physicist knows that it is one of the most difficult. I shall
mention here only the experiments of Tresca, which show that solids
subjected to high pressures behave exactly as liquids do; for example, may
be made to flow out in the form of jets from orifices in the bottoms of

vessels. The supposed difference of kind between liquids and solids is thus
shown to be a mere difference of degree.
The common inference that because the earth is oblate in form, it was
originally fluid, is an error, in the light of these facts. True, a rotating
sphere, a few inches in diameter will assume an oblate form only if it is
very soft, for example, is composed of freshly kneaded clay or some
viscous stuff. But the earth, even if it consisted of the rigidest stone, could
not help being crushed by its tremendous weight, and must perforce behave
as a fluid. Even our mountains could not extend beyond a certain height
without crumbling. The earth may once have been fluid, but this by no
means follows from its oblateness.
The particles of a liquid are displaced on the application of the slightest
pressure; a liquid conforms exactly to the shapes of the vessels in which it
is contained; it possesses no form of its own, as you have all learned in the
schools. Accommodating itself in the most trifling respects to the conditions
of the vessel in which it is placed, and showing, even on its surface, where
one would suppose it had the freest play, nothing but a polished, smiling,
expressionless countenance, it is the courtier par excellence of the natural
bodies.
Liquids have no form of their own! No, not for the superficial observer. But
persons who have observed that a raindrop is round and never angular, will
not be disposed to accept this dogma so unconditionally.
It is fair to suppose that every man, even the weakest, would possess a
character, if it were not too difficult in this world to keep it. So, too, we
must suppose that liquids would possess forms of their own, if the pressure
of the circumstances permitted it,—if they were not crushed by their own
weights.
An astronomer once calculated that human beings could not exist on the
sun, apart from its great heat, because they would be crushed to pieces there
by their own weight. The greater mass of this body would also make the
weight of the human body there much greater. But on the moon, because
here we should be much lighter, we could jump as high as the church-
steeples without any difficulty, with the same muscular power which we
now possess. Statues and "plaster" casts of syrup are undoubtedly things of
fancy, even on the moon, but maple-syrup would flow so slowly there that

we could easily build a maple-syrup man on the moon, for the fun of the
thing, just as our children here build snow-men.
Accordingly, if liquids have no form of their own with us on earth, they
have, perhaps, a form of their own on the moon, or on some smaller and
lighter heavenly body. The problem, then, simply is to get rid of the effects
of gravity; and, this done, we shall be able to find out what the peculiar
forms of liquids are.
The problem was solved by Plateau of Ghent, whose method was to
immerse the liquid in another of the same specific gravity.[1] He employed
for his experiments oil and a mixture of alcohol and water. By Archimedes's
well-known principle, the oil in this mixture loses its entire weight. It no
longer sinks beneath its weight; its formative forces, be they ever so weak,
are now in full play.
As a fact, we now see, to our surprise, that the oil, instead of spreading out
into a layer, or lying in a formless mass, assumes the shape of a beautiful
and perfect sphere, freely suspended in the mixture, as the moon is in space.
We can construct in this way a sphere of oil several inches in diameter.
If, now, we affix a thin plate to a wire and insert the plate in the oil sphere,
we can, by twisting the wire between our fingers, set the whole ball in
rotation. Doing this, the ball assumes an oblate shape, and we can, if we are
skilful enough, separate by such rotation a ring from the ball, like that
which surrounds Saturn. This ring is finally rent asunder, and, breaking up
into a number of smaller balls, exhibits to us a kind of model of the origin
of the planetary system according to the hypothesis of Kant and Laplace.
Still more curious are the phenomena exhibited when the formative forces
of the liquid are partly disturbed by putting in contact with the liquid's
surface some rigid body. If we immerse, for example, the wire framework
of a cube in our mass of oil, the oil will everywhere stick to the wire
framework. If the quantity of oil is exactly sufficient we shall obtain an oil
cube with perfectly smooth walls. If there is too much or too little oil, the
walls of the cube will bulge out or cave in. In this manner we can produce
all kinds of geometrical figures of oil, for example, a three-sided pyramid, a
cylinder (by bringing the oil between two wire rings), and so on. Interesting
is the change of form that occurs when we gradually suck out the oil by
means of a glass tube from the cube or pyramid. The wire holds the oil fast.

The figure grows smaller and smaller, until it is at last
quite thin. Ultimately it consists simply of a number of
thin, smooth plates of oil, which extend from the edges
of the cube to the centre, where they meet in a small
drop. The same is true of the pyramid.

Fig. 1.

Fig. 2.
The idea now suggests itself that liquid figures as thin as this, and
possessing, therefore, so slight a weight, cannot be crushed or deformed by

their weight; just as a small, soft ball of clay is not affected in this respect
by its weight. This being the case, we no longer need our mixture of alcohol
and water for the production of figures, but can construct them in the open
air. And Plateau, in fact, found that these thin figures, or at least very
similar ones, could be produced in the air, by dipping the wire nets
described in a solution of soap and water and quickly drawing them out
again. The experiment is not difficult. The figure is formed of itself. The
preceding drawing represents to the eye the forms obtained with cubical and
pyramidal nets. In the cube, thin, smooth films of soap-suds proceed from
the edges to a small, quadratic film in the centre. In the pyramid, a film
proceeds from each edge to the centre.
These figures are so beautiful that they hardly admit of appropriate
description. Their great regularity and geometrical exactness evokes
surprise from all who see them for the first time. Unfortunately, they are of
only short duration. They burst, on the drying of the solution in the air, but
only after exhibiting to us the most brilliant play of colors, such as is often
seen in soap-bubbles. Partly their beauty of form and partly our desire to
examine them more minutely induces us to conceive of methods of
endowing them with permanent form. This is very simply done.[2] Instead
of dipping the wire nets in solutions of soap, we dip them in pure melted
colophonium (resin). When drawn out the figure at once forms and
solidifies by contact with the air.
It is to be remarked that also solid fluid-figures can be constructed in the
open air, if their weight be light enough, or the wire nets of very small
dimensions. If we make, for example, of very fine wire a cubical net whose
sides measure about one-eighth of an inch in length, we need simply to dip
this net in water to obtain a small solid cube of water. With a piece of
blotting paper the superfluous water may be easily removed and the sides of
the cube made smooth.
Yet another simple method may be devised for observing these figures. A
drop of water on a greased glass plate will not run if it is small enough, but
will be flattened by its weight, which presses it against its support. The
smaller the drop the less the flattening. The smaller the drop the nearer it
approaches the form of a sphere. On the other hand, a drop suspended from
a stick is elongated by its weight. The undermost parts of a drop of water on
a support are pressed against the support, and the upper parts are pressed

against the lower parts because the latter cannot yield. But when a drop falls
freely downward all its parts move equally fast; no part is impeded by
another; no part presses against another. A freely falling drop, accordingly,
is not affected by its weight; it acts as if it were weightless; it assumes a
spherical form.
A moment's glance at the soap-film figures produced by our various wire
models, reveals to us a great multiplicity of form. But great as this
multiplicity is, the common features of the figures also are easily
discernible.

"All forms of Nature are allied, though none is the same as
the other;
Thus, their common chorus points to a hidden law."

This hidden law Plateau discovered. It may be expressed, somewhat prosily,
as follows:
1) If several plane liquid films meet in a figure they are always three in
number, and, taken in pairs, form, each with another, nearly equal angles.
2) If several liquid edges meet in a figure they are always four in number,
and, taken in pairs, form, each with another, nearly equal angles.
This is a strange law, and its reason is not evident. But we might apply this
criticism to almost all laws. It is not always that the motives of a law-maker
are discernible in the form of the law he constructs. But our law admits of
analysis into very simple elements or reasons. If we closely examine the
paragraphs which state it, we shall find that their meaning is simply this,
that the surface of the liquid assumes the shape of smallest area that is
possible under the circumstances.
If, therefore, some extraordinarily intelligent tailor, possessing a knowledge
of all the artifices of the higher mathematics, should set himself the task of
so covering the wire frame of a cube with cloth that every piece of cloth
should be connected with the wire and joined with the remaining cloth, and
should seek to accomplish this feat with the greatest saving of material, he
would construct no other figure than that which is here formed on the wire
frame in our solution of soap and water. Nature acts in the construction of
liquid figures on the principle of a covetous tailor, and gives no thought in

her work to the fashions. But, strange to say, in this work, the most
beautiful fashions are of themselves produced.
The two paragraphs which state our law apply primarily only to soap-film
figures, and are not applicable, of course, to solid oil-figures. But the
principle that the superficial area of the liquid shall be the least possible
under the circumstances, is applicable to all fluid figures. He who
understands not only the letter but also the reason of the law will not be at a
loss when confronted with cases to which the letter does not accurately
apply. And this is the case with the principle of least superficial area. It is a
sure guide for us even in cases in which the above-stated paragraphs are not
applicable.
Our first task will now be, to show by a palpable illustration the mode of
formation of liquid figures by the principle of least superficial area. The oil
on the wire pyramid in our mixture of alcohol and water, being unable to
leave the wire edges, clings to them, and the given mass of oil strives so to
shape itself that its surface shall have the least possible area. Suppose we
attempt to imitate this phenomenon. We take a wire pyramid, draw over it a
stout film of rubber, and in place of the wire handle insert a small tube
leading into the interior of the space enclosed by the rubber (Fig. 3).
Through this tube we can blow in or suck out air. The quantity of air in the
enclosure represents the quantity of oil. The stretched rubber film, which,
clinging to the wire edges, does its utmost to contract, represents the surface
of the oil endeavoring to decrease its area. By blowing in, and drawing out
the air, now, we actually obtain all the oil pyramidal figures, from those
bulged out to those hollowed in. Finally, when all the air is pumped or
sucked out, the soap-film figure is exhibited. The rubber films strike
together, assume the form of planes, and meet at four sharp edges in the
centre of the pyramid.
The tendency of soap-films to assume smaller forms may be directly
demonstrated by a method of Van der Mensbrugghe. If we dip a square wire
frame to which a handle is attached into a solution of soap and water, we
shall obtain on the frame a beautiful, plane film of soap-suds. (Fig. 4.) On
this we lay a thread having its two ends tied together. If, now, we puncture
the part enclosed by the thread, we shall obtain a soap-film having a
circular hole in it, whose circumference is the thread. The remainder of the
film decreasing in area as much as it can, the hole assumes the largest area

that it can. But
the figure of
largest area,
with a given
periphery, is
the circle.

Fig. 4.

Fig. 3.

Fig. 5.
Similarly, by the principle of least superficial area, a freely suspended mass
of oil assumes the shape of a sphere. The sphere is the form of least surface
for a given content. This is evident. The more we put into a travelling-bag,
the nearer its shape approaches the spherical form.
The connexion of the two above-mentioned paragraphs with the principle of
least superficial area may be shown by a yet simpler example. Picture to
yourselves four fixed pulleys, a, b, c, d, and two movable rings f, g (Fig. 5);
about the pulleys and through the rings imagine a smooth cord passed,
fastened at one extremity to a nail e, and loaded at the other with a weight
h. Now this weight always tends to sink, or, what is the same thing, always
tends to make the portion of the string e h as long as possible, and
consequently the remainder of the string, wound round the pulleys, as short
as possible. The strings must remain connected with the pulleys, and on

account of the rings also with each other. The conditions of the case,
accordingly, are similar to those of the liquid figures discussed. The result
also is a similar one. When, as in the right hand figure of the cut, four pairs
of strings meet, a different configuration must be established. The
consequence of the endeavor of the string to shorten itself is that the rings
separate from each other, and that now at all points only three pairs of
strings meet, every two at equal angles of one hundred and twenty degrees.
As a fact, by this arrangement the greatest possible shortening of the string
is attained; as can be easily proved by geometry.
This will help us to some extent to understand the creation of beautiful and
complicated figures by the simple tendency of liquids to assume surfaces of
least superficial area. But the question arises, Why do liquids seek surfaces
of least superficial area?
The particles of a liquid cling together. Drops brought into contact coalesce.
We can say, liquid particles attract each other. If so, they seek to come as
close as they can to each other. The particles at the surface will endeavor to
penetrate as far as they can into the interior. This process will not stop,
cannot stop, until the surface has become as small as under the
circumstances it possibly can become, until as few particles as possible
remain at the surface, until as many particles as possible have penetrated
into the interior, until the forces of attraction have no more work to perform.
[3]

The root of the principle of least surface is to be sought, accordingly, in
another and much simpler principle, which may be illustrated by some such
analogy as this. We can conceive of the natural forces of attraction and
repulsion as purposes or intentions of nature. As a matter of fact, that
interior pressure which we feel before an act and which we call an intention
or purpose, is not, in a final analysis, so essentially different from the
pressure of a stone on its support, or the pressure of a magnet on another,
that it is necessarily unallowable to use for both the same term—at least for
well-defined purposes.[4] It is the purpose of nature, accordingly, to bring
the iron nearer the magnet, the stone nearer the centre of the earth, and so
forth. If such a purpose can be realised, it is carried out. But where she
cannot realise her purposes, nature does nothing. In this respect she acts
exactly as a good man of business does.

It is a constant purpose of nature to bring weights lower. We can raise a
weight by causing another, larger weight to sink; that is, by satisfying
another, more powerful, purpose of nature. If we fancy we are making
nature serve our purposes in this, it will be found, upon closer examination,
that the contrary is true, and that nature has employed us to attain her
purposes.
Equilibrium, rest, exists only, but then always, when nature is brought to a
halt in her purposes, when the forces of nature are as fully satisfied as,
under the circumstances, they can be. Thus, for example, heavy bodies are
in equilibrium, when their so-called centre of gravity lies as low as it
possibly can, or when as much weight as the circumstances admit of has
sunk as low as it can.
The idea forcibly suggests itself that perhaps this principle also holds good
in other realms. Equilibrium exists also in the state when the purposes of
the parties are as fully satisfied as for the time being they can be, or, as we
may say, jestingly, in the language of physics, when the social potential is a
maximum.[5]
You see, our miserly mercantile principle is replete with consequences.[6]
The result of sober research, it has become as fruitful for physics as the dry
questions of Socrates for science generally. If the principle seems to lack in
ideality, the more ideal are the fruits which it bears.
But why, tell me, should science be ashamed of such a principle? Is
science[7] itself anything more than—a business? Is not its task to acquire
with the least possible work, in the least possible time, with the least
possible thought, the greatest possible part of eternal truth?

THE FIBRES OF CORTI.
Whoever has roamed through a beautiful country knows that the tourist's
delights increase with his progress. How pretty that wooded dell must look
from yonder hill! Whither does that clear brook flow, that hides itself in
yonder sedge? If I only knew how the landscape looked behind that
mountain! Thus even the child thinks in his first rambles. It is also true of
the natural philosopher.
The first questions are forced upon the attention of the inquirer by practical
considerations; the subsequent ones are not. An irresistible attraction draws
him to these; a nobler interest which far transcends the mere needs of life.
Let us look at a special case.
For a long time the structure of the organ of hearing has actively engaged
the attention of anatomists. A considerable number of brilliant discoveries
has been brought to light by their labors, and a splendid array of facts and
truths established. But with these facts a host of new enigmas has been
presented.
Whilst in the theory of the organisation and functions of the eye
comparative clearness has been attained; whilst, hand in hand with this,
ophthalmology has reached a degree of perfection which the preceding
century could hardly have dreamed of, and by the help of the
ophthalmoscope the observing physician penetrates into the profoundest
recesses of the eye, the theory of the ear is still much shrouded in
mysterious darkness, full of attraction for the investigator.
Look at this model of the ear. Even at that familiar part by whose extent we
measure the quantity of people's intelligence, even at the external ear, the
problems begin. You see here a succession of helixes or spiral windings, at
times very pretty, whose significance we cannot accurately state, yet for
which there must certainly be some reason.
The shell or concha of the ear, a in the annexed diagram, conducts the
sound into the curved auditory passage b, which is terminated by a thin
membrane, the so-called tympanic membrane, e. This membrane is set in

motion by the sound, and in its turn
sets in motion a series of little
bones of very peculiar formation, c.
At the end of all is the labyrinth d.
The labyrinth consists of a group of
cavities filled with a liquid, in
which the innumerable fibres of the
nerve of hearing are imbedded. By
the vibration of the chain of bones
c, the liquid of the labyrinth is
shaken, and the auditory nerve Fig. 6.
excited. Here the process of hearing
begins. So much is certain. But the details of the process are one and all
unanswered questions.
To these old puzzles, the Marchese Corti, as late as 1851, added a new
enigma. And, strange to say, it is this last enigma, which, perhaps, has first
received its correct solution. This will be the subject of our remarks to-day.
Corti found in the cochlea, or snail-shell of the labyrinth, a large number of
microscopic fibres placed side by side in geometrically graduated order.
According to Kölliker their number is three thousand. They were also the
subject of investigation at the hands of Max Schultze and Deiters.
A description of the details of this organ would only weary you, besides not
rendering the matter much clearer. I prefer, therefore, to state briefly what in
the opinion of prominent investigators like Helmholtz and Fechner is the
peculiar function of Corti's fibres. The cochlea, it seems, contains a large
number of elastic fibres of graduated lengths (Fig. 7), to which the branches
of the auditory nerve are attached. These fibres, called the fibres, pillars, or
rods of Corti, being of unequal length, must also be of unequal elasticity,
and, consequently, pitched to different notes. The cochlea, therefore, is a
species of pianoforte.
What, now, may be the office of this structure, which is found in no other
organ of sense? May it not be connected with some special property of the
ear? It is quite probable; for the ear possesses a very similar power. You
know that it is possible to follow the individual voices of a symphony.
Indeed, the feat is possible even in a fugue of Bach, where it is certainly no

inconsiderable achievement. The
ear can pick out the single
constituent tonal parts, not only of a
harmony, but of the wildest clash of
music imaginable. The musical ear
analyses every agglomeration of
tones.
The eye does not possess this
ability. Who, for example, could tell
from the mere sight of white,
without a previous experimental
knowledge of the fact, that white is
composed of a mixture of other
Fig. 7. colors? Could it be, now, that these
two facts, the property of the ear
just mentioned, and the structure
discovered by Corti, are really connected? It is very probable. The enigma
is solved if we assume that every note of definite pitch has its special string
in this pianoforte of Corti, and, therefore, its special branch of the auditory
nerve attached to that string. But before I can make this point perfectly
plain to you, I must ask you to follow me a few steps into the dry domain of
physics.
Look at this pendulum. Forced from its position of equilibrium by an
impulse, it begins to swing with a definite time of oscillation, dependent
upon its length. Longer pendulums swing more slowly, shorter ones more
quickly. We will suppose our pendulum to execute one to-and-fro
movement in a second.
This pendulum, now, can be thrown into violent vibration in two ways;
either by a single heavy impulse, or by a number of properly communicated
slight impulses. For example, we impart to the pendulum, while at rest in its
position of equilibrium, a very slight impulse. It will execute a very small
vibration. As it passes a third time its position of equilibrium, a second
having elapsed, we impart to it again a slight shock, in the same direction
with the first. Again after the lapse of a second, on its fifth passage through
the position of equilibrium, we strike it again in the same manner; and so
continue. You see, by this process the shocks imparted augment continually

the motion of the pendulum. After each slight impulse, the pendulum
reaches out a little further in its swing, and finally acquires a considerable
motion.[8]
But this is not the case under all circumstances. It is possible only when the
impulses imparted synchronise with the swings of the pendulum. If we
should communicate the second impulse at the end of half a second and in
the same direction with the first impulse, its effects would counteract the
motion of the pendulum. It is easily seen that our little impulses help the
motion of the pendulum more and more, according as their time accords
with the time of the pendulum. If we strike the pendulum in any other time
than in that of its vibration, in some instances, it is true, we shall augment
its vibration, but in others again, we shall obstruct it. Our impulses will be
less effective the more the motion of our own hand departs from the motion
of the pendulum.
What is true of the pendulum holds true of every vibrating body. A tuning-
fork when it sounds, also vibrates. It vibrates more rapidly when its sound is
higher; more slowly when it is deeper. The standard A of our musical scale
is produced by about four hundred and fifty vibrations in a second.
I place by the side of each other on this table two tuning-forks, exactly
alike, resting on resonant cases. I strike the first one a sharp blow, so that it
emits a loud note, and immediately grasp it again with my hand to quench
its note. Nevertheless, you still hear the note distinctly sounded, and by
feeling it you may convince yourselves that the other fork which was not
struck now vibrates.
I now attach a small bit of wax to one of the forks. It is thrown thus out of
tune; its note is made a little deeper. I now repeat the same experiment with
the two forks, now of unequal pitch, by striking one of them and again
grasping it with my hand; but in the present case the note ceases the very
instant I touch the fork.
What has happened here in these two experiments? Simply this. The
vibrating fork imparts to the air and to the table four hundred and fifty
shocks a second, which are carried over to the other fork. If the other fork is
pitched to the same note, that is to say, if it vibrates when struck in the same
time with the first, then the shocks first emitted, no matter how slight they
may be, are sufficient to throw the second fork into rapid sympathetic

vibration. But when the time of vibration of the two forks is slightly
different, this does not take place. We may strike as many forks as we will,
the fork tuned to A is perfectly indifferent to their notes; is deaf, in fact, to
all except its own; and if you strike three, or four, or five, or any number
whatsoever, of forks all at the same time, so as to make the shocks which
come from them ever so great, the A fork will not join in with their
vibrations unless another fork A is found in the collection struck. It picks
out, in other words, from all the notes sounded, that which accords with it.
The same is true of all bodies which can yield notes. Tumblers resound
when a piano is played, on the striking of certain notes, and so do window
panes. Nor is the phenomenon without analogy in other provinces. Take a
dog that answers to the name "Nero." He lies under your table. You speak of
Domitian, Vespasian, and Marcus Aurelius Antoninus, you call upon all the
names of the Roman Emperors that occur to you, but the dog does not stir,
although a slight tremor of his ear tells you of a faint response of his
consciousness. But the moment you call "Nero" he jumps joyfully towards
you. The tuning-fork is like your dog. It answers to the name A.
You smile, ladies. You shake your heads. The simile does not catch your
fancy. But I have another, which is very near to you: and for punishment
you shall hear it. You, too, are like tuning-forks. Many are the hearts that
throb with ardor for you, of which you take no notice, but are cold. Yet what
does it profit you! Soon the heart will come that beats in just the proper
rhythm, and then your knell, too, has struck. Then your heart, too, will beat
in unison, whether you will or no.
The law of sympathetic vibration, here propounded for sounding bodies,
suffers some modification for bodies incompetent to yield notes. Bodies of
this kind vibrate to almost every note. A high silk hat, we know, will not
sound; but if you will hold your hat in your hand when attending your next
concert you will not only hear the pieces played, but also feel them with
your fingers. It is exactly so with men. People who are themselves able to
give tone to their surroundings, bother little about the prattle of others. But
the person without character tarries everywhere: in the temperance hall, and
at the bar of the public-house—everywhere where a committee is formed.
The high silk hat is among bells what the weakling is among men of
conviction.

A sonorous body, therefore, always sounds when its special note, either
alone or in company with others, is struck. We may now go a step further.
What will be the behaviour of a group of sonorous bodies which in the pitch
of their notes form a scale? Let us picture to ourselves, for example (Fig. 8),
a series of rods or strings pitched to the notes c d e f g.... On a musical
instrument the accord c e g is struck. Every one of the rods of Fig. 8 will see
if its special note is contained in the accord, and if it finds it, it will respond.
The rod c will give at once the note c, the rod e the note e, the rod g the note
g. All the other rods will remain at rest, will not sound.
We need not look about us long for
such an instrument. Every piano is
an instrument of this kind, with
which the experiment mentioned
may be executed with splendid
success. Two pianos stand here by
the side of each other, both tuned
alike. We will employ the first for
exciting the notes, while we will
allow the second to respond; after
having first pressed upon the loud
pedal, so as to render all the strings
capable of motion.
Every harmony struck with vigor on
the first piano is distinctly repeated
on the second. To prove that it is the Fig. 8.
same strings that are sounded in
both pianos, we repeat the
experiment in a slightly changed form. We let go the loud pedal of the
second piano and pressing on the keys c e g of that instrument vigorously
strike the harmony c e g on the first piano. The harmony c e g is now also
sounded on the second piano. But if we press only on one key g of one
piano, while we strike c e g on the other, only g will be sounded on the
second. It is thus always the like strings of the two pianos that excite each
other.
The piano can reproduce any sound that is composed of its musical notes. It
will reproduce, for example, very distinctly, a vowel sound that is sung into

it. And in truth physics has proved that the vowels may be regarded as
composed of simple musical notes.
You see that by the exciting of definite tones in the air quite definite
motions are set up with mechanical necessity in the piano. The idea might
be made use of for the performance of some pretty pieces of wizardry.
Imagine a box in which is a stretched string of definite pitch. This is thrown
into motion as often as its note is sung or whistled. Now it would not be a
very difficult task for a skilful mechanic to so construct the box that the
vibrating cord would close a galvanic circuit and open the lock. And it
would not be a much more difficult task to construct a box which would
open at the whistling of a certain melody. Sesame! and the bolts fall. Truly,
we should have here a veritable puzzle-lock. Still another fragment rescued
from that old kingdom of fables, of which our day has realised so much,
that world of fairy-stories to which the latest contributions are Casselli's
telegraph, by which one can write at a distance in one's own hand, and Prof.
Elisha Gray's telautograph. What would the good old Herodotus have said
to these things who even in Egypt shook his head at much that he saw? ἐμοἱ
μἑνe ού πιστα, just as simple-heartedly as then, when he heard of the
circumnavigation of Africa.
A new puzzle-lock! But why invent one? Are not we human beings
ourselves puzzle-locks? Think of the stupendous groups of thoughts,
feelings, and emotions that can be aroused in us by a word! Are there not
moments in all our lives when a mere name drives the blood to our hearts?
Who that has attended a large mass-meeting has not experienced what
tremendous quantities of energy and motion can be evolved by the innocent
words, "Liberty, Equality, Fraternity."
But let us return to the subject proper of our discourse. Let us look again at
our piano, or what will do just as well, at some other contrivance of the
same character. What does this instrument do? Plainly, it decomposes, it
analyses every agglomeration of sounds set up in the air into its individual
component parts, each tone being taken up by a different string; it performs
a real spectral analysis of sound. A person completely deaf, with the help of
a piano, simply by touching the strings or examining their vibrations with a
microscope, might investigate the sonorous motion of the air, and pick out
the separate tones excited in it.

The ear has the same capacity as this piano. The ear performs for the mind
what the piano performs for a person who is deaf. The mind without the ear
is deaf. But a deaf person, with the piano, does hear after a fashion, though
much less vividly, and more clumsily, than with the ear. The ear, thus, also
decomposes sound into its component tonal parts. I shall now not be
deceived, I think, if I assume that you already have a presentiment of what
the function of Corti's fibres is. We can make the matter very plain to
ourselves. We will use the one piano for exciting the sounds, and we shall
imagine the second one in the ear of the observer in the place of Corti's
fibres, which is a model of such an instrument. To every string of the piano
in the ear we will suppose a special fibre of the auditory nerve attached, so
that this fibre and this alone, is irritated when the string is thrown into
vibration. If we strike now an accord on the external piano, for every tone
of that accord a definite string of the internal piano will sound and as many
different nervous fibres will be irritated as there are notes in the accord. The
simultaneous sense-impressions due to different notes can thus be preserved
unmingled and be separated by the attention. It is the same as with the five
fingers of the hand. With each finger I can touch something different. Now
the ear has three thousand such fingers, and each one is designed for the
touching of a different tone.[9] Our ear is a puzzle-lock of the kind
mentioned. It opens at the magic melody of a sound. But it is a
stupendously ingenious lock. Not only one tone, but every tone makes it
open; but each one differently. To each tone it replies with a different
sensation.
More than once it has happened in the history of science that a phenomenon
predicted by theory, has not been brought within the range of actual
observation until long afterwards. Leverrier predicted the existence and the
place of the planet Neptune, but it was not until sometime later that Galle
actually found the planet at the predicted spot. Hamilton unfolded
theoretically the phenomenon of the so-called conical refraction of light, but
it was reserved for Lloyd some time subsequently to observe the fact. The
fortunes of Helmholtz's theory of Corti's fibres have been somewhat similar.
This theory, too, received its substantial confirmation from the subsequent
observations of V. Hensen. On the free surface of the bodies of Crustacea,
connected with the auditory nerves, rows of little hairy filaments of varying
lengths and thicknesses are found, which to some extent are the analogues

of Corti's fibres. Hensen saw these hairs vibrate when sounds were excited,
and when different notes were struck different hairs were set in vibration.
I have compared the work of the physical inquirer to the journey of the
tourist. When the tourist ascends a new hill he obtains of the whole district
a different view. When the inquirer has found the solution of one enigma,
the solution of a host of others falls into his hands.
Surely you have often felt the strange impression experienced when in
singing through the scale the octave is reached, and nearly the same
sensation is produced as by the fundamental tone. The phenomenon finds its
explanation in the view here laid down of the ear. And not only this
phenomenon but all the laws of the theory of harmony may be grasped and
verified from this point of view with a clearness before undreamt of.
Unfortunately, I must content myself to-day with the simple indication of
these beautiful prospects. Their consideration would lead us too far aside
into the fields of other sciences.
The searcher of nature, too, must restrain himself in his path. He also is
drawn along from one beauty to another as the tourist from dale to dale, and
as circumstances generally draw men from one condition of life into others.
It is not he so much that makes the quests, as that the quests are made of
him. Yet let him profit by his time, and let not his glance rove aimlessly
hither and thither. For soon the evening sun will shine, and ere he has
caught a full glimpse of the wonders close by, a mighty hand will seize him
and lead him away into a different world of puzzles.
Respected hearers, science once stood in an entirely different relation to
poetry. The old Hindu mathematicians wrote their theorems in verses, and
lotus-flowers, roses, and lilies, beautiful sceneries, lakes, and mountains
figured in their problems.
"Thou goest forth on this lake in a boat. A lily juts forth, one palm above the
water. A breeze bends it downwards, and it vanishes two palms from its
previous spot beneath the surface. Quick, mathematician, tell me how deep
is the lake!"
Thus spoke an ancient Hindu scholar. This poetry, and rightly, has
disappeared from science, but from its dry leaves another poetry is wafted
aloft which cannot be described to him who has never felt it. Whoever will

fully enjoy this poetry must put his hand to the plough, must himself
investigate. Therefore, enough of this! I shall reckon myself fortunate if you
do not repent of this brief excursion into the flowered dale of physiology,
and if you take with yourselves the belief that we can say of science what
we say of poetry,

"Who the song would understand,
Needs must seek the song's own land;
Who the minstrel understand
Needs must seek the minstrel's land."

ON THE CAUSES OF HARMONY.
We are to speak to-day of a theme which is perhaps of somewhat more
general interest—the causes of the harmony of musical sounds. The first
and simplest experiences relative to harmony are very ancient. Not so the
explanation of its laws. These were first supplied by the investigators of a
recent epoch. Allow me an historical retrospect.
Pythagoras (586 B. C.) knew that the note yielded by a string of steady
tension was converted into its octave when the length of the string was
reduced one-half, and into its fifth when reduced two-thirds; and that then
the first fundamental tone was consonant with the two others. He knew
generally that the same string under fixed tension gives consonant tones
when successively divided into lengths that are in the proportions of the
simplest natural numbers; that is, in the proportions of 1:2, 2:3, 3:4, 4:5.
Pythagoras failed to reveal the causes of these laws. What have consonant
tones to do with the simple natural numbers? That is the question we should
ask to-day. But this circumstance must have appeared less strange than
inexplicable to Pythagoras. This philosopher sought for the causes of
harmony in the occult, miraculous powers of numbers. His procedure was
largely the cause of the upgrowth of a numerical mysticism, of which the
traces may still be detected in our oneirocritical books and among some
scientists, to whom marvels are more attractive than lucidity.
Euclid (300 B. C.) gives a definition of consonance and dissonance that
could hardly be improved upon, in point of verbal accuracy. The
consonance (συμφωνία) of two tones, he says, is the mixture, the blending
(κρᾶσις) of those two tones; dissonance (διαφωνία), on the other hand, is
the incapacity of the tones to blend (ἀμιξία), whereby they are made harsh
for the ear. The person who knows the correct explanation of the
phenomenon hears it, so to speak, reverberated in these words of Euclid.
Still, Euclid did not know the true cause of harmony. He had unwittingly
come very near to the truth, but without really grasping it.
Leibnitz (1646-1716 A. D.) resumed the question which his predecessors
had left unsolved. He, of course, knew that musical notes were produced by

vibrations, that twice as many vibrations corresponded to the octave as to
the fundamental tone, etc. A passionate lover of mathematics, he sought for
the cause of harmony in the secret computation and comparison of the
simple numbers of vibrations and in the secret satisfaction of the soul at this
occupation. But how, we ask, if one does not know that musical notes are
vibrations? The computation and the satisfaction at the computation must
indeed be pretty secret if it is unknown. What queer ideas philosophers
have! Could anything more wearisome be imagined than computation as a
principle of æsthetics? Yes, you are not utterly wrong in your conjecture,
yet you may be sure that Leibnitz's theory is not wholly nonsense, although
it is difficult to make out precisely what he meant by his secret
computation.
The great Euler (1707-1783) sought the cause of harmony, almost as
Leibnitz did, in the pleasure which the soul derives from the contemplation
of order in the numbers of the vibrations.[10]
Rameau and D'Alembert (1717-1783) approached nearer to the truth. They
knew that in every sound available in music besides the fundamental note
also the twelfth and the next higher third could be heard; and further that
the resemblance between a fundamental tone and its octave was always
strongly marked. Accordingly, the combination of the octave, fifth, third,
etc., with the fundamental tone appeared to them "natural." They possessed,
we must admit, the correct point of view; but with the simple naturalness of
a phenomenon no inquirer can rest content; for it is precisely this
naturalness for which he seeks his explanations.
Rameau's remark dragged along through the whole modern period, but
without leading to the full discovery of the truth. Marx places it at the head
of his theory of composition, but makes no further application of it. Also
Goethe and Zelter in their correspondence were, so to speak, on the brink of
the truth. Zelter knew of Rameau's view. Finally, you will be appalled at the
difficulty of the problem, when I tell you that till very recent times even
professors of physics were dumb when asked what were the causes of
harmony.
Not till quite recently did Helmholtz find the solution of the question. But
to make this solution clear to you I must first speak of some experimental
principles of physics and psychology.

1) In every process of perception, in every observation, the attention plays a
highly important part. We need not look about us long for proofs of this.
You receive, for example, a letter written in a very poor hand. Do your best,
you cannot make it out. You put together now these, now those lines, yet
you cannot construct from them a single intelligible character. Not until you
direct your attention to groups of lines which really belong together, is the
reading of the letter possible. Manuscripts, the letters of which are formed
of minute figures and scrolls, can only be read at a considerable distance,
where the attention is no longer diverted from the significant outlines to the
details. A beautiful example of this class is furnished by the famous
iconographs of Giuseppe Arcimboldo in the basement of the Belvedere
gallery at Vienna. These are symbolic representations of water, fire, etc.:
human heads composed of aquatic animals and of combustibles. At a short
distance one sees only the details, at a greater distance only the whole
figure. Yet a point can be easily found at which, by a simple voluntary
movement of the attention, there is no difficulty in seeing now the whole
figure and now the smaller forms of which it is composed. A picture is often
seen representing the tomb of Napoleon. The tomb is surrounded by dark
trees between which the bright heavens are visible as background. One can
look a long time at this picture without noticing anything except the trees,
but suddenly, on the attention being accidentally directed to the bright
background, one sees the figure of Napoleon between the trees. This case
shows us very distinctly the important part which attention plays. The same
sensuous object can, solely by the interposition of attention, give rise to
wholly different perceptions.
If I strike a harmony, or chord, on this piano, by a mere effort of attention
you can fix every tone of that harmony. You then hear most distinctly the
fixed tone, and all the rest appear as a mere addition, altering only the
quality, or acoustic color, of the primary tone. The effect of the same
harmony is essentially modified if we direct our attention to different tones.
Strike in succession two harmonies, for example, the two represented in the
annexed diagram, and first fix by the attention the upper note e, afterwards
the base e-a; in the two cases you will hear the same sequence of harmonies
differently. In the first case, you have the impression as if the fixed tone
remained unchanged and simply altered its timbre; in the second case, the
whole acoustic agglomeration seems to fall sensibly in depth. There is an

art of composition to guide the attention of the hearer. But there is also an
art of hearing, which is not the gift of every person.
The piano-player knows the remarkable effects
obtained when one of the keys of a chord that is
struck is let loose. Bar 1 played on the piano
sounds almost like bar 2. The note which lies
next to the key let loose resounds after its
release as if it were freshly struck. The attention
no longer occupied with the upper note is by
that very fact insensibly led to the upper note.

Fig. 9.

Fig. 10.
Any tolerably cultivated musical ear can perform the resolution of a
harmony into its component parts. By much practice we can go even
further. Then, every musical sound heretofore regarded as simple can be
resolved into a subordinate succession of musical tones. For example, if I
strike on the piano the note 1, (annexed diagram,) we shall hear, if we make
the requisite effort of attention, besides the loud fundamental note the
feebler, higher overtones, or harmonics, 2 ... 7, that is, the octave, the
twelfth, the double octave, and the third, the fifth, and the seventh of the
double octave.
The same is true of every musically available sound. Each yields, with
varying degrees of intensity, besides its fundamental note, also the octave,
the twelfth, the double octave, etc. The phenomenon is observable with
special facility on the open and closed flue-pipes of organs. According,
now, as certain overtones are more or less distinctly emphasised in a sound,

the timbre of the sound changes—that peculiar
quality of the sound by which we distinguish the
music of the piano from that of the violin, the
clarinet, etc.
On the piano these overtones can be very easily
rendered audible. If I strike, for example,
sharply note 1 of the foregoing series, whilst I
simply press down upon, one after another, the
keys 2, 3, ... 7, the notes 2, 3, ... 7 will continue
to sound after the striking of 1, because the
strings corresponding to these notes, now freed
from their dampers, are thrown into sympathetic Fig. 11.
vibration.
As you know, this sympathetic vibration of the like-pitched strings with the
overtones is really not to be conceived as sympathy, but rather as lifeless
mechanical necessity. We must not think of this sympathetic vibration as an
ingenious journalist pictured it, who tells a gruesome story of Beethoven's F
minor sonata, Op. 2, that I cannot withhold from you. "At the last London
Industrial Exhibition nineteen virtuosos played the F minor sonata on the
same piano. When the twentieth stepped up to the instrument to play by
way of variation the same production, to the terror of all present the piano
began to render the sonata of its own accord. The Archbishop of
Canterbury, who happened to be present, was set to work and forthwith
expelled the F minor devil."
Although, now, the overtones or harmonics which we have discussed are
heard only upon a special effort of the attention, nevertheless they play a
highly important part in the formation of musical timbre, as also in the
production of the consonance and dissonance of sounds. This may strike
you as singular. How can a thing which is heard only under exceptional
circumstances be of importance generally for audition?
But consider some familiar incidents of your every-day life. Think of how
many things you see which you do not notice, which never strike your
attention until they are missing. A friend calls upon you; you cannot
understand why he looks so changed. Not until you make a close
examination do you discover that his hair has been cut. It is not difficult to

tell the publisher of a work from its letter-press, and yet no one can state
precisely the points by which this style of type is so strikingly different
from that style. I have often recognised a book which I was in search of
from a simple piece of unprinted white paper that peeped out from
underneath the heap of books covering it, and yet I had never carefully
examined the paper, nor could I have stated its difference from other papers.
What we must remember, therefore, is that every sound that is musically
available yields, besides its fundamental note, its octave, its twelfth, its
double octave, etc., as overtones or harmonics, and that these are important
for the agreeable combination of several musical sounds.
2) One other fact still remains to be dealt with. Look at this tuning-fork. It
yields, when struck, a perfectly smooth tone. But if you strike in company
with it a second fork which is of slightly different pitch, and which alone
also gives a perfectly smooth tone, you will hear, if you set both forks on
the table, or hold both before your ear, a uniform tone no longer, but a
number of shocks of tones. The rapidity of the shocks increases with the
difference of the pitch of the forks. These shocks, which become very
disagreeable for the ear when they amount to thirty-three in a second, are
called "beats."
Always, when one of two like musical sounds is thrown out of unison with
the other, beats arise. Their number increases with the divergence from
unison, and simultaneously they grow more unpleasant. Their roughness
reaches its maximum at about thirty-three beats in a second. On a still
further departure from unison, and a consequent increase of the number of
beats, the unpleasant effect is diminished, so that tones which are widely
apart in pitch no longer produce offensive beats.
To give yourselves a clear idea of the production of beats, take two
metronomes and set them almost alike. You can, for that matter, set the two
exactly alike. You need not fear that they will strike alike. The metronomes
usually for sale in the shops are poor enough to yield, when set alike,
appreciably unequal strokes. Set, now, these two metronomes, which strike
at unequal intervals, in motion; you will readily see that their strokes
alternately coincide and conflict with each other. The alternation is quicker
the greater the difference of time of the two metronomes.

If metronomes are not to be had, the experiment may be performed with
two watches.
Beats arise in the same way. The rhythmical shocks of two sounding bodies,
of unequal pitch, sometimes coincide, sometimes interfere, whereby they
alternately augment and enfeeble each other's effects. Hence the shock-like,
unpleasant swelling of the tone.
Now that we have made ourselves acquainted with overtones and beats, we
may proceed to the answer of our main question, Why do certain relations
of pitch produce pleasant sounds, consonances, others unpleasant sounds,
dissonances? It will be readily seen that all the unpleasant effects of
simultaneous sound-combinations are the result of beats produced by those
combinations. Beats are the only sin, the sole evil of music. Consonance is
the coalescence of sounds without appreciable beats.
To make this perfectly clear to you I have constructed the model which you
see in Fig. 12. It represents a claviatur. At its top a movable strip of wood
aa with the marks 1, 2 ... 6 is placed. By setting this strip in any position,
for example, in that where the mark 1 is over the note c of the claviatur, the
marks 2, 3 ... 6, as you see, stand over the overtones of c. The same happens
when the strip is placed in any other position. A second, exactly similar
strip, bb, possesses the same properties. Thus, together, the two strips, in
any two positions, point out by their marks all the tones brought into play
upon the simultaneous sounding of the notes indicated by the marks 1.
The two strips, placed over the same fundamental note, show that also all
the overtones of those notes coincide. The first note is simply intensified by
the other. The single overtones of a sound lie too far apart to permit
appreciable beats. The second sound supplies nothing new, consequently,
also, no new beats. Unison is the most perfect consonance.
Moving one of the two strips along the other is equivalent to a departure
from unison. All the overtones of the one sound now fall alongside those of
the other; beats are at once produced; the combination of the tones becomes
unpleasant: we obtain a dissonance. If we move the strip further and further
along, we shall find that as a general rule the overtones always fall
alongside each other, that is, always produce beats and dissonances. Only in
a few quite definite positions do the overtones partially coincide. Such

positions, therefore, signify higher degrees of
euphony—they point out the consonant intervals.
These consonant intervals can be readily found
experimentally by cutting Fig. 12 out of paper and
moving bb lengthwise along aa. The most perfect
consonances are the octave and the twelfth, since
in these two cases the overtones of the one sound
coincide absolutely with those of the other. In the
octave, for example, 1b falls on 2a, 2b on 4a, 3b
on 6a. Consonances, therefore, are simultaneous
sound-combinations not accompanied by
disagreeable beats. This, by the way, is, expressed
in English, what Euclid said in Greek.
Only such sounds are consonant as possess in
common some portion of their partial tones.
Plainly we must recognise between such sounds,
also when struck one after another, a certain
affinity. For the second sound, by virtue of the
common overtones, will produce partly the same
sensation as the first. The octave is the most
striking exemplification of this. When we reach
the octave in the ascent of the scale we actually
fancy we hear the fundamental tone repeated. The
foundations of harmony, therefore, are the
foundations of melody.
Consonance is the coalescence of sounds without
appreciable beats! This principle is competent to
introduce wonderful order and logic into the
doctrines of the fundamental bass. The
compendiums of the theory of harmony which
(Heaven be witness!) have stood hitherto little
behind the cook-books in subtlety of logic, are
rendered extraordinarily clear and simple. And
what is more, all that the great masters, such as
Palestrina, Mozart, Beethoven, unconsciously got
Fig. 12.
right, and of which heretofore no text-book could

render just account, receives from the preceding principle its perfect
verification.
But the beauty of the theory is, that it bears upon its face the stamp of truth.
It is no phantom of the brain. Every musician can hear for himself the beats
which the overtones of his musical sounds produce. Every musician can
satisfy himself that for any given case the number and the harshness of the
beats can be calculated beforehand, and that they occur in exactly the
measure that theory determines.
This is the answer which Helmholtz gave to the question of Pythagoras, so
far as it can be explained with the means now at my command. A long
period of time lies between the raising and the solving of this question.
More than once were eminent inquirers nearer to the answer than they
dreamed of.
The inquirer seeks the truth. I do not know if the truth seeks the inquirer.
But were that so, then the history of science would vividly remind us of that
classical rendezvous, so often immortalised by painters and poets. A high
garden wall. At the right a youth, at the left a maiden. The youth sighs, the
maiden sighs! Both wait. Neither dreams how near the other is.
I like this simile. Truth suffers herself to be courted, but she has evidently
no desire to be won. She flirts at times disgracefully. Above all, she is
determined to be merited, and has naught but contempt for the man who
will win her too quickly. And if, forsooth, one breaks his head in his efforts
of conquest, what matter is it, another will come, and truth is always young.
At times, indeed, it really seems as if she were well disposed towards her
admirer, but that admitted—never! Only when Truth is in exceptionally
good spirits does she bestow upon her wooer a glance of encouragement.
For, thinks Truth, if I do not do something, in the end the fellow will not
seek me at all.
This one fragment of truth, then, we have, and it shall never escape us. But
when I reflect what it has cost in labor and in the lives of thinking men, how
it painfully groped its way through centuries, a half-matured thought, before
it became complete; when I reflect that it is the toil of more than two
thousand years that speaks out of this unobtrusive model of mine, then,
without dissimulation, I almost repent me of the jest I have made.

And think of how much we still lack! When, several thousand years hence,
boots, top-hats, hoops, pianos, and bass-viols are dug out of the earth, out of
the newest alluvium as fossils of the nineteenth century; when the scientists
of that time shall pursue their studies both upon these wonderful structures
and upon our modern Broadways, as we to-day make studies of the
implements of the stone age and of the prehistoric lake-dwellings—then,
too, perhaps, people will be unable to comprehend how we could come so
near to many great truths without grasping them. And thus it is for all time
the unsolved dissonance, for all time the troublesome seventh, that
everywhere resounds in our ears; we feel, perhaps, that it will find its
solution, but we shall never live to see the day of the pure triple accord, nor
shall our remotest descendants.
Ladies, if it is the sweet purpose of your life to sow confusion, it is the
purpose of mine to be clear; and so I must confess to you a slight
transgression that I have been guilty of. On one point I have told you an
untruth. But you will pardon me this falsehood, if in full repentance I make
it good. The model represented in Fig. 12 does not tell the whole truth, for it
is based upon the so-called "even temperament" system of tuning. The
overtones, however, of musical sounds are not tempered, but purely tuned.
By means of this slight inexactness the model is made considerably simpler.
In this form it is fully adequate for ordinary purposes, and no one who
makes use of it in his studies need be in fear of appreciable error.
If you should demand of me, however, the full truth, I could give you that
only by the help of a mathematical formula. I should have to take the chalk
into my hands and—think of it!—reckon in your presence. This you might
take amiss. Nor shall it happen. I have resolved to do no more reckoning for
to-day. I shall reckon now only upon your forbearance, and this you will
surely not gainsay me when you reflect that I have made only a limited use
of my privilege to weary you. I could have taken up much more of your
time, and may, therefore, justly close with Lessing's epigram:

"If thou hast found in all these pages naught that's worth the
thanks,
At least have gratitude for what I've spared thee."

THE VELOCITY OF LIGHT.
When a criminal judge has a right crafty knave before him, one well versed
in the arts of prevarication, his main object is to wring a confession from
the culprit by a few skilful questions. In almost a similar position the
natural philosopher seems to be placed with respect to nature. True, his
functions here are more those of the spy than the judge; but his object
remains pretty much the same. Her hidden motives and laws of action is
what nature must be made to confess. Whether a confession will be
extracted depends upon the shrewdness of the inquirer. Not without reason,
therefore, did Lord Bacon call the experimental method a questioning of
nature. The art consists in so putting our questions that they may not remain
unanswered without a breach of etiquette.
Look, too, at the countless tools, engines, and instruments of torture with
which man conducts his inquisitions of nature, and which mock the poet's
words:

"Mysterious even in open day,
Nature retains her veil, despite our clamors;
That which she doth not willingly display
Cannot be wrenched from her with levers, screws, and
hammers."

Look at these instruments and you will see that the comparison with torture
also is admissible.[11]
This view of nature, as of something designedly concealed from man, that
can be unveiled only by force or dishonesty, chimed in better with the
conceptions of the ancients than with modern notions. A Grecian
philosopher once said, in offering his opinion of the natural science of his
time, that it could only be displeasing to the gods to see men endeavoring to
spy out what the gods were not minded to reveal to them.[12] Of course all
the contemporaries of the speaker were not of his opinion.

Traces of this view may still be found to-day, but upon the whole we are
now not so narrow-minded. We believe no longer that nature designedly
hides herself. We know now from the history of science that our questions
are sometimes meaningless, and that, therefore, no answer can be
forthcoming. Soon we shall see how man, with all his thoughts and quests,
is only a fragment of nature's life.
Picture, then, as your fancy dictates, the tools of the physicist as instruments
of torture or as engines of endearment, at all events a chapter from the
history of those implements will be of interest to you, and it will not be
unpleasant to learn what were the peculiar difficulties that led to the
invention of such strange apparatus.
Galileo (born at Pisa in 1564, died at Arcetri in 1642) was the first who
asked what was the velocity of light, that is, what time it would take for a
light struck at one place to become visible at another, a certain distance
away.[13]
The method which Galileo devised was as simple as it was natural. Two
practised observers, with muffled lanterns, were to take up positions in a
dark night at a considerable distance from each other, one at A and one at B.
At a moment previously fixed upon, A was instructed to unmask his lantern;
while as soon as B saw the light of A's lantern he was to unmask his. Now it
is clear that the time which A counted from the uncovering of his lantern
until he caught sight of the light of B's would be the time which it would
take light to travel from A to B and from B back to A.
The experiment was not executed,
nor could it, in the nature of the
case, have been a success. As we
now know, light travels too rapidly
to be thus noted. The time elapsing
Fig. 13. between the arrival of the light at B
and its perception by the observer,
with that between the decision to uncover and the uncovering of the lantern,
is, as we now know, incomparably greater than the time which it takes light
to travel the greatest earthly distances. The great velocity of light will be
made apparent, if we reflect that a flash of lightning in the night illuminates
instantaneously a very extensive region, whilst the single reflected claps of

thunder arrive at the observer's ear very gradually and in appreciable
succession.
During his life, then, the efforts of Galileo to determine the velocity of light
remained uncrowned with success. But the subsequent history of the
measurement of the velocity of light is intimately associated with his name,
for with the telescope which he constructed he discovered the four satellites
of Jupiter, and these furnished the next occasion for the determination of the
velocity of light.
The terrestrial spaces were too small for Galileo's experiment. The
measurement was first executed when the spaces of the planetary system
were employed. Olaf Römer, (born at Aarhuus in 1644, died at Copenhagen
in 1710) accomplished the feat (1675-1676), while watching with Cassini at
the observatory of Paris the revolutions of Jupiter's moons.

Fig. 14.
Let AB (Fig. 14) be Jupiter's orbit. Let S stand for the sun, E for the earth, J
for Jupiter, and T for Jupiter's first satellite. When the earth is at E1 we see
the satellite enter regularly into Jupiter's shadow, and by watching the time
between two successive eclipses, can calculate its time of revolution. The
time which Römer noted was forty-two hours, twenty-eight minutes, and

thirty-five seconds. Now, as the earth passes along in its orbit towards E2,
the revolutions of the satellite grow apparently longer and longer: the
eclipses take place later and later. The greatest retardation of the eclipse,
which occurs when the earth is at E2, amounts to sixteen minutes and
twenty-six seconds. As the earth passes back again to E1, the revolutions
grow apparently shorter, and they occur in exactly the time that they first
did when the earth arrives at E1. It is to be remarked that Jupiter changes
only very slightly its position during one revolution of the earth. Römer
guessed at once that these periodical changes of the time of revolution of
Jupiter's satellite were not actual, but apparent changes, which were in some
way connected with the velocity of light.
Let us make this matter clear to ourselves by a simile. We receive regularly
by the post, news of the political status at our capital. However far away we
may be from the capital, we hear the news of every event, later it is true, but
of all equally late. The events reach us in the same succession of time as
that in which they took place. But if we are travelling away from the
capital, every successive post will have a greater distance to pass over, and
the events will reach us more slowly than they took place. The reverse will
be the case if we are approaching the capital.
At rest, we hear a piece of music played in the same tempo at all distances.
But the tempo will be seemingly accelerated if we are carried rapidly
towards the band, or retarded if we are carried rapidly away from it.[14]
Picture to yourself a cross, say the sails of a wind-mill
(Fig. 15), in uniform rotation about its centre. Clearly,
the rotation of the cross will appear to you more
slowly executed if you are carried very rapidly away
from it. For the post which in this case conveys to you
the light and brings to you the news of the successive
positions of the cross will have to travel in each
Fig. 15. successive instant over a longer path.
Now this must also be the case with the rotation (the
revolution) of the satellite of Jupiter. The greatest retardation of the eclipse
(16-1/2 minutes), due to the passage of the earth from E1 to E2, or to its
removal from Jupiter by a distance equal to the diameter of the orbit of the

earth, plainly corresponds to the time which it takes light to traverse a
distance equal to the diameter of the earth's orbit. The velocity of light, that
is, the distance described by light in a second, as determined by this
calculation, is 311,000 kilometres,[15] or 193,000 miles. A subsequent
correction of the diameter of the earth's orbit, gives, by the same method,
the velocity of light as approximately 186,000 miles a second.
The method is exactly that of Galileo; only better conditions are selected.
Instead of a short terrestrial distance we have the diameter of the earth's
orbit, three hundred and seven million kilometres; in place of the uncovered
and covered lanterns we have the satellite of Jupiter, which alternately
appears and disappears. Galileo, therefore, although he could not carry out
himself the proposed measurement, found the lantern by which it was
ultimately executed.
Physicists did not long remain satisfied with this beautiful discovery. They
sought after easier methods of measuring the velocity of light, such as
might be performed on the earth. This was possible after the difficulties of
the problem were clearly exposed. A measurement of the kind referred to
was executed in 1849 by Fizeau (born at Paris in 1819).
I shall endeavor to make the principle of Fizeau's apparatus clear to you.
Let s (Fig. 16) be a disk free to rotate about its centre, and perforated at its
rim with a series of holes. Let l be a luminous point casting its light on an
unsilvered glass, a, inclined at an angle of forty-five degrees to the axis of
the disk. The ray of light, reflected at this point, passes through one of the
holes of the disk and falls at right angles upon a mirror b, erected at a point
about five miles distant. From the mirror b the light is again reflected,
passes once more through the hole in s, and, penetrating the glass plate,
finally strikes the eye, o, of the observer. The eye, o, thus, sees the image of
the luminous point l through the glass plate and the hole of the disk in the
mirror b.

Fig. 16.
If, now, the disk be set in rotation, the unpierced spaces between the
apertures will alternately take the place of the apertures, and the eye o will
now see the image of the luminous point in b only at interrupted intervals.
On increasing the rapidity of the rotation, however, the interruptions for the
eye become again unnoticeable, and the eye sees the mirror b uniformly
illuminated.
But all this holds true only for relatively small velocities of the disk, when
the light sent through an aperture in s to b on its return strikes the aperture
at almost the same place and passes through it a second time. Conceive,
now, the speed of the disk to be so increased that the light on its return finds
before it an unpierced space instead of an aperture, it will then no longer be
able to reach the eye. We then see the mirror b only when no light is emitted
from it, but only when light is sent to it; it is covered when light comes
from it. In this case, accordingly, the mirror will always appear dark.
If the velocity of rotation at this point were still further increased, the light
sent through one aperture could not, of course, on its return pass through the
same aperture but might strike the next and reach the eye by that. Hence, by
constantly increasing the velocity of the rotation, the mirror b may be made
to appear alternately bright and dark. Plainly, now, if we know the number
of apertures of the disk, the number of rotations per second, and the
distance sb, we can calculate the velocity of light. The result agrees with
that obtained by Römer.
The experiment is not quite as simple as my exposition might lead you to
believe. Care must be taken that the light shall travel back and forth over

the miles of distance sb and bs undispersed. This difficulty is obviated by
means of telescopes.
If we examine Fizeau's apparatus closely, we shall recognise in it an old
acquaintance: the arrangement of Galileo's experiment. The luminous point
l is the lantern A, while the rotation of the perforated disk performs
mechanically the uncovering and covering of the lantern. Instead of the
unskilful observer B we have the mirror b, which is unfailingly illuminated
the instant the light arrives from s. The disk s, by alternately transmitting
and intercepting the reflected light, assists the observer o. Galileo's
experiment is here executed, so to speak, countless times in a second, yet
the total result admits of actual observation. If I might be pardoned the use
of a phrase of Darwin's in this field, I should say that Fizeau's apparatus
was the descendant of Galileo's lantern.
A still more refined and delicate method for the measurement of the
velocity of light was employed by Foucault, but a description of it here
would lead us too far from our subject.
The measurement of the velocity of sound is easily executed by the method
of Galileo. It was unnecessary, therefore, for physicists to rack their brains
further about the matter; but the idea which with light grew out of necessity
was applied also in this field. Koenig of Paris constructs an apparatus for
the measurement of the velocity of sound which is closely allied to the
method of Fizeau.
The apparatus is very simple. It consists of two electrical clock-works
which strike simultaneously, with perfect precision, tenths of seconds. If we
place the two clock-works directly side by side, we hear their strokes
simultaneously, wherever we stand. But if we take our stand by the side of
one of the works and place the other at some distance from us, in general a
coincidence of the strokes will now not be heard. The companion strokes of
the remote clock-work arrive, as sound, later. The first stroke of the remote
work is heard, for example, immediately after the first of the adjacent work,
and so on. But by increasing the distance we may produce again a
coincidence of the strokes. For example, the first stroke of the remote work
coincides with the second of the near work, the second of the remote work
with the third of the near work, and so on. If, now, the works strike tenths of
seconds and the distance between them is increased until the first

coincidence is noted, plainly that distance is travelled over by the sound in a
tenth of a second.
We meet frequently the phenomenon here presented, that a thought which
centuries of slow and painful endeavor are necessary to produce, when once
developed, fairly thrives. It spreads and runs everywhere, even entering
minds in which it could never have arisen. It simply cannot be eradicated.
The determination of the velocity of light is not the only case in which the
direct perception of the senses is too slow and clumsy for use. The usual
method of studying events too fleet for direct observation consists in putting
into reciprocal action with them other events already known, the velocities
of all of which are capable of comparison. The result is usually
unmistakable, and susceptible of direct inference respecting the character of
the event which is unknown. The velocity of electricity cannot be
determined by direct observation. But it was ascertained by Wheatstone,
simply by the expedient of watching an electric spark in a mirror rotating
with tremendous known velocity.
If we wave a staff
irregularly hither and
thither, simple observation
cannot determine how
quickly it moves at each
point of its course. But let
us look at the staff through
holes in the rim of a
rapidly rotating disk (Fig.
17). We shall then see the Fig. 18.
moving staff only in
Fig. 17. certain positions, namely,
when a hole passes in front of the eye. The single
pictures of the staff remain for a time impressed upon
the eye; we think we see several staffs, having some such disposition as that
represented in Fig. 18. If, now, the holes of the disk are equally far apart,
and the disk is rotated with uniform velocity, we see clearly that the staff
has moved slowly from a to b, more quickly from b to c, still more quickly
from c to d, and with its greatest velocity from d to e.

A jet of water flowing from an orifice in the bottom of a vessel has the
appearance of perfect quiet and uniformity, but if we illuminate it for a
second, in a dark room, by means of an electric flash we shall see that the
jet is composed of separate drops. By their quick descent the images of the
drops are obliterated and the jet appears uniform. Let us look at the jet
through the rotating disk. The disk is supposed to be rotated so rapidly that
while the second aperture passes into the place of the first, drop 1 falls into
the place of 2, 2 into the place of 3, and so on. We see drops then always in
the same places. The jet appears to be at rest. If we turn the disk a trifle
more slowly, then while the second aperture passes into the place of the
first, drop 1 will have fallen somewhat lower than 2, 2 somewhat lower than
3, etc. Through every successive aperture we shall see drops in successively
lower positions. The jet will appear to be flowing slowly downwards.
Now let us turn the disk more rapidly. Then while the
second aperture is passing into the place of the first,
drop 1 will not quite have reached the place of 2, but
will be found slightly above 2, 2 slightly above 3, etc.
Through the successive apertures we shall see the
drops at successively higher places. It will now look as
if the jet were flowing upwards, as if the drops were
rising from the lower vessel into the higher.
You see, physics grows gradually more and more
terrible. The physicist will soon have it in his power to Fig. 19.
play the part of the famous lobster chained to the
bottom of the Lake of Mohrin, whose direful mission, if ever liberated, the
poet Kopisch humorously describes as that of a reversal of all the events of
the world; the rafters of houses become trees again, cows calves, honey
flowers, chickens eggs, and the poet's own poem flows back into his
inkstand.

You will now allow me the privilege of a few general remarks. You have
seen that the same principle often lies at the basis of large classes of
apparatus designed for different purposes. Frequently it is some very
unobtrusive idea which is productive of so much fruit and of such extensive

transformations in physical technics. It is not otherwise here than in
practical life.
The wheel of a waggon appears to us a very simple and insignificant
creation. But its inventor was certainly a man of genius. The round trunk of
a tree perhaps first accidentally led to the observation of the ease with
which a load can be moved on a roller. Now, the step from a simple
supporting roller to a fixed roller, or wheel, appears a very easy one. At
least it appears very easy to us who are accustomed from childhood up to
the action of the wheel. But if we put ourselves vividly into the position of a
man who never saw a wheel, but had to invent one, we shall begin to have
some idea of its difficulties. Indeed, it is even doubtful whether a single
man could have accomplished this feat, whether perhaps centuries were not
necessary to form the first wheel from the primitive roller.[16]
History does not name the progressive minds who constructed the first
wheel; their time lies far back of the historic period. No scientific academy
crowned their efforts, no society of engineers elected them honorary
members. They still live only in the stupendous results which they called
forth. Take from us the wheel, and little will remain of the arts and
industries of modern life. All disappears. From the spinning-wheel to the
spinning-mill, from the turning-lathe to the rolling-mill, from the
wheelbarrow to the railway train, all vanishes.
In science the wheel is equally important. Whirling machines, as the
simplest means of obtaining quick motions with inconsiderable changes of
place, play a part in all branches of physics. You know Wheatstone's
rotating mirror, Fizeau's wheel, Plateau's perforated rotating disks, etc.
Almost the same principle lies at the basis of all these apparatus. They
differ from one another no more than the pen-knife differs, in the purposes
it serves, from the knife of the anatomist or the knife of the vine-dresser.
Almost the same might be said of the screw.
It will now perhaps be clear to you that new thoughts do not spring up
suddenly. Thoughts need their time to ripen, grow, and develop in, like
every natural product; for man, with his thoughts, is also a part of nature.
Slowly, gradually, and laboriously one thought is transformed into a
different thought, as in all likelihood one animal species is gradually
transformed into new species. Many ideas arise simultaneously. They fight

the battle for existence not otherwise than do the Ichthyosaurus, the
Brahman, and the horse.
A few remain to spread rapidly over all fields of knowledge, to be
redeveloped, to be again split up, to begin again the struggle from the start.
As many animal species long since conquered, the relicts of ages past, still
live in remote regions where their enemies cannot reach them, so also we
find conquered ideas still living on in the minds of many men. Whoever
will look carefully into his own soul will acknowledge that thoughts battle
as obstinately for existence as animals. Who will gainsay that many
vanquished modes of thought still haunt obscure crannies of his brain, too
faint-hearted to step out into the clear light of reason? What inquirer does
not know that the hardest battle, in the transformation of his ideas, is fought
with himself.
Similar phenomena meet the natural inquirer in all paths and in the most
trifling matters. The true inquirer seeks the truth everywhere, in his country-
walks and on the streets of the great city. If he is not too learned, he will
observe that certain things, like ladies' hats, are constantly subject to
change. I have not pursued special studies on this subject, but as long as I
can remember, one form has always gradually changed into another. First,
they wore hats with long projecting rims, within which, scarcely accessible
with a telescope, lay concealed the face of the beautiful wearer. The rim
grew smaller and smaller; the bonnet shrank to the irony of a hat. Now a
tremendous superstructure is beginning to grow up in its place, and the gods
only know what its limits will be. It is not otherwise with ladies' hats than
with butterflies, whose multiplicity of form often simply comes from a
slight excrescence on the wing of one species developing in a cognate
species to a tremendous fold. Nature, too, has its fashions, but they last
thousands of years. I could elucidate this idea by many additional examples;
for instance, by the history of the evolution of the coat, if I were not fearful
that my gossip might prove irksome to you.

We have now wandered through an odd corner of the history of science.
What have we learned? The solution of a small, I might almost say
insignificant, problem—the measurement of the velocity of light. And more

than two centuries have worked at its solution! Three of the most eminent
natural philosophers, Galileo, an Italian, Römer, a Dane, and Fizeau, a
Frenchman, have fairly shared its labors. And so it is with countless other
questions. When we contemplate thus the many blossoms of thought that
must wither and fall before one shall bloom, then shall we first truly
appreciate Christ's weighty but little consolatory words: "Many be called
but few are chosen."
Such is the testimony of every page of history. But is history right? Are
really only those chosen whom she names? Have those lived and battled in
vain, who have won no prize?
I doubt it. And so will every one who has felt the pangs of sleepless nights
spent in thought, at first fruitless, but in the end successful. No thought in
such struggles was thought in vain; each one, even the most insignificant,
nay, even the erroneous thought, that which apparently was the least
productive, served to prepare the way for those that afterwards bore fruit.
And as in the thought of the individual naught is in vain, so, also, it is in
that of humanity.
Galileo wished to measure the velocity of light. He had to close his eyes
before his wish was realised. But he at least found the lantern by which his
successor could accomplish the task.
And so I may maintain that we all, so far as inclination goes, are working at
the civilisation of the future. If only we all strive for the right, then are we
all called and all chosen!

WHY HAS MAN TWO EYES?
Why has man two eyes? That the pretty symmetry of his face may not be
disturbed, the artist answers. That his second eye may furnish a substitute
for his first if that be lost, says the far-sighted economist. That we may
weep with two eyes at the sins of the world, replies the religious enthusiast.
Odd opinions! Yet if you should approach a modern scientist with this
question you might consider yourself fortunate if you escaped with less than
a rebuff. "Pardon me, madam, or my dear sir," he would say, with stern
expression, "man fulfils no purpose in the possession of his eyes; nature is
not a person, and consequently not so vulgar as to pursue purposes of any
kind."
Still an unsatisfactory answer! I once knew a professor who would shut
with horror the mouths of his pupils if they put to him such an unscientific
question.
But ask a more tolerant person, ask me. I, I candidly confess, do not know
exactly why man has two eyes, but the reason partly is, I think, that I may
see you here before me to-night and talk with you upon this delightful
subject.
Again you smile incredulously. Now this is one of those questions that a
hundred wise men together could not answer. You have heard, so far, only
five of these wise men. You will certainly want to be spared the opinions of
the other ninety-five. To the first you will reply that we should look just as
pretty if we were born with only one eye, like the Cyclops; to the second we
should be much better off, according to his principle, if we had four or eight
eyes, and that in this respect we are vastly inferior to spiders; to the third,
that you are not just in the mood to weep; to the fourth, that the unqualified
interdiction of the question excites rather than satisfies your curiosity; while
of me you will dispose by saying that my pleasure is not as intense as I
think, and certainly not great enough to justify the existence of a double eye
in man since the fall of Adam.

But since you are not satisfied with my brief and obvious answer, you have
only yourselves to blame for the consequences. You must now listen to a
longer and more learned explanation, such as it is in my power to give.
As the church of science, however, debars the question "Why?" let us put
the matter in a purely orthodox way: Man has two eyes, what more can he
see with two than with one?
I will invite you to take a walk with me? We see before us a wood. What is
it that makes this real wood contrast so favorably with a painted wood, no
matter how perfect the painting may be? What makes the one so much more
lovely than the other? Is it the vividness of the coloring, the distribution of
the lights and the shadows? I think not. On the contrary, it seems to me that
in this respect painting can accomplish very much.
The cunning hand of the painter can conjure up with a few strokes of his
brush forms of wonderful plasticity. By the help of other means even more
can be attained. Photographs of reliefs are so plastic that we often imagine
we can actually lay hold of the elevations and depressions.
But one thing the painter never can give with the vividness that nature does
—the difference of near and far. In the real woods you see plainly that you
can lay hold of some trees, but that others are inaccessibly far. The picture
of the painter is rigid. The picture of the real woods changes on the slightest
movement. Now this branch is hidden behind that; now that behind this.
The trees are alternately visible and invisible.
Let us look at this matter a little more closely. For convenience sake we
shall remain upon the highway, I, II. (Fig. 20.) To the right and the left lies
the forest. Standing at I, we see, let us say, three trees (1, 2, 3) in a line, so
that the two remote ones are covered by the nearest. Moving further along,
this changes. At II we shall not have to look round so far to see the remotest
tree 3 as to see the nearer tree 2, nor so far to see this as to see 1. Hence, as
we move onward, objects that are near to us seem to lag behind as
compared with objects that are remote from us, the lagging increasing with
the proximity of the objects. Very remote objects, towards which we must
always look in the same direction as we proceed, appear to travel along
with us.

If we should see, therefore, jutting
above the brow of yonder hill the
tops of two trees whose distance
from us we were in doubt about, we
should have in our hands a very
easy means of deciding the
question. We should take a few
steps forward, say to the right, and
the tree-top which receded most to
the left would be the one nearer to
us. In truth, from the amount of the
recession a geometer could actually
determine the distance of the trees
from us without ever going near
them. It is simply the scientific
development of this perception that
enables us to measure the distances
of the stars.
Hence, from change of view in
forward motion the distances of
objects in our field of vision can be
measured.
Fig. 20. Rigorously, however, even forward
motion is not necessary. For every
observer is composed really of two observers. Man has two eyes. The right
eye is a short step ahead of the left eye in the right-hand direction. Hence,
the two eyes receive different pictures of the same woods. The right eye will
see the near trees displaced to the left, and the left eye will see them
displaced to the right, the displacement being greater, the greater the
proximity. This difference is sufficient for forming ideas of distance.
We may now readily convince ourselves of the following facts:
1. With one eye, the other being shut, you have a very uncertain judgment
of distances. You will find it, for example, no easy task, with one eye shut,
to thrust a stick through a ring hung up before you; you will miss the ring in
almost every instance.

2. You see the same object differently with the right eye from what you do
with the left.
Place a lamp-shade on the table in front of you with its broad opening
turned downwards, and look at it from above. (Fig. 21.) You will see with
your right eye the image 2, with your left eye the image 1. Again, place the
shade with its wide opening turned upwards; you will receive with your
right eye the image 4, with your left eye the image 3. Euclid mentions
phenomena of this character.
3. Finally, you know that it is easy to judge of distances with both eyes.
Accordingly your judgment must spring in some way from a co-operation
of the two eyes. In the preceding example the openings in the different
images received by the two eyes seem displaced with respect to one
another, and this displacement is sufficient for the inference that the one
opening is nearer than the other.

Fig. 21.

I have no doubt that you, ladies, have frequently received delicate
compliments upon your eyes, but I feel sure that no one has ever told you,
and I know not whether it will flatter you, that you have in your eyes, be
they blue or black, little geometricians. You say you know nothing of them?
Well, for that matter, neither do I. But the facts are as I tell you.
You understand little of geometry? I shall accept that confession. Yet with
the help of your two eyes you judge of distances? Surely that is a
geometrical problem. And what is more, you know the solution of this
problem: for you estimate distances correctly. If, then, you do not solve the
problem, the little geometricians in your eyes must do it clandestinely and
whisper the solution to you. I doubt not they are fleet little fellows.
What amazes me most here is, that you know nothing about these little
geometricians. But perhaps they also know nothing about you. Perhaps they
are models of punctuality, routine clerks who bother about nothing but their
fixed work. In that case we may be able to deceive the gentlemen.
If we present to our right eye an image which looks exactly like the lamp-
shade for the right eye, and to our left eye an image which looks exactly
like a lamp-shade for the left eye, we shall imagine that we see the whole
lamp-shade bodily before us.
You know the experiment. If you are practised in squinting, you can
perform it directly with the figure, looking with your right eye at the right
image, and with your left eye at the left image. In this way the experiment
was first performed by Elliott. Improved and perfected, its form is
Wheatstone's stereoscope, made so popular and useful by Brewster.
By taking two photographs of the same object from two different points,
corresponding to the two eyes, a very clear three-dimensional picture of
distant places or buildings can be produced by the stereoscope.
But the stereoscope accomplishes still more than this. It can visualise things
for us which we never see with equal clearness in real objects. You know
that if you move much while your photograph is being taken, your picture
will come out like that of a Hindu deity, with several heads or several arms,
which, at the spaces where they overlap, show forth with equal distinctness,
so that we seem to see the one picture through the other. If a person moves
quickly away from the camera before the impression is completed, the

objects behind him will also be imprinted upon the photograph; the person
will look transparent. Photographic ghosts are made in this way.
Some very useful applications may be made of this discovery. For example,
if we photograph a machine stereoscopically, successively removing during
the operation the single parts (where of course the impression suffers
interruptions), we obtain a transparent view, endowed with all the marks of
spatial solidity, in which is distinctly visualised the interaction of parts
normally concealed. I have employed this method for obtaining transparent
stereoscopic views of anatomical structures.
You see, photography is making stupendous advances, and there is great
danger that in time some malicious artist will photograph his innocent
patrons with solid views of their most secret thoughts and emotions. How
tranquil politics will then be! What rich harvests our detective force will
reap!

By the joint action of the two eyes, therefore, we arrive at our judgments of
distances, as also of the forms of bodies.
Permit me to mention here a few additional facts connected with this
subject, which will assist us in the comprehension of certain phenomena in
the history of civilisation.
You have often heard, and know from personal experience, that remote
objects appear perspectively dwarfed. In fact, it is easy to satisfy yourself
that you can cover the image of a man a few feet away from you simply by
holding up your finger a short distance in front of your eye. Still, as a
general rule, you do not notice this shrinkage of objects. On the contrary,
you imagine you see a man at the end of a large hall, as large as you see
him near by you. For your eye, in its measurement of the distances, makes
remote objects correspondingly larger. The eye, so to speak, is aware of this
perspective contraction and is not deceived by it, although its possessor is
unconscious of the fact. All persons who have attempted to draw from
nature have vividly felt the difficulty which this superior dexterity of the
eye causes the perspective conception. Not until one's judgment of distances

is made uncertain, by their size, or from lack of points of reference, or from
being too quickly changed, is the perspective rendered very prominent.
On sweeping round a curve on a rapidly moving railway train, where a wide
prospect is suddenly opened up, the men upon distant hills appear like dolls.
[17]
You have at the moment, here, no known references for the
measurement of distances. The stones at the entrance of a tunnel grow
visibly larger as we ride towards it; they shrink visibly in size as we ride
from it.
Usually both eyes work together. As certain views are frequently repeated,
and lead always to substantially the same judgments of distances, the eyes
in time must acquire a special skill in geometrical constructions. In the end,
undoubtedly, this skill is so increased that a single eye alone is often
tempted to exercise that office.
Permit me to elucidate this point by an example. Is any sight more familiar
to you than that of a vista down a long street? Who has not looked with
hopeful eyes time and again into a street and measured its depth. I will take
you now into an art-gallery where I will suppose you to see a picture
representing a vista into a street. The artist has not spared his rulers to get
his perspective perfect. The geometrician in your left eye thinks, "Ah ha! I
have computed that case a hundred times or more. I know it by heart. It is a
vista into a street," he continues; "where the houses are lower is the remote
end." The geometrician in the right eye, too much at his ease to question his
possibly peevish comrade in the matter, answers the same. But the sense of
duty of these punctual little fellows is at once rearoused. They set to work at
their calculations and immediately find that all the points of the picture are
equally distant from them, that is, lie all upon a plane surface.
What opinion will you now accept, the first or the second? If you accept the
first you will see distinctly the vista. If you accept the second you will see
nothing but a painted sheet of distorted images.
It seems to you a trifling matter to look at a picture and understand its
perspective. Yet centuries elapsed before humanity came fully to appreciate
this trifle, and even the majority of you first learned it from education.
I can remember very distinctly that at three years of age all perspective
drawings appeared to me as gross caricatures of objects. I could not

understand why artists made tables so broad at one end and so narrow at the
other. Real tables seemed to me just as broad at one end as at the other,
because my eye made and interpreted its calculations without my
intervention. But that the picture of the table on the plane surface was not to
be conceived as a plane painted surface but stood for a table and so was to
be imaged with all the attributes of extension was a joke that I did not
understand. But I have the consolation that whole nations have not
understood it.
Ingenuous people there are who take the mock murders of the stage for real
murders, the dissembled actions of the players for real actions, and who can
scarcely restrain themselves, when the characters of the play are sorely
pressed, from running in deep indignation to their assistance. Others, again,
can never forget that the beautiful landscapes of the stage are painted, that
Richard III. is only the actor, Mr. Booth, whom they have met time and
again at the clubs.
Both points of view are equally mistaken. To look at a drama or a picture
properly one must understand that both are shows, simply denoting
something real. A certain preponderance of the intellectual life over the
sensuous life is requisite for such an achievement, where the intellectual
elements are safe from destruction by the direct sensuous impressions. A
certain liberty in choosing one's point of view is necessary, a sort of humor,
I might say, which is strongly wanting in children and in childlike peoples.
Let us look at a few historical facts. I shall not take you as far back as the
stone age, although we possess sketches from this epoch which show very
original ideas of perspective. But let us begin our sight-seeing in the tombs
and ruined temples of ancient Egypt, where the numberless reliefs and
gorgeous colorings have defied the ravages of thousands of years.
A rich and motley life is here opened to us. We find the Egyptians
represented in all conditions of life. What at once strikes our attention in
these pictures is the delicacy of their technical execution. The contours are
extremely exact and distinct. But on the other hand only a few bright colors
are found, unblended and without trace of transition. Shadows are totally
wanting. The paint is laid on the surfaces in equal thicknesses.
Shocking for the modern eye is the perspective. All the figures are equally
large, with the exception of the king, whose form is unduly exaggerated.

Near and far appear equally large. Perspective contraction is nowhere
employed. A pond with water-fowl is represented flat, as if its surface were
vertical.
Human figures are portrayed as they are never seen, the legs from the side,
the face in profile. The breast lies in its full breadth across the plane of
representation. The heads of cattle appear in profile, while the horns lie in
the plane of the drawing. The principle which the Egyptians followed might
be best expressed by saying that their figures are pressed in the plane of the
drawing as plants are pressed in a herbarium.
The matter is simply explained. If the Egyptians were accustomed to
looking at things ingenuously with both eyes at once, the construction of
perspective pictures in space could not be familiar to them. They saw all
arms, all legs on real men in their natural lengths. The figures pressed into
the planes resembled more closely, of course, in their eyes the originals than
perspective pictures could.
This will be better understood if we reflect that painting was developed
from relief. The minor dissimilarities between the pressed figures and the
originals must gradually have compelled men to the adoption of perspective
drawing. But physiologically the painting of the Egyptians is just as much
justified as the drawings of our children are.
A slight advance beyond the Egyptians is shown by the Assyrians. The
reliefs rescued from the ruined mounds of Nimrod at Mossul are, upon the
whole, similar to the Egyptian reliefs. They were made known to us
principally by Layard.
Painting enters on a new phase among the Chinese. This people have a
marked feeling for perspective and correct shading, yet without being very
logical in the application of their principles. Here, too, it seems, they took
the first step but did not go far. In harmony with this immobility is their
constitution, in which the muzzle and the bamboo-rod play significant
functions. In accord with it, too, is their language, which like the language
of children has not yet developed into a grammar, or, rather, according to
the modern conception, has not yet degenerated into a grammar. It is the
same also with their music which is satisfied with the five-toned scale.

The mural paintings at Herculaneum and Pompeii are distinguished by
grace of representation, as also by a pronounced sense for perspective and
correct illumination, yet they are not at all scrupulous in construction. Here
still we find abbreviations avoided. But to offset this defect, the members of
the body are brought into unnatural positions, in which they appear in their
full lengths. Abridgements are more frequently observed in clothed than in
unclothed figures.
A satisfactory explanation of these phenomena first occurred to me on the
making of a few simple experiments which show how differently one may
see the same object, after some mastery of one's senses has been attained,
simply by the arbitrary movement of the attention.

Fig. 22.
Look at the annexed drawing (Fig. 22). It represents a folded sheet of paper
with either its depressed or its elevated side turned towards you, as you
wish. You can conceive the drawing in either sense, and in either case it will
appear to you differently.
If, now, you have a real folded sheet of paper on the table before you, with
its sharp edges turned towards you, you can, on looking at it with one eye,
see the sheet alternately elevated, as it really is, or depressed. Here,
however, a remarkable phenomenon is presented. When you see the sheet
properly, neither illumination nor form presents anything conspicuous.
When you see it bent back you see it perspectively distorted. Light and
shadow appear much brighter or darker, or as if overlaid thickly with bright
colors. Light and shadow now appear devoid of all cause. They no longer
harmonise with the body's form, and are thus rendered much more
prominent.

In common life we employ the perspective and illumination of objects to
determine their forms and position. Hence we do not notice the lights, the
shadows, and the distortions. They first powerfully enter consciousness
when we employ a different construction from the usual spatial one. In
looking at the planar image of a camera obscura we are amazed at the
plenitude of the light and the profundity of the shadows, both of which we
do not notice in real objects.
In my earliest youth the shadows and lights on pictures appeared to me as
spots void of meaning. When I began to draw I regarded shading as a mere
custom of artists. I once drew the portrait of our pastor, a friend of the
family, and shaded, from no necessity, but simply from having seen
something similar in other pictures, the whole half of his face black. I was
subjected for this to a severe criticism on the part of my mother, and my
deeply offended artist's pride is probably the reason that these facts
remained so strongly impressed upon my memory.
You see, then, that many strange things, not only in the life of individuals,
but also in that of humanity, and in the history of general civilisation, may
be explained from the simple fact that man has two eyes.
Change man's eye and you change his conception of the world. We have
observed the truth of this fact among our nearest kin, the Egyptians, the
Chinese, and the lake-dwellers; how must it be among some of our remoter
relatives,—with monkeys and other animals? Nature must appear totally
different to animals equipped with substantially different eyes from those of
men, as, for example, to insects. But for the present science must forego the
pleasure of portraying this appearance, as we know very little as yet of the
mode of operation of these organs.
It is an enigma even how nature appears to animals closely related to man;
as to birds, who see scarcely anything with two eyes at once, but since their
eyes are placed on opposite sides of their heads, have a separate field of
vision for each.[18]
The soul of man is pent up in the prison-house of his head; it looks at nature
through its two windows, the eyes. It would also fain know how nature
looks through other windows. A desire apparently never to be fulfilled. But
our love for nature is inventive, and here, too, much has been accomplished.

Placing before me an angular mirror, consisting of two plane mirrors
slightly inclined to each other, I see my face twice reflected. In the right-
hand mirror I obtain a view of the right side, and in the left-hand mirror a
view of the left side, of my face. Also I shall see the face of a person
standing in front of me, more to the right with my right eye, more to the left
with my left. But in order to obtain such widely different views of a face as
those shown in the angular mirror, my two eyes would have to be set much
further apart from each other than they actually are.

Squinting with my right eye at the image in
the right-hand mirror, with my left eye at the
image in the left-hand mirror, my vision will
be the vision of a giant having an enormous Fig. 23.
head with his two eyes set far apart. This,
also, is the impression which my own face
makes upon me. I see it now, single and solid. Fixing my gaze, the relief
from second to second is magnified, the eyebrows start forth prominently
from above the eyes, the nose seems to grow a foot in length, my mustache
shoots forth like a fountain from my lip, the teeth seem to retreat
immeasurably. But by far the most horrible aspect of the phenomenon is the
nose.
Interesting in this connexion is the telestereoscope of Helmholtz. In the
telestereoscope we view a landscape by looking with our right eye (Fig. 24)
through the mirror a into the mirror A, and with our left eye through the
mirror b into the mirror B. The mirrors A and B stand far apart. Again we
see with the widely separated eyes of a giant. Everything appears dwarfed
and near us. The distant mountains look like moss-covered stones at our
feet. Between, you see the reduced model of a city, a veritable Liliput. You
are tempted almost to stroke with your hand the soft forest and city, did you
not fear that you might prick your fingers on the sharp, needle-shaped
steeples, or that they might crackle and break off.
Liliput is no fable. We need
only Swift's eyes, the
telestereoscope, to see it.
Picture to yourself the reverse
case. Let us suppose
ourselves so small that we
could take long walks in a
forest of moss, and that our
eyes were correspondingly
Fig. 24. near each other. The moss-
fibres would appear like
trees. On them we should see
strange, unshapely monsters creeping about. Branches of the oak-tree, at

whose base our moss-forest lay, would seem to us dark, immovable,
myriad-branched clouds, painted high on the vault of heaven; just as the
inhabitants of Saturn, forsooth, might see their enormous ring. On the tree-
trunks of our mossy woodland we should find colossal globes several feet
in diameter, brilliantly transparent, swayed by the winds with slow, peculiar
motions. We should approach inquisitively and should find that these
globes, in which here and there animals were gaily sporting, were liquid
globes, in fact that they were water. A short, incautious step, the slightest
contact, and woe betide us, our arm is irresistibly drawn by an invisible
power into the interior of the sphere and held there unrelentingly fast! A
drop of dew has engulfed in its capillary maw a manikin, in revenge for the
thousands of drops that its big human counterparts have quaffed at
breakfast. Thou shouldst have known, thou pygmy natural scientist, that
with thy present puny bulk thou shouldst not joke with capillarity!
My terror at the accident brings me back to my senses. I see I have turned
idyllic. You must pardon me. A patch of greensward, a moss or heather
forest with its tiny inhabitants have incomparably more charms for me than
many a bit of literature with its apotheosis of human character. If I had the
gift of writing novels I should certainly not make John and Mary my
characters. Nor should I transfer my loving pair to the Nile, nor to the age
of the old Egyptian Pharaohs, although perhaps I should choose that time in
preference to the present. For I must candidly confess that I hate the rubbish
of history, interesting though it may be as a mere phenomenon, because we
cannot simply observe it but must also feel it, because it comes to us mostly
with supercilious arrogance, mostly unvanquished. The hero of my novel
would be a cockchafer, venturing forth in his fifth year for the first time
with his newly grown wings into the light, free air. Truly it could do no
harm if man would thus throw off his inherited and acquired narrowness of
mind by making himself acquainted with the world-view of allied creatures.
He could not help gaining incomparably more in this way than the
inhabitant of a small town would in circumnavigating the globe and getting
acquainted with the views of strange peoples.

I have now conducted you, by many paths and by-ways, rapidly over hedge
and ditch, to show you what wide vistas we may reach in every field by the
rigorous pursuit of a single scientific fact. A close examination of the two
eyes of man has conducted us not only into the dim recesses of humanity's
childhood, but has also carried us far beyond the bourne of human life.
It has surely often struck you as strange that the sciences are divided into
two great groups; that the so-called humanistic sciences, belonging to the
so-called "higher education," are placed in almost a hostile attitude to the
natural sciences.
I must confess I do not overmuch believe in this partition of the sciences. I
believe that this view will appear as childlike and ingenuous to a matured
age as the want of perspective in the old paintings of Egypt does to us. Can
it really be that "higher culture" is to be gotten only from a few old pots and
palimpsests, which are at best mere scraps of nature, or that more is to be
learned from them alone than from all the rest of nature? I believe that both
these sciences are simply parts of the same science, which have begun at
different ends. If these two ends still act towards each other as the
Montagues and Capulets, if their retainers still indulge in lively tilts, I
believe that after all they are not in earnest. On the one side there is surely a
Romeo, and on the other a Juliet, who, some day, it is hoped, will unite the
two houses with a less tragic sequel than that of the play.
Philology began with the unqualified reverence and apotheosis of the
Greeks. Now it has begun to draw other languages, other peoples and their
histories, into its sphere; it has, through the mediation of comparative
linguistics, already struck up, though as yet somewhat cautiously, a
friendship with physiology.
Physical science began in the witch's kitchen. It now embraces the organic
and inorganic worlds, and with the physiology of articulation and the theory
of the senses, has even pushed its researches, at times impertinently, into the
province of mental phenomena.
In short, we come to the understanding of much within us solely by
directing our glance without, and vice versa. Every object belongs to both
sciences. You, ladies, are very interesting and difficult problems for the
psychologist, but you are also extremely pretty phenomena of nature.
Church and State are objects of the historian's research, but not less

phenomena of nature, and in part, indeed, very curious phenomena. If the
historical sciences have inaugurated wide extensions of view by presenting
to us the thoughts of new and strange peoples, the physical sciences in a
certain sense do this in a still greater degree. In making man disappear in
the All, in annihilating him, so to speak, they force him to take an
unprejudiced position without himself, and to form his judgments by a
different standard from that of the petty human.
But if you should ask me now why man has two eyes, I should answer:
That he may look at nature justly and accurately; that he may come to
understand that he himself, with all his views, correct and incorrect, with all
his haute politique, is simply an evanescent shred of nature; that, to speak
with Mephistopheles, he is a part of the part, and that it is absolutely
unjustified,

"For man, the microcosmic fool, to see
Himself a whole so frequently."

ON SYMMETRY.
[19]

An ancient philosopher once remarked that people who cudgelled their
brains about the nature of the moon reminded him of men who discussed
the laws and institutions of a distant city of which they had heard no more
than the name. The true philosopher, he said, should turn his glance within,
should study himself and his notions of right and wrong; only thence could
he derive real profit.
This ancient formula for happiness might be restated in the familiar words
of the Psalm:

"Dwell in the land, and verily thou shalt be fed."

To-day, if he could rise from the dead and walk about among us, this
philosopher would marvel much at the different turn which matters have
taken.
The motions of the moon and the other heavenly bodies are accurately
known. Our knowledge of the motions of our own body is by far not so
complete. The mountains and natural divisions of the moon have been
accurately outlined on maps, but physiologists are just beginning to find
their way in the geography of the brain. The chemical constitution of many
fixed stars has already been investigated. The chemical processes of the
animal body are questions of much greater difficulty and complexity. We
have our Mécanique céleste. But a Mécanique sociale or a Mécanique
morale of equal trustworthiness remains to be written.
Our philosopher would indeed admit that we have made great progress. But
we have not followed his advice. The patient has recovered, but he took for
his recovery exactly the opposite of what the doctor prescribed.
Humanity is now returned, much wiser, from its journey in celestial space,
against which it was so solemnly warned. Men, after having become
acquainted with the great and simple facts of the world without, are now

beginning to examine critically the world within. It sounds absurd, but it is
true, that only after we have thought about the moon are we able to take up
ourselves. It was necessary that we should acquire simple and clear ideas in
a less complicated domain, before we entered the more intricate one of
psychology, and with these ideas astronomy principally furnished us.
To attempt any description of that stupendous movement, which, originally
springing out of the physical sciences, went beyond the domain of physics
and is now occupied with the problems of psychology, would be
presumptuous in this place. I shall only attempt here, to illustrate to you by
a few simple examples the methods by which the province of psychology
can be reached from the facts of the physical world—especially the adjacent
province of sense-perception. And I wish it to be remembered that my brief
attempt is not to be taken as a measure of the present state of such scientific
questions.

It is a well-known fact that some objects please us, while others do not.
Generally speaking, anything that is constructed according to fixed and
logically followed rules, is a product of tolerable beauty. We see thus nature
herself, who always acts according to fixed rules, constantly producing such
pretty things. Every day the physicist is confronted in his workshop with the
most beautiful vibration-figures, tone-figures, phenomena of polarisation,
and forms of diffraction.
A rule always presupposes a repetition. Repetitions, therefore, will probably
be found to play some important part in the production of agreeable effects.
Of course, the nature of agreeable effects is not exhausted by this.
Furthermore, the repetition of a physical event becomes the source of
agreeable effects only when it is connected with a repetition of sensations.
An excellent example that repetition of sensations is a source of agreeable
effects is furnished by the copy-book of every schoolboy, which is usually a
treasure-house of such things, and only in need of an Abbé Domenech to
become celebrated. Any figure, no matter how crude or poor, if several
times repeated, with the repetitions placed in line, will produce a tolerable
frieze.

Also the pleasant effect of
symmetry is due to the repetition of
sensations. Let us abandon
ourselves a moment to this thought,
yet not imagine when we have
developed it, that we have fully
exhausted the nature of the
agreeable, much less of the
beautiful.
First, let us get a clear conception Fig. 25.
of what symmetry is. And in
preference to a definition let us take a living picture. You know that the
reflexion of an object in a mirror has a great likeness to the object itself. All
its proportions and outlines are the same. Yet there is a difference between
the object and its reflexion in the mirror, which you will readily observe.
Hold your right hand before a mirror, and you will see in the mirror a left
hand. Your right glove will produce its mate in the glass. For you could
never use the reflexion of your right glove, if it were present to you as a real
thing, for covering your right hand, but only for covering your left.
Similarly, your right ear will give as its reflexion a left ear; and you will at
once perceive that the left half of your body could very easily be substituted
for the reflexion of your right half. Now just as in the place of a missing
right ear a left ear cannot be put, unless the lobule of the ear be turned
upwards, or the opening into the concha backwards, so, despite all
similarity of form, the reflexion of an object can never take the place of the
object itself.[20]
The reason of this difference between the object and its reflexion is simple.
The reflexion appears as far behind the mirror as the object is in front of it.
The parts of the object, accordingly, which are nearest the mirror will also
be nearest the mirror in the reflexion. Consequently, the succession of the
parts in the reflexion will be reversed, as may best be seen in the reflexion
of the face of a watch or of a manuscript.
It will also be readily seen, that if a point of the object be joined with its
reflexion in the image, the line of junction will cut the mirror at right angles

and be bisected by it. This holds true of all corresponding points of object
and image.
If, now, we can divide an object by a plane into two halves so that each half,
as seen in the reflecting plane of division, is a reproduction of the other
half, such an object is termed symmetrical, and the plane of division is
called the plane of symmetry.
If the plane of symmetry is vertical, we can say that the body is vertically
symmetrical. An example of vertical symmetry is a Gothic cathedral.
If the plane of symmetry is horizontal, we can say that the object is
horizontally symmetrical. A landscape on the shores of a lake with its
reflexion in the water, is a system of horizontal symmetry.
Exactly here is a noticeable difference. The vertical symmetry of a Gothic
cathedral strikes us at once, whereas we can travel up and down the whole
length of the Rhine or the Hudson without becoming aware of the
symmetry between objects and their reflexions in the water. Vertical
symmetry pleases us, whilst horizontal symmetry is indifferent, and is
noticed only by the experienced eye.
Whence arises this difference? I say from the fact that vertical symmetry
produces a repetition of the same sensation, while horizontal symmetry
does not. I shall now show that this is so.
Let us look at the following letters:
dbqp
It is a fact known to all mothers and teachers, that children in their first
attempts to read and write, constantly confound d and b, and q and p, but
never d and q, or b and p. Now d and b and q and p are the two halves of a
vertically symmetrical figure, while d and q, and b and p are two halves of a
horizontally symmetrical figure. The first two are confounded; but
confusion is only possible of things that excite in us the same or similar
sensations.
Figures of two flower-girls are frequently seen on the decorations of
gardens and of drawing-rooms, one of whom carries a flower-basket in her
right hand and the other a flower-basket in her left. All know how apt we
are, unless we are very careful, to confound these figures with one another.

While turning a thing round from right to left is scarcely noticed, the eye is
not at all indifferent to the turning of a thing upside down. A human face
which has been turned upside down is scarcely recognisable as a face, and
makes an impression which is altogether strange. The reason of this is not to
be sought in the unwontedness of the sight, for it is just as difficult to
recognise an arabesque that has been inverted, where there can be no
question of a habit. This curious fact is the foundation of the familiar jokes
played with the portraits of unpopular personages, which are so drawn that
in the upright position of the page an exact picture of the person is
presented, but on being inverted some popular animal is shown.
It is a fact, then, that the two halves of a vertically symmetrical figure are
easily confounded and that they therefore probably produce very nearly the
same sensations. The question, accordingly, arises, why do the two halves of
a vertically symmetrical figure produce the same or similar sensations? The
answer is: Because our apparatus of vision, which consists of our eyes and
of the accompanying muscular apparatus is itself vertically symmetrical.[21]
Whatever external resemblances one eye may have with another they are
still not alike. The right eye of a man cannot take the place of a left eye any
more than a left ear or left hand can take the place of a right one. By
artificial means, we can change the part which each of our eyes plays.
(Wheatstone's pseudoscope.) But we then find ourselves in an entirely new
and strange world. What is convex appears concave; what is concave,
convex. What is distant appears near, and what is near appears far.
The left eye is the reflexion of the right. And the light-feeling retina of the
left eye is a reflexion of the light-feeling retina of the right, in all its
functions.
The lense of the eye, like a magic lantern, casts images of objects on the
retina. And you may picture to yourself the light-feeling retina of the eye,
with its countless nerves, as a hand with innumerable fingers, adapted to
feeling light. The ends of the visual nerves, like our fingers, are endowed
with varying degrees of sensitiveness. The two retinæ act like a right and a
left hand; the sensation of touch and the sensation of light in the two
instances are similar.
Examine the right-hand portion of this letter T: namely, T. Instead of the
two retinæ on which this image falls, imagine feeling the object, my two

hands. The ┌, grasped with the right hand, gives a different sensation from
that which it gives when grasped with the left. But if we turn our character
about from right to left, thus: ┐, it will give the same sensation in the left
hand that it gave before in the right. The sensation is repeated.
If we take a whole T, the right half will produce in the right hand the same
sensation that the left half produces in the left, and vice versa.
The symmetrical figure gives the same sensation twice.
If we turn the T over thus: ├, or invert the half T thus: L, so long as we do
not change the position of our hands we can make no use of the foregoing
reasoning.
The retinæ, in fact, are exactly like our two hands. They, too, have their
thumbs and index fingers, though they are thousands in number; and we
may say the thumbs are on the side of the eye near the nose, and the
remaining fingers on the side away from the nose.
With this I hope to have made perfectly clear that the pleasing effect of
symmetry is chiefly due to the repetition of sensations, and that the effect in
question takes place in symmetrical figures, only where there is a repetition
of sensation. The pleasing effect of regular figures, the preference which
straight lines, especially vertical and horizontal straight lines, enjoy, is
founded on a similar reason. A straight line, both in a horizontal and in a
vertical position, can cast on the two retinæ the same image, which falls
moreover on symmetrically corresponding spots. This also, it would appear,
is the reason of our psychological preference of straight to curved lines, and
not their property of being the shortest distance between two points. The
straight line is felt, to put the matter briefly, as symmetrical to itself, which
is the case also with the plane. Curved lines are felt as deviations from
straight lines, that is, as deviations from symmetry.[22] The presence of a
sense for symmetry in people possessing only one eye from birth, is indeed
a riddle. Of course, the sense of symmetry, although primarily acquired by
means of the eyes, cannot be wholly limited to the visual organs. It must
also be deeply rooted in other parts of the organism by ages of practice and
can thus not be eliminated forthwith by the loss of one eye. Also, when an
eye is lost, the symmetrical muscular apparatus is left, as is also the
symmetrical apparatus of innervation.

It appears, however, unquestionable that the phenomena mentioned have, in
the main, their origin in the peculiar structure of our eyes. It will therefore
be seen at once that our notions of what is beautiful and ugly would
undergo a change if our eyes were different. Also, if this view is correct, the
theory of the so-called eternally beautiful is somewhat mistaken. It can
scarcely be doubted that our culture, or form of civilisation, which stamps
upon the human body its unmistakable traces, should not also modify our
conceptions of the beautiful. Was not formerly the development of all
musical beauty restricted to the narrow limits of a five-toned scale?
The fact that a repetition of sensations is productive of pleasant effects is
not restricted to the realm of the visible. To-day, both the musician and the
physicist know that the harmonic or the melodic addition of one tone to
another affects us agreeably only when the added tone reproduces a part of
the sensation which the first one excited. When I add an octave to a
fundamental tone, I hear in the octave a part of what was heard in the
fundamental tone. (Helmholtz.) But it is not my purpose to develop this
idea fully here.[23] We shall only ask to-day, whether there is anything
similar to the symmetry of figures in the province of sounds.
Look at the reflexion of your piano in the mirror.
You will at once remark that you have never seen such a piano in the actual
world, for it has its high keys to the left and its low ones to the right. Such
pianos are not manufactured.
If you could sit down at such a piano and play in your usual manner, plainly
every step which you imagined you were performing in the upward scale
would be executed as a corresponding step in the downward scale. The
effect would be not a little surprising.
For the practised musician who is always accustomed to hearing certain
sounds produced when certain keys are struck, it is quite an anomalous
spectacle to watch a player in the glass and to observe that he always does
the opposite of what we hear.
But still more remarkable would be the effect of attempting to strike a
harmony on such a piano. For a melody it is not indifferent whether we
execute a step in an upward or a downward scale. But for a harmony, so
great a difference is not produced by reversal. I always retain the same

consonance whether I add to a fundamental note an upper or a lower third.
Only the order of the intervals of the harmony is reversed. In point of fact,
when we execute a movement in a major key on our reflected piano, we
hear a sound in a minor key, and vice versa.
It now remains to execute the experiments indicated. Instead of playing
upon the piano in the mirror, which is impossible, or of having a piano of
this kind built, which would be somewhat expensive, we may perform our
experiments in a simpler manner, as follows:
1) We play on our own piano in our usual manner, look into the mirror, and
then repeat on our real piano what we see in the mirror. In this way we
transform all steps upwards into corresponding steps downwards. We play a
movement, and then another movement, which, with respect to the key-
board, is symmetrical to the first.
2) We place a mirror beneath the music in which the notes are reflected as
in a body of water, and play according to the notes in the mirror. In this way
also, all steps upwards are changed into corresponding, equal steps
downwards.
3) We turn the music upside down and read the notes from right to left and
from below upwards. In doing this, we must regard all sharps as flats and
all flats as sharps, because they correspond to half lines and spaces.
Besides, in this use of the music we can only employ the bass clef, as only
in this clef are the notes not changed by symmetrical reversal.
You can judge of the effect of these experiments from the examples which
appear in the annexed musical cut. (Page 102.) The movement which
appears in the upper lines is symmetrically reversed in the lower.
The effect of the experiments may be briefly formulated. The melody is
rendered unrecognisable. The harmony suffers a transposition from a major
into a minor key and vice versa. The study of these pretty effects, which
have long been familiar to physicists and musicians, was revived some
years ago by Von Oettingen.[24]

Fig. 26.

Listen to 1.
Listen to 2.
Listen to 3.
Listen to 4.
Listen to 5.
Listen to 6.
Listen to 7.
Listen to 8.
(See pages 101 and 103.)]
Now, although in all the preceding examples I have transposed steps
upward into equal and similar steps downward, that is, as we may justly
say, have played for every movement the movement which is symmetrical
to it, yet the ear notices either little or nothing of symmetry. The
transposition from a major to a minor key is the sole indication of symmetry
remaining. The symmetry is there for the mind, but is wanting for sensation.
No symmetry exists for the ear, because a reversal of musical sounds
conditions no repetition of sensations. If we had an ear for height and an ear
for depth, just as we have an eye for the right and an eye for the left, we
should also find that symmetrical sound-structures existed for our auditory
organs. The contrast of major and minor for the ear corresponds to
inversion for the eye, which is also only symmetry for the mind, but not for
sensation.
By way of supplement to what I have said, I will add a brief remark for my
mathematical readers.
Our musical notation is essentially a graphical representation of a piece of
music in the form of curves, where the time is the abscissæ, and the
logarithms of the number of vibrations the ordinates. The deviations of
musical notation from this principle are only such as facilitate
interpretation, or are due to historical accidents.
If, now, it be further observed that the sensation of pitch also is proportional
to the logarithm of the number of vibrations, and that the intervals between
the notes correspond to the differences of the logarithms of the numbers of
vibrations, the justification will be found in these facts of calling the

harmonies and melodies which appear in the mirror, symmetrical to the
original ones.

I simply wish to bring home to your minds by these fragmentary remarks
that the progress of the physical sciences has been of great help to those
branches of psychology that have not scorned to consider the results of
physical research. On the other hand, psychology is beginning to return, as
it were, in a spirit of thankfulness, the powerful stimulus which it received
from physics.
The theories of physics which reduce all phenomena to the motion and
equilibrium of smallest particles, the so-called molecular theories, have
been gravely threatened by the progress of the theory of the senses and of
space, and we may say that their days are numbered.
I have shown elsewhere[25] that the musical scale is simply a species of
space—a space, however, of only one dimension, and that, a one-sided one.
If, now, a person who could only hear, should attempt to develop a
conception of the world in this, his linear space, he would become involved
in many difficulties, as his space would be incompetent to comprehend the
many sides of the relations of reality. But is it any more justifiable for us, to
attempt to force the whole world into the space of our eye, in aspects in
which it is not accessible to the eye? Yet this is the dilemma of all
molecular theories.
We possess, however, a sense, which, with respect to the scope of the
relations which it can comprehend, is richer than any other. It is our reason.
This stands above the senses. It alone is competent to found a permanent
and sufficient view of the world. The mechanical conception of the world
has performed wonders since Galileo's time. But it must now yield to a
broader view of things. A further development of this idea is beyond the
limits of my present purpose.
One more point and I have done. The advice of our philosopher to restrict
ourselves to what is near at hand and useful in our researches, which finds a
kind of exemplification in the present cry of inquirers for limitation and
division of labor, must not be too slavishly followed. In the seclusion of our

closets, we often rack our brains in vain to fulfil a work, the means of
accomplishing which lies before our very doors. If the inquirer must be
perforce a shoemaker, tapping constantly at his last, it may perhaps be
permitted him to be a shoemaker of the type of Hans Sachs, who did not
deem it beneath him to take a look now and then at his neighbor's work and
to comment on the latter's doings.
Let this be my apology, therefore, if I have forsaken for a moment to-day
the last of my specialty.

ON THE FUNDAMENTAL CONCEPTS OF
ELECTROSTATICS.[26]
The task has been assigned me to develop before you in a popular manner
the fundamental quantitative concepts of electrostatics—"quantity of
electricity," "potential," "capacity," and so forth. It would not be difficult,
even within the brief limits of an hour, to delight the eye with hosts of
beautiful experiments and to fill the imagination with numerous and varied
conceptions. But we should, in such a case, be still far from a lucid and easy
grasp of the phenomena. The means would still fail us for reproducing the
facts accurately in thought—a procedure which for the theoretical and
practical man is of equal importance. These means are the metrical concepts
of electricity.
As long as the pursuit of the facts of a given province of phenomena is in
the hands of a few isolated investigators, as long as every experiment can
be easily repeated, the fixing of the collected facts by provisional
description is ordinarily sufficient. But the case is different when the whole
world must make use of the results reached by many, as happens when the
science acquires broader foundations and scope, and particularly so when it
begins to supply intellectual nourishment to an important branch of the
practical arts, and to draw from that province in return stupendous empirical
results. Then the facts must be so described that individuals in all places and
at all times can, from a few easily obtained elements, put the facts
accurately together in thought, and reproduce them from the description.
This is done with the help of the metrical concepts and the international
measures.
The work which was begun in this direction in the period of the purely
scientific development of the science, especially by Coulomb (1784), Gauss
(1833), and Weber (1846), was powerfully stimulated by the requirements
of the great technical undertakings manifested since the laying of the first
transatlantic cable, and brought to a brilliant conclusion by the labors of the
British Association, 1861, and of the Paris Congress, 1881, chiefly through
the exertions of Sir William Thomson.

It is plain, that in the time allotted to me I cannot conduct you over all the
long and tortuous paths which the science has actually pursued, that it will
not be possible at every step to remind you of all the little precautions for
the avoidance of error which the early steps have taught us. On the contrary,
I must make shift with the simplest and rudest tools. I shall conduct you by
the shortest paths from the facts to the ideas, in doing which, of course, it
will not be possible to anticipate all the stray and chance ideas which may
and must arise from prospects into the by-paths which we leave untrodden.

Here are two small, light bodies (Fig. 27) of equal size, freely suspended,
which we "electrify" either by friction with a third body or by contact with a
body already electrified. At once a repulsive force is set up which drives the
two bodies away from each other in opposition to the action of gravity. This
force could accomplish anew the same mechanical work which was
expended to produce it.[27]
Coulomb, now, by
means of delicate
experiments with the
torsion-balance,
satisfied himself that
if the bodies in
question, say at a
distance of two
centimetres, repelled
Fig. 28.
each other with the
same force with
which a milligramme-weight strives to fall to the
Fig. 27. ground, at half that distance, or at one centimetre, they
would repel each other with the force of four
milligrammes, and at double that distance, or at four centimetres, they
would repel each other with the force of only one-fourth of a milligramme.
He found that the electrical force acts inversely as the square of the
distance.

Let us imagine, now, that we possessed some means of measuring electrical
repulsion by weights, a means which would be supplied, for example, by
our electrical pendulums; then we could make the following observation.
The body A (Fig. 28) is repelled by the body K at a distance of two
centimetres with a force of one milligramme. If we touch A, now, with an
equal body B, the half of this force of repulsion will pass to the body B;
both A and B, now, at a distance of two centimetres from K, are repelled
only with the force of one-half a milligramme. But both together are
repelled still with the force of one milligramme. Hence, the divisibility of
electrical force among bodies in contact is a fact. It is a useful, but by no
means a necessary supplement to this fact, to imagine an electrical fluid
present in the body A, with the quantity of which the electrical force varies,
and half of which flows over to B. For, in the place of the new physical
picture, thus, an old, familiar one is substituted, which moves
spontaneously in its wonted courses.
Adhering to this idea, we define the unit of electrical quantity, according to
the now almost universally adopted centimetre-gramme-second (C. G. S.)
system, as that quantity which at a distance of one centimetre repels an
equal quantity with unit of force, that is, with a force which in one second
would impart to a mass of one gramme a velocity-increment of a
centimetre. As a gramme mass acquires through the action of gravity a
velocity-increment of about 981 centimetres in a second, accordingly, a
gramme is attracted to the earth with 981, or, in round numbers, 1000 units
of force of the centimetre-gramme-second system, while a milligramme-
weight would strive to fall to the earth with approximately the unit force of
this system.
We may easily obtain by this means a clear idea of what the unit quantity of
electricity is. Two small bodies, K, weighing each a gramme, are hung up
by vertical threads, five metres in length and almost weightless, so as to
touch each other. If the two bodies be equally electrified and move apart
upon electrification to a distance of one centimetre, their charge is
approximately equivalent to the electrostatic unit of electric quantity, for the
repulsion then holds in equilibrium a gravitational force-component of
approximately one milligramme, which strives to bring the bodies together.

Vertically beneath a small sphere suspended from the equilibrated beam of a
balance a second sphere is placed at a distance of a centimetre. If both be
equally electrified the sphere suspended from the balance will be rendered
apparently lighter by the repulsion. If by adding a weight of one
milligramme equilibrium is restored, each of the spheres contains in round
numbers the electrostatic unit of electrical quantity.
In view of the fact that the same electrical bodies exert at different distances
different forces upon one another, exception might be taken to the measure
of quantity here developed. What kind of a quantity is that which now
weighs more, and now weighs less, so to speak? But this apparent deviation
from the method of determination commonly used in practical life, that by
weight, is, closely considered, an agreement. On a high mountain a heavy
mass also is less powerfully attracted to the earth than at the level of the
sea, and if it is permitted us in our determinations to neglect the
consideration of level, it is only because the comparison of a body with
fixed conventional weights is invariably effected at the same level. In fact,
if we were to make one of the two weights equilibrated on our balance
approach sensibly to the centre of the earth, by suspending it from a very
long thread, as Prof. von Jolly of Munich suggested, we should make the
gravity of that weight, its heaviness, proportionately greater.
Let us picture to ourselves, now, two different electrical fluids, a positive
and a negative fluid, of such nature that the particles of the one attract the
particles of the other according to the law of the inverse squares, but the
particles of the same fluid repel each other by the same law; in non-
electrical bodies let us imagine the two fluids uniformly distributed in equal
quantities, in electric bodies one of the two in excess; in conductors, further,
let us imagine the fluids mobile, in non-conductors immobile; having
formed such pictures, we possess the conception which Coulomb developed
and to which he gave mathematical precision. We have only to give this
conception free play in our minds and we shall see as in a clear picture the
fluid particles, say of a positively charged conductor, receding from one
another as far as they can, all making for the surface of the conductor and
there seeking out the prominent parts and points until the greatest possible
amount of work has been performed. On increasing the size of the surface,
we see a dispersion, on decreasing its size we see a condensation of the
particles. In a second, non-electrified conductor brought into the vicinity of

the first, we see the two fluids immediately separate, the positive collecting
itself on the remote and the negative on the adjacent side of its surface. In
the fact that this conception reproduces, lucidly and spontaneously, all the
data which arduous research only slowly and gradually discovered, is
contained its advantage and scientific value. With this, too, its value is
exhausted. We must not seek in nature for the two hypothetical fluids which
we have added as simple mental adjuncts, if we would not go astray.
Coulomb's view may be replaced by a totally different one, for example, by
that of Faraday, and the most proper course is always, after the general
survey is obtained, to go back to the actual facts, to the electrical forces.
We will now make ourselves
familiar with the concept of
electrical quantity, and with the
method of measuring or estimating
it. Imagine a common Leyden jar
(Fig. 29), the inner and outer
coatings of which are connected
together by means of two common
metallic knobs placed about a
centimetre apart. If the inside
coating be charged with the
quantity of electricity +q, on the
outer coating a distribution of the
electricities will take place. A
positive quantity almost equal[28] to
Fig. 29.
the quantity +q flows off to the
earth, while a corresponding
quantity-q is still left on the outer coating. The knobs of the jar receive their
portion of these quantities and when the quantity q is sufficiently great a
rupture of the insulating air between the knobs, accompanied by the self-
discharge of the jar, takes place. For any given distance and size of the
knobs, a charge of a definite electric quantity q is always necessary for the
spontaneous discharge of the jar.
Let us insulate, now, the outer coating of a Lane's unit jar L, the jar just
described, and put in connexion with it the inner coating of a jar F
exteriorly connected with the earth (Fig. 30). Every time that L is charged

with +q, a like quantity +q is
collected on the inner coating of F,
and the spontaneous discharge of
the jar L, which is now again empty,
takes place. The number of the
discharges of the jar L furnishes us,
thus, with a measure of the quantity
collected in the jar F, and if after 1,
2, 3, ... spontaneous discharges of L
the jar F is discharged, it is evident
that the charge of F has been
proportionately augmented.
Let us supply now, to effect the
spontaneous discharge, the jar F Fig. 30.

with knobs of the same size and at
the same distance apart as those of
the jar L (Fig. 31). If we find, then,
that five discharges of the unit jar
take place before one spontaneous
discharge of the jar F occurs,
plainly the jar F, for equal distances
between the knobs of the two jars,
equal striking distances, is able to
hold five times the quantity of
electricity that L can, that is, has
five times the capacity of L.[29]
Fig. 31. We will now replace the unit jar L,
with which we measure electricity,
so to speak, into the jar F, by a Franklin's pane, consisting of two parallel
flat metal plates (Fig. 32), separated only by air. If here, for example, thirty
spontaneous discharges of the pane are sufficient to fill the jar, ten
discharges will be found sufficient if the air-space between the two plates
be filled with a cake of sulphur. Hence, the capacity of a Franklin's pane of
sulphur is about three times greater than that of one of the same shape and
size made of air, or, as it is the custom to say, the specific inductive capacity

of sulphur (that of air
being taken as the unit)
is about 3.[30] We are
here arrived at a very
simple fact, which
clearly shows us the
significance of the
number called
dielectric constant, or
specific inductive
capacity, the
knowledge of which is
so important for the
theory of submarine
cables.
Let us consider a jar A,
Fig. 32.
which is charged with a
certain quantity of
electricity. We can discharge the jar directly. But we can also discharge the
jar A (Fig. 33) partly into a jar B, by connecting the two outer coatings with
each other. In this operation a portion of the quantity of electricity passes,
accompanied by sparks, into the jar B, and we now find both jars charged.
It may be shown as follows that the conception of a constant quantity of
electricity can be regarded as the expression of a pure fact. Picture to
yourself any sort of electrical conductor (Fig. 34); cut it up into a large
number of small pieces, and place these pieces by means of an insulated rod
at a distance of one centimetre from an electrical body which acts with unit
of force on an equal and like-constituted body at the same distance. Take
the sum of the forces which this last body exerts on the single pieces of the
conductor. The sum of these forces will be the quantity of electricity on the
whole conductor. It remains the same, whether we change the form and the
size of the conductor, or whether we bring it near or move it away from a
second electrical conductor, so long as we keep it insulated, that is, do not
discharge it.
A basis of reality for the notion of electric quantity seems also to present
itself from another quarter. If a current, that is, in the usual view, a definite

quantity of electricity per second, is
sent through a column of acidulated
water; in the direction of the
positive stream, hydrogen, but in
the opposite direction, oxygen is
liberated at the extremities of the
column. For a given quantity of
electricity a given quantity of
oxygen appears. You may picture
the column of water as a column of
hydrogen and a column of oxygen,
fitted into each other, and may say
the electric current is a chemical
Fig. 33. current and vice versa. Although
this notion is more difficult to
adhere to in the field of statical
electricity and with non-
decomposable conductors, its
further development is by no means
hopeless.
The concept quantity of electricity,
thus, is not so aerial as might
appear, but is able to conduct us
with certainty through a multitude
of varied phenomena, and is
suggested to us by the facts in Fig. 34.
almost palpable form. We can
collect electrical force in a body, measure it out with one body into another,
carry it over from one body into another, just as we can collect a liquid in a
vessel, measure it out with one vessel into another, or pour it from one into
another.
For the analysis of mechanical phenomena, a metrical notion, derived from
experience, and bearing the designation work, has proved itself useful. A
machine can be set in motion only when the forces acting on it can perform
work.

Let us consider, for example,
a wheel and axle (Fig. 35)
having the radii 1 and 2
metres, loaded respectively
with the weights 2 and 1
kilogrammes. On turning the
wheel and axle, the 1
kilogramme-weight, let us
say, sinks two metres, while
the 2 kilogramme-weight
rises one metre. On both
sides the product
KGR. M. KGR. M.
1 × 2 = 2 × 1.
is equal. So long as this is so,
the wheel and axle will not
move of itself. But if we take
such loads, or so change the
radii of the wheels, that this
Fig. 35.
product (kgr. × metre) on
displacement is in excess on
one side, that side will sink. As we see, this product is characteristic for
mechanical events, and for this reason has been invested with a special
name, work.
In all mechanical processes, and as all physical processes present a
mechanical side, in all physical processes, work plays a determinative part.
Electrical forces, also, produce only changes in which work is performed.
To the extent that forces come into play in electrical phenomena, electrical
phenomena, be they what they may, extend into the domain of mechanics
and are subject to the laws which hold in this domain. The universally
adopted measure of work, now, is the product of the force into the distance
through which it acts, and in the C. G. S. system, the unit of work is the
action through one centimetre of a force which would impart in one second
to a gramme-mass a velocity-increment of one centimetre, that is, in round
numbers, the action through a centimetre of a pressure equal to the weight

of a milligramme. From a positively charged body, electricity, yielding to
the force of repulsion and performing work, flows off to the earth,
providing conducting connexions exist. To a negatively charged body, on
the other hand, the earth under the same circumstances gives off positive
electricity. The electrical work possible in the interaction of a body with the
earth, characterises the electrical condition of that body. We will call the
work which must be expended on the unit quantity of positive electricity to
raise it from the earth to the body K the potential of the body K.[31]
We ascribe to the body K in the C. G. S. system the potential +1, if we must
expend the unit of work to raise the positive electrostatic unit of electric
quantity from the earth to that body; the potential -1, if we gain in this
procedure the unit of work; the potential 0, if no work at all is performed in
the operation.
The different parts of one and the same electrical conductor in electrical
equilibrium have the same potential, for otherwise the electricity would
perform work and move about upon the conductor, and equilibrium would
not have existed. Different conductors of equal potential, put in connexion
with one another, do not exchange electricity any more than bodies of equal
temperature in contact exchange heat, or in connected vessels, in which the
same pressures exist, liquids flow from one vessel to the other. Exchange of
electricity takes place only between conductors of different potentials, but
in conductors of given form and position a definite difference of potential is
necessary for a spark, which pierces the insulating air, to pass between
them.
On being connected, every two conductors assume at once the same
potential. With this the means is given of determining the potential of a
conductor through the agency of a second conductor expressly adapted to
the purpose called an electrometer, just as we determine the temperature of
a body with a thermometer. The values of the potentials of bodies obtained
in this way simplify vastly our analysis of their electrical behavior, as will
be evident from what has been said.
Think of a positively charged conductor. Double all the electrical forces
exerted by this conductor on a point charged with unit quantity, that is,
double the quantity at each point, or what is the same thing, double the total
charge. Plainly, equilibrium still subsists. But carry, now, the positive

electrostatic unit towards the conductor. Everywhere we shall have to
overcome double the force of repulsion we did before, everywhere we shall
have to expend double the work. By doubling the charge of the conductor a
double potential has been produced. Charge and potential go hand in hand,
are proportional. Consequently, calling the total quantity of electricity of a
conductor Q and its potential V, we can write: Q = CV, where C stands for a
constant, the import of which will be understood simply from noting that C
= Q/V.[32] But the division of a number representing the units of quantity of
a conductor by the number representing its units of potential tells us the
quantity which falls to the share of the unit of potential. Now the number C
here we call the capacity of a conductor, and have substituted, thus, in the
place of the old relative determination of capacity, an absolute
determination.[33]
In simple cases the connexion between charge, potential, and capacity is
easily ascertained. Our conductor, let us say, is a sphere of radius r,
suspended free in a large body of air. There being no other conductors in the
vicinity, the charge q will then distribute itself uniformly upon the surface
of the sphere, and simple geometrical considerations yield for its potential
the expression V = q/r. Hence, q/V = r; that is, the capacity of a sphere is
measured by its radius, and in the C. G. S. system in centimetres.[34] It is
clear also, since a potential is a quantity divided by a length, that a quantity
divided by a potential must be a length.
Imagine (Fig. 36) a jar composed of two concentric conductive spherical
shells of the radii r and r1, having only air between them. Connecting the
outside sphere with the earth, and charging the inside sphere by means of a
thin, insulated wire passing through the first, with the quantity Q, we shall
have V = (r1-r)/(r1r)Q, and for the capacity in this case (r1r)/(r1-r), or, to
take a specific example, if r = 16 and r1 = 19, a capacity of about 100
centimetres.
We shall now use these simple cases for illustrating the principle by which
capacity and potential are determined. First, it is clear that we can use the
jar composed of concentric spheres with its known capacity as our unit jar
and by means of this ascertain, in the manner above laid down, the capacity
of any given jar F. We find, for example, that 37 discharges of this unit jar
of the capacity 100, just charges the jar investigated at the same striking

distance, that is, at the same
potential. Hence, the capacity
of the jar investigated is 3700
centimetres. The large battery
of the Prague physical
laboratory, which consists of
sixteen such jars, all of nearly
equal size, has a capacity,
therefore, of something like
50,000 centimetres, or the
capacity of a sphere, a
kilometre in diameter, freely
suspended in atmospheric
space. This remark distinctly
shows us the great superiority
which Leyden jars possess
for the storage of electricity Fig. 36.
as compared with common
conductors. In fact, as
Faraday pointed out, jars differ from simple conductors mainly by their
great capacity.

Fig. 37.
For determining potential, imagine the inner coating of a jar F, the outer
coating of which communicates with the ground, connected by a long, thin
wire with a conductive sphere K placed free in a large atmospheric space,
compared with whose dimensions the radius of the sphere vanishes. (Fig.
37.) The jar and the sphere assume at once the same potential. But on the
surface of the sphere, if that be sufficiently far removed from all other
conductors, a uniform layer of electricity will be found. If the sphere,
having the radius r, contains the charge q, its potential is V = q/r. If the
upper half of the sphere be severed from the lower half and equilibrated on
a balance with one of whose beams it is connected by silk threads, the upper
half will be repelled from the lower half with the force P = q2/8r2 = 1/8V2.
This repulsion P may be counter-balanced by additional weights placed on
the beam-end, and so ascertained. The potential is then V = √(8P).[35]
That the potential is proportional to the square root of the force is not
difficult to see. A doubling or trebling of the potential means that the charge
of all the parts is doubled or trebled; hence their combined power of
repulsion quadrupled or nonupled.
Let us consider a special case. I wish to produce the potential 40 on the
sphere. What additional weight must I give to the half sphere in grammes
that the force of repulsion shall maintain the balance in exact equilibrium?
As a gramme weight is approximately equivalent to 1000 units of force, we
have only the following simple example to work out: 40×40 = 8× 1000.x,
where x stands for the number of grammes. In round numbers we get x =
0.2 gramme. I charge the jar. The balance is deflected; I have reached, or
rather passed, the potential 40, and you see when I discharge the jar the
associated spark.[36]
The striking distance between the knobs of a machine increases with the
difference of the potential, although not proportionately to that difference.
The striking distance increases faster than the potential difference. For a
distance between the knobs of one centimetre on this machine the
difference of potential is 110. It can easily be increased tenfold. Of the
tremendous differences of potential which occur in nature some idea may
be obtained from the fact that the striking distances of lightning in thunder-
storms is counted by miles. The differences of potential in galvanic batteries

are considerably smaller than those of our machine, for it takes fully one
hundred elements to give a spark of microscopic striking distance.

We shall now employ the ideas reached to shed some light upon another
important relation between electrical and mechanical phenomena. We shall
investigate what is the potential energy, or the store of work, contained in a
charged conductor, for example, in a jar.
If we bring a quantity of electricity up to a conductor, or, to speak less
pictorially, if we generate by work electrical force in a conductor, this force
is able to produce anew the work by which it was generated. How great,
now, is the energy or capacity for work of a conductor of known charge Q
and known potential V?
Imagine the given charge Q divided into very small parts q, q1, q2 ..., and
these little parts successively carried up to the conductor. The first very
small quantity q is brought up without any appreciable work and produces
by its presence a small potential V'. To bring up the second quantity,
accordingly, we must do the work q'V', and similarly for the quantities
which follow the work q''V'', q'''V''', and so forth. Now, as the potential rises
proportionately to the quantities added until the value V is reached, we
have, agreeably to the graphical representation of Fig. 38, for the total work
performed,
W = 1/2QV,
which corresponds to the total energy of the charged conductor. Using the
equation Q = CV, where C stands for capacity, we also have,
W = 1/2CV2, or W = Q2/2C.
It will be helpful, perhaps, to elucidate this idea by an analogy from the
province of mechanics. If we pump a quantity of liquid, Q, gradually into a
cylindrical vessel (Fig. 39), the level of the liquid in the vessel will
gradually rise. The more we have pumped in, the greater the pressure we
must overcome, or the higher the level to which we must lift the liquid. The
stored-up work is rendered again available when the heavy liquid Q, which

reaches up to the level h, flows out. This work W corresponds to the fall of
the whole liquid weight Q, through the distance h/2 or through the altitude
of its centre of gravity. We have
W = 1/2Qh.
Further, since Q = Kh, or since the weight of the liquid and the height h are
proportional, we get also
W = 1/2Kh2 and W = Q2/2K.
As a special case let us consider our
jar. Its capacity is C = 3700, its
potential V = 110; accordingly, its
quantity Q = CV = 407,000
electrostatic units and its energy W
= 1/2QV = 22,385,000 C. G. S.
units of work.
The unit of work of the C. G. S.
system is not readily appreciable by
the senses, nor does it well admit of
Fig. 38. representation, as we are
accustomed to work with weights.
Let us adopt, therefore, as our unit
of work the gramme-centimetre, or
the gravitational pressure of a
gramme-weight through the
distance of a centimetre, which in
round numbers is 1000 times
greater than the unit assumed
above; in this case, our numerical
result will be approximately 1000
times smaller. Again, if we pass, as
more familiar in practice, to the
kilogramme-metre as our unit of
work, our unit, the distance being Fig. 39.
increased a hundred fold, and the
weight a thousand fold, will be

100,000 times larger. The numerical result expressing the work done is in
this case 100,000 times less, being in round numbers 0.22 kilogramme-
metre. We can obtain a clear idea of the work done here by letting a
kilogramme-weight fall 22 centimetres.
This amount of work, accordingly, is performed on the charging of the jar,
and on its discharge appears again, according to the circumstances, partly as
sound, partly as a mechanical disruption of insulators, partly as light and
heat, and so forth.
The large battery of the Prague physical laboratory, with its sixteen jars
charged to equal potentials, furnishes, although the effect of the discharge is
imposing, a total amount of work of only three kilogramme-metres.
In the development of the ideas above laid down we are not restricted to the
method there pursued; in fact, that method was selected only as one
especially fitted to familiarise us with the phenomena. On the contrary, the
connexion of the physical processes is so multifarious that we can come at
the same event from very different directions. Particularly are electrical
phenomena connected with all other physical events; and so intimate is this
connexion that we might justly call the study of electricity the theory of the
general connexion of physical processes.
With respect to the principle of the conservation of energy which unites
electrical with mechanical phenomena, I should like to point out briefly two
ways of following up the study of this connexion.
A few years ago Professor Rosetti, taking an influence-machine, which he
set in motion by means of weights alternately in the electrical and non-
electrical condition with the same velocities, determined the mechanical
work expended in the two cases and was thus enabled, after deducting the
work of friction, to ascertain the mechanical work consumed in the
development of the electricity.
I myself have made this experiment in a modified, and, as I think, more
advantageous form. Instead of determining the work of friction by special
trial, I arranged my apparatus so that it was eliminated of itself in the
measurement and could consequently be neglected. The so-called fixed disk
of the machine, the axis of which is placed vertically, is suspended
somewhat like a chandelier by three vertical threads of equal lengths l at a

distance r from the axis. Only when the machine is excited does this fixed
disk, which represents a Prony's brake, receive, through its reciprocal action
with the rotating disk, a deflexion α and a moment of torsion which is
expressed by D = (Pr2/l)α, where P is the weight of the disk.[37] The angle α
is determined by a mirror set in the disk. The work expended in n rotations
is given by 2nπD.
If we close the machine, as Rosetti did, we obtain a continuous current
which has all the properties of a very weak galvanic current; for example, it
produces a deflexion in a multiplier which we interpose, and so forth. We
can directly ascertain, now, the mechanical work expended in the
maintenance of this current.
If we charge a jar by means of a machine, the energy of the jar employed in
the production of sparks, in the disruption of the insulators, etc.,
corresponds to a part only of the mechanical work expended, a second part
of it being consumed in the arc which forms the circuit.[38] This machine,
with the interposed jar, affords in miniature a picture of the transference of
force, or more properly of work. And in fact nearly the same laws hold here
for the economical coefficient as obtain for large dynamo-machines.
Another means of investigating electrical energy is by its transformation
into heat. A long time ago (1838), before the mechanical theory of heat had
attained its present popularity, Riess performed experiments in this field
with the help of his electrical air-thermometer or thermo-electrometer.
If the discharge be conducted through a fine wire passing through the globe
of the air-thermometer, a development of heat is observed proportional to
the expression above-discussed W = 1/2QV. Although the total energy has
not yet been transformed into measurable heat by this means, in as much as
a portion is left behind in the spark in the air outside the thermometer, still
everything tends to show that the total heat developed in all parts of the
conductor and along all the paths of discharge is the equivalent of the work
1/2QV.
It is not important here whether the electrical energy is transformed all at
once or partly, by degrees. For example, if of two equal jars one is charged
with the quantity Q at the potential V the energy present is 1/2QV. If the
first jar be discharged into the second, V, since the capacity is now doubled,
falls to V/2. Accordingly, the energy 1/4QV remains, while 1/4QV is

transformed in the spark of discharge into heat.
The remainder, however, is equally distributed
between the two jars so that each on discharge is
still able to transform 1/8QV into heat.

Fig. 40.

We have here discussed electricity in the limited phenomenal form in which
it was known to the inquirers before Volta, and which has been called,
perhaps not very felicitously, "statical electricity." It is evident, however,
that the nature of electricity is everywhere one and the same; that a
substantial difference between statical and galvanic electricity does not
exist. Only the quantitative circumstances in the two provinces are so
widely different that totally new aspects of phenomena may appear in the
second, for example, magnetic effects, which in the first remained
unnoticed, whilst, vice versa, in the second field statical attractions and
repulsions are scarcely appreciable. As a fact, we can easily show the
magnetic effect of the current of discharge of an influence-machine on the
galvanoscope although we could hardly have made the original discovery of
the magnetic effects with this current. The statical distant action of the wire
poles of a galvanic element also would hardly have been noticed had not the
phenomenon been known from a different quarter in a striking form.
If we wished to characterise the two fields in their chief and most general
features, we should say that in the first, high potentials and small quantities
come into play, in the second small potentials and large quantities. A jar

which is discharging and a galvanic element deport themselves somewhat
like an air-gun and the bellows of an organ. The first gives forth suddenly
under a very high pressure a small quantity of air; the latter liberates
gradually under a very slight pressure a large quantity of air.
In point of principle, too, nothing prevents our retaining the electrostatical
units in the domain of galvanic electricity and in measuring, for example,
the strength of a current by the number of electrostatic units which flow per
second through its cross-section. But this would be in a double aspect
impractical. In the first place, we should totally neglect the magnetic
facilities for measurement so conveniently offered by the current, and
substitute for this easy means a method which can be applied only with
difficulty and is not capable of great exactness. In the second place our units
would be much too small, and we should find ourselves in the predicament
of the astronomer who attempted to measure celestial distances in metres
instead of in radii of the earth and the earth's orbit; for the current which by
the magnetic C. G. S. standard represents the unit, would require a flow of
some 30,000,000,000 electrostatic units per second through its cross-
section. Accordingly, different units must be adopted here. The
development of this point, however, lies beyond my present task.

ON THE PRINCIPLE OF THE
CONSERVATION OF ENERGY.[39]
In a popular lecture, distinguished for its charming simplicity and clearness,
which Joule delivered in the year 1847,[40] that famous physicist declares
that the living force which a heavy body has acquired by its descent through
a certain height and which it carries with it in the form of the velocity with
which it is impressed, is the equivalent of the attraction of gravity through
the space fallen through, and that it would be "absurd" to assume that this
living force could be destroyed without some restitution of that equivalent.
He then adds: "You will therefore be surprised to hear that until very
recently the universal opinion has been that living force could be absolutely
and irrevocably destroyed at any one's option." Let us add that to-day, after
forty-seven years, the law of the conservation of energy, wherever
civilisation exists, is accepted as a fully established truth and receives the
widest applications in all domains of natural science.
The fate of all momentous discoveries is similar. On their first appearance
they are regarded by the majority of men as errors. J. R. Mayer's work on
the principle of energy (1842) was rejected by the first physical journal of
Germany; Helmholtz's treatise (1847) met with no better success; and even
Joule, to judge from an intimation of Playfair, seems to have encountered
difficulties with his first publication (1843). Gradually, however, people are
led to see that the new view was long prepared for and ready for
enunciation, only that a few favored minds had perceived it much earlier
than the rest, and in this way the opposition of the majority is overcome.
With proofs of the fruitfulness of the new view, with its success, confidence
in it increases. The majority of the men who employ it cannot enter into a
deep-going analysis of it; for them, its success is its proof. It can thus
happen that a view which has led to the greatest discoveries, like Black's
theory of caloric, in a subsequent period in a province where it does not
apply may actually become an obstacle to progress by its blinding our eyes
to facts which do not fit in with our favorite conceptions. If a theory is to be

protected from this dubious rôle, the grounds and motives of its evolution
and existence must be examined from time to time with the utmost care.
The most multifarious physical changes, thermal, electrical, chemical, and
so forth, can be brought about by mechanical work. When such alterations
are reversed they yield anew the mechanical work in exactly the quantity
which was required for the production of the part reversed. This is the
principle of the conservation of energy; "energy" being the term which has
gradually come into use for that "indestructible something" of which the
measure is mechanical work.
How did we acquire this idea? What are the sources from which we have
drawn it? This question is not only of interest in itself, but also for the
important reason above touched upon. The opinions which are held
concerning the foundations of the law of energy still diverge very widely
from one another. Many trace the principle to the impossibility of a
perpetual motion, which they regard either as sufficiently proved by
experience, or as self-evident. In the province of pure mechanics the
impossibility of a perpetual motion, or the continuous production of work
without some permanent alteration, is easily demonstrated. Accordingly, if
we start from the theory that all physical processes are purely mechanical
processes, motions of molecules and atoms, we embrace also, by this
mechanical conception of physics, the impossibility of a perpetual motion
in the whole physical domain. At present this view probably counts the most
adherents. Other inquirers, however, are for accepting only a purely
experimental establishment of the law of energy.
It will appear, from the discussion to follow, that all the factors mentioned
have co-operated in the development of the view in question; but that in
addition to them a logical and purely formal factor, hitherto little
considered, has also played a very important part.

I. THE PRINCIPLE OF THE EXCLUDED PERPETUAL
MOTION.

The law of energy in its modern form is not identical with the principle of
the excluded perpetual motion, but it is very closely related to it. The latter
principle, however, is by no means new, for in the province of mechanics it

has controlled for centuries the thoughts and investigations of the greatest
thinkers. Let us convince ourselves of this by the study of a few historical
examples.
S. Stevinus, in his famous work
Hypomnemata mathematica, Tom.
IV, De statica, (Leyden, 1605, p.
34), treats of the equilibrium of
bodies on inclined planes.
Over a triangular prism ABC, one
side of which, AC, is horizontal, an
endless cord or chain is slung, to
which at equal distances apart
fourteen balls of equal weight are
attached, as represented in cross-
section in Figure 41. Since we can
imagine the lower symmetrical part
Fig. 41.
of the cord ABC taken away,
Stevinus concludes that the four
balls on AB hold in equilibrium the two balls on BC. For if the equilibrium
were for a moment disturbed, it could never subsist: the cord would keep
moving round forever in the same direction,—we should have a perpetual
motion. He says:

"But if this took place, our row or ring of balls would come once more
into their original position, and from the same cause the eight globes to
the left would again be heavier than the six to the right, and therefore
those eight would sink a second time and these six rise, and all the
globes would keep up, of themselves, a continuous and unending
motion, which is false."[41]

Stevinus, now, easily derives from this principle the laws of equilibrium on
the inclined plane and numerous other fruitful consequences.
In the chapter "Hydrostatics" of the same work, page 114, Stevinus sets up
the following principle: "Aquam datam, datum sibi intra aquam locum
servare,"—a given mass of water preserves within water its given place.

This principle is demonstrated as follows (see Fig. 42):

"For, assuming it to be possible by natural means,
let us suppose that A does not preserve the place
assigned to it, but sinks down to D. This being
posited, the water which succeeds A will, for the
same reason, also flow down to D; A will be forced
out of its place in D; and thus this body of water, for
the conditions in it are everywhere the same, will set
up a perpetual motion, which is absurd."[42] Fig. 42.

From this all the principles of hydrostatics are deduced. On this occasion
Stevinus also first develops the thought so fruitful for modern analytical
mechanics that the equilibrium of a system is not destroyed by the addition
of rigid connexions. As we know, the principle of the conservation of the
centre of gravity is now sometimes deduced from D'Alembert's principle
with the help of that remark. If we were to reproduce Stevinus's
demonstration to-day, we should have to change it slightly. We find no
difficulty in imagining the cord on the prism possessed of unending uniform
motion if all hindrances are thought away, but we should protest against the
assumption of an accelerated motion or even against that of a uniform
motion, if the resistances were not removed. Moreover, for greater precision
of proof, the string of balls might be replaced by a heavy homogeneous
cord of infinite flexibility. But all this does not affect in the least the
historical value of Stevinus's thoughts. It is a fact, Stevinus deduces
apparently much simpler truths from the principle of an impossible
perpetual motion.
In the process of thought which conducted Galileo to his discoveries at the
end of the sixteenth century, the following principle plays an important part,
that a body in virtue of the velocity acquired in its descent can rise exactly
as high as it fell. This principle, which appears frequently and with much
clearness in Galileo's thought, is simply another form of the principle of
excluded perpetual motion, as we shall see it is also in Huygens.
Galileo, as we know, arrived at the law of uniformly accelerated motion by
a priori considerations, as that law which was the "simplest and most
natural," after having first assumed a different law which he was compelled

to reject. To verify his law he executed experiments with falling bodies on
inclined planes, measuring the times of descent by the weights of the water
which flowed out of a small orifice in a large vessel. In this experiment he
assumes as a fundamental principle, that the velocity acquired in descent
down an inclined plane always corresponds to the vertical height descended
through, a conclusion which for him is the immediate outcome of the fact
that a body which has fallen down one inclined plane can, with the velocity
it has acquired, rise on another plane of any inclination only to the same
vertical height. This principle of the height of ascent also led him, as it
seems, to the law of inertia. Let us hear his own masterful words in the
Dialogo terzo (Opere, Padova, 1744, Tom. III). On page 96 we read:

"I take it for granted that the velocities acquired by a body in descent
down planes of different inclinations are equal if the heights of those
planes are equal."[43]

Then he makes Salviati say in the dialogue:[44]

"What you say seems very probable, but I wish to go further and by an
experiment so to increase the probability of it that it shall amount
almost to absolute demonstration. Suppose this sheet of paper to be a
vertical wall, and from a nail driven in it a ball of lead weighing two or
three ounces to hang by a very fine thread AB four or five feet long.
(Fig. 43.) On the wall mark a horizontal line DC perpendicular to the
vertical AB, which latter ought to hang about two inches from the wall.
If now the thread AB with the ball attached take the position AC and
the ball be let go, you will see the ball first descend through the arc CB
and passing beyond B rise through the arc BD almost to the level of the
line CD, being prevented from reaching it exactly by the resistance of
the air and of the thread. From this we may truly conclude that its
impetus at the point B, acquired by its descent through the arc CB, is
sufficient to urge it through a similar arc BD to the same height.
Having performed this experiment and repeated it several times, let us
drive in the wall, in the projection of the vertical AB, as at E or at F, a
nail five or six inches long, so that the thread AC, carrying as before
the ball through the arc CB, at the moment it reaches the position AB,
shall strike the nail E, and the ball be thus compelled to move up the

arc BG described about E as centre. Then we shall see what the same
impetus will here accomplish, acquired now as before at the same
point B, which then drove the same moving body through the arc BD
to the height of the horizontal CD. Now gentlemen, you will be
pleased to see the ball rise to the horizontal line at the point G, and the
same thing also happen if the nail be placed lower as at F, in which
case the ball would describe the arc BJ, always terminating its ascent
precisely at the line CD. If the nail be placed so low that the length of
thread below it does not reach to the height of CD (which would
happen if F were nearer B than to the intersection of AB with the
horizontal CD), then the thread will wind itself about the nail. This
experiment leaves no room for doubt as to the truth of the supposition.
For as the two arcs CB, DB are equal and similarly situated, the
momentum acquired in the descent of the arc CB is the same as that
acquired in the descent of the arc DB; but the momentum acquired at B
by the descent through the arc CB is capable of driving up the same
moving body through the arc BD; hence also the momentum acquired
in the descent DB is equal to that which drives the same moving body
through the same arc from B to D, so that in general every momentum
acquired in the descent of an arc is equal to that which causes the same
moving body to ascend through the same arc; but all the momenta
which cause the ascent of all the arcs BD, BG, BJ, are equal since they
are made by the same momentum acquired in the descent CB, as the
experiment shows: therefore all the momenta acquired in the descent
of the arcs DB, GB, JB are equal."

Fig. 43.
The remark relative to the pendulum may be applied to the inclined plane
and leads to the law of inertia. We read on page 124:[45]

"It is plain now that a movable body, starting from rest at A and
descending down the inclined plane AB, acquires a velocity
proportional to the increment of its time: the velocity possessed at B is
the greatest of the velocities acquired, and by its nature immutably
impressed, provided all causes of new acceleration or retardation are
taken away: I say acceleration, having in view its possible further
progress along the plane extended; retardation, in view of the
possibility of its being reversed and made to mount the ascending
plane BC. But in the horizontal plane GH its equable motion,
according to its velocity as acquired in the descent from A to B, will be
continued ad infinitum." (Fig. 44.)

Fig. 44.
Huygens, upon whose shoulders the mantel of Galileo fell, forms a sharper
conception of the law of inertia and generalises the principle respecting the
heights of ascent which was so fruitful in Galileo's hands. He employs the
latter principle in the solution of the problem of the centre of oscillation and
is perfectly clear in the statement that the principle respecting the heights of
ascent is identical with the principle of the excluded perpetual motion.
The following important passages then occur (Hugenii, Horologium
oscillatorium, pars secunda). Hypotheses:

"If gravity did not exist, nor the atmosphere obstruct the motions of
bodies, a body would keep up forever the motion once impressed upon
it, with equable velocity, in a straight line."[46]

In part four of the Horologium de centro oscillationis we read:

"If any number of weights be set in motion by the force of gravity, the
common centre of gravity of the weights as a whole cannot possibly
rise higher than the place which it occupied when the motion began.
"That this hypothesis of ours may arouse no scruples, we will state that
it simply imports, what no one has ever denied, that heavy bodies do
not move upwards.—And truly if the devisers of the new machines
who make such futile attempts to construct a perpetual motion would
acquaint themselves with this principle, they could easily be brought to
see their errors and to understand that the thing is utterly impossible by
mechanical means."[47]

There is possibly a Jesuitical mental reservation contained in the words
"mechanical means." One might be led to believe from them that Huygens
held a non-mechanical perpetual motion for possible.
The generalisation of Galileo's principle is still more clearly put in Prop. IV
of the same chapter:

"If a pendulum, composed of several weights, set in motion from rest,
complete any part of its full oscillation, and from that point onwards,
the individual weights, with their common connexions dissolved,
change their acquired velocities upwards and ascend as far as they can,
the common centre of gravity of all will be carried up to the same
altitude with that which it occupied before the beginning of the
oscillation."[48]

On this last principle now, which is a generalisation, applied to a system of
masses, of one of Galileo's ideas respecting a single mass and which from
Huygens's explanation we recognise as the principle of excluded perpetual
motion, Huygens grounds his theory of the centre of oscillation. Lagrange
characterises this principle as precarious and is rejoiced at James Bernoulli's
successful attempt, in 1681, to reduce the theory of the centre of oscillation
to the laws of the lever, which appeared to him clearer. All the great
inquirers of the seventeenth and eighteenth centuries broke a lance on this
problem, and it led ultimately, in conjunction with the principle of virtual

velocities, to the principle enunciated by D'Alembert in 1743 in his Traité
de dynamique, though previously employed in a somewhat different form
by Euler and Hermann.
Furthermore, the Huygenian principle respecting the heights of ascent
became the foundation of the "law of the conservation of living force," as
that was enunciated by John and Daniel Bernoulli and employed with such
signal success by the latter in his Hydrodynamics. The theorems of the
Bernoullis differ in form only from Lagrange's expression in the Analytical
Mechanics.
The manner in which Torricelli reached his famous law of efflux for liquids
leads again to our principle. Torricelli assumed that the liquid which flows
out of the basal orifice of a vessel cannot by its velocity of efflux ascend to
a greater height than its level in the vessel.
Let us next consider a point which belongs to pure mechanics, the history of
the principle of virtual motions or virtual velocities. This principle was not
first enunciated, as is usually stated, and as Lagrange also asserts, by
Galileo, but earlier, by Stevinus. In his Trochleostatica of the above-cited
work, page 72, he says:

"Observe that this axiom of statics holds good here:
"As the space of the body acting is to the space of the body acted upon,
so is the power of the body acted upon to the power of the body
acting."[49]

Galileo, as we know, recognised the truth of the principle in the
consideration of the simple machines, and also deduced the laws of the
equilibrium of liquids from it.
Torricelli carries the principle back to the properties of the centre of gravity.
The condition controlling equilibrium in a simple machine, in which power
and load are represented by weights, is that the common centre of gravity of
the weights shall not sink. Conversely, if the centre of gravity cannot sink
equilibrium obtains, because heavy bodies of themselves do not move
upwards. In this form the principle of virtual velocities is identical with
Huygens's principle of the impossibility of a perpetual motion.

John Bernoulli, in 1717, first perceived the universal import of the principle
of virtual movements for all systems; a discovery stated in a letter to
Varignon. Finally, Lagrange gives a general demonstration of the principle
and founds upon it his whole Analytical Mechanics. But this general
demonstration is based after all upon Huygens and Torricelli's remarks.
Lagrange, as is known, conceives simple pulleys arranged in the directions
of the forces of the system, passes a cord through these pulleys, and
appends to its free extremity a weight which is a common measure of all the
forces of the system. With no difficulty, now, the number of elements of
each pulley may be so chosen that the forces in question shall be replaced
by them. It is then clear that if the weight at the extremity cannot sink,
equilibrium subsists, because heavy bodies cannot of themselves move
upwards. If we do not go so far, but wish to abide by Torricelli's idea, we
may conceive every individual force of the system replaced by a special
weight suspended from a cord passing over a pulley in the direction of the
force and attached at its point of application. Equilibrium subsists then
when the common centre of gravity of all the weights together cannot sink.
The fundamental supposition of this demonstration is plainly the
impossibility of a perpetual motion.
Lagrange tried in every way to supply a proof free from extraneous
elements and fully satisfactory, but without complete success. Nor were his
successors more fortunate.
The whole of mechanics, thus, is based upon an idea, which, though
unequivocal, is yet unwonted and not coequal with the other principles and
axioms of mechanics. Every student of mechanics, at some stage of his
progress, feels the uncomfortableness of this state of affairs; every one
wishes it removed; but seldom is the difficulty stated in words. Accordingly,
the zealous pupil of the science is highly rejoiced when he reads in a master
like Poinsot (Théorie générale de l'équilibre et du mouvement des systèmes)
the following passage, in which that author is giving his opinion of the
Analytical Mechanics:

"In the meantime, because our attention in that work was first wholly
engrossed with the consideration of its beautiful development of
mechanics, which seemed to spring complete from a single formula,
we naturally believed that the science was completed or that it only

remained to seek the demonstration of the principle of virtual
velocities. But that quest brought back all the difficulties that we had
overcome by the principle itself. That law so general, wherein are
mingled the vague and unfamiliar ideas of infinitely small movements
and of perturbations of equilibrium, only grew obscure upon
examination; and the work of Lagrange supplying nothing clearer than
the march of analysis, we saw plainly that the clouds had only
appeared lifted from the course of mechanics because they had, so to
speak, been gathered at the very origin of that science.
"At bottom, a general demonstration of the principle of virtual
velocities would be equivalent to the establishment of the whole of
mechanics upon a different basis: for the demonstration of a law which
embraces a whole science is neither more nor less than the reduction of
that science to another law just as general, but evident, or at least more
simple than the first, and which, consequently, would render that
useless."[50]

According to Poinsot, therefore, a proof of the principle of virtual
movements is tantamount to a total rehabilitation of mechanics.
Another circumstance of discomfort to the mathematician is, that in the
historical form in which mechanics at present exists, dynamics is founded
on statics, whereas it is desirable that in a science which pretends to
deductive completeness the more special statical theorems should be
deducible from the more general dynamical principles.
In fact, a great master, Gauss, gave expression to this desire in his
presentment of the principle of least constraint (Crelle's Journal für reine
und angewandte Mathematik, Vol. IV, p. 233) in the following words:
"Proper as it is that in the gradual development of a science, and in the
instruction of individuals, the easy should precede the difficult, the simple
the complex, the special the general, yet the mind, when once it has reached
a higher point of view, demands the contrary course, in which all statics
shall appear simply as a special case of mechanics." Gauss's own principle,
now, possesses all the requisites of universality, but its difficulty is that it is
not immediately intelligible and that Gauss deduced it with the help of
D'Alembert's principle, a procedure which left matters where they were
before.

Whence, now, is derived this strange part which the principle of virtual
motion plays in mechanics? For the present I shall only make this reply. It
would be difficult for me to tell the difference of impression which
Lagrange's proof of the principle made on me when I first took it up as a
student and when I subsequently resumed it after having made historical
researches. It first appeared to me insipid, chiefly on account of the pulleys
and the cords which did not fit in with the mathematical view, and whose
action I would much rather have discovered from the principle itself than
have taken for granted. But now that I have studied the history of the
science I cannot imagine a more beautiful demonstration.
In fact, through all mechanics it is this self-same principle of excluded
perpetual motion which accomplishes almost all, which displeased
Lagrange, but which he still had to employ, at least tacitly, in his own
demonstration. If we give this principle its proper place and setting, the
paradox is explained.
The principle of excluded perpetual motion is thus no new discovery; it has
been the guiding idea, for three hundred years, of all the great inquirers. But
the principle cannot properly be based upon mechanical perceptions. For
long before the development of mechanics the conviction of its truth existed
and even contributed to that development. Its power of conviction,
therefore, must have more universal and deeper roots. We shall revert to this
point.

II. MECHANICAL PHYSICS.

It cannot be denied that an unmistakable tendency has prevailed, from
Democritus to the present day, to explain all physical events mechanically.
Not to mention earlier obscure expressions of that tendency we read in
Huygens the following:[51]

"There can be no doubt that light consists of the motion of a certain
substance. For if we examine its production, we find that here on earth
it is principally fire and flame which engender it, both of which
contain beyond doubt bodies which are in rapid movement, since they
dissolve and destroy many other bodies more solid than they: while if
we regard its effects, we see that when light is accumulated, say by

concave mirrors, it has the property of combustion just as fire has, that
is to say, it disunites the parts of bodies, which is assuredly a proof of
motion, at least in the true philosophy, in which the causes of all
natural effects are conceived as mechanical causes. Which in my
judgment must be accomplished or all hope of ever understanding
physics renounced."[52]

S. Carnot,[53] in introducing the principle of excluded perpetual motion into
the theory of heat, makes the following apology:

"It will be objected here, perhaps, that a perpetual motion proved
impossible for purely mechanical actions, is perhaps not so when the
influence of heat or of electricity is employed. But can phenomena of
heat or electricity be thought of as due to anything else than to certain
motions of bodies, and as such must they not be subject to the general
laws of mechanics?"[54]

These examples, which might be multiplied by quotations from recent
literature indefinitely, show that a tendency to explain all things
mechanically actually exists. This tendency is also intelligible. Mechanical
events as simple motions in space and time best admit of observation and
pursuit by the help of our highly organised senses. We reproduce
mechanical processes almost without effort in our imagination. Pressure as
a circumstance that produces motion is very familiar to us from daily
experience. All changes which the individual personally produces in his
environment, or humanity brings about by means of the arts in the world,
are effected through the instrumentality of motions. Almost of necessity,
therefore, motion appears to us as the most important physical factor.
Moreover, mechanical properties may be discovered in all physical events.
The sounding bell trembles, the heated body expands, the electrified body
attracts other bodies. Why, therefore, should we not attempt to grasp all
events under their mechanical aspect, since that is so easily apprehended
and most accessible to observation and measurement? In fact, no objection
is to be made to the attempt to elucidate the properties of physical events by
mechanical analogies.
But modern physics has proceeded very far in this direction. The point of
view which Wundt represents in his excellent treatise On the Physical

Axioms is probably shared by the majority of physicists. The axioms of
physics which Wundt sets up are as follows:
1. All natural causes are motional causes.
2. Every motional cause lies outside the object moved.
3. All motional causes act in the direction of the straight line of junction,
and so forth.
4. The effect of every cause persists.
5. Every effect involves an equal countereffect.
6. Every effect is equivalent to its cause.
These principles might be studied properly enough as fundamental
principles of mechanics. But when they are set up as axioms of physics,
their enunciation is simply tantamount to a negation of all events except
motion.
According to Wundt, all changes of nature are mere changes of place. All
causes are motional causes (page 26). Any discussion of the philosophical
grounds on which Wundt supports his theory would lead us deep into the
speculations of the Eleatics and the Herbartians. Change of place, Wundt
holds, is the only change of a thing in which a thing remains identical with
itself. If a thing changed qualitatively, we should be obliged to imagine that
something was annihilated and something else created in its place, which is
not to be reconciled with our idea of the identity of the object observed and
of the indestructibility of matter. But we have only to remember that the
Eleatics encountered difficulties of exactly the same sort in motion. Can we
not also imagine that a thing is destroyed in one place and in another an
exactly similar thing created? After all, do we really know more why a body
leaves one place and appears in another, than why a cold body grows
warm? Granted that we had a perfect knowledge of the mechanical
processes of nature, could we and should we, for that reason, put out of the
world all other processes that we do not understand? On this principle it
would really be the simplest course to deny the existence of the whole
world. This is the point at which the Eleatics ultimately arrived, and the
school of Herbart stopped little short of the same goal.

Physics treated in this sense supplies us simply with a diagram of the world,
in which we do not know reality again. It happens, in fact, to men who give
themselves up to this view for many years, that the world of sense from
which they start as a province of the greatest familiarity, suddenly becomes,
in their eyes, the supreme "world-riddle."
Intelligible as it is, therefore, that the efforts of thinkers have always been
bent upon the "reduction of all physical processes to the motions of atoms,"
it must yet be affirmed that this is a chimerical ideal. This ideal has often
played an effective part in popular lectures, but in the workshop of the
serious inquirer it has discharged scarcely the least function. What has
really been achieved in mechanical physics is either the elucidation of
physical processes by more familiar mechanical analogies, (for example,
the theories of light and of electricity,) or the exact quantitative
ascertainment of the connexion of mechanical processes with other physical
processes, for example, the results of thermodynamics.

III. THE PRINCIPLE OF ENERGY IN PHYSICS.

We can know only from experience that mechanical processes produce
other physical transformations, or vice versa. The attention was first
directed to the connexion of mechanical processes, especially the
performance of work, with changes of thermal conditions by the invention
of the steam-engine, and by its great technical importance. Technical
interests and the need of scientific lucidity meeting in the mind of S. Carnot
led to the remarkable development from which thermodynamics flowed. It
is simply an accident of history that the development in question was not
connected with the practical applications of electricity.
In the determination of the maximum quantity of work that, generally, a
heat-machine, or, to take a special case, a steam-engine, can perform with
the expenditure of a given amount of heat of combustion, Carnot is guided
by mechanical analogies. A body can do work on being heated, by
expanding under pressure. But to do this the body must receive heat from a
hotter body. Heat, therefore, to do work, must pass from a hotter body to a
colder body, just as water must fall from a higher level to a lower level to
put a mill-wheel in motion. Differences of temperature, accordingly,
represent forces able to do work exactly as do differences of height in heavy

bodies. Carnot pictures to himself an ideal process in which no heat flows
away unused, that is, without doing work. With a given expenditure of heat,
accordingly, this process furnishes the maximum of work. An analogue of
the process would be a mill-wheel which scooping its water out of a higher
level would slowly carry it to a lower level without the loss of a drop. A
peculiar property of the process is, that with the expenditure of the same
work the water can be raised again exactly to its original level. This
property of reversibility is also shared by the process of Carnot. His process
also can be reversed by the expenditure of the same amount of work, and
the heat again brought back to its original temperature level.
Suppose, now, we had two different reversible processes A, B, such that in A
a quantity of heat, Q, flowing off from the temperature t1 to the lower
temperature t2 should perform the work W, but in B under the same
circumstances it should perform a greater quantity of work W + W'; then,
we could join B in the sense assigned and A in the reverse sense into a
single process. Here A would reverse the transformation of heat produced
by B and would leave a surplus of work W', produced, so to speak, from
nothing. The combination would present a perpetual motion.
With the feeling, now, that it makes little difference whether the mechanical
laws are broken directly or indirectly (by processes of heat), and convinced
of the existence of a universal law-ruled connexion of nature, Carnot here
excludes for the first time from the province of general physics the
possibility of a perpetual motion. But it follows, then, that the quantity of
work W, produced by the passage of a quantity of heat Q from a
temperature t1 to a temperature t2, is independent of the nature of the
substances as also of the character of the process, so far as that is
unaccompanied by loss, but is wholly dependent upon the temperature t1, t2.
This important principle has been fully confirmed by the special researches
of Carnot himself (1824), of Clapeyron (1834), and of Sir William
Thomson (1849), now Lord Kelvin. The principle was reached without any
assumption whatever concerning the nature of heat, simply by the exclusion
of a perpetual motion. Carnot, it is true, was an adherent of the theory of
Black, according to which the sum-total of the quantity of heat in the world
is constant, but so far as his investigations have been hitherto considered the
decision on this point is of no consequence. Carnot's principle led to the

most remarkable results. W. Thomson (1848) founded upon it the ingenious
idea of an "absolute" scale of temperature. James Thomson (1849)
conceived a Carnot process to take place with water freezing under pressure
and, therefore, performing work. He discovered, thus, that the freezing
point is lowered 0·0075° Celsius by every additional atmosphere of
pressure. This is mentioned merely as an example.
About twenty years after the publication of Carnot's book a further advance
was made by J. R. Mayer and J. P. Joule. Mayer, while engaged as a
physician in the service of the Dutch, observed, during a process of
bleeding in Java, an unusual redness of the venous blood. In agreement with
Liebig's theory of animal heat he connected this fact with the diminished
loss of heat in warmer climates, and with the diminished expenditure of
organic combustibles. The total expenditure of heat of a man at rest must be
equal to the total heat of combustion. But since all organic actions, even the
mechanical actions, must be set down to the credit of the heat of
combustion, some connexion must exist between mechanical work and
expenditure of heat.
Joule started from quite similar convictions concerning the galvanic battery.
A heat of association equivalent to the consumption of the zinc can be made
to appear in the galvanic cell. If a current is set up, a part of this heat
appears in the conductor of the current. The interposition of an apparatus
for the decomposition of water causes a part of this heat to disappear, which
on the burning of the explosive gas formed, is reproduced. If the current
runs an electromotor, a portion of the heat again disappears, which, on the
consumption of the work by friction, again makes its appearance.
Accordingly, both the heat produced and the work produced, appeared to
Joule also as connected with the consumption of material. The thought was
therefore present, both to Mayer and to Joule, of regarding heat and work as
equivalent quantities, so connected with each other that what is lost in one
form universally appears in another. The result of this was a substantial
conception of heat and of work, and ultimately a substantial conception of
energy. Here every physical change of condition is regarded as energy, the
destruction of which generates work or equivalent heat. An electric charge,
for example, is energy.
In 1842 Mayer had calculated from the physical constants then universally
accepted that by the disappearance of one kilogramme-calorie 365

kilogramme-metres of work could be performed, and vice versa. Joule, on
the other hand, by a long series of delicate and varied experiments
beginning in 1843 ultimately determined the mechanical equivalent of the
kilogramme-calorie, more exactly, as 425 kilogramme-metres.
If we estimate every change of physical condition by the mechanical work
which can be performed upon the disappearance of that condition, and call
this measure energy, then we can measure all physical changes of condition,
no matter how different they may be, with the same common measure, and
say: the sum-total of all energy remains constant. This is the form that the
principle of excluded perpetual motion received at the hands of Mayer,
Joule, Helmholtz, and W. Thomson in its extension to the whole domain of
physics.
After it had been proved that heat must disappear if mechanical work was
to be done at its expense, Carnot's principle could no longer be regarded as
a complete expression of the facts. Its improved form was first given, in
1850, by Clausius, whom Thomson followed in 1851. It runs thus: "If a
quantity of heat Q' is transformed into work in a reversible process, another
quantity of heat Q of the absolute[55] temperature T1 is lowered to the
absolute temperature T2." Here Q' is dependent only on Q, T1, T2, but is
independent of the substances used and of the character of the process, so
far as that is unaccompanied by loss. Owing to this last fact, it is sufficient
to find the relation which obtains for some one well-known physical
substance, say a gas, and some definite simple process. The relation found
will be the one that holds generally. We get, thus,
Q'/(Q' + Q) = (T1-T2)/T1 (1)
that is, the quotient of the available heat Q' transformed into work divided
by the sum of the transformed and transferred heats (the total sum used), the
so-called economical coefficient of the process, is,
(T1-T2)/T1.

IV. THE CONCEPTIONS OF HEAT.

When a cold body is put in contact with a warm body it is observed that the
first body is warmed and that the second body is cooled. We may say that

the first body is warmed at the expense of the second body. This suggests
the notion of a thing, or heat-substance, which passes from the one body to
the other. If two masses of water m, m', of unequal temperatures, be put
together, it will be found, upon the rapid equalisation of the temperatures,
that the respective changes of temperatures u and u' are inversely
proportional to the masses and of opposite signs, so that the algebraical sum
of the products is,
mu + m'u' = 0.
Black called the products mu, m'u', which are decisive for our knowledge of
the process, quantities of heat. We may form a very clear picture of these
products by conceiving them with Black as measures of the quantities of
some substance. But the essential thing is not this picture but the constancy
of the sum of these products in simple processes of conduction. If a quantity
of heat disappears at one point, an equally large quantity will make its
appearance at some other point. The retention of this idea leads to the
discovery of specific heat. Black, finally, perceives that also something else
may appear for a vanished quantity of heat, namely: the fusion or
vaporisation of a definite quantity of matter. He adheres here still to this
favorite view, though with some freedom, and considers the vanished
quantity of heat as still present, but as latent.
The generally accepted notion of a caloric, or heat-stuff, was strongly
shaken by the work of Mayer and Joule. If the quantity of heat can be
increased and diminished, people said, heat cannot be a substance, but must
be a motion. The subordinate part of this statement has become much more
popular than all the rest of the doctrine of energy. But we may convince
ourselves that the motional conception of heat is now as unessential as was
formerly its conception as a substance. Both ideas were favored or impeded
solely by accidental historical circumstances. It does not follow that heat is
not a substance from the fact that a mechanical equivalent exists for
quantity of heat. We will make this clear by the following question which
bright students have sometimes put to me. Is there a mechanical equivalent
of electricity as there is a mechanical equivalent of heat? Yes, and no. There
is no mechanical equivalent of quantity of electricity as there is an
equivalent of quantity of heat, because the same quantity of electricity has a
very different capacity for work, according to the circumstances in which it
is placed; but there is a mechanical equivalent of electrical energy.

Let us ask another question. Is there a mechanical equivalent of water? No,
there is no mechanical equivalent of quantity of water, but there is a
mechanical equivalent of weight of water multiplied by its distance of
descent.
When a Leyden jar is discharged and work thereby performed, we do not
picture to ourselves that the quantity of electricity disappears as work is
done, but we simply assume that the electricities come into different
positions, equal quantities of positive and negative electricity being united
with one another.
What, now, is the reason of this difference of view in our treatment of heat
and of electricity? The reason is purely historical, wholly conventional, and,
what is still more important, is wholly indifferent. I may be allowed to
establish this assertion.
In 1785 Coulomb constructed his torsion balance, by which he was enabled
to measure the repulsion of electrified bodies. Suppose we have two small
balls, A, B, which over their whole extent are similarly electrified. These
two balls will exert on one another, at a certain distance r of their centres, a
certain repulsion p. We bring into contact with B now a ball C, suffer both
to be equally electrified, and then measure the repulsion of B from A and of
C from A at the same distance r. The sum of these repulsions is again p.
Accordingly something has remained constant. If we ascribe this effect to a
substance, then we infer naturally its constancy. But the essential point of
the exposition is the divisibility of the electric force p and not the simile of
substance.
In 1838 Riess constructed his electrical air-thermometer (the
thermoelectrometer). This gives a measure of the quantity of heat produced
by the discharge of jars. This quantity of heat is not proportional to the
quantity of electricity contained in the jar by Coulomb's measure, but if Q
be this quantity and C be the capacity, is proportional to Q2/2C, or, more
simply still, to the energy of the charged jar. If, now, we discharge the jar
completely through the thermometer, we obtain a certain quantity of heat,
W. But if we make the discharge through the thermometer into a second jar,
we obtain a quantity less than W. But we may obtain the remainder by
completely discharging both jars through the air-thermometer, when it will

again be proportional to the energy of the two jars. On the first, incomplete
discharge, accordingly, a part of the electricity's capacity for work was lost.
When the charge of a jar produces heat its energy is changed and its value
by Riess's thermometer is decreased. But by Coulomb's measure the
quantity remains unaltered.
Now let us imagine that Riess's thermometer had been invented before
Coulomb's torsion balance, which is not a difficult feat, since both
inventions are independent of each other; what would be more natural than
that the "quantity" of electricity contained in a jar should be measured by
the heat produced in the thermometer? But then, this so-called quantity of
electricity would decrease on the production of heat or on the performance
of work, whereas it now remains unchanged; in that case, therefore,
electricity would not be a substance but a motion, whereas now it is still a
substance. The reason, therefore, why we have other notions of electricity
than we have of heat, is purely historical, accidental, and conventional.
This is also the case with other physical things. Water does not disappear
when work is done. Why? Because we measure quantity of water with
scales, just as we do electricity. But suppose the capacity of water for work
were called quantity, and had to be measured, therefore, by a mill instead of
by scales; then this quantity also would disappear as it performed the work.
It may, now, be easily conceived that many substances are not so easily got
at as water. In that case we should be unable to carry out the one kind of
measurement with the scales whilst many other modes of measurement
would still be left us.
In the case of heat, now, the historically established measure of "quantity" is
accidentally the work-value of the heat. Accordingly, its quantity disappears
when work is done. But that heat is not a substance follows from this as
little as does the opposite conclusion that it is a substance. In Black's case
the quantity of heat remains constant because the heat passes into no other
form of energy.
If any one to-day should still wish to think of heat as a substance, we might
allow that person this liberty with little ado. He would only have to assume
that that which we call quantity of heat was the energy of a substance whose
quantity remained unaltered, but whose energy changed. In point of fact we

might much better say, in analogy with the other terms of physics, energy of
heat, instead of quantity of heat.
When we wonder, therefore, at the discovery that heat is motion, we wonder
at something that was never discovered. It is perfectly indifferent and
possesses not the slightest scientific value, whether we think of heat as a
substance or not. The fact is, heat behaves in some connexions like a
substance, in others not. Heat is latent in steam as oxygen is latent in water.

V. THE CONFORMITY IN THE DEPORTMENT OF THE
ENERGIES.

The foregoing reflexions will gain in lucidity from a consideration of the
conformity which obtains in the behavior of all energies, a point to which I
called attention long ago.[56]
A weight P at a height H1 represents an energy W1 = PH1. If we suffer the
weight to sink to a lower height H2, during which work is done, and the
work done is employed in the production of living force, heat, or an electric
charge, in short, is transformed, then the energy W2 = PH2 is still left. The
equation subsists
W1/H1 = W2/H2, (2) or, denoting the transformed energy by W' = W1-W2 and
the transferred energy, that transported to the lower level, by W = W2,
W'/(W' + W) = (H1-H2)/H1, (3)
an equation in all respects analogous to equation (1) at page 165. The
property in question, therefore, is by no means peculiar to heat. Equation
(2) gives the relation between the energy taken from the higher level and
that deposited on the lower level (the energy left behind); it says that these
energies are proportional to the heights of the levels. An equation analogous
to equation (2) may be set up for every form of energy; hence the equation
which corresponds to equation (3), and so to equation (1), may be regarded
as valid for every form. For electricity, for example, H1, H2 signify the
potentials.
When we observe for the first time the agreement here indicated in the
transformative law of the energies, it appears surprising and unexpected, for

we do not perceive at once its reason. But to him who pursues the
comparative historical method that reason will not long remain a secret.
Since Galileo, mechanical work, though long under a different name, has
been a fundamental concept of mechanics, as also a very important notion
in the applied sciences. The transformation of work into living force, and of
living force into work, suggests directly the notion of energy—the idea
having been first fruitfully employed by Huygens, although Thomas Young
first called it by the name of "energy." Let us add to this the constancy of
weight (really the constancy of mass) and we shall see that with respect to
mechanical energy it is involved in the very definition of the term that the
capacity for work or the potential energy of a weight is proportional to the
height of the level at which it is, in the geometrical sense, and that it
decreases on the lowering of the weight, on transformation, proportionally
to the height of the level. The zero level here is wholly arbitrary. With this,
equation (2) is given, from which all the other forms follow.
When we reflect on the tremendous start which mechanics had over the
other branches of physics, it is not to be wondered at that the attempt was
always made to apply the notions of that science wherever this was
possible. Thus the notion of mass, for example, was imitated by Coulomb in
the notion of quantity of electricity. In the further development of the theory
of electricity, the notion of work was likewise immediately introduced in
the theory of potential, and heights of electrical level were measured by the
work of unit of quantity raised to that level. But with this the preceding
equation with all its consequences is given for electrical energy. The case
with the other energies was similar.
Thermal energy, however, appears as a special case. Only by the peculiar
experiments mentioned could it be discovered that heat is an energy. But the
measure of this energy by Black's quantity of heat is the outcome of
fortuitous circumstances. In the first place, the accidental slight variability
of the capacity for heat c with the temperature, and the accidental slight
deviation of the usual thermometrical scales from the scale derived from the
tensions of gases, brings it about that the notion "quantity of heat" can be
set up and that the quantity of heat ct corresponding to a difference of
temperature t is nearly proportional to the energy of the heat. It is a quite
accidental historical circumstance that Amontons hit upon the idea of
measuring temperature by the tension of a gas. It is certain in this that he

did not think of the work of the heat.[57] But the numbers standing for
temperature, thus, are made proportional to the tensions of gases, that is, to
the work done by gases, with otherwise equal changes of volume. It thus
happens that temperature heights and level heights of work are proportional
to one another.
If properties of the thermal condition varying greatly from the tensions of
gases had been chosen, this relation would have assumed very complicated
forms, and the agreement between heat and the other energies above
considered would not subsist. It is very instructive to reflect upon this point.
A natural law, therefore, is not implied in the conformity of the behavior of
the energies, but this conformity is rather conditioned by the uniformity of
our modes of conception and is also partly a matter of good fortune.

VI. THE DIFFERENCES OF THE ENERGIES AND THE
LIMITS OF THE PRINCIPLE OF ENERGY.

Of every quantity of heat Q which does work in a reversible process (one
unaccompanied by loss) between the absolute temperatures T1, T2, only the
portion
(T1-T2)/T1
is transformed into work, while the remainder is transferred to the lower
temperature-level T2. This transferred portion can, upon the reversal of the
process, with the same expenditure of work, again be brought back to the
level T1. But if the process is not reversible, then more heat than in the
foregoing case flows to the lower level, and the surplus can no longer be
brought back to the higher level T2 without some special expenditure. W.
Thomson (1852), accordingly, drew attention to the fact, that in all non-
reversible, that is, in all real thermal processes, quantities of heat are lost for
mechanical work, and that accordingly a dissipation or waste of mechanical
energy is taking place. In all cases, heat is only partially transformed into
work, but frequently work is wholly transformed into heat. Hence, a
tendency exists towards a diminution of the mechanical energy and towards
an increase of the thermal energy of the world.

For a simple, closed cyclical process, accompanied by no loss, in which the
quantity of heat Q_{1} is taken from the level T_{1}, and the quantity
Q_{2} is deposited upon the level T_{2}, the following relation, agreeably
to equation (2), exists,
-(Q1/T1) + (Q2/T2) = 0.
Similarly, for any number of compound reversible cycles Clausius finds the
algebraical sum
ΣQ/T = 0,
and supposing the temperature to change continuously,
∫dQ/T = 0 (4)
Here the elements of the quantities of heat deducted from a given level are
reckoned negative, and the elements imparted to it, positive. If the process
is not reversible, then expression (4), which Clausius calls entropy,
increases. In actual practice this is always the case, and Clausius finds
himself led to the statement:
1. That the energy of the world remains constant.
2. That the entropy of the world tends toward a maximum.
Once we have noted the above-indicated conformity in the behavior of
different energies, the peculiarity of thermal energy here mentioned must
strike us. Whence is this peculiarity derived, for, generally every energy
passes only partly into another form, which is also true of thermal energy?
The explanation will be found in the following.
Every transformation of a special kind of energy A is accompanied with a
fall of potential of that particular kind of energy, including heat. But whilst
for the other kinds of energy a transformation and therefore a loss of energy
on the part of the kind sinking in potential is connected with the fall of the
potential, with heat the case is different. Heat can suffer a fall of potential
without sustaining a loss of energy, at least according to the customary
mode of estimation. If a weight sinks, it must create perforce kinetic energy,
or heat, or some other form of energy. Also, an electrical charge cannot
suffer a fall of potential without loss of energy, i. e., without transformation.
But heat can pass with a fall of temperature to a body of greater capacity

and the same thermal energy still be preserved, so long as we regard every
quantity of heat as energy. This it is that gives to heat, besides its property
of energy, in many cases the character of a material substance, or quantity.
If we look at the matter in an unprejudiced light, we must ask if there is any
scientific sense or purpose in still considering as energy a quantity of heat
that can no longer be transformed into mechanical work, (for example, the
heat of a closed equably warmed material system). The principle of energy
certainly plays in this case a wholly superfluous rôle, which is assigned to it
only from habit.[58] To maintain the principle of energy in the face of a
knowledge of the dissipation or waste of mechanical energy, in the face of
the increase of entropy is equivalent almost to the liberty which Black took
when he regarded the heat of liquefaction as still present but latent.[59] It is
to be remarked further, that the expressions "energy of the world" and
"entropy of the world" are slightly permeated with scholasticism. Energy
and entropy are metrical notions. What meaning can there be in applying
these notions to a case in which they are not applicable, in which their
values are not determinable?
If we could really determine the entropy of the world it would represent a
true, absolute measure of time. In this way is best seen the utter tautology of
a statement that the entropy of the world increases with the time. Time, and
the fact that certain changes take place only in a definite sense, are one and
the same thing.

VII. THE SOURCES OF THE PRINCIPLE OF ENERGY.

We are now prepared to answer the question, What are the sources of the
principle of energy? All knowledge of nature is derived in the last instance
from experience. In this sense they are right who look upon the principle of
energy as a result of experience.
Experience teaches that the sense-elements αβγδ... into which the world
may be decomposed, are subject to change. It tells us further, that certain of
these elements are connected with other elements, so that they appear and
disappear together; or, that the appearance of the elements of one class is
connected with the disappearance of the elements of the other class. We will
avoid here the notions of cause and effect because of their obscurity and

equivocalness. The result of experience may be expressed as follows: The
sensuous elements of the world (αβγδ...) show themselves to be
interdependent. This interdependence is best represented by some such
conception as is in geometry that of the mutual dependence of the sides and
angles of a triangle, only much more varied and complex.
As an example, we may take a mass of gas enclosed in a cylinder and
possessed of a definite volume (α), which we change by a pressure (β) on
the piston, at the same time feeling the cylinder with our hand and receiving
a sensation of heat (γ). Increase of pressure diminishes the volume and
increases the sensation of heat.
The various facts of experience are not in all respects alike. Their common
sensuous elements are placed in relief by a process of abstraction and thus
impressed upon the memory. In this way the expression is obtained of the
features of agreement of extensive groups of facts. The simplest sentence
which we can utter is, by the very nature of language, an abstraction of this
kind. But account must also be taken of the differences of related facts.
Facts may be so nearly related as to contain the same kind of a αβγ..., but
the relation be such that the αβγ... of the one differ from the αβγ... of the
other only by the number of equal parts into which they can be divided.
Such being the case, if rules can be given for deducing from one another the
numbers which are the measures of these αβγ..., then we possess in such
rules the most general expression of a group of facts, as also that expression
which corresponds to all its differences. This is the goal of quantitative
investigation.
If this goal be reached what we have found is that between the αβγ... of a
group of facts, or better, between the numbers which are their measures, a
number of equations exists. The simple fact of change brings it about that
the number of these equations must be smaller than the number of the
αβγ.... If the former be smaller by one than the latter, then one portion of the
αβγ... is uniquely determined by the other portion.
The quest of relations of this last kind is the most important function of
special experimental research, because we are enabled by it to complete in
thought facts that are only partly given. It is self-evident that only
experience can ascertain that between the αβγ... relations exist and of what
kind they are. Further, only experience can tell that the relations that exist

between the αβγ... are such that changes of them can be reversed. If this
were not the fact all occasion for the enunciation of the principle of energy,
as is easily seen, would be wanting. In experience, therefore, is buried the
ultimate well-spring of all knowledge of nature, and consequently, in this
sense, also the ultimate source of the principle of energy.
But this does not exclude the fact that the principle of energy has also a
logical root, as will now be shown. Let us assume on the basis of
experience that one group of sensuous elements αβγ... determines uniquely
another group λμν.... Experience further teaches that changes of αβγ... can
be reversed. It is then a logical consequence of this observation, that every
time that αβγ... assume the same values this is also the case with λμν.... Or,
that purely periodical changes of αβγ... can produce no permanent changes
of λμν.... If the group λμν... is a mechanical group, then a perpetual motion
is excluded.
It will be said that this is a vicious circle, which we will grant. But
psychologically, the situation is essentially different, whether I think simply
of the unique determination and reversibility of events, or whether I exclude
a perpetual motion. The attention takes in the two cases different directions
and diffuses light over different sides of the question, which logically of
course are necessarily connected.
Surely that firm, logical setting of the thoughts noticeable in the great
inquirers, Stevinus, Galileo, and the rest, which, consciously or
instinctively, was supported by a fine feeling for the slightest
contradictions, has no other purpose than to limit the bounds of thought and
so exempt it from the possibility of error. In this, therefore, the logical root
of the principle of excluded perpetual motion is given, namely, in that
universal conviction which existed even before the development of
mechanics and co-operated in that development.
It is perfectly natural that the principle of excluded perpetual motion should
have been first developed in the simple domain of pure mechanics. Towards
the transference of that principle into the domain of general physics the idea
contributed much that all physical phenomena are mechanical phenomena.
But the foregoing discussion shows how little essential this notion is. The
issue really involved is the recognition of a general interconnexion of

nature. This once established, we see with Carnot that it is indifferent
whether the mechanical laws are broken directly or circuitously.
The principle of the excluded perpetual motion is very closely related to the
modern principle of energy, but it is not identical with it, for the latter is to
be deduced from the former only by means of a definite formal conception.
As may be seen from the preceding exposition, the perpetual motion can be
excluded without our employing or possessing the notion of work. The
modern principle of energy results primarily from a substantial conception
of work and of every change of physical condition which by being reversed
produces work. The strong need of such a conception, which is by no means
necessary, but in a formal sense is very convenient and lucid, is exhibited in
the case of J. R. Mayer and Joule. It was before remarked that this
conception was suggested to both inquirers by the observation that both the
production of heat and the production of mechanical work were connected
with an expenditure of substance. Mayer says: "Ex nihilo nil fit," and in
another place, "The creation or destruction of a force (work) lies without the
province of human activity." In Joule we find this passage: "It is manifestly
absurd to suppose that the powers with which God has endowed matter can
be destroyed."
Some writers have observed in such statements the attempt at a
metaphysical establishment of the doctrine of energy. But we see in them
simply the formal need of a simple, clear, and living grasp of the facts,
which receives its development in practical and technical life, and which we
carry over, as best we can, into the province of science. As a fact, Mayer
writes to Griesinger: "If, finally, you ask me how I became involved in the
whole affair, my answer is simply this: Engaged during a sea voyage almost
exclusively with the study of physiology, I discovered the new theory for
the sufficient reason that I vividly felt the need of it."
The substantial conception of work (energy) is by no means a necessary
one. And it is far from true that the problem is solved with the recognition
of the need of such a conception. Rather let us see how Mayer gradually
endeavored to satisfy that need. He first regards quantity of motion, or
momentum, mv, as the equivalent of work, and did not light, until later, on
the notion of living force (mv2/2). In the province of electricity he was
unable to assign the expression which is the equivalent of work. This was

done later by Helmholtz. The formal need, therefore, is first present, and
our conception of nature is subsequently gradually adapted to it.
The laying bare of the experimental, logical, and formal root of the present
principle of energy will perhaps contribute much to the removal of the
mysticism which still clings to this principle. With respect to our formal
need of a very simple, palpable, substantial conception of the processes in
our environment, it remains an open question how far nature corresponds to
that need, or how far we can satisfy it. In one phase of the preceding
discussions it would seem as if the substantial notion of the principle of
energy, like Black's material conception of heat, has its natural limits in
facts, beyond which it can only be artificially adhered to.

THE ECONOMICAL NATURE OF PHYSICAL
INQUIRY.[60]
When the human mind, with its limited powers, attempts to mirror in itself
the rich life of the world, of which it is itself only a small part, and which it
can never hope to exhaust, it has every reason for proceeding economically.
Hence that tendency, expressed in the philosophy of all times, to compass
by a few organic thoughts the fundamental features of reality. "Life
understands not death, nor death life." So spake an old Chinese philosopher.
Yet in his unceasing desire to diminish the boundaries of the
incomprehensible, man has always been engaged in attempts to understand
death by life and life by death.
Among the ancient civilised peoples, nature was filled with demons and
spirits having the feelings and desires of men. In all essential features, this
animistic view of nature, as Tylor[61] has aptly termed it, is shared in
common by the fetish-worshipper of modern Africa and the most advanced
nations of antiquity. As a theory of the world it has never completely
disappeared. The monotheism of the Christians never fully overcame it, no
more than did that of the Jews. In the belief in witchcraft and in the
superstitions of the sixteenth and seventeenth centuries, the centuries of the
rise of natural science, it assumed frightful pathological dimensions. Whilst
Stevinus, Kepler, and Galileo were slowly rearing the fabric of modern
physical science, a cruel and relentless war was waged with firebrand and
rack against the devils that glowered from every corner. To-day even, apart
from all survivals of that period, apart from the traces of fetishism which
still inhere in our physical concepts,[62] those very ideas still covertly lurk
in the practices of modern spiritualism.
By the side of this animistic conception of the world, we meet from time to
time, in different forms, from Democritus to the present day, another view,
which likewise claims exclusive competency to comprehend the universe.
This view may be characterised as the physico-mechanical view of the
world. To-day, that view holds, indisputably, the first place in the thoughts

of men, and determines the ideals and the character of our times. The
coming of the mind of man into the full consciousness of its powers, in the
eighteenth century, was a period of genuine disillusionment. It produced the
splendid precedent of a life really worthy of man, competent to overcome
the old barbarism in the practical fields of life; it created the Critique of
Pure Reason, which banished into the realm of shadows the sham-ideas of
the old metaphysics; it pressed into the hands of the mechanical philosophy
the reins which it now holds.
The oft-quoted words of the great Laplace,[63] which I will now give, have
the ring of a jubilant toast to the scientific achievements of the eighteenth
century: "A mind to which were given for a single instant all the forces of
nature and the mutual positions of all its masses, if it were otherwise
powerful enough to subject these problems to analysis, could grasp, with a
single formula, the motions of the largest masses as well as of the smallest
atoms; nothing would be uncertain for it; the future and the past would lie
revealed before its eyes." In writing these words, Laplace, as we know, had
also in mind the atoms of the brain. That idea has been expressed more
forcibly still by some of his followers, and it is not too much to say that
Laplace's ideal is substantially that of the great majority of modern
scientists.
Gladly do we accord to the creator of the Mécanique céleste the sense of
lofty pleasure awakened in him by the great success of the Enlightenment,
to which we too owe our intellectual freedom. But to-day, with minds
undisturbed and before new tasks, it becomes physical science to secure
itself against self-deception by a careful study of its character, so that it can
pursue with greater sureness its true objects. If I step, therefore, beyond the
narrow precincts of my specialty in this discussion, to trespass on friendly
neighboring domains, I may plead in my excuse that the subject-matter of
knowledge is common to all domains of research, and that fixed, sharp lines
of demarcation cannot be drawn.
The belief in occult magic powers of nature has gradually died away, but in
its place a new belief has arisen, the belief in the magical power of science.
Science throws her treasures, not like a capricious fairy into the laps of a
favored few, but into the laps of all humanity, with a lavish extravagance
that no legend ever dreamt of! Not without apparent justice, therefore, do
her distant admirers impute to her the power of opening up unfathomable

abysses of nature, to which the senses cannot penetrate. Yet she who came
to bring light into the world, can well dispense with the darkness of
mystery, and with pompous show, which she needs neither for the
justification of her aims nor for the adornment of her plain achievements.
The homely beginnings of science will best reveal to us its simple,
unchangeable character. Man acquires his first knowledge of nature half-
consciously and automatically, from an instinctive habit of mimicking and
forecasting facts in thought, of supplementing sluggish experience with the
swift wings of thought, at first only for his material welfare. When he hears
a noise in the underbrush he constructs there, just as the animal does, the
enemy which he fears; when he sees a certain rind he forms mentally the
image of the fruit which he is in search of; just as we mentally associate a
certain kind of matter with a certain line in the spectrum or an electric spark
with the friction of a piece of glass. A knowledge of causality in this form
certainly reaches far below the level of Schopenhauer's pet dog, to whom it
was ascribed. It probably exists in the whole animal world, and confirms
that great thinker's statement regarding the will which created the intellect
for its purposes. These primitive psychical functions are rooted in the
economy of our organism not less firmly than are motion and digestion.
Who would deny that we feel in them, too, the elemental power of a long
practised logical and physiological activity, bequeathed to us as an heirloom
from our forefathers?
Such primitive acts of knowledge constitute to-day the solidest foundation
of scientific thought. Our instinctive knowledge, as we shall briefly call it,
by virtue of the conviction that we have consciously and intentionally
contributed nothing to its formation, confronts us with an authority and
logical power which consciously acquired knowledge even from familiar
sources and of easily tested fallibility can never possess. All so-called
axioms are such instinctive knowledge. Not consciously gained knowledge
alone, but powerful intellectual instinct, joined with vast conceptive
powers, constitute the great inquirer. The greatest advances of science have
always consisted in some successful formulation, in clear, abstract, and
communicable terms, of what was instinctively known long before, and of
thus making it the permanent property of humanity. By Newton's principle
of the equality of pressure and counterpressure, whose truth all before him
had felt, but which no predecessor had abstractly formulated, mechanics

was placed by a single stroke on a higher level. Our statement might also be
historically justified by examples from the scientific labors of Stevinus, S.
Carnot, Faraday, J. R. Mayer, and others.
All this, however, is merely the soil from which science starts. The first real
beginnings of science appear in society, particularly in the manual arts,
where the necessity for the communication of experience arises. Here,
where some new discovery is to be described and related, the compulsion is
first felt of clearly defining in consciousness the important and essential
features of that discovery, as many writers can testify. The aim of
instruction is simply the saving of experience; the labor of one man is made
to take the place of that of another.
The most wonderful economy of communication is found in language.
Words are comparable to type, which spare the repetition of written signs
and thus serve a multitude of purposes; or to the few sounds of which our
numberless different words are composed. Language, with its helpmate,
conceptual thought, by fixing the essential and rejecting the unessential,
constructs its rigid pictures of the fluid world on the plan of a mosaic, at a
sacrifice of exactness and fidelity but with a saving of tools and labor. Like
a piano-player with previously prepared sounds, a speaker excites in his
listener thoughts previously prepared, but fitting many cases, which respond
to the speaker's summons with alacrity and little effort.
The principles which a prominent political economist, E. Hermann,[64] has
formulated for the economy of the industrial arts, are also applicable to the
ideas of common life and of science. The economy of language is
augmented, of course, in the terminology of science. With respect to the
economy of written intercourse there is scarcely a doubt that science itself
will realise that grand old dream of the philosophers of a Universal Real
Character. That time is not far distant. Our numeral characters, the symbols
of mathematical analysis, chemical symbols, and musical notes, which
might easily be supplemented by a system of color-signs, together with
some phonetic alphabets now in use, are all beginnings in this direction.
The logical extension of what we have, joined with a use of the ideas which
the Chinese ideography furnishes us, will render the special invention and
promulgation of a Universal Character wholly superfluous.

The communication of scientific knowledge always involves description,
that is, a mimetic reproduction of facts in thought, the object of which is to
replace and save the trouble of new experience. Again, to save the labor of
instruction and of acquisition, concise, abridged description is sought. This
is really all that natural laws are. Knowing the value of the acceleration of
gravity, and Galileo's laws of descent, we possess simple and compendious
directions for reproducing in thought all possible motions of falling bodies.
A formula of this kind is a complete substitute for a full table of motions of
descent, because by means of the formula the data of such a table can be
easily constructed at a moment's notice without the least burdening of the
memory.
No human mind could comprehend all the individual cases of refraction.
But knowing the index of refraction for the two media presented, and the
familiar law of the sines, we can easily reproduce or fill out in thought
every conceivable case of refraction. The advantage here consists in the
disburdening of the memory; an end immensely furthered by the written
preservation of the natural constants. More than this comprehensive and
condensed report about facts is not contained in a natural law of this sort. In
reality, the law always contains less than the fact itself, because it does not
reproduce the fact as a whole but only in that aspect of it which is important
for us, the rest being either intentionally or from necessity omitted. Natural
laws may be likened to intellectual type of a higher order, partly movable,
partly stereotyped, which last on new editions of experience may become
downright impediments.
When we look over a province of facts for the first time, it appears to us
diversified, irregular, confused, full of contradictions. We first succeed in
grasping only single facts, unrelated with the others. The province, as we
are wont to say, is not clear. By and by we discover the simple, permanent
elements of the mosaic, out of which we can mentally construct the whole
province. When we have reached a point where we can discover
everywhere the same facts, we no longer feel lost in this province; we
comprehend it without effort; it is explained for us.
Let me illustrate this by an example. As soon as we have grasped the fact of
the rectilinear propagation of light, the regular course of our thoughts
stumbles at the phenomena of refraction and diffraction. As soon as we
have cleared matters up by our index of refraction we discover that a special

index is necessary for each color. Soon after we have accustomed ourselves
to the fact that light added to light increases its intensity, we suddenly come
across a case of total darkness produced by this cause. Ultimately, however,
we see everywhere in the overwhelming multifariousness of optical
phenomena the fact of the spatial and temporal periodicity of light, with its
velocity of propagation dependent on the medium and the period. This
tendency of obtaining a survey of a given province with the least
expenditure of thought, and of representing all its facts by some one single
mental process, may be justly termed an economical one.
The greatest perfection of mental economy is attained in that science which
has reached the highest formal development, and which is widely employed
in physical inquiry, namely, in mathematics. Strange as it may sound, the
power of mathematics rests upon its evasion of all unnecessary thought and
on its wonderful saving of mental operations. Even those arrangement-signs
which we call numbers are a system of marvellous simplicity and economy.
When we employ the multiplication-table in multiplying numbers of several
places, and so use the results of old operations of counting instead of
performing the whole of each operation anew; when we consult our table of
logarithms, replacing and saving thus new calculations by old ones already
performed; when we employ determinants instead of always beginning
afresh the solution of a system of equations; when we resolve new integral
expressions into familiar old integrals; we see in this simply a feeble
reflexion of the intellectual activity of a Lagrange or a Cauchy, who, with
the keen discernment of a great military commander, substituted for new
operations whole hosts of old ones. No one will dispute me when I say that
the most elementary as well as the highest mathematics are economically-
ordered experiences of counting, put in forms ready for use.
In algebra we perform, as far as possible, all numerical operations which
are identical in form once for all, so that only a remnant of work is left for
the individual case. The use of the signs of algebra and analysis, which are
merely symbols of operations to be performed, is due to the observation that
we can materially disburden the mind in this way and spare its powers for
more important and more difficult duties, by imposing all mechanical
operations upon the hand. One result of this method, which attests its
economical character, is the construction of calculating machines. The
mathematician Babbage, the inventor of the difference-engine, was

probably the first who clearly perceived this fact, and he touched upon it,
although only cursorily, in his work, The Economy of Manufactures and
Machinery.
The student of mathematics often finds it hard to throw off the
uncomfortable feeling that his science, in the person of his pencil, surpasses
him in intelligence,—an impression which the great Euler confessed he
often could not get rid of. This feeling finds a sort of justification when we
reflect that the majority of the ideas we deal with were conceived by others,
often centuries ago. In great measure it is really the intelligence of other
people that confronts us in science. The moment we look at matters in this
light, the uncanniness and magical character of our impressions cease,
especially when we remember that we can think over again at will any one
of those alien thoughts.
Physics is experience, arranged in economical order. By this order not only
is a broad and comprehensive view of what we have rendered possible, but
also the defects and the needful alterations are made manifest, exactly as in
a well-kept household. Physics shares with mathematics the advantages of
succinct description and of brief, compendious definition, which precludes
confusion, even in ideas where, with no apparent burdening of the brain,
hosts of others are contained. Of these ideas the rich contents can be
produced at any moment and displayed in their full perceptual light. Think
of the swarm of well-ordered notions pent up in the idea of the potential. Is
it wonderful that ideas containing so much finished labor should be easy to
work with?
Our first knowledge, thus, is a product of the economy of self-preservation.
By communication, the experience of many persons, individually acquired
at first, is collected in one. The communication of knowledge and the
necessity which every one feels of managing his stock of experience with
the least expenditure of thought, compel us to put our knowledge in
economical forms. But here we have a clue which strips science of all its
mystery, and shows us what its power really is. With respect to specific
results it yields us nothing that we could not reach in a sufficiently long
time without methods. There is no problem in all mathematics that cannot
be solved by direct counting. But with the present implements of
mathematics many operations of counting can be performed in a few
minutes which without mathematical methods would take a lifetime. Just as

a single human being, restricted wholly to the fruits of his own labor, could
never amass a fortune, but on the contrary the accumulation of the labor of
many men in the hands of one is the foundation of wealth and power, so,
also, no knowledge worthy of the name can be gathered up in a single
human mind limited to the span of a human life and gifted only with finite
powers, except by the most exquisite economy of thought and by the careful
amassment of the economically ordered experience of thousands of co-
workers. What strikes us here as the fruits of sorcery are simply the rewards
of excellent housekeeping, as are the like results in civil life. But the
business of science has this advantage over every other enterprise, that from
its amassment of wealth no one suffers the least loss. This, too, is its
blessing, its freeing and saving power.
The recognition of the economical character of science will now help us,
perhaps, to understand better certain physical notions.
Those elements of an event which we call "cause and effect" are certain
salient features of it, which are important for its mental reproduction. Their
importance wanes and the attention is transferred to fresh characters the
moment the event or experience in question becomes familiar. If the
connexion of such features strikes us as a necessary one, it is simply
because the interpolation of certain intermediate links with which we are
very familiar, and which possess, therefore, higher authority for us, is often
attended with success in our explanations. That ready experience fixed in
the mosaic of the mind with which we meet new events, Kant calls an
innate concept of the understanding (Verstandesbegriff).
The grandest principles of physics, resolved into their elements, differ in no
wise from the descriptive principles of the natural historian. The question,
"Why?" which is always appropriate where the explanation of a
contradiction is concerned, like all proper habitudes of thought, can
overreach itself and be asked where nothing remains to be understood.
Suppose we were to attribute to nature the property of producing like effects
in like circumstances; just these like circumstances we should not know
how to find. Nature exists once only. Our schematic mental imitation alone
produces like events. Only in the mind, therefore, does the mutual
dependence of certain features exist.

All our efforts to mirror the world in thought would be futile if we found
nothing permanent in the varied changes of things. It is this that impels us
to form the notion of substance, the source of which is not different from
that of the modern ideas relative to the conservation of energy. The history
of physics furnishes numerous examples of this impulse in almost all fields,
and pretty examples of it may be traced back to the nursery. "Where does
the light go to when it is put out?" asks the child. The sudden shrivelling up
of a hydrogen balloon is inexplicable to a child; it looks everywhere for the
large body which was just there but is now gone.
Where does heat come from? Where does heat go to? Such childish
questions in the mouths of mature men shape the character of a century.
In mentally separating a body from the changeable environment in which it
moves, what we really do is to extricate a group of sensations on which our
thoughts are fastened and which is of relatively greater stability than the
others, from the stream of all our sensations. Absolutely unalterable this
group is not. Now this, now that member of it appears and disappears, or is
altered. In its full identity it never recurs. Yet the sum of its constant
elements as compared with the sum of its changeable ones, especially if we
consider the continuous character of the transition, is always so great that
for the purpose in hand the former usually appear sufficient to determine the
body's identity. But because we can separate from the group every single
member without the body's ceasing to be for us the same, we are easily led
to believe that after abstracting all the members something additional would
remain. It thus comes to pass that we form the notion of a substance distinct
from its attributes, of a thing-in-itself, whilst our sensations are regarded
merely as symbols or indications of the properties of this thing-in-itself. But
it would be much better to say that bodies or things are compendious mental
symbols for groups of sensations—symbols that do not exist outside of
thought. Thus, the merchant regards the labels of his boxes merely as
indexes of their contents, and not the contrary. He invests their contents, not
their labels, with real value. The same economy which induces us to analyse
a group and to establish special signs for its component parts, parts which
also go to make up other groups, may likewise induce us to mark out by
some single symbol a whole group.
On the old Egyptian monuments we see objects represented which do not
reproduce a single visual impression, but are composed of various

impressions. The heads and the legs of the figures appear in profile, the
head-dress and the breast are seen from the front, and so on. We have here,
so to speak, a mean view of the objects, in forming which the sculptor has
retained what he deemed essential, and neglected what he thought
indifferent. We have living exemplifications of the processes put into stone
on the walls of these old temples, in the drawings of our children, and we
also observe a faithful analogue of them in the formation of ideas in our
own minds. Only in virtue of some such facility of view as that indicated,
are we allowed to speak of a body. When we speak of a cube with trimmed
corners—a figure which is not a cube—we do so from a natural instinct of
economy, which prefers to add to an old familiar conception a correction
instead of forming an entirely new one. This is the process of all judgment.
The crude notion of "body" can no more stand the test of analysis than can
the art of the Egyptians or that of our little children. The physicist who sees
a body flexed, stretched, melted, and vaporised, cuts up this body into
smaller permanent parts; the chemist splits it up into elements. Yet even an
element is not unalterable. Take sodium. When warmed, the white, silvery
mass becomes a liquid, which, when the heat is increased and the air shut
out, is transformed into a violet vapor, and on the heat being still more
increased glows with a yellow light. If the name sodium is still retained, it
is because of the continuous character of the transitions and from a
necessary instinct of economy. By condensing the vapor, the white metal
may be made to reappear. Indeed, even after the metal is thrown into water
and has passed into sodium hydroxide, the vanished properties may by
skilful treatment still be made to appear; just as a moving body which has
passed behind a column and is lost to view for a moment may make its
appearance after a time. It is unquestionably very convenient always to
have ready the name and thought for a group of properties wherever that
group by any possibility can appear. But more than a compendious
economical symbol for these phenomena, that name and thought is not. It
would be a mere empty word for one in whom it did not awaken a large
group of well-ordered sense-impressions. And the same is true of the
molecules and atoms into which the chemical element is still further
analysed.
True, it is customary to regard the conservation of weight, or, more
precisely, the conservation of mass, as a direct proof of the constancy of

matter. But this proof is dissolved, when we go to the bottom of it, into such
a multitude of instrumental and intellectual operations, that in a sense it will
be found to constitute simply an equation which our ideas in imitating facts
have to satisfy. That obscure, mysterious lump which we involuntarily add
in thought, we seek for in vain outside the mind.
It is always, thus, the crude notion of substance that is slipping unnoticed
into science, proving itself constantly insufficient, and ever under the
necessity of being reduced to smaller and smaller world-particles. Here, as
elsewhere, the lower stage is not rendered indispensable by the higher
which is built upon it, no more than the simplest mode of locomotion,
walking, is rendered superfluous by the most elaborate means of
transportation. Body, as a compound of light and touch sensations, knit
together by sensations of space, must be as familiar to the physicist who
seeks it, as to the animal who hunts its prey. But the student of the theory of
knowledge, like the geologist and the astronomer, must be permitted to
reason back from the forms which are created before his eyes to others
which he finds ready made for him.
All physical ideas and principles are succinct directions, frequently
involving subordinate directions, for the employment of economically
classified experiences, ready for use. Their conciseness, as also the fact that
their contents are rarely exhibited in full, often invests them with the
semblance of independent existence. Poetical myths regarding such ideas,
—for example, that of Time, the producer and devourer of all things,—do
not concern us here. We need only remind the reader that even Newton
speaks of an absolute time independent of all phenomena, and of an
absolute space—views which even Kant did not shake off, and which are
often seriously entertained to-day. For the natural inquirer, determinations
of time are merely abbreviated statements of the dependence of one event
upon another, and nothing more. When we say the acceleration of a freely
falling body is 9·810 metres per second, we mean the velocity of the body
with respect to the centre of the earth is 9·810 metres greater when the earth
has performed an additional 86400th part of its rotation—a fact which itself
can be determined only by the earth's relation to other heavenly bodies.
Again, in velocity is contained simply a relation of the position of a body to
the position of the earth.[65] Instead of referring events to the earth we may
refer them to a clock, or even to our internal sensation of time. Now,

because all are connected, and each may be made the measure of the rest,
the illusion easily arises that time has significance independently of all.[66]
The aim of research is the discovery of the equations which subsist between
the elements of phenomena. The equation of an ellipse expresses the
universal conceivable relation between its co-ordinates, of which only the
real values have geometrical significance. Similarly, the equations between
the elements of phenomena express a universal, mathematically conceivable
relation. Here, however, for many values only certain directions of change
are physically admissible. As in the ellipse only certain values satisfying the
equation are realised, so in the physical world only certain changes of value
occur. Bodies are always accelerated towards the earth. Differences of
temperature, left to themselves, always grow less; and so on. Similarly, with
respect to space, mathematical and physiological researches have shown
that the space of experience is simply an actual case of many conceivable
cases, about whose peculiar properties experience alone can instruct us. The
elucidation which this idea diffuses cannot be questioned, despite the
absurd uses to which it has been put.
Let us endeavor now to summarise the results of our survey. In the
economical schematism of science lie both its strength and its weakness.
Facts are always represented at a sacrifice of completeness and never with
greater precision than fits the needs of the moment. The incongruence
between thought and experience, therefore, will continue to subsist as long
as the two pursue their course side by side; but it will be continually
diminished.
In reality, the point involved is always the completion of some partial
experience; the derivation of one portion of a phenomenon from some
other. In this act our ideas must be based directly upon sensations. We call
this measuring.[67] The condition of science, both in its origin and in its
application, is a great relative stability of our environment. What it teaches
us is interdependence. Absolute forecasts, consequently, have no
significance in science. With great changes in celestial space we should lose
our co-ordinate systems of space and time.
When a geometer wishes to understand the form of a curve, he first resolves
it into small rectilinear elements. In doing this, however, he is fully aware
that these elements are only provisional and arbitrary devices for

comprehending in parts what he cannot comprehend as a whole. When the
law of the curve is found he no longer thinks of the elements. Similarly, it
would not become physical science to see in its self-created, changeable,
economical tools, molecules and atoms, realities behind phenomena,
forgetful of the lately acquired sapience of her older sister, philosophy, in
substituting a mechanical mythology for the old animistic or metaphysical
scheme, and thus creating no end of suppositious problems. The atom must
remain a tool for representing phenomena, like the functions of
mathematics. Gradually, however, as the intellect, by contact with its
subject-matter, grows in discipline, physical science will give up its mosaic
play with stones and will seek out the boundaries and forms of the bed in
which the living stream of phenomena flows. The goal which it has set itself
is the simplest and most economical abstract expression of facts.

The question now remains, whether the same method of research which till
now we have tacitly restricted to physics, is also applicable in the psychical
domain. This question will appear superfluous to the physical inquirer. Our
physical and psychical views spring in exactly the same manner from
instinctive knowledge. We read the thoughts of men in their acts and facial
expressions without knowing how. Just as we predict the behavior of a
magnetic needle placed near a current by imagining Ampère's swimmer in
the current, similarly we predict in thought the acts and behavior of men by
assuming sensations, feelings, and wills similar to our own connected with
their bodies. What we here instinctively perform would appear to us as one
of the subtlest achievements of science, far outstripping in significance and
ingenuity Ampère's rule of the swimmer, were it not that every child
unconsciously accomplished it. The question simply is, therefore, to grasp
scientifically, that is, by conceptional thought, what we are already familiar
with from other sources. And here much is to be accomplished. A long
sequence of facts is to be disclosed between the physics of expression and
movement and feeling and thought.
We hear the question, "But how is it possible to explain feeling by the
motions of the atoms of the brain?" Certainly this will never be done, no
more than light or heat will ever be deduced from the law of refraction. We

need not deplore, therefore, the lack of ingenious solutions of this question.
The problem is not a problem. A child looking over the walls of a city or of
a fort into the moat below sees with astonishment living people in it, and
not knowing of the portal which connects the wall with the moat, cannot
understand how they could have got down from the high ramparts. So it is
with the notions of physics. We cannot climb up into the province of
psychology by the ladder of our abstractions, but we can climb down into it.
Let us look at the matter without bias. The world consists of colors, sounds,
temperatures, pressures, spaces, times, and so forth, which now we shall not
call sensations, nor phenomena, because in either term an arbitrary, one-
sided theory is embodied, but simply elements. The fixing of the flux of
these elements, whether mediately or immediately, is the real object of
physical research. As long as, neglecting our own body, we employ
ourselves with the interdependence of those groups of elements which,
including men and animals, make up foreign bodies, we are physicists. For
example, we investigate the change of the red color of a body as produced
by a change of illumination. But the moment we consider the special
influence on the red of the elements constituting our body, outlined by the
well-known perspective with head invisible, we are at work in the domain
of physiological psychology. We close our eyes, and the red together with
the whole visible world disappears. There exists, thus, in the perspective
field of every sense a portion which exercises on all the rest a different and
more powerful influence than the rest upon one another. With this, however,
all is said. In the light of this remark, we call all elements, in so far as we
regard them as dependent on this special part (our body), sensations. That
the world is our sensation, in this sense, cannot be questioned. But to make
a system of conduct out of this provisional conception, and to abide its
slaves, is as unnecessary for us as would be a similar course for a
mathematician who, in varying a series of variables of a function which
were previously assumed to be constant, or in interchanging the
independent variables, finds his method to be the source of some very
surprising ideas for him.[68]
If we look at the matter in this unbiassed light it will appear indubitable that
the method of physiological psychology is none other than that of physics;
what is more, that this science is a part of physics. Its subject-matter is not
different from that of physics. It will unquestionably determine the relations

the sensations bear to the physics of our body. We have already learned
from a member of this academy (Hering) that in all probability a sixfold
manifoldness of the chemical processes of the visual substance corresponds
to the sixfold manifoldness of color-sensation, and a threefold manifoldness
of the physiological processes to the threefold manifoldness of space-
sensations. The paths of reflex actions and of the will are followed up and
disclosed; it is ascertained what region of the brain subserves the function
of speech, what region the function of locomotion, etc. That which still
clings to our body, namely, our thoughts, will, when those investigations are
finished, present no difficulties new in principle. When experience has once
clearly exhibited these facts and science has marshalled them in economic
and perspicuous order, there is no doubt that we shall understand them. For
other "understanding" than a mental mastery of facts never existed. Science
does not create facts from facts, but simply orders known facts.
Let us look, now, a little more closely into the modes of research of
physiological psychology. We have a very clear idea of how a body moves
in the space encompassing it. With our optical field of sight we are very
familiar. But we are unable to state, as a rule, how we have come by an
idea, from what corner of our intellectual field of sight it has entered, or by
what region the impulse to a motion is sent forth. Moreover, we shall never
get acquainted with this mental field of view from self-observation alone.
Self-observation, in conjunction with physiological research, which seeks
out physical connexions, can put this field of vision in a clear light before
us, and will thus first really reveal to us our inner man.
Primarily, natural science, or physics, in its widest sense, makes us
acquainted with only the firmest connexions of groups of elements.
Provisorily, we may not bestow too much attention on the single
constituents of those groups, if we are desirous of retaining a
comprehensible whole. Instead of equations between the primitive
variables, physics gives us, as much the easiest course, equations between
functions of those variables. Physiological psychology teaches us how to
separate the visible, the tangible, and the audible from bodies—a labor
which is subsequently richly requited, as the division of the subjects of
physics well shows. Physiology further analyses the visible into light and
space sensations; the first into colors, the last also into their component
parts; it resolves noises into sounds, these into tones, and so on.

Unquestionably this analysis can be carried much further than it has been. It
will be possible in the end to exhibit the common elements at the basis of
very abstract but definite logical acts of like form,—elements which the
acute jurist and mathematician, as it were, feels out, with absolute certainty,
where the uninitiated hears only empty words. Physiology, in a word, will
reveal to us the true real elements of the world. Physiological psychology
bears to physics in its widest sense a relation similar to that which
chemistry bears to physics in its narrowest sense. But far greater than the
mutual support of physics and chemistry will be that which natural science
and psychology will render each other. And the results that shall spring
from this union will, in all likelihood, far outstrip those of the modern
mechanical physics.
What those ideas are with which we shall comprehend the world when the
closed circuit of physical and psychological facts shall lie complete before
us, (that circuit of which we now see only two disjoined parts,) cannot be
foreseen at the outset of the work. The men will be found who will see what
is right and will have the courage, instead of wandering in the intricate
paths of logical and historical accident, to enter on the straight ways to the
heights from which the mighty stream of facts can be surveyed. Whether
the notion which we now call matter will continue to have a scientific
significance beyond the crude purposes of common life, we do not know.
But we certainly shall wonder how colors and tones which were such
innermost parts of us could suddenly get lost in our physical world of
atoms; how we could be suddenly surprised that something which outside
us simply clicked and beat, in our heads should make light and music; and
how we could ask whether matter can feel, that is to say, whether a mental
symbol for a group of sensations can feel?
We cannot mark out in hard and fast lines the science of the future, but we
can foresee that the rigid walls which now divide man from the world will
gradually disappear; that human beings will not only confront each other,
but also the entire organic and so-called lifeless world, with less selfishness
and with livelier sympathy. Just such a presentiment as this perhaps
possessed the great Chinese philosopher Licius some two thousand years
ago when, pointing to a heap of mouldering human bones, he said to his
scholars in the rigid, lapidary style of his tongue: "These and I alone have
the knowledge that we neither live nor are dead."

ON TRANSFORMATION AND ADAPTATION
IN SCIENTIFIC THOUGHT.[69]
It was towards the close of the sixteenth century that Galileo with a superb
indifference to the dialectic arts and sophistic subtleties of the Schoolmen
of his time, turned the attention of his brilliant mind to nature. By nature his
ideas were transformed and released from the fetters of inherited prejudice.
At once the mighty revolution was felt, that was therewith effected in the
realm of human thought—felt indeed in circles far remote and wholly
unrelated to the sphere of science, felt in strata of society that hitherto had
only indirectly recognised the influence of scientific thought.
And how great and how far-reaching that revolution was! From the
beginning of the seventeenth century till its close we see arising, at least in
embryo, almost all that plays a part in the natural and technical science of
to-day, almost all that in the two centuries following so wonderfully
transformed the facial appearance of the earth, and all that is moving
onward in process of such mighty evolution to-day. And all this, the direct
result of Galilean ideas, the direct outcome of that freshly awakened sense
for the investigation of natural phenomena which taught the Tuscan
philosopher to form the concept and the law of falling bodies from the
observation of a falling stone! Galileo began his investigations without an
implement worthy of the name; he measured time in the most primitive
way, by the efflux of water. Yet soon afterwards the telescope, the
microscope, the barometer, the thermometer, the air-pump, the steam
engine, the pendulum, and the electrical machine were invented in rapid
succession. The fundamental theorems of dynamical science, of optics, of
heat, and of electricity were all disclosed in the century that followed
Galileo.
Of scarcely less importance, it seems, was that movement which was
prepared for by the illustrious biologists of the hundred years just past, and
formally begun by the late Mr. Darwin. Galileo quickened the sense for the
simpler phenomena of inorganic nature. And with the same simplicity and
frankness that marked the efforts of Galileo, and without the aid of

technical or scientific instruments, without physical or chemical
experiment, but solely by the power of thought and observation, Darwin
grasps a new property of organic nature—which we may briefly call its
plasticity.[70] With the same directness of purpose, Darwin, too, pursues his
way. With the same candor and love of truth, he points out the strength and
the weakness of his demonstrations. With masterly equanimity he holds
aloof from the discussion of irrelevant subjects and wins alike the
admiration of his adherents and of his adversaries.
Scarcely thirty years have elapsed[71] since Darwin first propounded the
principles of his theory of evolution. Yet, already we see his ideas firmly
rooted in every branch of human thought, however remote. Everywhere, in
history, in philosophy, even in the physical sciences, we hear the
watchwords: heredity, adaptation, selection. We speak of the struggle for
existence among the heavenly bodies and of the struggle for existence in the
world of molecules.[72]
The impetus given by Galileo to scientific thought was marked in every
direction; thus, his pupil, Borelli, founded the school of exact medicine,
from whence proceeded even distinguished mathematicians. And now
Darwinian ideas, in the same way, are animating all provinces of research.
It is true, nature is not made up of two distinct parts, the inorganic and the
organic; nor must these two divisions be treated perforce by totally distinct
methods. Many sides, however, nature has. Nature is like a thread in an
intricate tangle, which must be followed and traced, now from this point,
now from that. But we must never imagine,—and this physicists have
learned from Faraday and J. R. Mayer,—that progress along paths once
entered upon is the only means of reaching the truth.
It will devolve upon the specialists of the future to determine the relative
tenability and fruitfulness of the Darwinian ideas in the different provinces.
Here I wish simply to consider the growth of natural knowledge in the light
of the theory of evolution. For knowledge, too, is a product of organic
nature. And although ideas, as such, do not comport themselves in all
respects like independent organic individuals, and although violent
comparisons should be avoided, still, if Darwin reasoned rightly, the general
imprint of evolution and transformation must be noticeable in ideas also.

I shall waive here the consideration of the fruitful topic of the transmission
of ideas or rather of the transmission of the aptitude for certain ideas.[73]
Nor would it come within my province to discuss psychical evolution in
any form, as Spencer[74] and many other modern psychologists have done,
with varying success. Neither shall I enter upon a discussion of the struggle
for existence and of natural selection among scientific theories.[75] We shall
consider here only such processes of transformation as every student can
easily observe in his own mind.

The child of the forest picks out and pursues with marvellous acuteness the
trails of animals. He outwits and overreaches his foes with surpassing
cunning. He is perfectly at home in the sphere of his peculiar experience.
But confront him with an unwonted phenomenon; place him face to face
with a technical product of modern civilisation, and he will lapse into
impotency and helplessness. Here are facts which he does not comprehend.
If he endeavors to grasp their meaning, he misinterprets them. He fancies
the moon, when eclipsed, to be tormented by an evil spirit. To his mind a
puffing locomotive is a living monster. The letter accompanying a
commission with which he is entrusted, having once revealed his
thievishness, is in his imagination a conscious being, which he must hide
beneath a stone, before venturing to commit a fresh trespass. Arithmetic to
him is like the art of the geomancers in the Arabian Nights,—an art which is
able to accomplish every imaginable impossibility. And, like Voltaire's
ingénu, when placed in our social world, he plays, as we think, the maddest
pranks.
With the man who has made the achievements of modern science and
civilisation his own, the case is quite different. He sees the moon pass
temporarily into the shadow of the earth. He feels in his thoughts the water
growing hot in the boiler of the locomotive; he feels also the increase of the
tension which pushes the piston forward. Where he is not able to trace the
direct relation of things he has recourse to his yard-stick and table of
logarithms, which aid and facilitate his thought without predominating over
it. Such opinions as he cannot concur in, are at least known to him, and he
knows how to meet them in argument.

Now, wherein does the difference between these two men consist? The train
of thought habitually employed by the first one does not correspond to the
facts that he sees. He is surprised and nonplussed at every step. But the
thoughts of the second man follow and anticipate events, his thoughts have
become adapted or accommodated to the larger field of observation and
activity in which he is located; he conceives things as they are. The Indian's
sphere of experience, however, is quite different; his bodily organs of sense
are in constant activity; he is ever intensely alert and on the watch for his
foes; or, his entire attention and energy are engaged in procuring
sustenance. Now, how can such a creature project his mind into futurity,
foresee or prophesy? This is not possible until our fellow-beings have, in a
measure, relieved us of our concern for existence. It is then that we acquire
freedom for observation, and not infrequently too that narrowness of
thought which society helps and teaches us to disregard.
If we move for a time within a fixed circle of phenomena which recur with
unvarying uniformity, our thoughts gradually adapt themselves to our
environment; our ideas reflect unconsciously our surroundings. The stone
we hold in our hand, when dropped, not only falls to the ground in reality; it
also falls in our thoughts. Iron-filings dart towards a magnet in imagination
as well as in fact, and, when thrown into a fire, they grew hot in conception
as well.
The impulse to complete mentally a phenomenon that has been only
partially observed, has not its origin in the phenomenon itself; of this fact,
we are fully sensible. And we well know that it does not lie within the
sphere of our volition. It seems to confront us rather as a power and a law
imposed from without and controlling both thought and facts.
The fact that we are able by the help of this law to prophesy and forecast,
merely proves a sameness or uniformity of environment sufficient to effect
a mental adaptation of this kind. A necessity of fulfilment, however, is not
contained in this compulsory principle which controls our thoughts; nor is it
in any way determined by the possibility of prediction. We are always
obliged, in fact, to await the completion of what has been predicted. Errors
and departures are constantly discernible, and are slight only in provinces of
great rigid constancy, as in astronomy.

In cases where our thoughts follow the connexion of events with ease, and
in instances where we positively forefeel the course of a phenomenon, it is
natural to fancy that the latter is determined by and must conform to our
thoughts. But the belief in that mysterious agency called causality, which
holds thought and event in unison, is violently shaken when a person first
enters a province of inquiry in which he has previously had no experience.
Take for instance the strange interaction of electric currents and magnets, or
the reciprocal action of currents, which seem to defy all the resources of
mechanical science. Let him be confronted with such phenomena and he
will immediately feel himself forsaken by his power of prediction; he will
bring nothing with him into this strange field of events but the hope of soon
being able to adapt his ideas to the new conditions there presented.
A person constructs from a bone the remaining anatomy of an animal; or
from the visible part of a half-concealed wing of a butterfly he infers and
reconstructs the part concealed. He does so with a feeling of highest
confidence in the accuracy of his results; and in these processes we find
nothing preternatural or transcendent. But when physicists adapt their
thoughts to conform to the dynamical course of events in time, we
invariably surround their investigations with a metaphysical halo; yet these
latter adaptations bear quite the same character as the former, and our only
reason for investing them with a metaphysical garb, perhaps, is their high
practical value.[76]
Let us consider for a moment what takes place when the field of
observation to which our ideas have been adapted and now conform,
becomes enlarged. We had, let us say, always seen heavy bodies sink when
their support was taken away; we had also seen, perhaps, that the sinking of
heavier bodies forced lighter bodies upwards. But now we see a lever in
action, and we are suddenly struck with the fact that a lighter body is lifting
another of much greater weight. Our customary train of thought demands its
rights; the new and unwonted event likewise demands its rights. From this
conflict between thought and fact the problem arises; out of this partial
contrariety springs the question, "Why?" With the new adaptation to the
enlarged field of observation, the problem disappears, or, in other words, is
solved. In the instance cited, we must adopt the habit of always considering
the mechanical work performed.

The child just awakening into consciousness of the world, knows no
problem. The bright flower, the ringing bell, are all new to it; yet it is
surprised at nothing. The out and out Philistine, whose only thoughts lie in
the beaten path of his every-day pursuits, likewise has no problems.
Everything goes its wonted course, and if perchance a thing go wrong at
times, it is at most a mere object of curiosity and not worth serious
consideration. In fact, the question "Why?" loses all warrant in relations
where we are familiar with every aspect of events. But the capable and
talented young man has his head full of problems; he has acquired, to a
greater or less degree, certain habitudes of thought, and at the same time he
is constantly observing what is new and unwonted, and in his case there is
no end to the questions, "Why?"
Thus, the factor which most promotes scientific thought is the gradual
widening of the field of experience. We scarcely notice events we are
accustomed to; the latter do not really develop their intellectual significance
until placed in contrast with something to which we are unaccustomed.
Things that at home are passed by unnoticed, delight us when abroad,
though they may appear in only slightly different forms. The sun shines
with heightened radiance, the flowers bloom in brighter colors, our fellow-
men accost us with lighter and happier looks. And, returning home, we find
even the old familiar scenes more inspiring and suggestive than before.
Every motive that prompts and stimulates us to modify and transform our
thoughts, proceeds from what is new, uncommon, and not understood.
Novelty excites wonder in persons whose fixed habits of thought are shaken
and disarranged by what they see. But the element of wonder never lies in
the phenomenon or event observed; its place is in the person observing.
People of more vigorous mental type aim at once at an adaptation of
thought that will conform to what they have observed. Thus does science
eventually become the natural foe of the wonderful. The sources of the
marvellous are unveiled, and surprise gives way to calm interpretation.
Let us consider such a mental transformative process in detail. The
circumstance that heavy bodies fall to the earth appears perfectly natural
and regular. But when a person observes that wood floats upon water, and
that flames and smoke rise in the air, then the contrary of the first
phenomenon is presented. An olden theory endeavors to explain these facts
by imputing to substances the power of volition, as that attribute which is

most familiar to man. It asserted that every substance seeks its proper place,
heavy bodies tending downwards and light ones upwards. It soon turned
out, however, that even smoke had weight, that it, too, sought its place
below, and that it was forced upwards only because of the downward
tendency of the air, as wood is forced to the surface of water because the
water exerts the greater downward pressure.
Again, we see a body thrown into the air. It ascends. How is it that it does
not seek its proper place? Why does the velocity of its "violent" motion
decrease as it rises, while that of its "natural" fall increases as it descends. If
we mark closely the relation between these two facts, the problem will
solve itself. We shall see, as Galileo did, that the decrease of velocity in
rising and the increase of velocity in falling are one and the same
phenomenon, viz., an increase of velocity towards the earth. Accordingly, it
is not a place that is assigned to the body, but an increase of velocity
towards the earth.
By this idea the movements of heavy bodies are rendered perfectly familiar.
Newton, now, firmly grasping this new way of thinking, sees the moon and
the planets moving in their paths upon principles similar to those which
determine the motion of a projectile thrown into the air. Yet the movements
of the planets were marked by peculiarities which compelled him once
more to modify slightly his customary mode of thought. The heavenly
bodies, or rather the parts composing them, do not move with constant
accelerations towards each other, but "attract each other," directly as the
mass and inversely as the square of the distance.
This latter notion, which includes the one applying to terrestrial bodies as a
special case, is, as we see, quite different from the conception from which
we started. How limited in scope was the original idea and to what a
multitude of phenomena is not the present one applicable! Yet there is a
trace, after all, of the "search for place" in the expression "attraction." And
it would be folly, indeed, for us to avoid, with punctilious dread, this
conception of "attraction" as bearing marks of its pedigree. It is the
historical base of the Newtonian conception and it still continues to direct
our thoughts in the paths so long familiar to us. Thus, the happiest ideas do
not fall from heaven, but spring from notions already existing.

Similarly, a ray of light was first regarded as a continuous and
homogeneous straight line. It then became the path of projection for minute
missiles; then an aggregate of the paths of countless different kinds of
missiles. It became periodic; it acquired various sides; and ultimately it
even lost its motion in a straight line.
The electric current was conceived originally as the flow of a hypothetical
fluid. To this conception was soon added the notion of a chemical current,
the notion of an electric, magnetic, and anisotropic optical field, intimately
connected with the path of the current. And the richer a conception becomes
in following and keeping pace with facts, the better adapted it is to
anticipate them.
Adaptive processes of this kind have no assignable beginning, inasmuch as
every problem that incites to new adaptation, presupposes a fixed habitude
of thought. Moreover, they have no visible end; in so far as experience
never ceases. Science, accordingly, stands midway in the evolutionary
process; and science may advantageously direct and promote this process,
but it can never take its place. That science is inconceivable the principles
of which would enable a person with no experience to construct the world
of experience, without a knowledge of it. One might just as well expect to
become a great musician, solely by the aid of theory, and without musical
experience; or to become a painter by following the directions of a text-
book.
In glancing over the history of an idea with which we have become
perfectly familiar, we are no longer able to appreciate the full significance
of its growth. The deep and vital changes that have been effected in the
course of its evolution, are recognisable only from the astounding
narrowness of view with which great contemporary scientists have
occasionally opposed each other. Huygens's wave-theory of light was
incomprehensible to Newton, and Newton's idea of universal gravity was
unintelligible to Huygens. But a century afterwards both notions were
reconcilable, even in ordinary minds.
On the other hand, the original creations of pioneer intellects,
unconsciously formed, do not assume a foreign garb; their form is their
own. In them, childlike simplicity is joined to the maturity of manhood, and
they are not to be compared with processes of thought in the average mind.

The latter are carried on as are the acts of persons in the state of
mesmerism, where actions involuntarily follow the images which the words
of other persons suggest to their minds.
The ideas that have become most familiar through long experience, are the
very ones that intrude themselves into the conception of every new fact
observed. In every instance, thus, they become involved in a struggle for
self-preservation, and it is just they that are seized by the inevitable process
of transformation.
Upon this process rests substantially the method of explaining by
hypothesis new and uncomprehended phenomena. Thus, instead of forming
entirely new notions to explain the movements of the heavenly bodies and
the phenomena of the tides, we imagine the material particles composing
the bodies of the universe to possess weight or gravity with respect to one
another. Similarly, we imagine electrified bodies to be freighted with fluids
that attract and repel, or we conceive the space between them to be in a
state of elastic tension. In so doing, we substitute for new ideas distinct and
more familiar notions of old experience—notions which to a great extent
run unimpeded in their courses, although they too must suffer partial
transformation.
The animal cannot construct new members to perform every new function
that circumstances and fate demand of it. On the contrary it is obliged to
make use of those it already possesses. When a vertebrate animal chances
into an environment where it must learn to fly or swim, an additional pair of
extremities is not grown for the purpose. On the contrary, the animal must
adapt and transform a pair that it already has.
The construction of hypotheses, therefore, is not the product of artificial
scientific methods. This process is unconsciously carried on in the very
infancy of science. Even later, hypotheses do not become detrimental and
dangerous to progress except when more reliance is placed on them than on
the facts themselves; when the contents of the former are more highly
valued than the latter, and when, rigidly adhering to hypothetical notions,
we overestimate the ideas we possess as compared with those we have to
acquire.
The extension of our sphere of experience always involves a transformation
of our ideas. It matters not whether the face of nature becomes actually

altered, presenting new and strange phenomena, or whether these
phenomena are brought to light by an intentional or accidental turn of
observation. In fact, all the varied methods of scientific inquiry and of
purposive mental adaptation enumerated by John Stuart Mill, those of
observation as well as those of experiment, are ultimately recognisable as
forms of one fundamental method, the method of change, or variation. It is
through change of circumstances that the natural philosopher learns. This
process, however, is by no means confined to the investigator of nature. The
historian, the philosopher, the jurist, the mathematician, the artist, the
æsthetician,[77] all illuminate and unfold their ideas by producing from the
rich treasures of memory similar, but different, cases; thus, they observe
and experiment in their thoughts. Even if all sense-experience should
suddenly cease, the events of the days past would meet in different attitudes
in the mind and the process of adaptation would still continue—a process
which, in contradistinction to the adaptation of thoughts to facts in practical
spheres, would be strictly theoretical, being an adaptation of thoughts to
thoughts.
The method of change or variation brings before us like cases of
phenomena, having partly the same and partly different elements. It is only
by comparing different cases of refracted light at changing angles of
incidence that the common factor, the constancy of the refractive index, is
disclosed. And only by comparing the refractions of light of different
colors, does the difference, the inequality of the indices of refraction, arrest
the attention. Comparison based upon change leads the mind
simultaneously to the highest abstractions and to the finest distinctions.
Undoubtedly, the animal also is able to distinguish between the similar and
dissimilar of two cases. Its consciousness is aroused by a noise or a rustling,
and its motor centre is put in readiness. The sight of the creature causing the
disturbance, will, according to its size, provoke flight or prompt pursuit;
and in the latter case, the more exact distinctions will determine the mode
of attack. But man alone attains to the faculty of voluntary and conscious
comparison. Man alone can, by his power of abstraction, rise, in one
moment, to the comprehension of principles like the conservation of mass
or the conservation of energy, and in the next observe and mark the
arrangement of the iron lines in the spectrum. In thus dealing with the
objects of his conceptual life, his ideas unfold and expand, like his nervous

system, into a widely ramified and organically articulated tree, on which he
may follow every limb to its farthermost branches, and, when occasion
demands, return to the trunk from which he started.
The English philosopher Whewell has remarked that two things are
requisite to the formation of science: facts and ideas. Ideas alone lead to
empty speculation; mere facts can yield no organic knowledge. We see that
all depends upon the capacity of adapting existing notions to fresh facts.
Over-readiness to yield to every new fact prevents fixed habits of thought
from arising. Excessively rigid habits of thought impede freedom of
observation. In the struggle, in the compromise between judgment and
prejudgment (prejudice), if we may use the term, our understanding of
things broadens.
Habitual judgment, applied to a new case without antecedent tests, we call
prejudgment or prejudice. Who does not know its terrible power! But we
think less often of the importance and utility of prejudice. Physically, no
one could exist, if he had to guide and regulate the circulation, respiration,
and digestion of his body by conscious and purposive acts. So, too, no one
could exist intellectually if he had to form judgments on every passing
experience, instead of allowing himself to be controlled by the judgments
he has already formed. Prejudice is a sort of reflex motion in the province
of intelligence.
On prejudices, that is, on habitual judgments not tested in every case to
which they are applied, reposes a goodly portion of the thought and work of
the natural scientist. On prejudices reposes most of the conduct of society.
With the sudden disappearance of prejudice society would hopelessly
dissolve. That prince displayed a deep insight into the power of intellectual
habit, who quelled the loud menaces and demands of his body-guard for
arrears of pay and compelled them to turn about and march, by simply
pronouncing the regular word of command; he well knew that they would
be unable to resist that.
Not until the discrepancy between habitual judgments and facts becomes
great is the investigator implicated in appreciable illusion. Then tragic
complications and catastrophes occur in the practical life of individuals and
nations—crises where man, placing custom above life, instead of pressing it
into the service of life, becomes the victim of his error. The very power

which in intellectual life advances, fosters, and sustains us, may in other
circumstances delude and destroy us.

Ideas are not all of life. They are only momentary efflorescences of light,
designed to illuminate the paths of the will. But as delicate reagents on our
organic evolution our ideas are of paramount importance. No theory can
gainsay the vital transformation which we feel taking place within us
through their agency. Nor is it necessary that we should have a proof of this
process. We are immediately assured of it.
The transformation of ideas thus appears as a part of the general evolution
of life, as a part of its adaptation to a constantly widening sphere of action.
A granite boulder on a mountain-side tends towards the earth below. It must
abide in its resting-place for thousands of years before its support gives
way. The shrub that grows at its base is farther advanced; it accommodates
itself to summer and winter. The fox which, overcoming the force of
gravity, creeps to the summit where he has scented his prey, is freer in his
movements than either. The arm of man reaches further still; and scarcely
anything of note happens in Africa or Asia that does not leave an imprint
upon his life. What an immense portion of the life of other men is reflected
in ourselves; their joys, their affections, their happiness and misery! And
this too, when we survey only our immediate surroundings, and confine our
attention to modern literature. How much more do we experience when we
travel through ancient Egypt with Herodotus, when we stroll through the
streets of Pompeii, when we carry ourselves back to the gloomy period of
the crusades or to the golden age of Italian art, now making the
acquaintance of a physician of Molière, and now that of a Diderot or of a
D'Alembert. What a great part of the life of others, of their character and
their purpose, do we not absorb through poetry and music! And although
they only gently touch the chords of our emotions, like the memory of
youth softly breathing upon the spirit of an aged man, we have nevertheless
lived them over again in part. How great and comprehensive does self
become in this conception; and how insignificant the person! Egoistical
systems both of optimism and pessimism perish with their narrow standard
of the import of intellectual life. We feel that the real pearls of life lie in the

ever changing contents of consciousness, and that the person is merely an
indifferent symbolical thread on which they are strung.[78]
We are prepared, thus, to regard ourselves and every one of our ideas as a
product and a subject of universal evolution; and in this way we shall
advance sturdily and unimpeded along the paths which the future will throw
open to us.[79]

ON THE PRINCIPLE OF COMPARISON IN
PHYSICS.[80]
Twenty years ago when Kirchhoff defined the object of mechanics as the
"description, in complete and very simple terms, of the motions occurring
in nature," he produced by the statement a peculiar impression. Fourteen
years subsequently, Boltzmann, in the life-like picture which he drew of the
great inquirer, could still speak of the universal astonishment at this novel
method of treating mechanics, and we meet with epistemological treatises
to-day, which plainly show how difficult is the acceptance of this point of
view. A modest and small band of inquirers there were, however, to whom
Kirchhoff's few words were tidings of a welcome and powerful ally in the
epistemological field.
Now, how does it happen that we yield our assent so reluctantly to the
philosophical opinion of an inquirer for whose scientific achievements we
have only words of praise? One reason probably is that few inquirers can
find time and leisure, amid the exacting employments demanded for the
acquisition of new knowledge, to inquire closely into that tremendous
psychical process by which science is formed. Further, it is inevitable that
much should be put into Kirchhoff's rigid words that they were not
originally intended to convey, and that much should be found wanting in
them that had always been regarded as an essential element of scientific
knowledge. What can mere description accomplish? What has become of
explanation, of our insight into the causal connexion of things?

Permit me, for a moment, to contemplate not the results of science, but the
mode of its growth, in a frank and unbiassed manner. We know of only one
source of immediate revelation of scientific facts—our senses. Restricted to
this source alone, thrown wholly upon his own resources, obliged to start
always anew, what could the isolated individual accomplish? Of a stock of
knowledge so acquired the science of a distant negro hamlet in darkest

Africa could hardly give us a sufficiently humiliating conception. For there
that veritable miracle of thought-transference has already begun its work,
compared with which the miracles of the spiritualists are rank monstrosities
—communication by language. Reflect, too, that by means of the magical
characters which our libraries contain we can raise the spirits of the "the
sovereign dead of old" from Faraday to Galileo and Archimedes, through
ages of time—spirits who do not dismiss us with ambiguous and derisive
oracles, but tell us the best they know; then shall we feel what a stupendous
and indispensable factor in the formation of science communication is. Not
the dim, half-conscious surmises of the acute observer of nature or critic of
humanity belong to science, but only that which they possess clearly
enough to communicate to others.
But how, now, do we go about this communication of a newly acquired
experience, of a newly observed fact? As the different calls and battle-cries
of gregarious animals are unconsciously formed signs for a common
observation or action, irrespective of the causes which produce such action
—a fact that already involves the germ of the concept; so also the words of
human language, which is only more highly specialised, are names or signs
for universally known facts, which all can observe or have observed. If the
mental representation, accordingly, follows the new fact at once and
passively, then that new fact must, of itself, immediately be constituted and
represented in thought by facts already universally known and commonly
observed. Memory is always ready to put forward for comparison known
facts which resemble the new event, or agree with it in certain features, and
so renders possible that elementary internal judgment which the mature and
definitively formulated judgment soon follows.
Comparison, as the fundamental condition of communication, is the most
powerful inner vital element of science. The zoölogist sees in the bones of
the wing-membranes of bats, fingers; he compares the bones of the cranium
with the vertebræ, the embryos of different organisms with one another, and
the different stages of development of the same organism with one another.
The geographer sees in Lake Garda a fjord, in the Sea of Aral a lake in
process of drying up. The philologist compares different languages with one
another, and the formations of the same language as well. If it is not
customary to speak of comparative physics in the same sense that we speak
of comparative anatomy, the reason is that in a science of such great

experimental activity the attention is turned away too much from the
contemplative element. But like all other sciences, physics lives and grows
by comparison.

The manner in which the result of the comparison finds expression in the
communication, varies of course very much. When we say that the colors of
the spectrum are red, yellow, green, blue, and violet, the designations
employed may possibly have been derived from the technology of tattooing,
or they may subsequently have acquired the significance of standing for the
colors of the rose, the lemon, the leaf, the corn-flower, and the violet. From
the frequent repetition of such comparisons, however, made under the most
manifold circumstances, the inconstant features, as compared with the
permanent congruent features, get so obliterated that the latter acquire a
fixed significance independent of every object and connexion, or take on as
we say an abstract or conceptual import. No one thinks at the word "red" of
any other agreement with the rose than that of color, or at the word
"straight" of any other property of a stretched cord than the sameness of
direction. Just so, too, numbers, originally the names of the fingers of the
hands and feet, from being used as arrangement-signs for all kinds of
objects, were lifted to the plane of abstract concepts. A verbal report
(communication) of a fact that uses only these purely abstract implements,
we call a direct description.
The direct description of a fact of any great extent is an irksome task, even
where the requisite notions are already completely developed. What a
simplification it involves if we can say, the fact A now considered comports
itself, not in one, but in many or in all its features, like an old and well-
known fact B. The moon comports itself as a heavy body does with respect
to the earth; light like a wave-motion or an electric vibration; a magnet, as if
it were laden with gravitating fluids, and so on. We call such a description,
in which we appeal, as it were, to a description already and elsewhere
formulated, or perhaps still to be precisely formulated, an indirect
description. We are at liberty to supplement this description, gradually, by
direct description, to correct it, or to replace it altogether. We see, thus,

without difficulty, that what is called a theory or a theoretical idea, falls
under the category of what is here termed indirect description.

What, now, is a theoretical idea? Whence do we get it? What does it
accomplish for us? Why does it occupy a higher place in our judgment than
the mere holding fast to a fact or an observation? Here, too, memory and
comparison alone are in play. But instead of a single feature of resemblance
culled from memory, in this case a great system of resemblances confronts
us, a well-known physiognomy, by means of which the new fact is
immediately transformed into an old acquaintance. Besides, it is in the
power of the idea to offer us more than we actually see in the new fact, at
the first moment; it can extend the fact, and enrich it with features which we
are first induced to seek from such suggestions, and which are often actually
found. It is this rapidity in extending knowledge that gives to theory a
preference over simple observation. But that preference is wholly
quantitative. Qualitatively, and in real essential points, theory differs from
observation neither in the mode of its origin nor in its last results.
The adoption of a theory, however, always involves a danger. For a theory
puts in the place of a fact A in thought, always a different, but simpler and
more familiar fact B, which in some relations can mentally represent A, but
for the very reason that it is different, in other relations cannot represent it.
If now, as may readily happen, sufficient care is not exercised, the most
fruitful theory may, in special circumstances, become a downright obstacle
to inquiry. Thus, the emission-theory of light, in accustoming the physicist
to think of the projectile path of the "light-particles" as an undifferentiated
straight-line, demonstrably impeded the discovery of the periodicity of
light. By putting in the place of light the more familiar phenomena of
sound, Huygens renders light in many of its features a familiar event, but
with respect to polarisation, which lacks the longitudinal waves with which
alone he was acquainted, it had for him a doubly strange aspect. He is
unable thus to grasp in abstract thought the fact of polarisation, which is
before his eyes, whilst Newton, merely by adapting to the observation his
thoughts, and putting this question, "Annon radiorum luminis diversa sunt
latera?" abstractly grasped polarisation, that is, directly described it, a

century before Malus. On the other hand, if the agreement of the fact with
the idea theoretically representing it, extends further than its inventor
originally anticipated, then we may be led by it to unexpected discoveries,
of which conical refraction, circular polarisation by total reflexion, Hertz's
waves offer ready examples, in contrast to the illustrations given above.
Our insight into the conditions indicated will be improved, perhaps, by
contemplating the development of some theory or other more in detail. Let
us consider a magnetised bar of steel by the side of a second unmagnetised
bar, in all other respects the same. The second bar gives no indication of the
presence of iron-filings; the first attracts them. Also, when the iron-filings
are absent, we must think of the magnetised bar as in a different condition
from that of the unmagnetised. For, that the mere presence of the iron-
filings does not induce the phenomenon of attraction is proved by the
second unmagnetised bar. The ingenuous man, who finds in his will, as his
most familiar source of power, the best facilities for comparison, conceives
a species of spirit in the magnet. The behavior of a warm body or of an
electrified body suggests similar ideas. This is the point of view of the
oldest theory, fetishism, which the inquirers of the early Middle Ages had
not yet overcome, and which in its last vestiges, in the conception of forces,
still flourishes in modern physics. We see, thus, the dramatic element need
no more be absent in a scientific description, than in a thrilling novel.
If, on subsequent examination, it be observed that a cold body, in contact
with a hot body, warms itself, so to speak, at the expense of the hot body;
further, that when the substances are the same, the cold body, which, let us
say, has twice the mass of the other, gains only half the number of degrees
of temperature that the other loses, a wholly new impression arises. The
demoniac character of the event vanishes, for the supposed spirit acts not by
caprice, but according to fixed laws. In its place, however, instinctively the
notion of a substance is substituted, part of which flows over from the one
body to the other, but the total amount of which, representable by the sum
of the products of the masses into the respective changes of temperature,
remains constant. Black was the first to be powerfully struck with this
resemblance of thermal processes to the motion of a substance, and under
its guidance discovered the specific heat, the heat of fusion, and the heat of
vaporisation of bodies. Gaining strength and fixity, however, from these
successes, this notion of substance subsequently stood in the way of

scientific advancement. It blinded the eyes of the successors of Black, and
prevented them from seeing the manifest fact, which every savage knows,
that heat is produced by friction. Fruitful as that notion was for Black,
helpful as it still is to the learner to-day in Black's special field, permanent
and universal validity as a theory it could never maintain. But what is
essential, conceptually, in it, viz., the constancy of the product-sum above
mentioned, retains its value and may be regarded as a direct description of
Black's facts.
It stands to reason that those theories which push themselves forward
unsought, instinctively, and wholly of their own accord, should have the
greatest power, should carry our thoughts most with them, and exhibit the
staunchest powers of self-preservation. On the other hand, it may also be
observed that when critically scrutinised such theories are extremely apt to
lose their cogency. We are constantly busied with "substance," its modes of
action have stamped themselves indelibly upon our thoughts, our vividest
and clearest reminiscences are associated with it. It should cause us no
surprise, therefore, that Robert Mayer and Joule, who gave the final blow to
Black's substantial conception of heat, should have re-introduced the same
notion of substance in a more abstract and modified form, only applying to
a much more extensive field.
Here, too, the psychological circumstances which impart to the new
conception its power, lie clearly before us. By the unusual redness of the
venous blood in tropical climates Mayer's attention is directed to the
lessened expenditure of internal heat and to the proportionately lessened
consumption of material by the human body in those climates. But as every
effort of the human organism, including its mechanical work, is connected
with the consumption of material, and as work by friction can engender
heat, therefore heat and work appear in kind equivalent, and between them
a proportional relation must subsist. Not every quantity, but the
appropriately calculated sum of the two, as connected with a proportionate
consumption of material, appears substantial.
By exactly similar considerations, relative to the economy of the galvanic
element, Joule arrived at his view; he found experimentally that the sum of
the heat evolved in the circuit, of the heat consumed in the combustion of
the gas developed, of the electro-magnetic work of the current, properly
calculated,—in short, the sum of all the effects of the battery,—is connected

with a proportionate consumption of zinc. Accordingly, this sum itself has a
substantial character.
Mayer was so absorbed with the view attained, that the indestructibility of
force, in our phraseology work, appeared to him a priori evident. "The
creation or annihilation of a force," he says, "lies without the province of
human thought and power." Joule expressed himself to a similar effect: "It
is manifestly absurd to suppose that the powers with which God has
endowed matter can be destroyed." Strange to say, on the basis of such
utterances, not Joule, but Mayer, was stamped as a metaphysician. We may
be sure, however, that both men were merely giving expression, and that
half-unconsciously, to a powerful formal need of the new simple view, and
that both would have been extremely surprised if it had been proposed to
them that their principle should be submitted to a philosophical congress or
ecclesiastical synod for a decision upon its validity. But with all
agreements, the attitude of these two men, in other respects, was totally
different. Whilst Mayer represented this formal need with all the
stupendous instinctive force of genius, we might say almost with the ardor
of fanaticism, yet was withal not wanting in the conceptive ability to
compute, prior to all other inquirers, the mechanical equivalent of heat from
old physical constants long known and at the disposal of all, and so to set
up for the new doctrine a programme embracing all physics and physiology;
Joule, on the other hand, applied himself to the exact verification of the
doctrine by beautifully conceived and masterfully executed experiments,
extending over all departments of physics. Soon Helmholtz too attacked the
problem, in a totally independent and characteristic manner. After the
professional virtuosity with which this physicist grasped and disposed of all
the points unsettled by Mayer's programme and more besides, what
especially strikes us is the consummate critical lucidity of this young man
of twenty-six years. In his exposition is wanting that vehemence and
impetuosity which marked Mayer's. The principle of the conservation of
energy is no self-evident or a priori proposition for him. What follows, on
the assumption that that proposition obtains? In this hypothetical form, he
subjugates his matter.
I must confess, I have always marvelled at the æsthetic and ethical taste of
many of our contemporaries who have managed to fabricate out of this
relation of things, odious national and personal questions, instead of

praising the good fortune that made several such men work together and of
rejoicing at the instructive diversity and idiosyncrasies of great minds
fraught with such rich consequences for us.
We know that still another theoretical conception played a part in the
development of the principle of energy, which Mayer held aloof from,
namely, the conception that heat, as also the other physical processes, are
due to motion. But once the principle of energy has been reached, these
auxiliary and transitional theories discharge no essential function, and we
may regard the principle, like that which Black gave, as a contribution to
the direct description of a widely extended domain of facts.
It would appear from such considerations not only advisable, but even
necessary, with all due recognition of the helpfulness of theoretic ideas in
research, yet gradually, as the new facts grow familiar, to substitute for
indirect description direct description, which contains nothing that is
unessential and restricts itself absolutely to the abstract apprehension of
facts. We might almost say, that the descriptive sciences, so called with a
tincture of condescension, have, in respect of scientific character,
outstripped the physical expositions lately in vogue. Of course, a virtue has
been made of necessity here.
We must admit, that it is not in our power to describe directly every fact, on
the moment. Indeed, we should succumb in utter despair if the whole
wealth of facts which we come step by step to know, were presented to us
all at once. Happily, only detached and unusual features first strike us, and
such we bring nearer to ourselves by comparison with every-day events.
Here the notions of the common speech are first developed. The
comparisons then grow more manifold and numerous, the fields of facts
compared more extensive, the concepts that make direct description
possible, proportionately more general and more abstract.
First we become familiar with the motion of freely falling bodies. The
concepts of force, mass, and work are then carried over, with appropriate
modifications, to the phenomena of electricity and magnetism. A stream of
water is said to have suggested to Fourier the first distinct picture of
currents of heat. A special case of vibrations of strings investigated by
Taylor, cleared up for him a special case of the conduction of heat. Much in
the same way that Daniel Bernoulli and Euler constructed the most diverse

forms of vibrations of strings from Taylor's cases, so Fourier constructs out
of simple cases of conduction the most multifarious motions of heat; and
that method has extended itself over the whole of physics. Ohm forms his
conception of the electric current in imitation of Fourier's. The latter, also,
adopts Fick's theory of diffusion. In an analogous manner a conception of
the magnetic current is developed. All sorts of stationary currents are thus
made to exhibit common features, and even the condition of complete
equilibrium in an extended medium shares these features with the
dynamical condition of equilibrium of a stationary current. Things as
remote as the magnetic lines of force of an electric current and the stream-
lines of a frictionless liquid vortex enter in this way into a peculiar
relationship of similarity. The concept of potential, originally enunciated for
a restricted province, acquires a wide-reaching applicability. Things as
dissimilar as pressure, temperature, and electromotive force, now show
points of agreement in relation to ideas derived by definite methods from
that concept: viz., fall of pressure, fall of temperature, fall of potential, as
also with the further notions of liquid, thermal, and electric strength of
current. That relationship between systems of ideas in which the
dissimilarity of every two homologous concepts as well as the agreement in
logical relations of every two homologous pairs of concepts, is clearly
brought to light, is called an analogy. It is an effective means of mastering
heterogeneous fields of facts in unitary comprehension. The path is plainly
shown in which a universal physical phenomenology embracing all
domains, will be developed.
In the process described we attain for the first time to what is indispensable
in the direct description of broad fields of fact—the wide-reaching abstract
concept. And now I must put a question smacking of the school-master, but
unavoidable: What is a concept? Is it a hazy representation, admitting
withal of mental visualisation? No. Mental visualisation accompanies it
only in the simplest cases, and then merely as an adjunct. Think, for
example, of the "coefficient of self-induction," and seek for its visualised
mental image. Or is, perhaps, the concept a mere word? The adoption of
this forlorn idea, which has been actually proposed of late by a reputed
mathematician would only throw us back a thousand years into the deepest
scholasticism. We must, therefore, reject it.

The solution is not far to seek. We must not think that sensation, or
representation, is a purely passive process. The lowest organisms respond to
it with a simple reflex motion, by engulfing the prey which approaches
them. In higher organisms the centripetal stimulus encounters in the
nervous system obstacles and aids which modify the centrifugal process. In
still higher organisms, where prey is pursued and examined, the process in
question may go through extensive paths of circular motions before it
comes to relative rest. Our own life, too, is enacted in such processes; all
that we call science may be regarded as parts, or middle terms, of such
activities.
It will not surprise us now if I say: the definition of a concept, and, when it
is very familiar, even its name, is an impulse to some accurately determined,
often complicated, critical, comparative, or constructive activity, the usually
sense-perceptive result of which is a term or member of the concept's scope.
It matters not whether the concept draws the attention only to one certain
sense (as sight) or to a phase of a sense (as color, form), or is the starting-
point of a complicated action; nor whether the activity in question
(chemical, anatomical, and mathematical operations) is muscular or
technical, or performed wholly in the imagination, or only intimated. The
concept is to the physicist what a musical note is to a piano-player. A
trained physicist or mathematician reads a memoir like a musician reads a
score. But just as the piano-player must first learn to move his fingers
singly and collectively, before he can follow his notes without effort, so the
physicist or mathematician must go through a long apprenticeship before he
gains control, so to speak, of the manifold delicate innervations of his
muscles and imagination. Think of how frequently the beginner in physics
or mathematics performs more, or less, than is required, or of how
frequently he conceives things differently from what they are! But if, after
having had sufficient discipline, he lights upon the phrase "coefficient of
self-induction," he knows immediately what that term requires of him. Long
and thoroughly practised actions, which have their origin in the necessity of
comparing and representing facts by other facts, are thus the very kernel of
concepts. In fact, positive and philosophical philology both claim to have
established that all roots represent concepts and stood originally for
muscular activities alone. The slow assent of physicists to Kirchhoff's
dictum now becomes intelligible. They best could feel the vast amount of

individual labor, theory, and skill required before the ideal of direct
description could be realised.

Suppose, now, the ideal of a given province of facts is reached. Does
description accomplish all that the inquirer can ask? In my opinion, it does.
Description is a building up of facts in thought, and this building up is, in
the experimental sciences, often the condition of actual execution. For the
physicist, to take a special case, the metrical units are the building-stones,
the concepts the directions for building, and the facts the result of the
building. Our mental imagery is almost a complete substitute for the fact,
and by means of it we can ascertain all the fact's properties. We do not
know that worst which we ourselves have made.
People require of science that it should prophesy, and Hertz uses that
expression in his posthumous Mechanics. But, natural as it is, the
expression is too narrow. The geologist and the palæontologist, at times the
astronomer, and always the historian and the philologist, prophesy, so to
speak, backwards. The descriptive sciences, like geometry and
mathematics, prophesy neither forward or backwards, but seek from given
conditions the conditioned. Let us say rather: Science completes in thought
facts that are only partly given. This is rendered possible by description, for
description presupposes the interdependence of the descriptive elements:
otherwise nothing would be described.
It is said, description leaves the sense of causality unsatisfied. In fact, many
imagine they understand motions better when they picture to themselves the
pulling forces; and yet the accelerations, the facts, accomplish more,
without superfluous additions. I hope that the science of the future will
discard the idea of cause and effect, as being formally obscure; and in my
feeling that these ideas contain a strong tincture of fetishism, I am certainly
not alone. The more proper course is, to regard the abstract determinative
elements of a fact as interdependent, in a purely logical way, as the
mathematician or geometer does. True, by comparison with the will, forces
are brought nearer to our feeling; but it may be that ultimately the will itself
will be made clearer by comparison with the accelerations of masses.

If we are asked, candidly, when is a fact clear to us, we must say "when we
can reproduce it by very simple and very familiar intellectual operations,
such as the construction of accelerations, or the geometrical summation of
accelerations, and so forth." The requirement of simplicity is of course to
the expert a different matter from what it is to the novice. For the first,
description by a system of differential equations is sufficient; for the
second, a gradual construction out of elementary laws is required. The first
discerns at once the connexion of the two expositions. Of course, it is not
disputed that the artistic value of materially equivalent descriptions may not
be different.
Most difficult is it to persuade strangers that the grand universal laws of
physics, such as apply indiscriminately to material, electrical, magnetic, and
other systems, are not essentially different from descriptions. As compared
with many sciences, physics occupies in this respect a position of vantage
that is easily explained. Take, for example, anatomy. As the anatomist in his
quest for agreements and differences in animals ascends to ever higher and
higher classifications, the individual facts that represent the ultimate terms
of the system, are still so different that they must be singly noted. Think, for
example, of the common marks of the Vertebrates, of the class-characters of
Mammals and Birds on the one hand and of Fishes on the other, of the
double circulation of the blood on the one hand and of the single on the
other. In the end, always isolated facts remain, which show only a slight
likeness to one another.
A science still more closely allied to physics, chemistry, is often in the same
strait. The abrupt change of the qualitative properties, in all likelihood
conditioned by the slight stability of the intermediate states, the remote
resemblance of the co-ordinated facts of chemistry render the treatment of
its data difficult. Pairs of bodies of different qualitative properties unite in
different mass-ratios; but no connexion between the first and the last is to
be noted, at first.
Physics, on the other hand, reveals to us wide domains of qualitatively
homogeneous facts, differing from one another only in the number of equal
parts into which their characteristic marks are divisible, that is, differing
only quantitatively. Even where we have to deal with qualities (colors and
sounds), quantitative characters of those qualities are at our disposal. Here
the classification is so simple a task that it rarely impresses us as such,

whilst in infinitely fine gradations, in a continuum of facts, our number-
system is ready beforehand to follow as far as we wish. The co-ordinated
facts are here extremely similar and very closely affined, as are also their
descriptions which consist in the determination of the numerical measures
of one given set of characters from those of a different set by means of
familiar mathematical operations—methods of derivation. Thus, the
common characteristics of all descriptions can be found here; and with them
a succinct, comprehensive description, or a rule for the construction of all
single descriptions, is assigned,—and this we call law. Well-known
examples are the formulæ for freely falling bodies, for projectiles, for
central motion, and so forth. If physics apparently accomplishes more by its
methods than other sciences, we must remember that in a sense it has
presented to it much simpler problems.
The remaining sciences, whose facts also present a physical side, need not
be envious of physics for this superiority; for all its acquisitions ultimately
redound to their benefit as well. But also in other ways this mutual help
shall and must change. Chemistry has advanced very far in making the
methods of physics her own. Apart from older attempts, the periodical
series of Lothar Meyer and Mendelejeff are a brilliant and adequate means
of producing an easily surveyed system of facts, which by gradually
becoming complete, will take the place almost of a continuum of facts.
Further, by the study of solutions, of dissociation, in fact generally of
phenomena which present a continuum of cases, the methods of
thermodynamics have found entrance into chemistry. Similarly we may
hope that, at some future day, a mathematician, letting the fact-continuum
of embryology play before his mind, which the palæontologists of the
future will supposedly have enriched with more intermediate and derivative
forms between Saurian and Bird than the isolated Pterodactyl,
Archæopteryx, Ichthyornis, and so forth, which we now have—that such a
mathematician shall transform, by the variation of a few parameters, as in a
dissolving view, one form into another, just as we transform one conic
section into another.
Reverting now to Kirchhoff's words, we can come to some agreement
regarding their import. Nothing can be built without building-stones,
mortar, scaffolding, and a builder's skill. Yet assuredly the wish is well
founded, that will show to posterity the complete structure in its finished

form, bereft of unsightly scaffolding. It is the pure logical and æsthetic
sense of the mathematician that speaks out of Kirchhoff's words. Modern
expositions of physics aspire after his ideal; that, too, is intelligible. But it
would be a poor didactic shift, for one whose business it was to train
architects, to say: "Here is a splendid edifice; if thou wouldst really build,
go thou and do likewise".
The barriers between the special sciences, which make division of work and
concentration possible, but which appear to us after all as cold and
conventional restrictions, will gradually disappear. Bridge upon bridge is
thrown over the gaps. Contents and methods, even of the remotest branches,
are compared. When the Congress of Natural Scientists shall meet a
hundred years hence, we may expect that they will represent a unity in a
higher sense than is possible to-day, not in sentiment and aim alone, but in
method also. In the meantime, this great change will be helped by our
keeping constantly before our minds the fact of the intrinsic relationship of
all research, which Kirchhoff characterised with such classical simplicity.

THE PART PLAYED BY ACCIDENT IN
INVENTION AND DISCOVERY.[81]
It is characteristic of the naïve and sanguine beginnings of thought in
youthful men and nations, that all problems are held to be soluble and
fundamentally intelligible on the first appearance of success. The sage of
Miletus, on seeing plants take their rise from moisture, believed he had
comprehended the whole of nature, and he of Samos, on discovering that
definite numbers corresponded to the lengths of harmonic strings, imagined
he could exhaust the nature of the world by means of numbers. Philosophy
and science in such periods are blended. Wider experience, however,
speedily discloses the error of such a course, gives rise to criticism, and
leads to the division and ramification of the sciences.
At the same time, the necessity of a broad and general view of the world
remains; and to meet this need philosophy parts company with special
inquiry. It is true, the two are often found united in gigantic personalities.
But as a rule their ways diverge more and more widely from each other.
And if the estrangement of philosophy from science can reach a point
where data unworthy of the nursery are not deemed too scanty as
foundations of the world, on the other hand the thorough-paced specialist
may go to the extreme of rejecting point-blank the possibility of a broader
view, or at least of deeming it superfluous, forgetful of Voltaire's
apophthegm, nowhere more applicable than here, Le superflu—chose très
nécessaire.
It is true, the history of philosophy, owing to the insufficiency of its
constructive data, is and must be largely a history of error. But it would be
the height of ingratitude on our part to forget that the seeds of thoughts
which still fructify the soil of special research, such as the theory of
irrationals, the conceptions of conservation, the doctrine of evolution, the
idea of specific energies, and so forth, may be traced back in distant ages to
philosophical sources. Furthermore, to have deferred or abandoned the
attempt at a broad philosophical view of the world from a full knowledge of
the insufficiency of our materials, is quite a different thing from never

having undertaken it at all. The revenge of its neglect, moreover, is
constantly visited upon the specialist by his committal of the very errors
which philosophy long ago exposed. As a fact, in physics and physiology,
particularly during the first half of this century, are to be met intellectual
productions which for naïve simplicity are not a jot inferior to those of the
Ionian school, or to the Platonic ideas, or to that much reviled ontological
proof.
Latterly, there has been evidence of a gradual change in the situation.
Recent philosophy has set itself more modest and more attainable ends; it is
no longer inimical to special inquiry; in fact, it is zealously taking part in
that inquiry. On the other hand, the special sciences, mathematics and
physics, no less than philology, have become eminently philosophical. The
material presented is no longer accepted uncritically. The glance of the
inquirer is bent upon neighboring fields, whence that material has been
derived. The different special departments are striving for closer union, and
gradually the conviction is gaining ground that philosophy can consist only
of mutual, complemental criticism, interpenetration, and union of the
special sciences into a consolidated whole. As the blood in nourishing the
body separates into countless capillaries, only to be collected again and to
meet in the heart, so in the science of the future all the rills of knowledge
will gather more and more into a common and undivided stream.
It is this view—not an unfamiliar one to the present generation—that I
purpose to advocate. Entertain no hope, or rather fear, that I shall construct
systems for you. I shall remain a natural inquirer. Nor expect that it is my
intention to skirt all the fields of natural inquiry. I can attempt to be your
guide only in that branch which is familiar to me, and even there I can assist
in the furtherment of only a small portion of the allotted task. If I shall
succeed in rendering plain to you the relations of physics, psychology, and
the theory of knowledge, so that you may draw from each profit and light,
redounding to the advantage of each, I shall regard my work as not having
been in vain. Therefore, to illustrate by an example how, consonantly with
my powers and views, I conceive such inquiries should be conducted, I
shall treat to-day, in the form of a brief sketch, of the following special and
limited subject—of the part which accidental circumstances play in the
development of inventions and discoveries.

When we Germans say of a man that he was not the inventor of gunpowder,
[82]
we impliedly cast a grave suspicion on his abilities. But the expression
is not a felicitous one, as there is probably no invention in which deliberate
thought had a smaller, and pure luck a larger, share than in this. It is well to
ask, Are we justified in placing a low estimate on the achievement of an
inventor because accident has assisted him in his work? Huygens, whose
discoveries and inventions are justly sufficient to entitle him to an opinion
in such matters, lays great emphasis on this factor. He asserts that a man
capable of inventing the telescope without the concurrence of accident must
have been gifted with superhuman genius.[83]
A man living in the midst of civilisation finds himself surrounded by a host
of marvellous inventions, considering none other than the means of
satisfying the needs of daily life. Picture such a man transported to the
epoch preceding the invention of these ingenious appliances, and imagine
him undertaking in a serious manner to comprehend their origin. At first the
intellectual power of the men capable of producing such marvels will strike
him as incredible, or, if we adopt the ancient view, as divine. But his
astonishment is considerably allayed by the disenchanting yet elucidative
revelations of the history of primitive culture, which to a large extent prove
that these inventions took their rise very slowly and by imperceptible
degrees.
A small hole in the ground with fire kindled in it constituted the primitive
stove. The flesh of the quarry, wrapped with water in its skin, was boiled by
contact with heated stones. Cooking by stones was also done in wooden
vessels. Hollow gourds were protected from the fire by coats of clay. Thus,
from the burned clay accidentally originated the enveloping pot, which
rendered the gourd superfluous, although for a long time thereafter the clay
was still spread over the gourd, or pressed into woven wicker-work before
the potter's art assumed its final independence. Even then the wicker-work
ornament was retained, as a sort of attest of its origin.
We see, thus, it is by accidental circumstances, or by such as lie without our
purpose, foresight, and power, that man is gradually led to the acquaintance
of improved means of satisfying his wants. Let the reader picture to himself
the genius of a man who could have foreseen without the help of accident

that clay handled in the ordinary manner would produce a useful cooking
utensil! The majority of the inventions made in the early stages of
civilisation, including language, writing, money, and the rest, could not
have been the product of deliberate methodical reflexion for the simple
reason that no idea of their value and significance could have been had
except from practical use. The invention of the bridge may have been
suggested by the trunk of a tree which had fallen athwart a mountain-
torrent; that of the tool by the use of a stone accidentally taken into the hand
to crack nuts. The use of fire probably started in and was disseminated from
regions where volcanic eruptions, hot springs, and burning jets of natural
gas afforded opportunity for quietly observing and turning to practical
account the properties of fire. Only after that had been done could the
significance of the fire-drill be appreciated, an instrument which was
probably discovered from boring a hole through a piece of wood. The
suggestion of a distinguished inquirer that the invention of the fire-drill
originated on the occasion of a religious ceremony is both fantastic and
incredible. And as to the use of fire, we should no more attempt to derive
that from the invention of the fire-drill than we should from the invention of
sulphur matches. Unquestionably the opposite course was the real one.[84]
Similar phenomena, though still largely veiled in obscurity, mark the initial
transition of nations from a hunting to a nomadic life and to agriculture.[85]
We shall not multiply examples, but content ourselves with the remark that
the same phenomena recur in historical times, in the ages of great technical
inventions, and, further, that regarding them the most whimsical notions
have been circulated—notions which ascribe to accident an unduly
exaggerated part, and one which in a psychological respect is absolutely
impossible. The observation of steam escaping from a tea-kettle and of the
clattering of the lid is supposed to have led to the invention of the steam-
engine. Just think of the gap between this spectacle and the conception of
the performance of great mechanical work by steam, for a man totally
ignorant of the steam-engine! Let us suppose, however, that an engineer,
versed in the practical construction of pumps, should accidentally dip into
water an inverted bottle that had been filled with steam for drying and still
retained its steam. He would see the water rush violently into the bottle, and
the idea would very naturally suggest itself of founding on this experience a
convenient and useful atmospheric steam-pump, which by imperceptible

degrees, both psychologically possible and immediate, would then undergo
a natural and gradual transformation into Watt's steam-engine.
But granting that the most important inventions are brought to man's notice
accidentally and in ways that are beyond his foresight, yet it does not follow
that accident alone is sufficient to produce an invention. The part which
man plays is by no means a passive one. Even the first potter in the
primeval forest must have felt some stirrings of genius within him. In all
such cases, the inventor is obliged to take note of the new fact, he must
discover and grasp its advantageous feature, and must have the power to
turn that feature to account in the realisation of his purpose. He must isolate
the new feature, impress it upon his memory, unite and interweave it with
the rest of his thought; in short, he must possess the capacity to profit by
experience.
The capacity to profit by experience might well be set up as a test of
intelligence. This power varies considerably in men of the same race, and
increases enormously as we advance from the lower animals to man. The
former are limited in this regard almost entirely to the reflex actions which
they have inherited with their organism, they are almost totally incapable of
individual experience, and considering their simple wants are scarcely in
need of it. The ivory-snail (Eburna spirata) never learns to avoid the
carnivorous Actinia, no matter how often it may wince under the latter's
shower of needles, apparently having no memory for pain whatever.[86] A
spider can be lured forth repeatedly from its hole by touching its web with a
tuning-fork. The moth plunges again and again into the flame which has
burnt it. The humming-bird hawk-moth[87] dashes repeatedly against the
painted roses of the wall-paper, like the unhappy and desperate thinker who
never wearies of attacking the same insoluble chimerical problem. As
aimlessly almost as Maxwell's gaseous molecules and in the same
unreasoning manner common flies in their search for light and air stream
against the glass pane of a half-opened window and remain there from sheer
inability to find their way around the narrow frame. But a pike separated
from the minnows of his aquarium by a glass partition, learns after the lapse
of a few months, though only after having butted himself half to death, that
he cannot attack these fishes with impunity. What is more, he leaves them
in peace even after the removal of the partition, though he will bolt a
strange fish at once. Considerable memory must be attributed to birds of

passage, a memory which, probably owing to the absence of disturbing
thoughts, acts with the precision of that of some idiots. Finally, the
susceptibility to training evinced by the higher vertebrates is indisputable
proof of the ability of these animals to profit by experience.
A powerfully developed mechanical memory, which recalls vividly and
faithfully old situations, is sufficient for avoiding definite particular
dangers, or for taking advantage of definite particular opportunities. But
more is required for the development of inventions. More extensive chains
of images are necessary here, the excitation by mutual contact of widely
different trains of ideas, a more powerful, more manifold, and richer
connexion of the contents of memory, a more powerful and impressionable
psychical life, heightened by use. A man stands on the bank of a mountain-
torrent, which is a serious obstacle to him. He remembers that he has
crossed just such a torrent before on the trunk of a fallen tree. Hard by trees
are growing. He has often moved the trunks of fallen trees. He has also
felled trees before, and then moved them. To fell trees he has used sharp
stones. He goes in search of such a stone, and as the old situations that
crowd into his memory and are held there in living reality by the definite
powerful interest which he has in crossing just this torrent,—as these
impressions are made to pass before his mind in the inverse order in which
they were here evoked, he invents the bridge.
There can be no doubt but the higher vertebrates adapt their actions in some
moderate degree to circumstances. The fact that they give no appreciable
evidence of advance by the accumulation of inventions, is satisfactorily
explained by a difference of degree or intensity of intelligence as compared
with man; the assumption of a difference of kind is not necessary. A person
who saves a little every day, be it ever so little, has an incalculable
advantage over him who daily squanders that amount, or is unable to keep
what he has accumulated. A slight quantitative difference in such things
explains enormous differences of advancement.
The rules which hold good in prehistoric times also hold good in historical
times, and the remarks made on invention may be applied almost without
modification to discovery; for the two are distinguished solely by the use to
which the new knowledge is put. In both cases the investigator is concerned
with some newly observed relation of new or old properties, abstract or
concrete. It is observed, for example, that a substance which gives a

chemical reaction A is also the cause of a chemical reaction B. If this
observation fulfils no purpose but that of furthering the scientist's insight, or
of removing a source of intellectual discomfort, we have a discovery; but an
invention, if in using the substance giving the reaction A to produce the
desired reaction B, we have a practical end in view, and seek to remove a
source of material discomfort. The phrase, disclosure of the connexion of
reactions, is broad enough to cover discoveries and inventions in all
departments. It embraces the Pythagorean proposition, which is a
combination of a geometrical and an arithmetical reaction, Newton's
discovery of the connexion of Kepler's motions with the law of the inverse
squares, as perfectly as it does the detection of some minute but appropriate
alteration in the construction of a tool, or of some appropriate change in the
methods of a dyeing establishment.
The disclosure of new provinces of facts before unknown can only be
brought about by accidental circumstances, under which are remarked facts
that commonly go unnoticed. The achievement of the discoverer here
consists in his sharpened attention, which detects the uncommon features of
an occurrence and their determining conditions from their most evanescent
marks,[88] and discovers means of submitting them to exact and full
observation. Under this head belong the first disclosures of electrical and
magnetic phenomena, Grimaldi's observation of interference, Arago's
discovery of the increased check suffered by a magnetic needle vibrating in
a copper envelope as compared with that observed in a bandbox, Foucault's
observation of the stability of the plane of vibration of a rod accidentally
struck while rotating in a turning-lathe, Mayer's observation of the
increased redness of venous blood in the tropics, Kirchhoff's observation of
the augmentation of the D-line in the solar spectrum by the interposition of
a sodium lamp, Schönbein's discovery of ozone from the phosphoric smell
emitted on the disruption of air by electric sparks, and a host of others. All
these facts, of which unquestionably many were seen numbers of times
before they were noticed, are examples of the inauguration of momentous
discoveries by accidental circumstances, and place the importance of
strained attention in a brilliant light.
But not only is a significant part played in the beginning of an inquiry by
co-operative circumstances beyond the foresight of the investigator; their
influence is also active in its prosecution. Dufay, thus, whilst following up

the behavior of one electrical state which he had assumed, discovers the
existence of two. Fresnel learns by accident that the interference-bands
received on ground glass are seen to better advantage in the open air. The
diffraction-phenomenon of two slits proved to be considerably different
from what Fraunhofer had anticipated, and in following up this
circumstance he was led to the important discovery of grating-spectra.
Faraday's induction-phenomenon departed widely from the initial
conception which occasioned his experiments, and it is precisely this
deviation that constitutes his real discovery.
Every man has pondered on some subject. Every one of us can multiply the
examples cited, by less illustrious ones from his own experience. I shall cite
but one. On rounding a railway curve once, I accidentally remarked a
striking apparent inclination of the houses and trees. I inferred that the
direction of the total resultant physical acceleration of the body reacts
physiologically as the vertical. Afterwards, in attempting to inquire more
carefully into this phenomenon, and this only, in a large whirling machine,
the collateral phenomena conducted me to the sensation of angular
acceleration, vertigo, Flouren's experiments on the section of the semi-
circular canals etc., from which gradually resulted views relating to
sensations of direction which are also held by Breuer and Brown, which
were at first contested on all hands, but are now regarded on many sides as
correct, and which have been recently enriched by the interesting inquiries
of Breuer concerning the macula acustica, and Kreidel's experiments with
magnetically orientable crustacea.[89] Not disregard of accident but a direct
and purposeful employment of it advances research.
The more powerful the psychical connexion of the memory pictures is,—
and it varies with the individual and the mood,—the more apt is the same
accidental observation to be productive of results. Galileo knows that the air
has weight; he also knows of the "resistance to a vacuum," expressed both
in weight and in the height of a column of water. But the two ideas dwelt
asunder in his mind. It remained for Torricelli to vary the specific gravity of
the liquid measuring the pressure, and not till then was the air included in
the list of pressure-exerting fluids. The reversal of the lines of the spectrum
was seen repeatedly before Kirchhoff, and had been mechanically
explained. But it was left for his penetrating vision to discern the evidence
of the connexion of this phenomenon with questions of heat, and to him

alone through persistent labor was revealed the sweeping significance of the
fact for the mobile equilibrium of heat. Supposing, then, that such a rich
organic connexion of the elements of memory exists, and is the prime
distinguishing mark of the inquirer, next in importance certainly is that
intense interest in a definite object, in a definite idea, which fashions
advantageous combinations of thought from elements before disconnected,
and obtrudes that idea into every observation made, and into every thought
formed, making it enter into relationship with all things. Thus Bradley,
deeply engrossed with the subject of aberration, is led to its solution by an
exceedingly unobtrusive experience in crossing the Thames. It is
permissible, therefore, to ask whether accident leads the discoverer, or the
discoverer accident, to a successful outcome in scientific quests.
No man should dream of solving a great problem unless he is so thoroughly
saturated with his subject that everything else sinks into comparative
insignificance. During a hurried meeting with Mayer in Heidelberg once,
Jolly remarked, with a rather dubious implication, that if Mayer's theory
were correct water could be warmed by shaking. Mayer went away without
a word of reply. Several weeks later, and now unrecognised by Jolly, he
rushed into the latter's presence exclaiming: "Es ischt aso!" (It is so, it is
so!) It was only after considerable explanation that Jolly found out what
Mayer wanted to say. The incident needs no comment.[90]
A person deadened to sensory impressions and given up solely to the
pursuit of his own thoughts, may also light on an idea that will divert his
mental activity into totally new channels. In such cases it is a psychical
accident, an intellectual experience, as distinguished from a physical
accident, to which the person owes his discovery—a discovery which is
here made "deductively" by means of mental copies of the world, instead of
experimentally. Purely experimental inquiry, moreover, does not exist, for,
as Gauss says, virtually we always experiment with our thoughts. And it is
precisely that constant, corrective interchange or intimate union of
experiment and deduction, as it was cultivated by Galileo in his Dialogues
and by Newton in his Optics, that is the foundation of the benign
fruitfulness of modern scientific inquiry as contrasted with that of antiquity,
where observation and reflexion ofttimes pursued their respective courses
like two strangers.

We have to wait for the appearance of a favorable physical accident. The
movement of our thoughts obeys the law of association. In the case of
meagre experience the result of this law is simply the mechanical
reproduction of definite sensory experiences. On the other hand, if the
psychical life is subjected to the incessant influences of a powerful and rich
experience, then every representative element in the mind is connected with
so many others that the actual and natural course of the thoughts is easily
influenced and determined by insignificant circumstances, which
accidentally are decisive. Hereupon, the process termed imagination
produces its protean and infinitely diversified forms. Now what can we do
to guide this process, seeing that the combinatory law of the images is
without our reach? Rather let us ask, what influence can a powerful and
constantly recurring idea exert on the movement of our thoughts?
According to what has preceded, the answer is involved in the question
itself. The idea dominates the thought of the inquirer, not the latter the
former.
Let us see, now, if we can acquire a profounder insight into the process of
discovery. The condition of the discoverer is, as James has aptly remarked,
not unlike the situation of a person who is trying to remember something
that he has forgotten. Both are sensible of a gap, and have only a remote
presentiment of what is missing. Suppose I meet in a company a well-
known and affable gentleman whose name I have forgotten, and who to my
horror asks to be introduced to some one. I set to work according to
Lichtenberg's rule, and run down the alphabet in search of the initial letter
of his name. A vague sympathy holds me at the letter G. Tentatively I add
the second letter and am arrested at e, and long before I have tried the third
letter r, the name "Gerson" sounds sonorously upon my ear, and my anguish
is gone. While taking a walk I meet a gentleman from whom I receive a
communication. On returning home, and in attending to weightier affairs,
the matter slips my mind. Moodily, but in vain, I ransack my memory.
Finally I observe that I am going over my walk again in thought. On the
street corner in question the self-same gentleman stands before me and
repeats his communication. In this process are successively recalled to
consciousness all the percepts which were connected with the percept that
was lost, and with them, finally, that, too, is brought to light. In the first
case—where the experience had already been made and is permanently
impressed on our thought—a systematic procedure is both possible and

easy, for we know that a name must be composed of a limited number of
sounds. But at the same time it should be observed that the labor involved
in such a combinatorial task would be enormous if the name were long and
the responsiveness of the mind weaker.
It is often said, and not wholly without justification, that the scientist has
solved a riddle. Every problem in geometry may be clothed in the garb of a
riddle. Thus: "What thing is that M which has the properties A, B, C?"
"What circle is that which touches the straight lines A, B, but touches B in
the point C?" The first two conditions marshal before the imagination the
group of circles whose centres lie in the line of symmetry of A, B. The third
condition reminds us of all the circles having centres in the straight line that
stands at right angles to B in C. The common term, or common terms, of the
two groups of images solves the riddle—satisfies the problem. Puzzles
dealing with things or words induce similar processes, but the memory in
such cases is exerted in many directions and more varied and less clearly
ordered provinces of ideas are surveyed. The difference between the
situation of a geometer who has a construction to make, and that of an
engineer, or a scientist, confronted with a problem, is simply this, that the
first moves in a field with which he is thoroughly acquainted, whereas the
two latter are obliged to familiarise themselves with this field subsequently,
and in a measure far transcending what is commonly required. In this
process the mechanical engineer has at least always a definite goal before
him and definite means to accomplish his aim, whilst in the case of the
scientist that aim is in many instances presented only in vague and general
outlines. Often the very formulation of the riddle devolves on him.
Frequently it is not until the aim has been reached that the broader outlook
requisite for systematic procedure is obtained. By far the larger portion of
his success, therefore, is contingent on luck and instinct. It is immaterial, so
far as its character is concerned, whether the process in question is brought
rapidly to a conclusion in the brain of one man, or whether it is spun out for
centuries in the minds of a long succession of thinkers. The same relation
that a word solving a riddle bears to that riddle is borne by the modern
conception of light to the facts discovered by Grimaldi, Römer, Huygens,
Newton, Young, Malus, and Fresnel, and only by the help of this slowly
developed conception is our mental vision enabled to embrace the broad
domain of facts in question.

A welcome complement to the discoveries which the history of civilisation
and comparative psychology have furnished, is to be found in the
confessions of great scientists and artists. Scientists and artists, we might
say, for Liebig boldly declared there was no essential difference between
the two. Are we to regard Leonardo da Vinci as a scientist or as an artist? If
the artist builds up his work from a few motives, the scientist discovers the
motives which permeate reality. If scientists like Lagrange or Fourier are in
a certain measure artists in the presentation of their results, on the other
hand, artists like Shakespeare or Ruysdael are scientists in the insight which
must have preceded their creations.
Newton, when questioned about his methods of work, could give no other
answer but that he was wont to ponder again and again on a subject; and
similar utterances are accredited to D'Alembert and Helmholtz. Scientists
and artists both recommend persistent labor. After the repeated survey of a
field has afforded opportunity for the interposition of advantageous
accidents, has rendered all the traits that suit with the mood or the dominant
thought more vivid, and has gradually relegated to the background all
things that are inappropriate, making their future appearance impossible;
then from the teeming, swelling host of fancies which a free and high-flown
imagination calls forth, suddenly that particular form arises to the light
which harmonises perfectly with the ruling idea, mood, or design. Then it is
that that which has resulted slowly as the result of a gradual selection,
appears as if it were the outcome of a deliberate act of creation. Thus are to
be explained the statements of Newton, Mozart, Richard Wagner, and
others, when they say that thoughts, melodies, and harmonies had poured in
upon them, and that they had simply retained the right ones. Undoubtedly,
the man of genius, too, consciously or instinctively, pursues systematic
methods wherever it is possible; but in his delicate presentiment he will
omit many a task or abandon it after a hasty trial on which a less endowed
man would squander his energies in vain. Thus, the genius accomplishes[91]
in a brief space of time undertakings for which the life of an ordinary man
would far from suffice. We shall hardly go astray if we regard genius as
only a slight deviation from the average mental endowment—as possessing
simply a greater sensitiveness of cerebral reaction and a greater swiftness of
reaction. The men who, obeying their inner impulses, make sacrifices for an
idea instead of advancing their material welfare, may appear to the full-
blooded Philistine as fools; yet we shall scarcely adopt Lombroso's view,

that genius is to be regarded as a disease, although it is unfortunately true
that the sensitive brains and fragile constitutions succumb most readily to
sickness.
The remark of C. G. J. Jacobi that mathematics is slow of growth and only
reaches the truth by long and devious paths, that the way to its discovery
must be prepared for long beforehand, and that then the truth will make its
long-deferred appearance as if impelled by some divine necessity[92]—all
this holds true of every science. We are astounded often to note that it
required the combined labors of many eminent thinkers for a full century to
reach a truth which it takes us only a few hours to master and which once
acquired seems extremely easy to reach under the right sort of
circumstances. To our humiliation we learn that even the greatest men are
born more for life than for science. The extent to which even they are
indebted to accident—to that singular conflux of the physical and the
psychical life in which the continuous but yet imperfect and never-ending
adaptation of the latter to the former finds its distinct expression—that has
been the subject of our remarks to-day. Jacobi's poetical thought of a divine
necessity acting in science will lose none of its loftiness for us if we
discover in this necessity the same power that destroys the unfit and fosters
the fit. For loftier, nobler, and more romantic than poetry is the truth and the
reality.

ON SENSATIONS OF ORIENTATION.[93]
Through the co-operation of a succession of inquirers, among whom are
particularly to be mentioned Goltz of Strassburg and Breuer of Vienna,
considerable advances have been made during the last twenty-five years in
our knowledge of the means by which we ascertain our position in space
and the direction of our motion, or orient ourselves, as the phrase goes. I
presume that you are already acquainted with the physiological part of the
processes with which our sensations of movement, or, more generally
speaking, our sensations of orientation, are connected. Here I shall consider
more particularly the physical side of the matter. In fact, I was originally led
to the consideration of these questions by the observation of extremely
simple and perfectly well-known physical facts, before I had any great
acquaintance with physiology and while pursuing unbiasedly my natural
thoughts; and I am of the conviction that the way which I have pursued, and
which is entirely free from hypotheses, will, if you will follow my
exposition, be that of easiest acquisition for the most of you.
No man of sound common sense could ever have doubted that a pressure or
force is requisite to set a body in motion in a given direction and that a
contrary pressure is required to stop suddenly a body in motion. Though the
law of inertia was first formulated with anything like exactness by Galileo,
the facts at the basis of it were known long previously to men of the stamp
of Leonardo da Vinci, Rabelais, and others, and were illustrated by them
with appropriate experiments. Leonardo knew that by a swift stroke with a
ruler one can knock out from a vertical column of checkers a single checker
without over-throwing the column. The experiment with a coin resting on a
piece of pasteboard covering a goblet, which falls into the goblet when the
pasteboard is jerked away, like all experiments of the kind, is certainly very
old.
With Galileo the experience in question assumes greater clearness and
force. In the famous dialogue on the Copernican system which cost him his
freedom, he explains the tides in an unfelicitous, though in principle correct
manner, by the analogue of a platter of water swung to and fro. In

opposition to the Aristotelians of his time, who believed the descent of a
heavy body could be accelerated by the superposition of another heavy
body, he asserted that a body could never be accelerated by one lying upon
it unless the first in some way impeded the superposed body in its descent.
To seek to press a falling body by means of another placed upon it, is as
senseless as trying to prod a man with a lance when the man is speeding
away from one with the same velocity as the lance. Even this little
excursion into physics can explain much to us. You know the peculiar
sensation which one has in falling, as when one jumps from a high
springboard into the water, and which is also experienced in some measure
at the beginning of the descent of elevators and swings. The reciprocal
gravitational pressure of the different parts of our body, which is certainly
felt in some manner, vanishes in free descent, or, in the case of the elevator,
is diminished on the beginning of the descent. A similar sensation would be
experienced if we were suddenly transported to the moon where the
acceleration of gravity is much less than upon the earth. I was led to these
considerations in 1866 by a suggestion in physics, and having also taken
into account the alterations of the blood-pressure in the cases in question, I
found I coincided without knowing it with Wollaston and Purkinje. The first
as early as 1810 in his Croonian lecture had touched on the subject of sea-
sickness and explained it by alterations of the blood-pressure, and later had
laid similar considerations at the basis of his explanation of vertigo (1820-
1826).[94]
Newton was the first to enunciate with perfect generality that a body can
change the velocity and direction of its motion only by the action of a force,
or the action of a second body. A corollary of this law which was first
expressly deduced by Euler is that a body can never be set rotating or made
to cease rotating of itself but only by forces and other bodies. For example,
turn an open watch which has run down freely backwards and forwards in
your hand. The balance-wheel will not fully catch the rapid rotations, it
does not even respond fully to the elastic force of the spring which proves
too weak to carry the wheel entirely with it.
Let us consider now that whether we move ourselves by means of our legs,
or whether we are moved by a vehicle or a boat, at first only a part of our
body is directly moved and the rest of it is afterwards set in motion by the
first part. We see that pressures, pulls, and tensions are always produced

between the parts of the body in this action, which pressures, pulls, and
tensions give rise to sensations by which the forward or rotary movements
in which we are engaged are made perceptible.[95] But it is quite natural that
sensations so familiar should be little noticed and that attention should be
drawn to them only under special circumstances when they occur
unexpectedly or with unusual strength.

Fig. 45.
Thus my attention was drawn to this point by the sensation of falling and
subsequently by another singular occurrence. I was rounding a sharp
railway curve once when I suddenly saw all the trees, houses, and factory
chimneys along the track swerve from the vertical and assume a strikingly
inclined position. What had hitherto appeared to me perfectly natural,
namely, the fact that we distinguish the vertical so perfectly and sharply
from every other direction, now struck me as enigmatical. Why is it that the
same direction can now appear vertical to me and now cannot? By what is
the vertical distinguished for us? (Compare Figure 45.)
The rails are raised on the convex or outward side of the track in order to
insure the stability of the carriage as against the action of the centrifugal
force, the whole being so arranged that the combination of the force of
gravity with the centrifugal force of the train shall give rise to a force
perpendicular to the plane of the rails.

Let us assume, now, that under all circumstances we somehow sense the
direction of the total resultant mass-acceleration whencesoever it may arise
as the vertical. Then both the ordinary and the extraordinary phenomena
will be alike rendered intelligible.[96]
I was now desirous of putting the view I had reached to a more convenient
and exact test than was possible on a railway journey where one has no
control over the determining circumstances and cannot alter them at will. I
accordingly had the simple apparatus constructed which is represented in
Figure 46.
In a large frame BB, which is fastened to the walls, rotates about a vertical
axis AA a second frame RR, and within the latter a third one rr, which can
be set at any distance and position from the axis, made stationary or
movable, and is provided with a chair for the observer.

Fig. 46.
From Mach's Bewegungsempfindungen, Leipsic, Engelmann, 1875.]
The observer takes his seat in the chair and to prevent disturbances of
judgment is enclosed in a paper box. If the observer together with the frame
rr be then set in uniform rotation, he will feel and see the beginning of the

rotation both as to direction and amount very distinctly although every
outward visible or tangible point of reference is wanting. If the motion be
uniformly continued the sensation of rotation will gradually cease entirely
and the observer will imagine himself at rest. But if rr be placed outside the
axis of rotation, at once on the rotation beginning, a strikingly apparent,
palpable, actually visible inclination of the entire paper box is produced,
slight when the rotation is slow, strong when the rotation is rapid, and
continuing as long as the rotation lasts. It is absolutely impossible for the
observer to escape perceiving the inclination, although here also all outward
points of reference are wanting. If the observer, for example, is seated so as
to look towards the axis, he will feel the box strongly tipped backwards, as
it necessarily must be if the direction of the total resultant force is perceived
as the vertical. For other positions of the observer the situation is similar.[97]
Once, while performing one of these experiments, and after rotating so long
that I was no longer conscious of the movement, I suddenly caused the
apparatus to be stopped, whereupon I immediately felt and saw myself with
the whole box rapidly flung round in rotation in the opposite direction,
although I knew that the whole apparatus was at rest and every outward
point of reference for the perception of motion was wanting. Every one who
disbelieves in sensations of movement should be made acquainted with
these phenomena. Had Newton known them and had he ever observed how
we may actually imagine ourselves turned and displaced in space without
the assistance of stationary bodies as points of reference, he would certainly
have been confirmed more than ever in his unfortunate speculations
regarding absolute space.
The sensation of rotation in the opposite direction after the apparatus has
been stopped, slowly and gradually ceases. But on accidentally inclining
my head once during this occurrence, the axis of apparent rotation was also
observed to incline in exactly the same manner both as to direction and as
to amount. It is accordingly clear that the acceleration or retardation of
rotation is felt. The acceleration operates as a stimulus. The sensation,
however, like almost all sensations, though it gradually decreases, lasts
perceptibly longer than the stimulus. Hence the long continued apparent
rotation after the stopping of the apparatus. The organ, however, which
causes the persistence of this sensation must have its seat in the head, since

otherwise the axis of apparent rotation could not assume the same motion as
the head.
If I were to say, now, that a light had flashed upon me in making these last
observations, the expression would be a feeble one. I ought to say I
experienced a perfect illumination. My juvenile experiences of vertigo
occurred to me. I remembered Flourens's experiments relative to the section
of the semi-circular canals of the labyrinths of doves and rabbits, where this
inquirer had observed phenomena similar to vertigo, but which he preferred
to interpret, from his bias to the acoustic theory of the labyrinth, as the
expression of painful auditive disturbances. I saw that Goltz had nearly but
not quite hit the bull's eye with his theory of the semi-circular canals. This
inquirer, who, from his happy habit of following his own natural thoughts
without regard for tradition, has cleared up so much in science, spoke, as
early as 1870, on the ground of experiments, as follows: "It is uncertain
whether the semi-circular canals are auditive organs or not. In any event
they form an apparatus which serves for the preservation of equilibrium.
They are, so to speak, the sense-organs of equilibrium of the head and
indirectly of the whole body." I remembered the galvanic dizziness which
had been observed by Ritter and Purkinje on the passage of a current
through the head, when the persons experimented upon imagined they were
falling towards the cathode. The experiment was immediately repeated, and
sometime later (1874) I was enabled to demonstrate the same objectively
with fishes, all of which placed themselves sidewise and in the same
direction in the field of the current as if at command.[98] Müller's doctrine of
specific energies now appeared to me to bring all these new and old
observations into a simple, connected unity.

Fig. 47.
The labyrinth of a dove (stereoscopically reproduced), from R. Ewald,
Nervus Octavus, Wiesbaden, Bergmann, 1892.]
Let us picture to ourselves the labyrinth of the ear with its three semi-
circular canals lying in three mutually perpendicular planes (Comp. Fig.
47), the mysterious position of which inquirers have endeavored to explain
in every possible and impossible way. Let us conceive the nerves of the
ampullæ, or the dilated extensions of the semi-circular canals, equipped
with a capacity for responding to every imaginable stimulus with a
sensation of rotation just as the nerves of the retina of the eye when excited
by pressures, by electrical or chemical stimuli always respond with the
sensation of light; let us picture to ourselves, further, that the usual
excitation of the ampullæ nerves is produced by the inertia of the contents
of the semi-circular canals, which contents on suitable rotations in the plane
of the semi-circular canal are left behind in the motion, or at least have a
tendency to remain behind and consequently exert a pressure. It will be seen
that on this supposition all the single facts which without the theory appear

as so many different individual phenomena, become from this single point
of view clear and intelligible.
I had the satisfaction, immediately after the communication in which I set
forth this idea,[99] of seeing a paper by Breuer appear[100] in which this
author had arrived by entirely different methods at results that agreed in all
essential points with my own. A few weeks later appeared the researches of
Crum Brown of Edinburgh, whose methods were even still nearer mine.
Breuer's paper was far richer in physiological respects than mine, and he
had particularly gone into greater detail in his investigation of the collateral
effects of the reflex motions and orientation of the eyes in the phenomena
under consideration.[101] In addition certain experiments which I had
suggested in my paper as a test of the correctness of the view in question
had already been performed by Breuer. Breuer has also rendered services of
the highest order in the further elaboration of this field. But in a physical
regard, my paper was, of course, more complete.
In order to portray to the eye the behavior of the semi-circular canals, I have
constructed here a little apparatus. (See Fig. 48.) The large rotatable disc
represents the osseous semi-circular canal, which is continuous with the
bones of the head; the small disc, which is free to rotate on the axis of the
first, represents the mobile and partly liquid contents of the semi-circular
canal. On rotating the large disc, the small disc as you see remains behind. I
have to turn some time before the small disc is carried along with the large
one by friction. But if I now stop the large disc the small disc as you see
continues to rotate.

Fig. 48.
Model representing the action of the semi-circular canals.]
Simply assume now that the rotation of the small disc, say in the direction
of the hands of a watch, would give rise to a sensation of rotation in the
opposite direction, and conversely, and you already understand a good
portion of the facts above set forth. The explanation still holds, even if the
small disc does not perform appreciable rotations but is checked by a
contrivance similar to an elastic spring, the tension of which disengages a
sensation. Conceive, now, three such contrivances with their mutually
perpendicular planes of rotation joined together so as to form a single
apparatus; then to this apparatus as a whole, no rotation can be imparted
without its being indicated by the small mobile discs or by the springs
which are attached to them. Conceive both the right and the left ear
equipped with such an apparatus, and you will find that it answers all the

purposes of the semi-circular canals, which you see represented
stereoscopically in Fig. 47 for the ear of a dove.
Of the many experiments which I have made on my own person, and the
results of which could be predicted by the new view according to the
behavior of the model and consequently according to the rules of
mechanics, I shall cite but one. I fasten a horizontal board in the frame RR
of my rotatory apparatus, lie down upon the same with my right ear upon
the board, and cause the apparatus to be uniformly rotated. As soon as I no
longer perceive the rotation, I turn around upon my left ear and immediately
the sensation of rotation again starts up with marked vividness. The
experiment can be repeated as often as one wishes. A slight turn of the head
even is sufficient for reviving the sensation of rotation which in the
perfectly quiescent state at once disappears altogether.
We will imitate the experiment on the model. I turn the large disc until
finally the small disc is carried along with it. If, now, while the rotation
continues uniform, I burn off a little thread which you see here, the small
disc will be flipped round by a spring into its own plane 180°, so as now to
present its opposite side to you, when the rotation at once begins in the
opposite direction.
We have consequently a very simple means for determining whether one is
actually the subject or not of uniform and imperceptible rotations. If the
earth rotated much more rapidly than it really does, or if our semi-circular
canals were much more sensitive, a Nansen sleeping at the North Pole
would be waked by a sensation of rotation every time he turned over.
Foucault's pendulum experiment as a demonstration of the earth's rotation
would be superfluous under such circumstances. The only reason we cannot
prove the rotation of the earth with the help of our model, lies in the small
angular velocity of the earth and in the consequent liability to great
experimental errors.[102]
Aristotle has said that "The sweetest of all things is knowledge." And he is
right. But if you were to suppose that the publication of a new view were
productive of unbounded sweetness, you would be mightily mistaken. No
one disturbs his fellow-men with a new view unpunished. Nor should the
fact be made a subject of reproach to these fellow-men. To presume to
revolutionise the current way of thinking with regard to any question, is no

pleasant task, and above all not an easy one. They who have advanced new
views know best what serious difficulties stand in their way. With honest
and praiseworthy zeal, men set to work in search of everything that does not
suit with them. They seek to discover whether they cannot explain the facts
better or as well, or approximately as well, by the traditional views. And
that, too, is justified. But at times some extremely artless animadversions
are heard that almost nonplus us. "If a sixth sense existed it could not fail to
have been discovered thousands of years ago." Indeed; there was a time,
then, when only seven planets could have existed! But I do not believe that
any one will lay any weight on the philological question whether the set of
phenomena which we have been considering should be called a sense. The
phenomena will not disappear when the name disappears. It was further
said to me that animals exist which have no labyrinth, but which can yet
orientate themselves, and that consequently the labyrinth has nothing to do
with orientation. We do not walk forsooth with our legs, because snakes
propel themselves without them!
But if the promulgator of a new idea cannot hope for any great pleasure
from its publication, yet the critical process which his views undergo is
extremely helpful to the subject-matter of them. All the defects which
necessarily adhere to the new view are gradually discovered and eliminated.
Over-rating and exaggeration give way to more sober estimates. And so it
came about that it was found unpermissible to attribute all functions of
orientation exclusively to the labyrinth. In these critical labors Delage,
Aubert, Breuer, Ewald, and others have rendered distinguished services. It
can also not fail to happen that fresh facts become known in this process
which could have been predicted by the new view, which actually were
predicted in part, and which consequently furnish a support for the new
view. Breuer and Ewald succeeded in electrically and mechanically exciting
the labyrinth, and even single parts of the labyrinth, and thus in producing
the movements that belong to such stimuli. It was shown that when the
semi-circular canals were absent vertigo could not be produced, when the
entire labyrinth was removed the orientation of the head was no longer
possible, that without the labyrinth galvanic vertigo could not be induced. I
myself constructed as early as 1875 an apparatus for observing animals in
rotation, which was subsequently reinvented in various forms and has since
received the name of "cyclostat."[103] In experiments with the most varied
kinds of animals it was shown that, for example, the larvæ of frogs are not

subject to vertigo until their semi-circular canals which at the start are
wanting are developed (K. Schäfer). A large percentage of the deaf and
dumb are afflicted with grave affections of the labyrinth. The American
psychologist, William James, has made whirling experiments with many
deaf and dumb subjects, and in a large number of them found that
susceptibility to giddiness is wanting. He also found that many deaf and
dumb people on being ducked under water, whereby they lose their weight
and consequently have no longer the full assistance of their muscular sense,
utterly lose their sense of position in space, do not know which is up and
which is down, and are thrown into the greatest consternation,—results
which do not occur in normal men. Such facts are convincing proof that we
do not orientate ourselves entirely by means of the labyrinth, important as it
is for us. Dr. Kreidl has made experiments similar to those of James and
found that not only is vertigo absent in deaf and dumb people when whirled
about, but that also the reflex movements of the eyes which are normally
induced by the labyrinth are wanting. Finally, Dr. Pollak has found that
galvanic vertigo does not exist in a large percentage of the deaf and dumb.
Neither the jerking movements nor the uniform movements of the eyes
were observed which normal human beings exhibit in the Ritter and
Purkinje experiment.
After the physicist has arrived at the idea that the semi-circular canals are
the organ of sensation of rotation or of angular acceleration, he is next
constrained to ask for the organs that mediate the sensation of acceleration
noticed in forward movements. In searching for an organ for this function,
he of course is not apt to select one that stands in no anatomical and spatial
relation with the semi-circular canals. And in addition there are
physiological considerations to be weighed. The preconceived opinion once
having been abandoned that the entire labyrinth is auditory in its function,
there remains after the cochlea is reserved for sensations of tone and the
semi-circular canals for the sensation of angular acceleration, the vestibule
for the discharge of additional functions. The vestibule, particularly the part
of it known as the sacculus, appeared to me, by reason of the so-called
otoliths which it contains, eminently adapted for being the organ of
sensation of forward acceleration or of the position of the head. In this
conjecture I again closely coincided with Breuer.

That a sensation of position, of direction and amount of mass-acceleration
exists, our experience in elevators as well as of movement in curved paths
is sufficient proof. I have also attempted to produce and destroy suddenly
great velocities of forward movement by means of various contrivances of
which I shall mention only one here. If, while enclosed in the paper box of
my large whirling apparatus at some distance from the axis, my body is in
uniform rotation which I no longer feel, and I then loosen the connexions of
the frame rr with R thus making the former moveable and I then suddenly
stop the larger frame, my forward motion is abruptly impeded while the
frame rr continues to rotate. I imagine now that I am speeding on in a
straight line in a direction opposite to that of the checked motion.
Unfortunately, for many reasons it cannot be proved convincingly that the
organ in question has its seat in the head. According to the opinion of
Delage, the labyrinth has nothing to do with this particular sensation of
movement. Breuer, on the other hand, is of the opinion that the organ of
forward movement in man is stunted and the persistence of the sensation in
question is too brief to permit our instituting experiments as obvious as in
the case of rotation. In fact, Crum Brown once observed while in an
irritated condition peculiar vertical phenomena in his own person, which
were all satisfactorily explained by an abnormally long persistence of the
sensation of rotation, and I myself in an analogous case on the stopping of a
railway train felt the apparent backward motion in striking intensity and for
an unusual length of time.
There is no doubt whatever that we feel changes of vertical acceleration,
and it will appear from the following extremely probable that the otoliths of
the vestibule are the sense-organ for the direction of the mass-acceleration.
It will then be incompatible with a really logical view to regard the latter as
incapable of sensing horizontal accelerations.
In the lower animals the analogue of the labyrinth is shrunk to a little
vesicle filled with a liquid and containing tiny crystals, auditive stones, or
otoliths, of greater specific gravity, suspended on minute hairs. These
crystals appear physically well adapted for indicating both the direction of
gravity and the direction of incipient movements. That they discharge the
former function, Delage was the first to convince himself by experiments
with lower animals which on the removal of the otoliths utterly lost their
bearings and could no longer regain their normal position. Loeb also found

that fishes without labyrinths swim now on their bellies and now on their
backs. But the most remarkable, most beautiful, and most convincing
experiment is that which Dr. Kreidl instituted with crustaceans. According
to Hensen, certain Crustacea on sloughing spontaneously introduce fine
grains of sand as auditive stones into their otolith vesicle. At the ingenious
suggestion of S. Exner, Dr. Kreidl constrained some of these animals to put
up with iron filings (ferrum limatum). If the pole of an electro-magnet be
brought near the animal, it will at once turn its back away from the pole
accompanying the movement with appropriate reflex motions of the eye the
moment the current is closed, exactly as if gravity had been brought to bear
upon the animal in the same direction as the magnetic force.[104] This, in
fact, is what should be expected from the function ascribed to the otoliths. If
the eyes be covered with asphalt varnish, and the auditive sacs removed, the
crustaceans lose their sense of direction utterly, tumble head over heels, lie
on their side or their back indifferently. This does not happen when the eyes
only are covered. For vertebrates, Breuer has demonstrated by searching
investigations that the otoliths, or better, statoliths, slide in three planes
parallel to the planes of the semi-circular canals, and are consequently
perfectly adapted for indicating changes both in the amount and the
direction of the mass-acceleration.[105]
I have already remarked that not every function of orientation can be
ascribed exclusively to the labyrinth. The deaf and dumb who have to be
immersed in water, and the crustaceans who must have their eyes closed if
they are to be perfectly disorientated, are proof of this fact. I saw a blind cat
at Hering's laboratory which to one who was not a very attentive observer
behaved exactly like a seeing cat. It played nimbly with objects rolling on
the floor, stuck its head inquisitively into open drawers, sprang dexterously
upon chairs, ran with perfect accuracy through open doors, and never
bumped against closed ones. The visual sense had here been rapidly
replaced by the tactual and auditive senses. And it appears from Ewald's
investigations that even after the labyrinths have been removed, animals
gradually learn to move about again quite in the normal fashion,
presumably because the eliminated function of the labyrinth is now
performed by some part of the brain. A certain peculiar weakness of the
muscles alone is perceptible which Ewald ascribes to the absence of the
stimulus which is otherwise constantly emitted by the labyrinth (the

labyrinth-tonus). But if the part of the brain which discharges the deputed
function be removed, the animals are again completely disorientated and
absolutely helpless.
It may be said that the views enunciated by Breuer, Crum Brown and
myself in 1873 and 1874, and which are substantially a fuller and richer
development of Goltz's idea, have upon the whole been substantiated. At
least they have exercised a helpful and stimulative influence. New problems
have of course arisen in the course of the investigation which still await
solution, and much work remains to be done. At the same time we see how
fruitful the renewed co-operation of the various special departments of
science may become after a period of isolation and invigorating labor apart.
I may be permitted, therefore, to consider the relation between hearing and
orientation from another and more general point of view. What we call the
auditive organ is in the lower animals simply a sac containing auditive
stones. As we ascend the scale, 1, 2, 3 semi-circular canals gradually
develop from them, whilst the structure of the otolith organ itself becomes
more complicated. Finally, in the higher vertebrates, and particularly in the
mammals, a part of the latter organ (the lagena) becomes the cochlea, which
Helmholtz explained as the organ for sensations of tone. In the belief that
the entire labyrinth was an auditive organ, Helmholtz, contrary to the results
of his own masterly analysis, originally sought to interpret another part of
the labyrinth as the organ of noises. I showed a long time ago (1873) that
every tonal stimulus by shortening the duration of the excitation to a few
vibrations, gradually loses its character of pitch and takes on that of a sharp,
dry report or noise.[106] All the intervening stages between tones and noises
can be exhibited. Such being the case, it will hardly be assumed that one
organ is suddenly and at some given point replaced in function by another.
On the basis of different experiments and reasonings S. Exner also regards
the assumption of a special organ for the sensing of noises as unnecessary.
If we will but reflect how small a portion of the labyrinth of higher animals
is apparently in the service of the sense of hearing, and how large, on the
other hand, the portion is which very likely serves the purposes of
orientation, how much the first anatomical beginnings of the auditive sac of
lower animals resemble that part of the fully developed labyrinth which
does not hear, the view is irresistibly suggested which Breuer and I (1874,
1875) expressed, that the auditive organ took its development from an organ

for sensing movements by adaptation to weak periodic motional stimuli,
and that many apparatuses in the lower animals which are held to be organs
of hearing are not auditive organs at all.[107]
This view appears to be perceptibly gaining ground. Dr. Kreidl by skilfully-
planned experiments has arrived at the conclusion that even fishes do not
hear, whereas E. H. Weber, in his day, regarded the ossicles which unite the
air-bladder of fishes with the labyrinth as organs expressly designed for
conducting sound from the former to the latter.[108] Störensen has
investigated the excitation of sounds by the air-bladder of fishes, as also the
conduction of shocks through Weber's ossicles. He regards the air-bladder
as particularly adapted for receiving the noises made by other fishes and
conducting them to the labyrinth. He has heard the loud grunting tones of
the fishes in South American rivers, and is of the opinion that they allure
and find each other in this manner. According to these views certain fishes
are neither deaf nor dumb.[109] The question here involved might be solved
perhaps by sharply distinguishing between the sensation of hearing proper,
and the perception of shocks. The first-mentioned sensation may, even in
the case of many vertebrates, be extremely restricted, or perhaps even
absolutely wanting. But besides the auditive function, Weber's ossicles may
perfectly well discharge some other function. Although, as Moreau has
shown, the air-bladder itself is not an organ of equilibrium in the simple
physical sense of Borelli, yet doubtless some function of this character is
still reserved for it. The union with the labyrinth favors this conception, and
so a host of new problems rises here before us.
I should like to close with a reminiscence from the year 1863. Helmholtz's
Sensations of Tone had just been published and the function of the cochlea
now appeared clear to the whole world. In a private conversation which I
had with a physician, the latter declared it to be an almost hopeless
undertaking to seek to fathom the function of the other parts of the
labyrinth, whereas I in youthful boldness maintained that the question could
hardly fail to be solved, and that very soon, although of course I had then no
glimmering of how it was to be done. Ten years later the question was
substantially solved.
To-day, after having tried my powers frequently and in vain on many
questions, I no longer believe that we can make short work of the problems

of science. Nevertheless, I should not consider an "ignorabimus" as an
expression of modesty, but rather as the opposite. That expression is a
suitable one only with regard to problems which are wrongly formulated
and which are therefore not problems at all. Every real problem can and
will be solved in due course of time without supernatural divination,
entirely by accurate observation and close, searching thought.

ON SOME PHENOMENA ATTENDING THE
FLIGHT OF PROJECTILES.[110]
"I have led my ragamuffins where they were peppered."—Falstaff.
"He goes but to see a noise that he heard."—Midsummer Night's
Dream.

To shoot, in the shortest time possible, as many holes as possible in one
another's bodies, and not always for exactly pardonable objects and ideals,
seems to have risen to the dignity of a duty with modern men, who, by a
singular inconsistency, and in subservience to a diametrically contrary ideal,
are bound by the equally holy obligation of making these holes as small as
possible, and, when made, of stopping them up and of healing them as
speedily as possible. Since, then, shooting and all that appertains thereto, is
a very important, if not the most important, affair of modern life, you will
doubtless not be averse to giving your attention for an hour to some
experiments which have been undertaken, not for advancing the ends of
war, but for promoting the ends of science, and which throw some light on
the phenomena attending the flight of projectiles.
Modern science strives to construct its picture of the world not from
speculations but so far as possible from facts. It verifies its constructs by
recourse to observation. Every newly observed fact completes its world-
picture, and every divergence of a construct from observation points to
some imperfection, to some lacuna in it. What is seen is put to the test of,
and supplemented by, what is thought, which is again naught but the result
of things previously seen. It is always peculiarly fascinating, therefore, to
subject to direct verification by observation, that is, to render palpable to
the senses, something which we have only theoretically excogitated or
theoretically surmised.
In 1881, on hearing in Paris the lecture of the Belgian artillerist Melsens,
who hazarded the conjecture that projectiles travelling at a high rate of
speed carry masses of compressed air before them which are instrumental in

producing in bodies struck by the projectiles certain well-known facts of the
nature of explosions, the desire arose in me of experimentally testing his
conjecture and of rendering the phenomenon, if it really existed,
perceptible. The desire was the stronger as I could say that all the means for
realising it existed, and that I had in part already used and tested them for
other purposes.
And first let us get clear regarding the difficulties which have to be
surmounted. Our task is that of observing a bullet or other projectile which
is rushing through space at a velocity of many hundred yards a second,
together with the disturbances which the bullet causes in the surrounding
atmosphere. Even the opaque solid body itself, the projectile, is only
exceptionally visible under such circumstances—only when it is of
considerable size and when we see its line of flight in strong perspective
abridgement so that the velocity is apparently diminished. We see a large
projectile quite clearly when we stand behind the cannon and look steadily
along its line of flight or in the less pleasant case when the projectile is
speeding towards us. There is, however, a very simple and effective method
of observing swiftly moving bodies with as little trouble as if they were
held at rest at some point in their path. The method is that of illumination by
a brilliant electric spark of extremely short duration in a dark room. But
since, for the full intellectual comprehension of a picture presented to the
eye, a certain, not inconsiderable interval of time is necessary, the method
of instantaneous photography will naturally also be employed. The pictures,
which are of extremely minute duration, are thus permanently recorded and
can be examined and analysed at one's convenience and leisure.
With the difficulty just mentioned is associated still another and greater
difficulty which is due to the air. The atmosphere in its usual condition is
generally not visible even when at rest. But the task presented to us is to
render visible masses of air which in addition are moving with a high
velocity.
To be visible, a body must either emit light itself, must shine, or must affect
in some way the light which falls upon it, must take up that light entirely or
partly, absorb it, or must have a deflective effect upon it, that is, reflect or
refract it. We cannot see the air as we can a flame, for it shines only
exceptionally, as in a Geissler's tube. The atmosphere is extremely
transparent and colorless; it cannot be seen, therefore, as a dark or colored

body can, or as chlorine gas can, or vapor of bromine or iodine. Air, finally,
has so small an index of refraction and so small a deflective influence upon
light, that the refractive effect is commonly imperceptible altogether.
A glass rod is visible in air or in water, but it is almost invisible in a mixture
of benzol and bisulphuret of carbon, which has the same mean index of
refraction as the glass. Powdered glass in the same mixture has a vivid
coloring, because owing to the decomposition of the colors the indices are
the same for only one color which traverses the mixture unimpeded, whilst
the other colors undergo repeated reflexions.[111]
Water is invisible in water, alcohol in alcohol. But if alcohol be mixed with
water the flocculent streaks of the alcohol in the water will be seen at once
and vice versa. And in like manner the air, too, under favorable
circumstances, may be seen. Over a roof heated by the burning sun, a
tremulous wavering of objects is noticeable, as there is also over red-hot
stoves, radiators, and registers. In all these cases tiny flocculent masses of
hot and cold air, of slightly differing refrangibility, are mingled together.
In like manner the more highly refracting parts of non-homogeneous
masses of glass, the so-called striæ or imperfections of the glass, are readily
detectible among the less refracting parts which constitute the bulk of the
same. Such glasses are unserviceable for optical purposes, and special
attention has been devoted to the investigation of the methods for
eliminating or avoiding these defects. The result has been the development
of an extremely delicate method for detecting optical faults—the so-called
method of Foucault and Toepler—which is suitable also for our present
purpose.

Fig. 49.
Even Huygens when trying to detect the presence of striæ in polished glasses viewed
them under oblique illumination, usually at a considerable distance, so as to give full
scope to the aberrations, and had recourse for greater exactitude to a telescope. But the
method was carried to its highest pitch of perfection in 1867 by Toepler who
employed the following procedure: A small luminous source a (Fig. 49) illuminates a
lens L which throws an image b of the luminous source. If the eye be so placed that
the image falls on the pupil, the entire lens, if perfect, will appear equally illuminated,
for the reason that all points of it send out rays to the eye. Coarse imperfections of
form or of homogeneity are rendered visible only in case the aberrations are so large
that the light from many spots passes by the pupil of the eye. But if the image b be
partly intercepted by the edge of a small slide, then those spots in the lens as thus
partly darkened will appear brighter whose light by its greater aberrations still reaches
the eye in spite of the intercepting slide, while those spots will appear darker which in
consequence of aberration in the other direction throw their light entirely upon the
slide. This artifice of the intercepting slide which had previously been employed by
Foucault for the investigation of the optical imperfections of mirrors enhances
enormously the delicacy of the method, which is still further augmented by Toepler's
employment of a telescope behind the slide. Toepler's method, accordingly, enjoys all
the advantages of the Huygens and the Foucault procedure combined. It is so delicate
that the minutest irregularities in the air surrounding the lens can be rendered
distinctly visible, as I shall show by an example. I place a candle before the lens L
(Fig. 50) and so arrange a second lens M that the flame of the candle is imaged upon
the screen S. As soon as the intercepting slide is pushed into the focus, b, of the light
issuing from a, you see the images of the changes of density and the images of the
movements induced in the air by the flame quite distinctly upon the screen. The
distinctness of the phenomenon as a whole depends upon the position of the
intercepting slide b. The removal of b increases the illumination but decreases the
distinctness. If the luminous source a be removed, we see the image of the candle
flame only upon the screen S. If we remove the flame and allow a to continue shining,
the screen S will appear uniformly illuminated.

Fig. 50.
After Toepler had sought long and in vain to render the irregularities produced in air
by sound-waves visible by this principle, he was at last conducted to his goal by the
favorable circumstances attending the production of electric sparks. The waves
generated in the air by electric sparks and accompanying the explosive snapping of the
same, are of sufficiently short period and sufficiently powerful to be rendered visible
by these methods. Thus we see how by a careful regard for the merest and most
shadowy indications of a phenomenon and by slight progressive and appropriate
alterations of the circumstances and the methods, ultimately the most astounding
results can be attained. Consider, for example, two such phenomena as the rubbing of
amber and the electric lighting of modern streets. A person ignorant of the myriad
minute links that join these two things together, will be absolutely nonplussed at their
connexion, and will comprehend it no more than the ordinary observer who is
unacquainted with embryology, anatomy, and paleontology will understand the
connexion between a saurian and a bird. The high value and significance of the co-
operation of inquirers through centuries, where each has but to take up the thread of
work of his predecessors and spin it onwards, is rendered forcibly evident by such
examples. And such knowledge destroys, too, in the clearest manner imaginable that
impression of the marvellous which the spectator may receive from science, and at the
same time is a most salutary admonishment to the worker in science against
superciliousness. I have also to add the sobering remark that all our art would be in
vain did not nature herself afford at least some slight guiding threads leading from a
hidden phenomenon into the domain of the observable. And so it need not surprise us
that once under particularly favorable circumstances an extremely powerful sound-
wave which had been caused by the explosion of several hundred pounds of dynamite
threw a directly visible shadow in the sunlight, as Boys has recently told us. If the
sound-waves were absolutely without influence upon the light, this could not have
occurred, and all our artifices would then, too, be in vain. And so, similarly, the
phenomenon accompanying projectiles which I am about to show you was once in a
very imperfect manner incidentally seen by a French artillerist, Journée, while that

observer was simply following the line of flight of a projectile with a telescope, just as
also the undulations produced by candle flames are in a weak degree directly visible
and in the bright sunlight are imaged in shadowy waves upon a uniform white
background.
Instantaneous illumination by the electric spark, the method of rendering visible small
optical differences or striæ, which may hence be called the striate, or differential,
method,[112] invented by Foucault and Toepler, and finally the recording of the image
by a photographic plate,—these therefore are the chief means which are to lead us to
our goal.
I instituted my first experiments in the summer of 1884 with a target-pistol, shooting
the bullet through a striate field as described above, and taking care that the projectile
whilst in the field should disengage an illuminating electric spark from a Leyden jar or
Franklin's pane, which spark produced a photographic impression of the projectile
upon a plate, especially arranged for the purpose. I obtained the image of the
projectile at once and without difficulty. I also readily obtained, with the still rather
defective dry plate which I was using, exceedingly delicate images of the sound-
waves (spark-waves). But no atmospheric condensation produced by the projectile
was visible. I now determined the velocity of my projectile and found it to be only
240 metres per second, or considerably less than the velocity of sound (which is 340
metres per second). I saw immediately that under such circumstances no noticeable
compression of the air could be produced, for any atmospheric compression must of
necessity travel forward at the same speed with sound (340 metres per second) and
consequently would be always ahead of and speeding away from the projectile.
I was so thoroughly convinced, however, of the existence of the supposed
phenomenon at a velocity exceeding 340 metres per second, that I requested Professor
Salcher, of Fiume, an Austrian port on the Gulf of Quarnero, to undertake the
experiment with projectiles travelling at a high rate of speed. In the summer of 1886
Salcher in conjunction with Professor Riegler conducted in a spacious and suitable
apartment placed at their disposal by the Directors of the Royal Imperial Naval
Academy, experiments of the kind indicated and conforming in method exactly to
those which I had instituted, with the precise results expected. The phenomenon, in
fact, accorded perfectly with the a priori sketch of it which I had drafted previously to
the experiment. As the experimenting was continued, new and unforeseen features
made their appearance.
It would be unfair, of course, to expect from the very first experiments faultless and
highly distinct photographs. It was sufficient that success was secured and that I had
convinced myself that further labor and expenditure would not be vain. And on this
score I am greatly indebted to the two gentlemen above mentioned.
The Austrian Naval Department subsequently placed a cannon at Salcher's disposal in
Pola, an Adriatic seaport, and I myself, together with my son, then a student of

medicine, having received and accepted a courteous invitation from Krupp, repaired to
Meppen, a town in Hanover, where we conducted with only the necessary apparatus
several experiments on the open artillery range. All these experiments furnished
tolerably good and complete pictures. Some little progress, too, was made. The
outcome of our experience on both artillery ranges, however, was the settled
conviction that really good results could be obtained only by the most careful conduct
of the experiments in a laboratory especially adapted to the purpose. The
expensiveness of the experiments on a large scale was not the determining
consideration here, for the size of the projectile is indifferent. Given the same velocity
and the results are quite similar, whether the projectiles are large or small. On the
other hand, in a laboratory the experimenter has perfect control over the initial
velocity, which, provided the proper equipment is at hand, can be altered at will
simply by altering the charge and the weight of the projectile. The requisite
experiments were accordingly conducted by me in my laboratory at Prague, partly in
conjunction with my son and partly afterwards by him alone. The latter are the most
perfect and I shall accordingly speak in detail here of these only.

Fig. 51.
Picture to yourself an apparatus for detecting optical striæ set up in a dark room. In
order not to make the description too complicated, I shall give the essential features
only of the apparatus, leaving out of account altogether the minuter details which are
rather of consequence for the technical performance of the experiment than for its
understanding. We suppose the projectile speeding on its path, accordingly, through
the field of our differential optical apparatus. On reaching the centre of the field (Fig.
51) the projectile disengages an illuminating electric spark a, and the image of the
projectile, so produced, is photographically impressed upon the plate of the camera
behind the intercepting slide b. In the last and best experiments the lens L was

replaced by a spherical silvered-glass mirror made by K. Fritsch (formerly Prokesch)
of Vienna, whereby the apparatus was naturally more complicated than it appears in
our diagram. The projectile having been carefully aimed passes in crossing the
differential field between two vertical isolated wires which are connected with the two
coatings of a Leyden jar, and completely filling the space between the wires
discharges the jar. In the axis of the differential apparatus the circuit has a second gap
a which furnishes the illuminating spark, the image of which falls on the intercepting
slide b. The wires in the differential field having occasioned manifold disturbances
were subsequently done away with. In the new arrangement the projectile passes
through a ring (see dotted line, Fig. 51), to the air in which it imparts a sharp impulse
which travels forward in the tube r as a sound-wave having the approximate velocity
of 340 metres per second, topples over through the aperture of an electric screen the
flame of a candle situated at the other opening of the tube, and so discharges the jar.
The length of the tube r is so adjusted that the discharge occurs the moment the
projectile enters the centre of the now fully clear and free field of vision. We will also
leave out of account the fact that to secure fully the success of the experiment, a large
jar is first discharged by the flame, and that by the agency of this first discharge the
discharge of a second small jar having a spark of very short period which furnishes the
spark really illuminating the projectile is effected. Sparks from large jars have an
appreciable duration, and owing to the great velocity of the projectiles furnish blurred
photographs only. By carefully husbanding the light of the differential apparatus, and
owing to the fact that much more light reaches the photographic plate in this way than
would otherwise reach it, we can obtain beautiful, strong, and sharp photographs with
incredibly small sparks. The contours of the pictures appear as very delicate and very
sharp, closely adjacent double lines. From their distance from one another, and from
the velocity of the projectile, the duration of the illumination, or of the spark, is found
to be 1/800000 of a second. It is evident, therefore, that experiments with mechanical
snap slides can furnish no results worthy of the name.

Fig. 52.
Let us consider now first the picture of a projectile in the rough, as represented in
Figure 52, and then let us examine it in its photographic form as seen in Figure 53.
The latter picture is of a shot from an Austrian Mannlicher rifle. If I were not to tell
you what the picture represented you would very likely imagine it to be a bird's eye
view of a boat b moving swiftly through the water. In front you see the bow-wave and
behind the body a phenomenon k which closely resembles the eddies formed in the
wake of a ship. And as a matter of fact the dark hyperboloid arc which streams from
the tip of the projectile really is a compressed wave of air exactly analogous to the
bow-wave produced by a ship moving through the water, with the exception that the
wave of air is not a surface-wave. The air-wave is produced in atmospheric space and
encompasses the projectile in the form of a shell on all sides. The wave is visible for
the same reason that the heated shell of air surrounding the candle flame of our former
experiments is visible. And the cylinder of friction-heated air which the projectile
throws off in the form of vortex rings really does answer to the water in the wake of a
vessel.
Now just as a slowly moving boat produces no bow-wave, but the bow-wave is seen
only when the boat moves with a speed which is greater than the velocity of

propagation of surface-waves in water, so, in
like manner, no wave of compression is
visible in front of a projectile so long as the
speed of the projectile is less than the velocity
of sound. But if the speed of the projectile
reaches and exceeds the velocity of sound,
then the head-wave, as we shall call it,
augments noticeably in power, and is more
and more extended, that is, the angle made by
the contours of the wave with the direction of
flight is more and more diminished, just as
when the speed of a boat is increased a similar
phenomenon is noticed in connexion with the
bow-wave. In fact, we can from an
instantaneous photograph so taken
approximately estimate the speed with which
the projectile is travelling.
Fig. 53. Photograph of a blunted
projectile.] The explanation of the bow-wave of a ship
and that of the head-wave of a body travelling
in atmospheric space both repose upon the
same principle, long ago employed by Huygens. Conceive a number of pebbles to be
cast into a pond of water at regular intervals in such wise that all the spots struck are
situate in the same straight line, and that every spot subsequently struck lies a short
space farther to the right. The spots first struck will furnish then the wave-circles
which are widest, and all of them together will, at the points where they are thickest,
form a sort of cornucopia closely resembling the bow-wave. (Fig. 54.) The
resemblance is greater the smaller the pebbles are, and the more quickly they succeed
each other. If a rod be dipped into the water and quickly carried along its surface, the
falling of the pebbles will then take place, so to speak, uninterruptedly, and we shall
have a real bow-wave. If we put the compressed air-wave in the place of the surface-
waves of the water, we shall have the head-wave of the projectile.

Fig. 54.
You may be disposed to say now, it is all very pretty and interesting to observe a
projectile in its flight, but of what practical use is it?
It is true, I reply, one cannot wage war with photographed projectiles. And I have
likewise often had to say to medical students attending my lectures on physics, when
they inquired for the practical value of some physical observation, "You cannot,
gentlemen, cure diseases with it." I had also once to give my opinion regarding how
much physics should be taught at a school for millers, supposing the instruction there
to be confined exactly to what was necessary for a miller. I was obliged to reply: "A
miller always needs exactly as much physics as he knows." Knowledge which one
does not possess one cannot use.
Let us forego entirely the consideration that as a general thing every scientific
advance, every new problem elucidated, every extension or enrichment of our
knowledge of facts, affords a better foundation for practical pursuits. Let us rather put
the special question, Is it not possible to derive some really practical knowledge from
our theoretical acquaintance with the phenomena which take place in the space
surrounding a projectile?
No physicist who has ever studied waves of sound or photographed them will have the
least doubt regarding the sound-wave character of the atmospheric condensation
encompassing the head of a flying projectile. We have therefore, without ado, called
this condensation the head-wave.
Knowing this, it follows that the view of Melsens according to which the projectile
carries along with it masses of air which it forces into the bodies struck, is untenable.
A forward-moving sound-wave is not a forward-moving mass of matter but a forward-
moving form of motion, just as a water-wave or the waves of a field of wheat are only
forward-moving forms of motion and not movements of masses of water or masses of
wheat.

By interference-experiments, on which I cannot touch here but which will be found
roughly represented in Figure 55, it was found that the bell-shaped head-wave in
question is an extremely thin shell and that the condensations of the same are quite
moderate, scarcely exceeding two-tenths of an atmosphere. There can be no question,
therefore, of explosive effects in the body struck by the projectile through so slight a
degree of atmospheric compression. The phenomena attending wounds from rifle
balls, for example, are not to be explained as Melsens and Busch explain them, but are
due, as Kocher and Reger maintain, to the effects of the impact of the projectile itself.
A simple experiment will show
how insignificant is the part
played by the friction of the air,
or the supposed conveyance of
the air along with the moving
projectile. If the photograph of
the projectile be taken while
passing through a flame, i. e., a
visible gas, the flame will be seen
to be, not torn and deformed, but
smoothly and cleanly perforated,
like any solid body. Within and
around the flame the contours of
the head-wave will be seen. The
flickering, the extinction of the
flame, etc., take place only after
the projectile has travelled on a
considerable distance in its path,
and is then affected by the Fig. 55.
powder gases which hurry after
the bullet or by the air preceding
the powder-gases.
The physicist who examines the head-wave and recognises its sound-wave character
also sees that the wave in question is of the same kind with the short sharp waves
produced by electric sparks, that it is a noise-wave. Hence, whenever any portion of
the head-wave strikes the ear it will be heard as a report. Appearances point to the
conclusion that the projectile carries this report along with it. In addition to this report,
which advances with the velocity of the projectile and so usually travels at a speed
greater than the velocity of sound, there is also to be heard the report of the exploding
powder which travels forward with the ordinary velocity of sound. Hence two
explosions will be heard, each distinct in time. The circumstance that this fact was
long misconstrued by practical observers but when actually noticed frequently
received grotesque explanations and that ultimately my view was accepted as the

correct one, appears to me in itself a sufficient justification that researches such as we
are here speaking of are not utterly superfluous even in practical directions. That the
flashes and sounds of discharging artillery are used for estimating the distances of
batteries is well known, and it stands to reason that any unclear theoretical conception
of the facts here involved will seriously affect the correctness of practical calculations.
It may appear astonishing to a person hearing it for the first time, that a single shot has
a double report due to two different velocities of propagation. But the reflexion that
projectiles whose velocity is less than the velocity of sound produce no head-waves
(because every impulse imparted to the air travels forward, that is, ahead, with exactly
the velocity of sound), throws full light when logically developed upon the peculiar
circumstance above mentioned. If the projectile moves faster than sound, the air ahead
of it cannot recede from it quickly enough. The air is condensed and warmed, and
thereupon, as all know, the velocity of sound is augmented until the head-wave travels
forward as rapidly as the projectile itself, so that there is no need whatever of any
additional augmentation of the velocity of propagation. If such a wave were left
entirely to itself, it would increase in length and soon pass into an ordinary sound-
wave, travelling with less velocity. But the projectile is always behind it and so
maintains it at its proper density and velocity. Even if the projectile penetrates a piece
of cardboard or a board of wood, which catches and obstructs the head-wave, there
will, as Figure 56 shows, immediately appear at the emerging apex a newly formed,
not to say newly born, head-wave. We may observe on the cardboard the reflexion and
diffraction of the head-wave, and by means of a flame its refraction, so that no doubt
as to its nature can remain.
Permit me, now, to illustrate the most essential of the points that I have just adduced,
by means of a few rough drawings taken from older and less perfect photographs.
In the sketch of Figure 57 you see the projectile, which has just left the barrel of the
rifle, touch a wire and disengage the illuminating spark. At the apex of the projectile
you already see the beginnings of a powerful head-wave, and in front of the wave a
transparent fungiform cluster. This latter is the air which has been forced out of the
barrel by the projectile. Circular sound-waves, noise-waves, which are soon overtaken
by the projectile, also issue from the barrel. But behind the projectile opaque puffs of
powder-gas rush forth. It is scarcely necessary to add that many other questions in
ballistics may be studied by this method, as, for example, the movement of the gun-
carriage.
A distinguished French artillerist, M. Gossot, has applied the views of the head-wave
here given in quite a different manner. The practice in measuring the velocity of
projectiles is to cause the projectile to pass through wire screens placed at different
points in its path, and by the tearing of these screens to give rise to electro-magnetic
time-signals on falling slabs or rotating drums. Gossot caused these signals to be made
directly by the impact of the head-wave, did away thus with the wire screens, and

carried the method so far as to be able
to measure the velocities of projectiles
travelling in high altitudes, where the
use of wire screens was quite out of the
question.
The laws of the resistance of fluids and
of air to bodies travelling in them form
an extremely complicated problem,
which can be reasoned out very simply
and prettily as a matter of pure
philosophy but practice offers not a few
difficulties. The same body having the
velocity 2, 3, 4 ... displaces in the same
interval 2, 3, 4 ... times the same mass
of air, or the same mass of fluid, and
imparts to it in addition 2, 3, 4 ... times
the same velocity. But for this, plainly,
4, 9, 16 ... times the original force is
required. Hence, the resistance, it is
Fig. 56.
said, increases with the square of the
velocity. This is all very
pretty and simple and
obvious. But practice and
theory are at daggers'
points here. Practice tells
us that when we increase
the velocity, the law of the
resistance changes. For
every portion of the
velocity the law is
different.
The studies of the talented
English naval architect,
Froude, have thrown light
upon this question. Froude
has shown that the
Fig. 57. resistance is conditioned by
a combination of the most
multifarious phenomena. A ship in motion is subjected to the friction of the water. It
causes eddies and it generates in addition waves which radiate outward from it. Every
one of these phenomena are dependent upon the velocity in some different manner,

and it is consequently not astonishing that the law of the resistance should be a
complicated one.
The preceding observations suggest quite analogous reflexions for projectiles. Here
also we have friction, the formation of eddies, and the generation of waves. Here, also,
therefore, we should not be surprised at finding the law of the resistance of the air a
complicated one, nor puzzled at learning that in actuality the law of resistance changes
as soon as the speed of the projectile exceeds the velocity of sound, for this is the
precise point at which one important element of the resistance, namely, the formation
of waves, first comes into play.
No one doubts that a pointed bullet pierces the air with less resistance than a blunt
bullet. The photographs themselves show that the head-wave is weaker for a pointed
projectile. It is not impossible, similarly, that forms of bullets will be invented which
generate fewer eddies, etc., and that we shall study these phenomena also by
photography. I am of opinion from the few experiments which I have made in this
direction that not much more can be done by changing the form of the projectile when
the velocity is very great, but I have not gone into the question thoroughly. Researches
of the kind we are considering can certainly not be detrimental to practical artillery,
and it is no less certain that experiments by artillerists on a large scale will be of
undoubted benefit to physics.
No one who has had the opportunity of studying modern guns and projectiles in their
marvellous perfection, their power and precision, can help confessing that a high
technical and scientific achievement has found its incarnation in these objects. We
may surrender ourselves so completely to this impression as to forget for a moment
the terrible purposes they serve.
Permit me, therefore, before we separate, to say a few words on this glaring contrast.
The greatest man of war and of silence which the present age has produced once
asserted that perpetual peace is a dream, and not a beautiful dream at that. We may
accord to this profound student of mankind a judgment in these matters and can also
appreciate the soldier's horror of stagnation from all too lengthy peace. But it requires
a strong belief in the insuperableness of mediæval barbarism to hope for and to expect
no great improvement in international relations. Think of our forefathers and of the
times when club law ruled supreme, when within the same country and the same state
brutal assaults and equally brutal self-defence were universal and self-evident. This
state of affairs grew so oppressive that finally a thousand and one circumstances
compelled people to put an end to it, and the cannon had most to say in accomplishing
the work. Yet the rule of club law was not abolished so quickly after all. It had simply
passed to other clubs. We must not abandon ourselves to dreams of the Rousseau type.
Questions of law will in a sense forever remain questions of might. Even in the United
States where every one is as a matter of principle entitled to the same privileges, the
ballot according to Stallo's pertinent remark is but a milder substitute for the club. Nor

need I tell you that many of our own fellow-citizens are still enamored of the old
original methods. Very, very gradually, however, as civilisation progresses, the
intercourse of men takes on gentler forms, and no one who really knows the good old
times will ever honestly wish them back again, however beautifully they may be
painted and rhymed about.
In the intercourse of the nations, however, the old club law still reigns supreme. But
since its rule is taxing the intellectual, the moral, and the material resources of the
nations to the utmost and constitutes scarcely less a burden in peace than in war,
scarcely less a yoke for the victor than for the vanquished, it must necessarily grow
more and more unendurable. Reason, fortunately, is no longer the exclusive
possession of those who modestly call themselves the upper ten thousand. Here, as
everywhere, the evil itself will awaken the intellectual and ethical forces which are
destined to mitigate it. Let the hate of races and of nationalities run riot as it may, the
intercourse of nations will still increase and grow more intimate. By the side of the
problems which separate nations, the great and common ideals which claim the
exclusive powers of the men of the future appear one after another in greater
distinctness and in greater might.

ON INSTRUCTION IN THE CLASSICS AND THE
SCIENCES.[113]
Perhaps the most fantastic proposition that Maupertuis,[114] the renowned president of
the Berlin Academy, ever put forward for the approval of his contemporaries was that
of founding a city in which, to instruct and discipline young students, only Latin
should be spoken. Maupertuis's Latin city remained an idle wish. But for centuries
Latin and Greek institutions exist in which our children spend a goodly portion of
their days, and whose atmosphere constantly surrounds them, even when without their
walls.
For centuries instruction in the ancient languages has been zealously cultivated. For
centuries its necessity has been alternately championed and contested. More strongly
than ever are authoritative voices now raised against the preponderance of instruction
in the classics and in favor of an education more suited to the needs of the time,
especially for a more generous treatment of mathematics and the natural sciences.
In accepting your invitation to speak here on the relative educational value of the
classical and the mathematico-physical sciences in colleges and high schools, I find
my justification in the duty and the necessity laid upon every teacher of forming from
his own experiences an opinion upon this important question, as partly also in the
special circumstance that in my youth I was personally under the influence of school-
life for only a short time, just previous to my entering the university, and had,
therefore, ample opportunity to observe the effects of widely different methods upon
my own person.
Passing now, to a review of the arguments which the advocates of instruction in the
classics advance, and of what the adherents of instruction in the physical sciences in
their turn adduce, we find ourselves in rather a perplexing position with respect to the
arguments of the first named. For these have been different at different times, and they
are even now of a very multifarious character, as must be where men advance, in
favor of an institution that exists and which they are determined to retain at any cost,
everything they can possibly think of. We shall find here much that has evidently been
brought forward only to impress the minds of the ignorant; much, too, that was
advanced in good faith and which is not wholly without foundation. We shall get a fair
idea of the reasoning employed by considering, first, the arguments that have grown
out of the historical circumstances connected with the original introduction of the
classics, and, lastly, those which were subsequently adduced as accidental
afterthoughts.

Instruction in Latin, as Paulsen[115] has minutely shown, was introduced by the
Roman Church along with Christianity. With the Latin language were also transmitted
the scant and meagre remnants of ancient science. Whoever wished to acquire this
ancient education, then the only one worthy of the name, for him the Latin language
was the only and indispensable means; such a person had to learn Latin to rank among
educated people.
The wide-spread influence of the Roman Church wrought many and various results.
Among those for which all are glad, we may safely count the establishment of a sort
of uniformity among the nations and of a regular international intercourse by means of
the Latin language, which did much to unite the nations in the common work of
civilisation, carried on from the fifteenth to the eighteenth century. The Latin language
was thus long the language of scholars, and instruction in Latin the road to a liberal
education—a shibboleth still employed, though long inappropriate.
For scholars as a class, it is to be regretted, perhaps, that Latin has ceased to be the
medium of international communication. But the attributing of the loss of this
function by the Latin language to its incapacity to accommodate itself to the numerous
new ideas and conceptions which have arisen in the course of the development of
science is, in my opinion, wholly erroneous. It would be difficult to find a modern
scientist who had enriched science with as many new ideas as Newton has, yet
Newton knew how to express those ideas very correctly and precisely in the Latin
language. If this view were correct, it would also hold true of every living language.
Originally every language has to adapt itself to new ideas.
It is far more likely that Latin was displaced as the literary vehicle of science by the
influence of the nobility. By their desire to enjoy the fruits of literature and science,
through a less irksome medium than Latin, the nobility performed for the people at
large an undeniable service. For the days were now past when acquaintance with the
language and literature of science was restricted to a caste, and in this step, perhaps,
was made the most important advance of modern times. To-day, when international
intercourse is firmly established in spite of the many languages employed, no one
would think of reintroducing Latin.[116]
The facility with which the ancient languages lend themselves to the expression of
new ideas is evidenced by the fact that the great majority of our scientific ideas, as
survivals of this period of Latin intercourse, bear Latin and Greek designations, while
in great measure scientific ideas are even now invested with names from these
sources. But to deduce from the existence and use of such terms the necessity of still
learning Latin and Greek on the part of all who employ them is carrying the
conclusion too far. All terms, appropriate and inappropriate,—and there are a large
number of inappropriate and monstrous combinations in science,—rest on convention.

The essential thing is, that people should associate with the sign the precise idea that
is designated by it. It matters little whether a person can correctly derive the words
telegraph, tangent, ellipse, evolute, etc., if the correct idea is present in his mind when
he uses them. On the other hand, no matter how well he may know their etymology,
his knowledge will be of little use to him if the correct idea is absent. Ask the average
and fairly educated classical scholar to translate a few lines for you from Newton's
Principia, or from Huygens's Horologium, and you will discover at once what an
extremely subordinate rôle the mere knowledge of language plays in such things.
Without its associated thought a word remains a mere sound. The fashion of
employing Greek and Latin designations—for it can be termed nothing else—has a
natural root in history; it is impossible for the practice to disappear suddenly, but it has
fallen of late considerably into disuse. The terms gas, ohm, Ampère, volt, etc., are in
international use, but they are not Latin nor Greek. Only the person who rates the
unessential and accidental husk higher than its contents, can speak of the necessity of
learning Latin or Greek for such reasons, to say nothing of spending eight or ten years
on the task. Will not a dictionary supply in a few seconds all the information we wish
on such subjects?[117]
It is indisputable that our modern civilisation took up the threads of the ancient
civilisation, that at many points it begins where the latter left off, and that centuries
ago the remains of the ancient culture were the only culture existing in Europe. Then,
of course, a classical education really was the liberal education, the higher education,
the ideal education, for it was the sole education. But when the same claim is now
raised in behalf of a classical education, it must be uncompromisingly contested as
bereft of all foundation. For our civilisation has gradually attained its independence; it
has lifted itself far above the ancient civilisation, and has entered generally new
directions of progress. Its note, its characteristic feature, is the enlightenment that has
come from the great mathematical and physical researches of the last centuries, and
which has permeated not only the practical arts and industries but is also gradually
finding its way into all fields of thought, including philosophy and history, sociology
and linguistics. Those traces of ancient views that are still discoverable in philosophy,
law, art, and science, operate more as hindrances than helps, and will not long stand
before the development of independent and more natural views.
It ill becomes classical scholars, therefore, to regard themselves, at this day, as the
educated class par excellence, to condemn as uneducated all persons who do not
understand Latin and Greek, to complain that with such people profitable
conversations are not to be carried on, etc. The most delectable stories have got into
circulation, illustrative of the defective education of scientists and engineers. A
renowned inquirer, for example, is said to have once announced his intention of
holding a free course of university lectures, with the word "frustra"; an engineer who
spent his leisure hours in collecting insects is said to have declared that he was
studying "etymology." It is true, incidents of this character make us shudder or smile,

according to our mood or temperament. But we must admit, the next moment, that in
giving way to such feelings we have merely succumbed to a childish prejudice. A lack
of tact but certainly no lack of education is displayed in the use of such half-
understood expressions. Every candid person will confess that there are many
branches of knowledge about which he had better be silent. We shall not be so
uncharitable as to turn the tables and discuss the impression that classical scholars
might make on a scientist or engineer, in speaking of science. Possibly many ludicrous
stories might be told of them, and of far more serious import, which should fully
compensate for the blunders of the other party.
The mutual severity of judgment which we have here come upon, may also forcibly
bring home to us how really scarce a true liberal culture is. We may detect in this
mutual attitude, too, something of that narrow, mediæval arrogance of caste, where a
man began, according to the special point of view of the speaker, with the scholar, the
soldier, or the nobleman. Little sense or appreciation is to be found in it for the
common task of humanity, little feeling for the need of mutual assistance in the great
work of civilisation, little breadth of mind, little truly liberal culture.
A knowledge of Latin, and partly, also, a knowledge of Greek, is still a necessity for
the members of a few professions by nature more or less directly concerned with the
civilisations of antiquity, as for lawyers, theologians, philologists, historians, and
generally for a small number of persons, among whom from time to time I count
myself, who are compelled to seek for information in the Latin literature of the
centuries just past.[118] But that all young persons in search of a higher education
should pursue for this reason Latin and Greek to such excess; that persons intending to
become physicians and scientists should come to the universities defectively educated,
or even miseducated; and that they should be compelled to come only from schools
that do not supply them with the proper preparatory knowledge is going a little bit too
far.

After the conditions which had given to the study of Latin and Greek their high import
had ceased to exist, the traditional curriculum, naturally, was retained. Then, the
different effects of this method of education, good and bad, which no one had thought
of at its introduction, were realised and noted. As natural, too, was it that those who
had strong interests in the preservation of these studies, from knowing no others or
from living by them, or for still other reasons, should emphasise the good results of
such instruction. They pointed to the good effects as if they had been consciously
aimed at by the method and could be attained only through its agency.
One real benefit that students might derive from a rightly conducted course in the
classics would be the opening up of the rich literary treasures of antiquity, and

intimacy with the conceptions and views of the world held by two advanced nations.
A person who has read and understood the Greek and Roman authors has felt and
experienced more than one who is restricted to the impressions of the present. He sees
how men placed in different circumstances judge quite differently of the same things
from what we do to-day. His own judgments will be rendered thus more independent.
Again, the Greek and Latin authors are indisputably a rich fountain of recreation, of
enlightenment, and of intellectual pleasure after the day's toil, and the individual, not
less than civilised humanity generally, will remain grateful to them for all time. Who
does not recall with pleasure the wanderings of Ulysses, who does not listen joyfully
to the simple narratives of Herodotus, who would ever repent of having made the
acquaintance of Plato's Dialogues, or of having tasted Lucian's divine humor? Who
would give up the glances he has obtained into the private life of antiquity from
Cicero's letters, from Plautus or Terence? To whom are not the portraits of Suetonius
undying reminiscences? Who, in fact, would throw away any knowledge he had once
gained?
Yet people who draw from these sources only, who know only this culture, have
surely no right to dogmatise about the value of some other culture. As objects of
research for individuals, this literature is extremely valuable, but it is a different
question whether it is equally valuable as the almost exclusive means of education of
our youth.
Do not other nations and other literatures exist from which we ought to learn? Is not
nature herself our first school-mistress? Are our highest models always to be the
Greeks, with their narrow provinciality of mind, that divided the world into "Greeks
and barbarians," with their superstitions, with their eternal questioning of oracles?
Aristotle with his incapacity to learn from facts, with his word-science; Plato with his
heavy, interminable dialogues, with his barren, at times childish, dialectics—are they
unsurpassable?[119] The Romans with their apathy, their pompous externality, set off
by fulsome and bombastic phrases, with their narrow-minded, philistine philosophy,
with their frenzied sensuality, with their cruel and bestial indulgence in animal and
man baiting, with their outrageous maltreatment and plundering of their subjects—are
they patterns worthy of imitation? Or shall, perhaps, our science edify itself with the
works of Pliny who cites midwives as authorities and himself stands on their point of
view?
Besides, if an acquaintance with the ancient world really were attained, we might
come to some settlement with the advocates of classical education. But it is words and
forms, and forms and words only, that are supplied to our youth; and even collateral
subjects are forced into the strait-jacket of the same rigid method and made a science
of words, sheer feats of mechanical memory. Really, we feel ourselves set back a
thousand years into the dull cloister-cells of the Middle Ages.

This must be changed. It is possible to get acquainted with the views of the Greeks
and Romans by a shorter road than the intellect deadening process of eight or ten
years of declining, conjugating, analysing, and extemporisation. There are to-day
plenty of educated persons who have acquired through good translations vivider,
clearer, and more just views of classical antiquity than the graduates of our
gymnasiums and colleges.[120]
For us moderns, the Greeks and the Romans are simply two objects of archæological
and historical research like all others. If we put them before our youth in fresh and
living pictures, and not merely in words and syllables, the effect will be assured. We
derive a totally different enjoyment from the Greeks when we approach them after a
study of the results of modern research in the history of civilisation. We read many a
chapter of Herodotus differently when we attack his works equipped with a
knowledge of natural science, and with information about the stone age and the lake-
dwellers. What our classical institutions pretend to give can and actually will be given
to our youth with much more fruitful results by competent historical instruction,
which must supply, not names and numbers alone, nor the mere history of dynasties
and wars, but be in every sense of the word a true history of civilisation.
The view still widely prevails that although all "higher, ideal culture," all extension of
our view of the world, is acquired by philological and in a lesser degree by historical
studies, still the mathematics and natural sciences should not be neglected on account
of their usefulness. This is an opinion to which I must refuse my assent. It were
strange if man could learn more, could draw more intellectual nourishment, from the
shards of a few old broken jugs, from inscribed stones, or yellow parchments, than
from all the rest of nature. True, man is man's first concern, but he is not his sole
concern.
In ceasing to regard man as the centre of the world; in discovering that the earth is a
top whirled about the sun, which speeds off with it into infinite space; in finding that
in the fixed stars the same elements exist as on earth; in meeting everywhere the same
processes of which the life of man is merely a vanishingly small part—in such things,
too, is a widening of our view of the world, and edification, and poetry. There are here
perhaps grander and more significant facts than the bellowing of the wounded Ares, or
the charming island of Calypso, or the ocean-stream engirdling the earth. He only
should speak of the relative value of these two domains of thought, of their poetry,
who knows both.
The "utility" of physical science is, in a measure, only a collateral product of that
flight of the intellect which produced science. No one, however, should underrate the
utility of science who has shared in the realisation by modern industrial art of the
Oriental world of fables, much less one upon whom those treasures have been poured,
as it were, from the fourth dimension, without his aid or understanding.

Nor may we believe that science is useful only to the practical man. Its influence
permeates all our affairs, our whole life; everywhere its ideas are decisive. How
differently does the jurist, the legislator, or the political economist think, who knows,
for example, that a square mile of the most fertile soil can support with the solar heat
annually consumed only a definite number of human beings, which no art or science
can increase. Many economical theories, which open new air-paths of progress, air-
paths in the literal sense of the word, would be made impossible by such knowledge.

The eulogists of classical education love to emphasise the cultivation of taste which
comes from employment with the ancient models. I candidly confess that there is
something absolutely revolting in this to me. To form the taste, then, our youths must
sacrifice ten years of their life! Luxury takes precedence over necessity. Have the
future generations, in the face of the difficult problems, the great social questions,
which they must meet, and that with strengthened mind and heart, no more important
duties to fulfil than these?
But let us assume that this end were desirable. Can taste be formed by rules and
precepts? Do not ideals of beauty change? Is it not a stupendous absurdity to force
one's self artificially to admire things which, with all their historical interest, with all
their beauty in individual points, are for the most part foreign to the rest of our
thoughts and feelings, provided we have such of our own. A nation that is truly such,
has its own taste and will not go to others for it. And every individual perfect man has
his own taste.[121]
And what, after all, does this cultivation of taste consist in? In the acquisition of the
personal literary style of a few select authors! What should we think of a people that
would force its youth a thousand years from now, by years of practice, to master the
tortuous or bombastic style of some successful lawyer or politician of to-day? Should
we not justly accuse them of a woful lack of taste?
The evil effects of this imagined cultivation of the taste find expression often enough.
The young savant who regards the composition of a scientific essay as a rhetorical
exercise instead of a simple and unadorned presentation of the facts and the truth, still
sits unconsciously on the school-bench, and still unwittingly represents the point of
view of the Romans, by whom the elaboration of speeches was regarded as a serious
scientific (!) employment.

Far be it from me to underrate the value of the development of the instinct of speech
and of the increased comprehension of our own language which comes from
philological studies. By the study of a foreign language, especially of one which
differs widely from ours, the signs and forms of words are first clearly distinguished
from the thoughts which they express. Words of the closest possible correspondence
in different languages never coincide absolutely with the ideas they stand for, but
place in relief slightly different aspects of the same thing, and by the study of
language the attention is directed to these shades of difference. But it would be far
from admissible to contend that the study of Latin and Greek is the most fruitful and
natural, let alone the only, means of attaining this end. Any one who will give himself
the pleasure of a few hours' companionship with a Chinese grammar; who will seek to
make clear to himself the mode of speech and thought of a people who never
advanced as far as the analysis of articulate sounds, but stopped at the analysis of
syllables, to whom our alphabetical characters, therefore, are an inexplicable puzzle,
and who express all their rich and profound thoughts by means of a few syllables with
variable emphasis and position,—such a person, perhaps, will acquire new, and
extremely elucidative ideas upon the relation of language and thought. But should our
children, therefore, study Chinese? Certainly not. No more, then, should they be
burdened with Latin, at least in the measure they are.
It is a beautiful achievement to reproduce a Latin thought in a modern language with
the maximum fidelity of meaning and expression—for the translator. Moreover, we
shall be very grateful to the translator for his performance. But to demand this feat of
every educated man, without consideration of the sacrifice of time and labor which it
entails, is unreasonable. And for this very reason, as classical teachers admit, that ideal
is never perfectly attained, except in rare cases with scholars possessed of special
talents and great industry. Without slurring, therefore, the high importance of the study
of the ancient languages as a profession, we may yet feel sure that the instinct for
speech which is part of every liberal education can, and must, be acquired in a
different way. Should we, indeed, be forever lost if the Greeks had not lived before
us?
The fact is, we must carry our demands further than the representatives of classical
philology. We must ask of every educated man a fair scientific conception of the
nature and value of language, of the formation of language, of the alteration of the
meaning of roots, of the degeneration of fixed forms of speech to grammatical forms,
in brief, of all the main results of modern comparative philology. We should judge that
this were attainable by a careful study of our mother tongue and of the languages next
allied to it, and subsequently of the more ancient tongues from which the former are
derived. If any one object that this is too difficult and entails too much labor, I should
advise such a person to place side by side an English, a Dutch, a Danish, a Swedish,
and a German Bible, and to compare a few lines of them; he will be amazed at the
multitude of suggestions that offer themselves.[122] In fact, I believe that a really

progressive, fruitful, rational, and instructive study of languages can be conducted
only on this plan. Many of my audience will remember, perhaps, the bright and
encouraging effect, like that of a ray of sunlight on a gloomy day, which the meagre
and furtive remarks on comparative philology in Curtius's Greek grammar wrought in
that barren and lifeless desert of verbal quibbles.

The principal result obtained by the present method of studying the ancient languages
is that which comes from the student's employment with their complicated grammars.
It consists in the sharpening of the attention and in the exercise of the judgment by the
practice of subsuming special cases under general rules, and of distinguishing between
different cases. Obviously, the same result can be reached by many other methods; for
example, by difficult games of cards. Every science, the mathematics and the physical
sciences included, accomplish as much, if not more, in this disciplining of the
judgment. In addition, the matter treated by those sciences has a much higher intrinsic
interest for young people, and so engages spontaneously their attention; while on the
other hand they are elucidative and useful in other directions in which grammar can
accomplish nothing.
Who cares, so far as the matter of it is concerned, whether we say hominum or
hominorum in the genitive plural, interesting as the fact may be for the philologist?
And who would dispute that the intellectual need of causal insight is awakened not by
grammar but by the natural sciences?
It is not our intention, therefore, to gainsay in the least the good influence which the
study of Latin and Greek grammar also exercises on the sharpening of the judgment.
In so far as the study of words as such must greatly promote lucidity and accuracy of
expression, in so far as Latin and Greek are not yet wholly indispensable to many
branches of knowledge, we willingly concede to them a place in our schools, but
would demand that the disproportionate amount of time allotted to them, wrongly
withdrawn from other useful studies, should be considerably curtailed. That in the end
Latin and Greek will not be employed as the universal means of education, we are
fully convinced. They will be relegated to the closet of the scholar or professional
philologist, and gradually make way for the modern languages and the modern science
of language.
Long ago Locke reduced to their proper limits the exaggerated notions which obtained
of the close connexion of thought and speech, of logic and grammar, and recent
investigators have established on still surer foundations his views. How little a
complicated grammar is necessary for expressing delicate shades of thought is
demonstrated by the Italians and French, who, although they have almost totally
discarded the grammatical redundancies of the Romans, are yet not surpassed by the

latter in accuracy of thought, and whose poetical, but especially whose scientific
literature, as no one will dispute, can bear favorable comparison with the Roman.
Reviewing again the arguments advanced in favor of the study of the ancient
languages, we are obliged to say that in the main and as applied to the present, they
are wholly devoid of force. In so far as the aims which this study theoretically pursues
are still worthy of attainment, they appear to us as altogether too narrow, and are
surpassed in this only by the means employed. As almost the sole, indisputable result
of this study we must count the increase of the student's skill and precision in
expression. One inclined to be uncharitable might say that our gymnasiums and
classical academies turn out men who can speak and write, but, unfortunately, have
little to write or speak about. Of that broad, liberal view, of that famed universal
culture, which the classical curriculum is supposed to yield, serious words need not be
lost. This culture might, perhaps, more properly be termed the contracted or lopsided
culture.

While considering the study of languages we threw a few side glances at mathematics
and the natural sciences. Let us now inquire whether these, as branches of study,
cannot accomplish much that is to be attained in no other way. I shall meet with no
contradiction when I say that without at least an elementary mathematical and
scientific education a man remains a total stranger in the world in which he lives, a
stranger in the civilisation of the time that bears him. Whatever he meets in nature, or
in the industrial world, either does not appeal to him at all, from his having neither eye
nor ear for it, or it speaks to him in a totally unintelligible language.
A real understanding of the world and its civilisation, however, is not the only result
of the study of mathematics and the physical sciences. Much more essential for the
preparatory school is the formal cultivation which comes from these studies, the
strengthening of the reason and the judgment, the exercise of the imagination.
Mathematics, physics, chemistry, and the so-called descriptive sciences are so much
alike in this respect, that, apart from a few points, we need not separate them in our
discussion.
Logical sequence and continuity of ideas, so necessary for fruitful thought, are par
excellence the results of mathematics; the ability to follow facts with thoughts, that is,
to observe or collect experiences, is chiefly developed by the natural sciences.
Whether we notice that the sides and the angles of a triangle are connected in a
definite way, that an equilateral triangle possesses certain definite properties of
symmetry, or whether we notice the deflexion of a magnetic needle by an electric
current, the dissolution of zinc in diluted sulphuric acid, whether we remark that the
wings of a butterfly are slightly colored on the under, and the fore-wings of the moth

on the upper, surface: indiscriminately here we proceed from observations, from
individual acts of immediate intuitive knowledge. The field of observation is more
restricted and lies closer at hand in mathematics; it is more varied and broader but
more difficult to compass in the natural sciences. The essential thing, however, is for
the student to learn to make observations in all these fields. The philosophical
question whether our acts of knowledge in mathematics are of a special kind is here of
no importance for us. It is true, of course, that the observation can be practised by
languages also. But no one, surely, will deny, that the concrete, living pictures
presented in the fields just mentioned possess different and more powerful attractions
for the mind of the youth than the abstract and hazy figures which language offers,
and on which the attention is certainly not so spontaneously bestowed, nor with such
good results.[123]
Observation having revealed the different properties of a given geometrical or
physical object, it is discovered that in many cases these properties depend in some
way upon one another. This interdependence of properties (say that of equal sides and
equal angles at the base of a triangle, the relation of pressure to motion,) is nowhere so
distinctly marked, nowhere is the necessity and permanency of the interdependence so
plainly noticeable, as in the fields mentioned. Hence the continuity and logical
consequence of the ideas which we acquire in those fields. The relative simplicity and
perspicuity of geometrical and physical relations supply here the conditions of natural
and easy progress. Relations of equal simplicity are not met with in the fields which
the study of language opens up. Many of you, doubtless, have often wondered at the
little respect for the notions of cause and effect and their connexion that is sometimes
found among professed representatives of the classical studies. The explanation is
probably to be sought in the fact that the analogous relation of motive and action
familiar to them from their studies, presents nothing like the clear simplicity and
determinateness that the relation of cause and effect does.
That perfect mental grasp of all possible cases, that economical order and organic
union of the thoughts which comes from it, which has grown for every one who has
ever tasted it a permanent need which he seeks to satisfy in every new province, can
be developed only by employment with the relative simplicity of mathematical and
scientific investigations.
When a set of facts comes into apparent conflict with another set of facts, and a
problem is presented, its solution consists ordinarily in a more refined distinction or in
a more extended view of the facts, as may be aptly illustrated by Newton's solution of
the problem of dispersion. When a new mathematical or scientific fact is
demonstrated, or explained, such demonstration also rests simply upon showing the
connexion of the new fact with the facts already known; for example, that the radius
of a circle can be laid off as chord exactly six times in the circle is explained or
proved by dividing the regular hexagon inscribed in the circle into equilateral
triangles. That the quantity of heat developed in a second in a wire conveying an

electric current is quadrupled on the doubling of the strength of the current, we
explain from the doubling of the fall of the potential due to the doubling of the
current's intensity, as also from the doubling of the quantity flowing through, in a
word, from the quadrupling of the work done. In point of principle, explanation and
direct proof do not differ much.
He who solves scientifically a geometrical, physical, or technical problem, easily
remarks that his procedure is a methodical mental quest, rendered possible by the
economical order of the province—a simplified purposeful quest as contrasted with
unmethodical, unscientific guess-work. The geometer, for example, who has to
construct a circle touching two given straight lines, casts his eye over the relations of
symmetry of the desired construction, and seeks the centre of his circle solely in the
line of symmetry of the two straight lines. The person who wants a triangle of which
two angles and the sum of the sides are given, grasps in his mind the determinateness
of the form of this triangle and restricts his search for it to a certain group of triangles
of the same form. Under very different circumstances, therefore, the simplicity, the
intellectual perviousness, of the subject-matter of mathematics and natural science is
felt, and promotes both the discipline and the self-confidence of the reason.
Unquestionably, much more will be attained by instruction in the mathematics and the
natural sciences than now is, when more natural methods are adopted. One point of
importance here is that young students should not be spoiled by premature abstraction,
but should be made acquainted with their material from living pictures of it before
they are made to work with it by purely ratiocinative methods. A good stock of
geometrical experience could be obtained, for example, from geometrical drawing and
from the practical construction of models. In the place of the unfruitful method of
Euclid, which is only fit for special, restricted uses, a broader and more conscious
method must be adopted, as Hankel has pointed out.[124] Then, if, on reviewing
geometry, and after it presents no substantial difficulties, the more general points of
view, the principles of scientific method are placed in relief and brought to
consciousness, as Von Nagel,[125] J. K. Becker,[126] Mann,[127] and others have well
done, fruitful results will be surely attained. In the same way, the subject-matter of the
natural sciences should be made familiar by pictures and experiment before a
profounder and reasoned grasp of these subjects is attempted. Here the emphasis of
the more general points of view is to be postponed.
Before my present audience it would be superfluous for me to contend further that
mathematics and natural science are justified constituents of a sound education,—a
claim that even philologists, after some resistance, have conceded. Here I may count
upon assent when I say that mathematics and the natural sciences pursued alone as
means of instruction yield a richer education in matter and form, a more general
education, an education better adapted to the needs and spirit of the time,—than the
philological branches pursued alone would yield.

But how shall this idea be realised in the curricula of our intermediate educational
institutions? It is unquestionable in my mind that the German Realschulen and
Realgymnasien, where the exclusive classical course is for the most part replaced by
mathematics, science, and modern languages, give the average man a more timely
education than the gymnasium proper, although they are not yet regarded as fit
preparatory schools for future theologians and professional philologists. The German
gymnasiums are too one-sided. With these the first changes are to be made; of these
alone we shall speak here. Possibly a single preparatory school, suitably planned,
might serve all purposes.
Shall we, then, in our gymnasiums fill out the hours of study which stand at our
disposal, or are still to be wrested from the classicists, with as great and as varied a
quantity of mathematical and scientific matter as possible? Expect no such proposition
from me. No one will suggest such a course who has himself been actively engaged in
scientific thought. Thoughts can be awakened and fructified as a field is fructified by
sunshine and rain. But thoughts cannot be juggled out and worried out by heaping up
materials and the hours of instruction, nor by any sort of precepts: they must grow
naturally of their own free accord. Furthermore, thoughts cannot be accumulated
beyond a certain limit in a single head, any more than the produce of a field can be
increased beyond certain limits.
I believe that the amount of matter necessary for a useful education, such as should be
offered to all the pupils of a preparatory school, is very small. If I had the requisite
influence, I should, in all composure, and fully convinced that I was doing what was
best, first greatly curtail in the lower classes the amount of matter in both the classical
and the scientific courses; I should cut down considerably the number of the school
hours and the work done outside the school. I am not with many teachers of opinion
that ten hours work a day for a child is not too much. I am convinced that the mature
men who offer this advice so lightly are themselves unable to give their attention
successfully for as long a time to any subject that is new to them, (for example, to
elementary mathematics or physics,) and I would ask every one who thinks the
contrary to make the experiment upon himself. Learning and teaching are not routine
office-work that can be kept up mechanically for long periods. But even such work
tires in the end. If our young men are not to enter the universities with blunted and
impoverished minds, if they are not to leave in the preparatory schools their vital
energy, which they should there gather, great changes must be made. Waiving the
injurious effects of overwork upon the body, the consequences of it for the mind seem
to me positively dreadful.
I know of nothing more terrible than the poor creatures who have learned too much.
Instead of that sound powerful judgment which would probably have grown up if they
had learned nothing, their thoughts creep timidly and hypnotically after words,
principles, and formulæ, constantly by the same paths. What they have acquired is a

spider's web of thoughts too weak to furnish sure supports, but complicated enough to
produce confusion.
But how shall better methods of mathematical and scientific education be combined
with the decrease of the subject-matter of instruction? I think, by abandoning
systematic instruction altogether, at least in so far as that is required of all young
pupils. I see no necessity whatever that the graduates of our high schools and
preparatory schools should be little philologists, and at the same time little
mathematicians, physicists, and botanists; in fact, I do not see the possibility of such a
result. I see in the endeavor to attain this result, in which every instructor seeks for his
own branch a place apart from the others, the main mistake of our whole system. I
should be satisfied if every young student could come into living contact with and
pursue to their ultimate logical consequences merely a few mathematical or scientific
discoveries. Such instruction would be mainly and naturally associated with selections
from the great scientific classics. A few powerful and lucid ideas could thus be made
to take root in the mind and receive thorough elaboration. This accomplished, our
youth would make a different showing from what they do to-day.[128]
What need is there, for example, of burdening the head of a young student with all the
details of botany? The student who has botanised under the guidance of a teacher
finds on all hands, not indifferent things, but known or unknown things, by which he
is stimulated, and his gain made permanent. I express here, not my own, but the
opinion of a friend, a practical teacher. Again, it is not at all necessary that all the
matter that is offered in the schools should be learned. The best that we have learned,
that which has remained with us for life, outlived the test of examination. How can the
mind thrive when matter is heaped on matter, and new materials piled constantly on
old, undigested materials? The question here is not so much that of the accumulation
of positive knowledge as of intellectual discipline. It seems also unnecessary that all
branches should be treated at school, and that exactly the same studies should be
pursued in all schools. A single philological, a single historical, a single mathematical,
a single scientific branch, pursued as common subjects of instruction for all pupils, are
sufficient to accomplish all that is necessary for the intellectual development. On the
other hand, a wholesome mutual stimulus would be produced by this greater variety in
the positive culture of men. Uniforms are excellent for soldiers, but they will not fit
heads. Charles V. learned this, and it should never be forgotten. On the contrary,
teachers and pupils both need considerable latitude, if they are to yield good results.
With John Karl Becker I am of the opinion that the utility and amount for individuals
of every study should be precisely determined. All that exceeds this amount should be
unconditionally banished from the lower classes. With respect to mathematics,
Becker,[129] in my judgment, has admirably solved this question.
With respect to the upper classes the demand assumes a different form. Here also the
amount of matter obligatory on all pupils ought not to exceed a certain limit. But in

the great mass of knowledge that a young man must acquire to-day for his profession
it is no longer just that ten years of his youth should be wasted with mere preludes.
The upper classes should supply a truly useful preparation for the professions, and
should not be modelled upon the wants merely of future lawyers, ministers, and
philologists. Again, it would be both foolish and impossible to attempt to prepare the
same person properly for all the different professions. In such case the function of the
schools would be, as Lichtenberg feared, simply to select the persons best fitted for
being drilled, whilst precisely the finest special talents, which do not submit to
indiscriminate discipline, would be excluded from the contest. Hence, a certain
amount of liberty in the choice of studies must be introduced in the upper classes, by
means of which it will be free for every one who is clear about the choice of his
profession to devote his chief attention either to the study of the philologico-historical
or to that of the mathematico-scientific branches. Then the matter now treated could
be retained, and in some branches, perhaps, judiciously extended,[130] without
burdening the scholar with many branches or increasing the number of the hours of
study. With more homogeneous work the student's capacity for work increases, one
part of his labor supporting the other instead of obstructing it. If, however, a young
man should subsequently choose a different profession, then it is his business to make
up what he has lost. No harm certainly will come to society from this change, nor
could it be regarded as a misfortune if philologists and lawyers with mathematical
educations or physical scientists with classical educations should now and then
appear.

The view is now wide-spread that a Latin and Greek education no longer meets the
general wants of the times, that a more opportune, a more "liberal" education exists.
The phrase, "a liberal education," has been greatly misused. A truly liberal education
is unquestionably very rare. The schools can hardly offer such; at best they can only
bring home to the student the necessity of it. It is, then, his business to acquire, as best
he can, a more or less liberal education. It would be very difficult, too, at any one time
to give a definition of a "liberal" education which would satisfy every one, still more
difficult to give one which would hold good for a hundred years. The educational
ideal, in fact, varies much. To one, a knowledge of classical antiquity appears not too
dearly bought "with early death." We have no objection to this person, or to those who
think like him, pursuing their ideal after their own fashion. But we may certainly
protest strongly against the realisation of such ideals on our own children. Another,—
Plato, for example,—puts men ignorant of geometry on a level with animals.[131] If
such narrow views had the magical powers of the sorceress Circe, many a man who
perhaps justly thought himself well educated would become conscious of a not very

flattering transformation of himself. Let us seek, therefore, in our educational system
to meet the wants of the present, and not establish prejudices for the future.
But how does it come, we must ask, that institutions so antiquated as the German
gymnasiums could subsist so long in opposition to public opinion? The answer is
simple. The schools were first organised by the Church; since the Reformation they
have been in the hands of the State. On so large a scale, the plan presents many
advantages. Means can be placed at the disposal of education such as no private
source, at least in Europe, could furnish. Work can be conducted upon the same plan
in many schools, and so experiments made of extensive scope which would be
otherwise impossible. A single man with influence and ideas can under such
circumstances do great things for the promotion of education.
But the matter has also its reverse aspect. The party in power works for its own
interests, uses the schools for its special purposes. Educational competition is
excluded, for all successful attempts at improvement are impossible unless undertaken
or permitted by the State. By the uniformity of the people's education, a prejudice
once in vogue is permanently established. The highest intelligences, the strongest
wills cannot overthrow it suddenly. In fact, as everything is adapted to the view in
question, a sudden change would be physically impossible. The two classes which
virtually hold the reins of power in the State, the jurists and theologians, know only
the one-sided, predominantly classical culture which they have acquired in the State
schools, and would have this culture alone valued. Others accept this opinion from
credulity; others, underestimating their true worth for society, bow before the power
of the prevalent opinion; others, again, affect the opinion of the ruling classes even
against their better judgment, so as to abide on the same plane of respect with the
latter. I will make no charges, but I must confess that the deportment of medical men
with respect to the question of the qualification of graduates of your Realschulen has
frequently made that impression upon me. Let us remember, finally, that an influential
statesman, even within the boundaries which the law and public opinion set him, can
do serious harm to the cause of education by considering his own one-sided views
infallible, and in enforcing them recklessly and inconsiderately—which not only can
happen, but has, repeatedly, happened.[132] The monopoly of education by the
State[133] thus assumes in our eyes a somewhat different aspect. And to revert to the
question above asked, there is not the slightest doubt that the German gymnasiums in
their present form would have ceased to exist long ago if the State had not supported
them.
All this must be changed. But the change will not be made of itself, nor without our
energetic interference, and it will be made slowly. But the path is marked out for us,
the will of the people must acquire and exert upon our school legislation a greater and
more powerful influence. Furthermore, the questions at issue must be publicly and
candidly discussed that the views of the people may be clarified. All who feel the
insufficiency of the existing régime must combine into a powerful organisation that

their views may acquire impressiveness and the opinions of the individual not die
away unheard.
I recently read, gentlemen, in an excellent book of travels, that the Chinese speak with
unwillingness of politics. Conversations of this sort are usually cut short with the
remark that they may bother about such things whose business it is and who are paid
for it. Now it seems to me that it is not only the business of the State, but a very
serious concern of all of us, how our children shall be educated in the public schools
at our cost.

APPENDIX.
I.

A CONTRIBUTION TO THE HISTORY OF ACOUSTICS.[134]

While searching for papers by Amontons, several volumes of the Memoirs of the Paris
Academy for the first years of the eighteenth century, fell into my hands. It is difficult
to portray the delight which one experiences in running over the leaves of these
volumes. One sees as an actual spectator almost the rise of the most important
discoveries and witnesses the progress of many fields of knowledge from almost total
ignorance to relatively perfect clearness.
I propose to discuss here the fundamental researches of Sauveur in Acoustics. It is
astonishing how extraordinarily near Sauveur was to the view which Helmholtz was
the first to adopt in its full extent a hundred and fifty years later.
The Histoire de l'Académie for 1700, p. 131, tells us that Sauveur had succeeded in
making music an object of scientific research, and that he had invested the new
science with the name of "acoustics." On five successive pages a number of
discoveries are recorded which are more fully discussed in the volume for the year
following.
Sauveur regards the simplicity of the ratios obtaining between the rates of vibration of
consonances as something universally known.[135] He is in hope, by further research,
of determining the chief rules of musical composition and of fathoming the
"metaphysics of the agreeable," the main law of which he asserts to be the union of
"simplicity with multiplicity." Precisely as Euler[136] did a number of years later, he
regards a consonance as more perfect according as the ratio of its vibrational rates is
expressed in smaller whole numbers, because the smaller these whole numbers are the
oftener the vibrations of the two tones coincide, and hence the more readily they are
apprehended. As the limit of consonance, he takes the ratio 5:6, although he does not
conceal the fact that practice, sharpened attention, habit, taste, and even prejudice play
collateral rôles in the matter, and that consequently the question is not a purely
scientific one.
Sauveur's ideas took their development from his having instituted at all points more
exact quantitative investigations than his predecessors. He is first desirous of
determining as the foundation of musical tuning a fixed note of one hundred
vibrations which can be reproduced at any time; the fixing of the notes of musical
instruments by the common tuning pipes then in use with rates of vibration unknown,

appearing to him inadequate. According to Mersenne (Harmonie Universelle, 1636), a
given cord seventeen feet long and weighted with eight pounds executes eight visible
vibrations in a second. By diminishing its length then in a given proportion we obtain
a proportionately augmented rate of vibration. But this procedure appears too
uncertain to Sauveur, and he employs for his purpose the beats (battemens), which
were known to the organ-makers of his day, and which he correctly explains as due to
the alternate coincidence and non-coincidence of the same vibrational phases of
differently pitched notes.[137] At every coincidence there is a swelling of the sound,
and hence the number of beats per second will be equal to the difference of the rates
of vibration. If we tune two of three organ-pipes to the remaining one in the ratio of
the minor and major third, the mutual ratio of the rates of vibration of the first two
will be as 24: 25, that is to say, for every 24 vibrations to the lower note there will be
25 to the higher, and one beat. If the two pipes give together four beats in a second,
then the higher has the fixed tone of 100 vibrations. The open pipe in question will
consequently be five feet in length. We also determine by this procedure the absolute
rates of vibration of all the other notes.
It follows at once that a pipe eight times as long or 40 feet in length will yield a
vibrational rate of 12-1/2, which Sauveur ascribes to the lowest audible tone, and
further also that a pipe 64 times as small will execute 6,400 vibrations, which Sauveur
took for the highest audible limit. The author's delight at his successful enumeration of
the "imperceptible vibrations" is unmistakably asserted here, and it is justified when
we reflect that to-day even Sauveur's principle, slightly modified, constitutes the
simplest and most delicate means we have for exactly determining rates of vibration.
Far more important still, however, is a second observation which Sauveur made while
studying beats, and to which we shall revert later.
Strings whose lengths can be altered by movable bridges are much easier to handle
than pipes in such investigations, and it was natural that Sauveur should soon resort to
their use.
One of his bridges accidentally not having been brought into full and hard contact
with the string, and consequently only imperfectly impeding the vibrations, Sauveur
discovered the harmonic overtones of the string, at first by the unaided ear, and
concluded from this fact that the string was divided into aliquot parts. The string when
plucked, and when the bridge stood at the third division for example, yielded the
twelfth of its fundamental note. At the suggestion of some academician[138] probably,
variously colored paper riders were placed at the nodes (noeuds) and ventral segments
(ventres), and the division of the string due to the excitation of the overtones (sons
harmoniques) belonging to its fundamental note (son fondamental) thus rendered
visible. For the clumsy bridge the more convenient feather or brush was soon
substituted. . While engaged in these investigations Sauveur also observed the
sympathetic vibration of a string induced by the excitation of a second one in unison
with it. He also discovered that the overtone of a string can respond to another string

tuned to its note. He even went further and discovered that on exciting one string the
overtone which it has in common with another, differently pitched string can be
produced on that other; for example, on strings having for their vibrational ratio 3:4,
the fourth of the lower and the third of the higher may be made to respond. It follows
indisputably from this that the excited string yields overtones simultaneously with its
fundamental tone. Previously to this Sauveur's attention had been drawn by other
observers to the fact that the overtones of musical instruments can be picked out by
attentive listening, particularly in the night.[139] He himself mentions the simultaneous
sounding of the overtones and the fundamental tone.[140] That he did not give the
proper consideration to this circumstance was, as will afterwards be seen, fatal to his
theory.
While studying beats Sauveur makes the remark that they are displeasing to the ear.
He held the beats were distinctly audible only when less than six occurred in a second.
Larger numbers were not distinctly perceptible and gave rise accordingly to no
disturbance. He then attempts to reduce the difference between consonance and
dissonance to a question of beats. Let us hear his own words.[141]

"Beats are unpleasing to the ear because of the unevenness of the sound, and it
may be held with much plausibility that the reason why octaves are so pleasing is
that we never hear their beats.[142]
"In following out this idea, we find that the chords whose beats we cannot hear
are precisely those which the musicians call consonances and that those whose
beats are heard are the dissonances, and that when a chord is a dissonance in one
octave and a consonance in another, it beats in the one and does not beat in the
other. Consequently it is called an imperfect consonance. It is very easy by the
principles of M. Sauveur, here established, to ascertain what chords beat and in
what octaves, above or below the fixed note. If this hypothesis be correct, it will
disclose the true source of the rules of composition, hitherto unknown to science,
and given over almost entirely to judgment by the ear. These sorts of natural
judgment, marvellous though they may sometimes appear, are not so but have
very real causes, the knowledge of which belongs to science, provided it can gain
possession thereof."[143]

Sauveur thus correctly discerns in beats the cause of the disturbance of consonance, to
which all disharmony is "probably" to be referred. It will be seen, however, that
according to his view all distant intervals must necessarily be consonances and all
near intervals dissonances. He also overlooks the absolute difference in point of
principle between his old view, mentioned at the outset, and his new view, rather
attempting to obliterate it.