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Title: Memorabilia Mathematica; or, the Philomath's Quotation-Book

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MEMORABILIA MATHEMATICA

THE MACMILLAN COMPANY
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MEMORABILIA
MATHEMATICA
OR

THE PHILOMATH’S QUOTATION-BOOK

BY

ROBERT EDOUARD MORITZ, Ph. D., Ph. N. D.
PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF WASHINGTON

New York
THE MACMILLAN COMPANY
1914

All rights reserved

Copyright, 1914, by
ROBERT EDOUARD MORITZ

PREFACE
Every one knows that the fine phrase “God geometrizes” is attributed
to Plato, but few know where this famous passage is found, or the exact
words in which it was first expressed. Those who, like the author, have
spent hours and even days in the search of the exact statements, or the exact
references, of similar famous passages, will not question the timeliness and
usefulness of a book whose distinct purpose it is to bring together into a
single volume exact quotations, with their exact references, bearing on one
of the most time-honored, and even today the most active and most fruitful
of all the sciences, the queen-mother of all the sciences, that is,
mathematics.
It is hoped that the present volume will prove indispensable to every
teacher of mathematics, to every writer on mathematics, and that the student
of mathematics and the related sciences will find its perusal not only a
source of pleasure but of encouragement and inspiration as well. The
layman will find it a repository of useful information covering a field of
knowledge which, owing to the unfamiliar and hence repellant character of
the language employed by mathematicians, is peculiarly inaccessible to the
general reader. No technical processes or technical facility is required to
understand and appreciate the wealth of ideas here set forth in the words of
the world’s great thinkers.
No labor has been spared to make the present volume worthy of a
place among collections of a like kind in other fields. Ten years have been
devoted to its preparation, years, which if they could have been more
profitably, could scarcely have been more pleasurably employed. As a result
there have been brought together over one thousand more or less familiar
passages pertaining to mathematics, by poets, philosophers, historians,
statesmen, scientists, and mathematicians. These have been gathered from
over three hundred authors, and have been grouped under twenty heads, and
cross indexed under nearly seven hundred topics.

The author’s original plan was to give foreign quotations both in the
original and in translation, but with the growth of material this plan was
abandoned as infeasible. It was thought to serve the best interest of the
greater number of English readers to give translations only, while
preserving the references to the original sources, so that the student or
critical reader may readily consult the original of any given extract. In cases
where the translation is borrowed the translator’s name is inserted in
brackets [] immediately after the author’s name. Brackets are also used to
indicate inserted words or phrases made necessary to bring out the context.
The absence of similar English works has made the author’s work
largely that of the pioneer. Rebière’s “Mathématiques et Mathématiciens”
and Ahrens’ “Scherz und Ernst in der Mathematik” have indeed been
frequently consulted but rather with a view to avoid overlapping than to
receive aid. Thus certain topics as the correspondence of German and
French mathematicians, so excellently treated by Ahrens, have purposely
been omitted. The repetitions are limited to a small number of famous
utterances whose absence from a work of this kind could scarcely be
defended on any grounds.
No one can be more keenly aware of the shortcomings of a work than
its author, for none can have so intimate an acquaintance with it. Among
those of the present work is its incompleteness, but it should be borne in
mind that incompleteness is a necessary concomitant of every collection of
whatever kind. Much less can completeness be expected in a first
collection, made by a single individual, in his leisure hours, and in a field
which is already boundless and is yet expanding day by day. A collection of
great thoughts, even if complete today, would be incomplete tomorrow.
Again, if some authors are quoted more frequently than others of greater
fame and authority, the reason may be sought not only in the fact that the
writings of some authors peculiarly lent themselves to quotation, a quality
singularly absent in other writers of the greatest merit and authority, but
also in this, that the greatest freedom has been exercised in the choice of
selections. The author has followed the bent of his own fancy in collecting
whatever seemed to him sufficiently valuable because of its content, its
beauty, its originality, or its terseness, to deserve a place in a
“Memorabilia.”

Great pains has been taken to furnish exact readings and references. In
some cases where a passage could not be traced to its first source, the
secondary source has been given rather than the reputed source. For the
same reason many references are to later editions rather than to inaccessible
first editions.
The author feels confident that this work will be of assistance to his
co-workers in the field of mathematics and allied fields. If in addition it
should aid in a better appreciation of mathematicians and their work on the
part of laymen and students in other fields, the author’s foremost aim in the
preparation of this work will have been achieved.
Robert Edouard Moritz,
September, 1913.

CONTENTS

Chapter Page
I. Definitions and Object of Mathematics 1
II. The Nature of Mathematics 10
III. Estimates of Mathematics 39
IV. The Value of Mathematics 49
V. The Teaching of Mathematics 72
VI. Study and Research in Mathematics 86
VII. Modern Mathematics 108
VIII. The Mathematician 121
IX. Persons and Anecdotes (A-M) 135
X. Persons and Anecdotes (N-Z) 166
XI. Mathematics as a Fine Art 181
XII. Mathematics as a Language 194
XIII. Mathematics and Logic 201
XIV. Mathematics and Philosophy 209
XV. Mathematics and Science 224
XVI. Arithmetic 261
XVII. Algebra 275
XVIII. Geometry 292

XIX. The Calculus and Allied Topics 323
XX. The Fundamental Concepts of Time and 345
Space
XXI. Paradoxes and Curiosities 364
Index 385

Alles Gescheite ist schon gedacht worden; man muss nur versuchen, es
noch einmal zu denken.—Goethe.
Sprüche in Prosa, Ethisches, I. 1.

A great man quotes bravely, and will not draw on his invention when his
memory serves him with a word as good.—Emerson.
Letters and Social Aims, Quotation and
Originality.

MEMORABILIA MATHEMATICA

MEMORABILIA MATHEMATICA

CHAPTER I
DEFINITIONS AND OBJECT OF MATHEMATICS

101. I think it would be desirable that this form of word [mathematics]
should be reserved for the applications of the science, and that we should
use mathematic in the singular to denote the science itself, in the same way
as we speak of logic, rhetoric, or (own sister to algebra) music.—Sylvester,
J. J.
Presidential Address to the British
Association, Exeter British Association
Report (1869); Collected Mathematical
Papers, Vol. 2, p. 659.

102. ... all the sciences which have for their end investigations concerning
order and measure, are related to mathematics, it being of small importance
whether this measure be sought in numbers, forms, stars, sounds, or any
other object; that, accordingly, there ought to exist a general science which
should explain all that can be known about order and measure, considered
independently of any application to a particular subject, and that, indeed,
this science has its own proper name, consecrated by long usage, to wit,
mathematics. And a proof that it far surpasses in facility and importance the
sciences which depend upon it is that it embraces at once all the objects to
which these are devoted and a great many others besides; ....—Descartes.
Rules for the Direction of the Mind,
Philosophy of D. [Torrey] (New York, 1892),
p. 72.

103. [Mathematics] has for its object the indirect measurement of
magnitudes, and it purposes to determine magnitudes by each other,
according to the precise relations which exist between them.—Comte.

Positive Philosophy [Martineau], Bk. 1,
chap. 1.

104. The business of concrete mathematics is to discover the equations
which express the mathematical laws of the phenomenon under
consideration; and these equations are the starting-point of the calculus,
which must obtain from them certain quantities by means of others.—
Comte.
Positive Philosophy [Martineau], Bk. 1,
chap. 2.

105. Mathematics is the science of the connection of magnitudes.
Magnitude is anything that can be put equal or unequal to another thing.
Two things are equal when in every assertion each may be replaced by the
other.—Grassmann, Hermann.
Stücke aus dem Lehrbuche der Arithmetik,
Werke (Leipzig, 1904), Bd. 2, p. 298.

106. Mathematic is either Pure or Mixed: To Pure Mathematic belong those
sciences which handle Quantity entirely severed from matter and from
axioms of natural philosophy. These are two, Geometry and Arithmetic; the
one handling quantity continued, the other dissevered.... Mixed Mathematic
has for its subject some axioms and parts of natural philosophy, and
considers quantity in so far as it assists to explain, demonstrate and actuate
these.—Bacon, Francis.
De Augmentis, Bk. 3; Advancement of
Learning, Bk. 2.

107. The ideas which these sciences, Geometry, Theoretical Arithmetic and
Algebra involve extend to all objects and changes which we observe in the
external world; and hence the consideration of mathematical relations forms
a large portion of many of the sciences which treat of the phenomena and
laws of external nature, as Astronomy, Optics, and Mechanics. Such
sciences are hence often termed Mixed Mathematics, the relations of space

and number being, in these branches of knowledge, combined with
principles collected from special observation; while Geometry, Algebra, and
the like subjects, which involve no result of experience, are called Pure
Mathematics.—Whewell, William.
The Philosophy of the Inductive Sciences,
Part 1, Bk. 2, chap. I, sect. 4. (London,
1858).

108. Higher Mathematics is the art of reasoning about numerical relations
between natural phenomena; and the several sections of Higher
Mathematics are different modes of viewing these relations.—Mellor, J.
W.
Higher Mathematics for Students of
Chemistry and Physics (New York, 1902),
Prologue

109. Number, place, and combination ... the three intersecting but distinct
spheres of thought to which all mathematical ideas admit of being referred.
—Sylvester, J. J.
Philosophical Magazine, Vol. 24 (1844), p.
285; Collected Mathematical Papers, Vol. 1,
p. 91.

110. There are three ruling ideas, three so to say, spheres of thought, which
pervade the whole body of mathematical science, to some one or other of
which, or to two or all three of them combined, every mathematical truth
admits of being referred; these are the three cardinal notions, of Number,
Space and Order.
Arithmetic has for its object the properties of number in the abstract. In
algebra, viewed as a science of operations, order is the predominating idea.
The business of geometry is with the evolution of the properties of space, or
of bodies viewed as existing in space.—Sylvester, J. J.
A Probationary Lecture on Geometry, York
British Association Report (1844), Part 2;

Collected Mathematical Papers, Vol. 2, p. 5.

111. The object of pure mathematics is those relations which may be
conceptually established among any conceived elements whatsoever by
assuming them contained in some ordered manifold; the law of order of this
manifold must be subject to our choice; the latter is the case in both of the
only conceivable kinds of manifolds, in the discrete as well as in the
continuous.—Papperitz, E.
über das System der rein mathematischen
Wissenschaften, Jahresbericht der Deutschen
Mathematiker-Vereinigung, Bd. 1, p. 36.

112. Pure mathematics is not concerned with magnitude. It is merely the
doctrine of notation of relatively ordered thought operations which have
become mechanical.—Novalis.
Schriften (Berlin, 1901), Zweiter Teil, p. 282.

113. Any conception which is definitely and completely determined by
means of a finite number of specifications, say by assigning a finite number
of elements, is a mathematical conception. Mathematics has for its function
to develop the consequences involved in the definition of a group of
mathematical conceptions. Interdependence and mutual logical consistency
among the members of the group are postulated, otherwise the group would
either have to be treated as several distinct groups, or would lie beyond the
sphere of mathematics.—Chrystal, George.
Encyclopedia Britannica (9th edition),
Article “Mathematics.”

114. The purely formal sciences, logic and mathematics, deal with those
relations which are, or can be, independent of the particular content or the
substance of objects. To mathematics in particular fall those relations
between objects which involve the concepts of magnitude, of measure and
of number.—Hankel, Hermann.

Theorie der Complexen Zahlensysteme,
(Leipzig, 1867), p. 1.

115. Quantity is that which is operated with according to fixed mutually
consistent laws. Both operator and operand must derive their meaning from
the laws of operation. In the case of ordinary algebra these are the three
laws already indicated [the commutative, associative, and distributive laws],
in the algebra of quaternions the same save the law of commutation for
multiplication and division, and so on. It may be questioned whether this
definition is sufficient, and it may be objected that it is vague; but the reader
will do well to reflect that any definition must include the linear algebras of
Peirce, the algebra of logic, and others that may be easily imagined,
although they have not yet been developed. This general definition of
quantity enables us to see how operators may be treated as quantities, and
thus to understand the rationale of the so called symbolical methods.—
Chrystal, George.
Encyclopedia Britannica (9th edition),
Article “Mathematics.”

116. Mathematics—in a strict sense—is the abstract science which
investigates deductively the conclusions implicit in the elementary
conceptions of spatial and numerical relations.—Murray, J. A. H.
A New English Dictionary.

117. Everything that the greatest minds of all times have accomplished
toward the comprehension of forms by means of concepts is gathered into
one great science, mathematics.—Herbart, J. F.
Pestalozzi’s Idee eines A B C der
Anschauung, Werke [Kehrbach],
(Langensalza, 1890), Bd. 1, p. 163.

118. Perhaps the least inadequate description of the general scope of
modern Pure Mathematics—I will not call it a definition—would be to say
that it deals with form, in a very general sense of the term; this would

include algebraic form, functional relationship, the relations of order in any
ordered set of entities such as numbers, and the analysis of the peculiarities
of form of groups of operations.—Hobson, E. W.
Presidential Address British Association for
the Advancement of Science (1910); Nature,
Vol. 84, p. 287.

119. The ideal of mathematics should be to erect a calculus to facilitate
reasoning in connection with every province of thought, or of external
experience, in which the succession of thoughts, or of events can be
definitely ascertained and precisely stated. So that all serious thought which
is not philosophy, or inductive reasoning, or imaginative literature, shall be
mathematics developed by means of a calculus.—Whitehead, A. N.
Universal Algebra (Cambridge, 1898),
Preface.

120. Mathematics is the science which draws necessary conclusions.—
Peirce, Benjamin.
Linear Associative Algebra, American
Journal of Mathematics, Vol. 4 (1881), p. 97.

121. Mathematics is the universal art apodictic.—Smith, W. B.
Quoted by Keyser, C. J. in Lectures on
Science, Philosophy and Art (New York,
1908), p. 13.

122. Mathematics in its widest signification is the development of all types
of formal, necessary, deductive reasoning.—Whitehead, A. N.
Universal Algebra (Cambridge, 1898),
Preface, p. vi.

123. Mathematics in general is fundamentally the science of self-evident
things.—Klein, Felix.

Anwendung der Differential- und
Integralrechnung auf Geometrie (Leipzig,
1902), p. 26.

124. A mathematical science is any body of propositions which is capable
of an abstract formulation and arrangement in such a way that every
proposition of the set after a certain one is a formal logical consequence of
some or all the preceding propositions. Mathematics consists of all such
mathematical sciences.—Young, Charles Wesley.
Fundamental Concepts of Algebra and
Geometry (New York, 1911), p. 222.

125. Pure mathematics is a collection of hypothetical, deductive theories,
each consisting of a definite system of primitive, undefined, concepts or
symbols and primitive, unproved, but self-consistent assumptions
(commonly called axioms) together with their logically deducible
consequences following by rigidly deductive processes without appeal to
intuition.—Fitch, G. D.
The Fourth Dimension simply Explained
(New York, 1910), p. 58.

126. The whole of Mathematics consists in the organization of a series of
aids to the imagination in the process of reasoning.—Whitehead, A. N.
Universal Algebra (Cambridge, 1898), p. 12.

127. Pure mathematics consists entirely of such asseverations as that, if
such and such a proposition is true of anything, then such and such another
proposition is true of that thing. It is essential not to discuss whether the
first proposition is really true, and not to mention what the anything is of
which it is supposed to be true.... If our hypothesis is about anything and
not about some one or more particular things, then our deductions constitute
mathematics. Thus mathematics may be defined as the subject in which we
never know what we are talking about, nor whether what we are saying is
true.—Russell, Bertrand.

Recent Work on the Principles of
Mathematics, International Monthly, Vol. 4
(1901), p. 84.

128. Pure Mathematics is the class of all propositions of the form “p implies
q,” where p and q are propositions containing one or more variables, the
same in the two propositions, and neither p nor q contains any constants
except logical constants. And logical constants are all notions definable in
terms of the following: Implication, the relation of a term to a class of
which it is a member, the notion of such that, the notion of relation, and
such further notions as may be involved in the general notion of
propositions of the above form. In addition to these, Mathematics uses a
notion which is not a constituent of the propositions which it considers—
namely, the notion of truth.—Russell, Bertrand.
Principles of Mathematics (Cambridge,
1903), p. 1.

129. The object of pure Physic is the unfolding of the laws of the
intelligible world; the object of pure Mathematic that of unfolding the laws
of human intelligence.—Sylvester, J. J.
On a theorem, connected with Newton’s Rule,
etc., Collected Mathematical Papers, Vol. 3,
p. 424.

130. First of all, we ought to observe, that mathematical propositions,
properly so called, are always judgments a priori, and not empirical,
because they carry along with them necessity, which can never be deduced
from experience. If people should object to this, I am quite willing to
confine my statements to pure mathematics, the very concept of which
implies that it does not contain empirical, but only pure knowledge a priori.
—Kant, Immanuel.
Critique of Pure Reason [Müller], (New
York, 1900), p. 720.

131. Mathematics, the science of the ideal, becomes the means of
investigating, understanding and making known the world of the real. The
complex is expressed in terms of the simple. From one point of view
mathematics may be defined as the science of successive substitutions of
simpler concepts for more complex....—White, William F.
A Scrap-book of Elementary Mathematics,
(Chicago, 1908), p. 215.

132. The critical mathematician has abandoned the search for truth. He no
longer flatters himself that his propositions are or can be known to him or to
any other human being to be true; and he contents himself with aiming at
the correct, or the consistent. The distinction is not annulled nor even
blurred by the reflection that consistency contains immanently a kind of
truth. He is not absolutely certain, but he believes profoundly that it is
possible to find various sets of a few propositions each such that the
propositions of each set are compatible, that the propositions of each such
set imply other propositions, and that the latter can be deduced from the
former with certainty. That is to say, he believes that there are systems of
coherent or consistent propositions, and he regards it his business to
discover such systems. Any such system is a branch of mathematics.—
Keyser, C. J.
Science, New Series, Vol. 35, p. 107.

133. [Mathematics is] the study of ideal constructions (often applicable to
real problems), and the discovery thereby of relations between the parts of
these constructions, before unknown.—Peirce, C. S.
Century Dictionary, Article “Mathematics.”

134. Mathematics is that form of intelligence in which we bring the objects
of the phenomenal world under the control of the conception of quantity.
[Provisional definition.]—Howison, G. H.
The Departments of Mathematics, and their
Mutual Relations; Journal of Speculative
Philosophy, Vol. 5, p. 164.

135. Mathematics is the science of the functional laws and transformations
which enable us to convert figured extension and rated motion into number.
—Howison, G. H.
The Departments of Mathematics, and their
Mutual Relations; Journal of Speculative
Philosophy, Vol. 5, p. 170.

CHAPTER II
THE NATURE OF MATHEMATICS

201. Mathematics, from the earliest times to which the history of human
reason can reach, has followed, among that wonderful people of the Greeks,
the safe way of science. But it must not be supposed that it was as easy for
mathematics as for logic, in which reason is concerned with itself alone, to
find, or rather to make for itself that royal road. I believe, on the contrary,
that there was a long period of tentative work (chiefly still among the
Egyptians), and that the change is to be ascribed to a revolution, produced
by the happy thought of a single man, whose experiments pointed
unmistakably to the path that had to be followed, and opened and traced out
for the most distant times the safe way of a science. The history of that
intellectual revolution, which was far more important than the passage
round the celebrated Cape of Good Hope, and the name of its fortunate
author, have not been preserved to us.... A new light flashed on the first man
who demonstrated the properties of the isosceles triangle (whether his name
was Thales or any other name), for he found that he had not to investigate
what he saw in the figure, or the mere concepts of that figure, and thus to
learn its properties; but that he had to produce (by construction) what he
had himself, according to concepts a priori, placed into that figure and
represented in it, so that, in order to know anything with certainty a priori,
he must not attribute to that figure anything beyond what necessarily
follows from what he has himself placed into it, in accordance with the
concept.—Kant, Immanuel.
Critique of Pure Reason, Preface to the
Second Edition [Müller], (New York, 1900),
p. 690.

202. [When followed in the proper spirit], there is no study in the world
which brings into more harmonious action all the faculties of the mind than

the one [mathematics] of which I stand here as the humble representative
and advocate. There is none other which prepares so many agreeable
surprises for its followers, more wonderful than the transformation scene of
a pantomime, or, like this, seems to raise them, by successive steps of
initiation to higher and higher states of conscious intellectual being.—
Sylvester, J. J.
A Plea for the Mathematician, Nature, Vol. 1,
p. 261.

203. Thought-economy is most highly developed in mathematics, that
science which has reached the highest formal development, and on which
natural science so frequently calls for assistance. Strange as it may seem,
the strength of mathematics lies in the avoidance of all unnecessary
thoughts, in the utmost economy of thought-operations. The symbols of
order, which we call numbers, form already a system of wonderful
simplicity and economy. When in the multiplication of a number with
several digits we employ the multiplication table and thus make use of
previously accomplished results rather than to repeat them each time, when
by the use of tables of logarithms we avoid new numerical calculations by
replacing them by others long since performed, when we employ
determinants instead of carrying through from the beginning the solution of
a system of equations, when we decompose new integral expressions into
others that are familiar,—we see in all this but a faint reflection of the
intellectual activity of a Lagrange or Cauchy, who with the keen
discernment of a military commander marshalls a whole troop of completed
operations in the execution of a new one.—Mach, E.
Populär-wissenschafliche Vorlesungen
(1908), pp. 224-225.

204. Pure mathematics proves itself a royal science both through its content
and form, which contains within itself the cause of its being and its methods
of proof. For in complete independence mathematics creates for itself the
object of which it treats, its magnitudes and laws, its formulas and symbols.
—Dillmann, E.
Die Mathematik die Fackelträgerin einer
neuen Zeit (Stuttgart, 1889), p. 94.

205. The essence of mathematics lies in its freedom.—Cantor, George.
Mathematische Annalen, Bd. 21, p. 564.

206. Mathematics pursues its own course unrestrained, not indeed with an
unbridled licence which submits to no laws, but rather with the freedom
which is determined by its own nature and in conformity with its own
being.—Hankel, Hermann.
Die Entwickelung der Mathematik in den
letzten Jahrhunderten (Tübingen, 1884), p.
16.

207. Mathematics is perfectly free in its development and is subject only to
the obvious consideration, that its concepts must be free from contradictions
in themselves, as well as definitely and orderly related by means of
definitions to the previously existing and established concepts.—Cantor,
George.
Grundlagen einer allgemeinen
Manigfaltigkeitslehre (Leipzig, 1883), Sect.
8.

208. Mathematicians assume the right to choose, within the limits of logical
contradiction, what path they please in reaching their results.—Adams,
Henry.
A Letter to American Teachers of History
(Washington, 1910), Introduction, p. v.

209. Mathematics is the predominant science of our time; its conquests
grow daily, though without noise; he who does not employ it for himself,
will some day find it employed against himself.—Herbart, J. F.
Werke [Kehrbach] (Langensalza, 1890), Bd.
5, p. 105.

210. Mathematics is not the discoverer of laws, for it is not induction;
neither is it the framer of theories, for it is not hypothesis; but it is the judge
over both, and it is the arbiter to which each must refer its claims; and
neither law can rule nor theory explain without the sanction of mathematics.
—Peirce, Benjamin.
Linear Associative Algebra, American
Journal of Mathematics, Vol. 4 (1881), p. 97.

211. Mathematics is a science continually expanding; and its growth, unlike
some political and industrial events, is attended by universal acclamation.—
White, H. S.
Congress of Arts and Sciences (Boston and
New York, 1905), Vol. 1, p. 455.

212. Mathematics accomplishes really nothing outside of the realm of
magnitude; marvellous, however, is the skill with which it masters
magnitude wherever it finds it. We recall at once the network of lines which
it has spun about heavens and earth; the system of lines to which azimuth
and altitude, declination and right ascension, longitude and latitude are
referred; those abscissas and ordinates, tangents and normals, circles of
curvature and evolutes; those trigonometric and logarithmic functions
which have been prepared in advance and await application. A look at this
apparatus is sufficient to show that mathematicians are not magicians, but
that everything is accomplished by natural means; one is rather impressed
by the multitude of skilful machines, numerous witnesses of a manifold and
intensely active industry, admirably fitted for the acquisition of true and
lasting treasures.—Herbart, J. F.
Werke [Kehrbach] (Langensalza, 1890), Bd.
5, p. 101.

213. They [mathematicians] only take those things into consideration, of
which they have clear and distinct ideas, designating them by proper,
adequate, and invariable names, and premising only a few axioms which are
most noted and certain to investigate their affections and draw conclusions
from them, and agreeably laying down a very few hypotheses, such as are

in the highest degree consonant with reason and not to be denied by anyone
in his right mind. In like manner they assign generations or causes easy to
be understood and readily admitted by all, they preserve a most accurate
order, every proposition immediately following from what is supposed and
proved before, and reject all things howsoever specious and probable which
can not be inferred and deduced after the same manner.—Barrow, Isaac.
Mathematical Lectures (London, 1734), p.
66.

214. The dexterous management of terms and being able to fend and prove
with them, I know has and does pass in the world for a great part of
learning; but it is learning distinct from knowledge, for knowledge consists
only in perceiving the habitudes and relations of ideas one to another, which
is done without words; the intervention of sounds helps nothing to it. And
hence we see that there is least use of distinction where there is most
knowledge: I mean in mathematics, where men have determined ideas with
known names to them; and so, there being no room for equivocations, there
is no need of distinctions.—Locke, John.
Conduct of the Understanding, Sect. 31.

215. In mathematics it [sophistry] had no place from the beginning:
Mathematicians having had the wisdom to define accurately the terms they
use, and to lay down, as axioms, the first principles on which their
reasoning is grounded. Accordingly we find no parties among
mathematicians, and hardly any disputes.—Reid, Thomas.
Essays on the Intellectual Powers of Man,
Essay 1, chap. 1.

216. In most sciences one generation tears down what another has built and
what one has established another undoes. In Mathematics alone each
generation builds a new story to the old structure.—Hankel, Hermann.
Die Entwickelung der Mathematik in den
letzten Jahrhunderten (Tübingen, 1884), p.
25.

217. Mathematics, the priestess of definiteness and clearness.—Herbart, J.
F.
Werke [Kehrbach] (Langensalza, 1890), Bd.
1, p. 171.

218. ... mathematical analysis is co-extensive with nature itself, it defines
all perceivable relations, measures times, spaces, forces, temperatures; it is
a difficult science which forms but slowly, but preserves carefully every
principle once acquired; it increases and becomes stronger incessantly
amidst all the changes and errors of the human mind.
Its chief attribute is clearness; it has no means for expressing confused
ideas. It compares the most diverse phenomena and discovers the secret
analogies which unite them. If matter escapes us, as that of air and light
because of its extreme tenuity, if bodies are placed far from us in the
immensity of space, if man wishes to know the aspect of the heavens at
successive periods separated by many centuries, if gravity and heat act in
the interior of the solid earth at depths which will forever be inaccessible,
mathematical analysis is still able to trace the laws of these phenomena. It
renders them present and measurable, and appears to be the faculty of the
human mind destined to supplement the brevity of life and the imperfection
of the senses, and what is even more remarkable, it follows the same course
in the study of all phenomena; it explains them in the same language, as if
in witness to the unity and simplicity of the plan of the universe, and to
make more manifest the unchangeable order which presides over all natural
causes.—Fourier, J.
Théorie Analytique de la Chaleur, Discours
Préliminaire.

219. Let us now declare the means whereby our understanding can rise to
knowledge without fear of error. There are two such means: intuition and
deduction. By intuition I mean not the varying testimony of the senses, nor
the deductive judgment of imagination naturally extravagant, but the
conception of an attentive mind so distinct and so clear that no doubt
remains to it with regard to that which it comprehends; or, what amounts to
the same thing, the self-evidencing conception of a sound and attentive

mind, a conception which springs from the light of reason alone, and is
more certain, because more simple, than deduction itself....
It may perhaps be asked why to intuition we add this other mode of
knowing, by deduction, that is to say, the process which, from something of
which we have certain knowledge, draws consequences which necessarily
follow therefrom. But we are obliged to admit this second step; for there are
a great many things which, without being evident of themselves,
nevertheless bear the marks of certainty if only they are deduced from true
and incontestable principles by a continuous and uninterrupted movement
of thought, with distinct intuition of each thing; just as we know that the last
link of a long chain holds to the first, although we can not take in with one
glance of the eye the intermediate links, provided that, after having run over
them in succession, we can recall them all, each as being joined to its
fellows, from the first up to the last. Thus we distinguish intuition from
deduction, inasmuch as in the latter case there is conceived a certain
progress or succession, while it is not so in the former;... whence it follows
that primary propositions, derived immediately from principles, may be said
to be known, according to the way we view them, now by intuition, now by
deduction; although the principles themselves can be known only by
intuition, the remote consequences only by deduction.—Descartes.
Rules for the Direction of the Mind,
Philosophy of D. [Torrey] (New York, 1892),
pp. 64, 65.

220. Analysis and natural philosophy owe their most important discoveries
to this fruitful means, which is called induction. Newton was indebted to it
for his theorem of the binomial and the principle of universal gravity.—
Laplace.
A Philosophical Essay on Probabilities
[Truscott and Emory] (New York 1902), p.
176.

221. There is in every step of an arithmetical or algebraical calculation a
real induction, a real inference from facts to facts, and what disguises the

induction is simply its comprehensive nature, and the consequent extreme
generality of its language.—Mill, J. S.
System of Logic, Bk. 2, chap. 6, 2.

222. It would appear that Deductive and Demonstrative Sciences are all,
without exception, Inductive Sciences: that their evidence is that of
experience, but that they are also, in virtue of the peculiar character of one
indispensable portion of the general formulae according to which their
inductions are made, Hypothetical Sciences. Their conclusions are true only
upon certain suppositions, which are, or ought to be, approximations to the
truth, but are seldom, if ever, exactly true; and to this hypothetical character
is to be ascribed the peculiar certainty, which is supposed to be inherent in
demonstration.—Mill, J. S.
System of Logic, Bk. 2, chap. 6, 1.

223. The peculiar character of mathematical truth is, that it is necessarily
and inevitably true; and one of the most important lessons which we learn
from our mathematical studies is a knowledge that there are such truths, and
a familiarity with their form and character.
This lesson is not only lost, but read backward, if the student is taught
that there is no such difference, and that mathematical truths themselves are
learned by experience.—Whewell, W.
Thoughts on the Study of Mathematics.
Principles of English University Education
(London, 1838).

224. These sciences, Geometry, Theoretical Arithmetic and Algebra, have
no principles besides definitions and axioms, and no process of proof but
deduction; this process, however, assuming a most remarkable character;
and exhibiting a combination of simplicity and complexity, of rigour and
generality, quite unparalleled in other subjects.—Whewell, W.
The Philosophy of the Inductive Sciences,
Part 1, Bk. 2, chap. 1, sect. 2 (London, 1858).

225. The apodictic quality of mathematical thought, the certainty and
correctness of its conclusions, are due, not to a special mode of
ratiocination, but to the character of the concepts with which it deals. What
is that distinctive characteristic? I answer: precision, sharpness,
completeness,1 of definition. But how comes your mathematician by such
completeness? There is no mysterious trick involved; some ideas admit of
such precision, others do not; and the mathematician is one who deals with
those that do.—Keyser, C. J.
The Universe and Beyond; Hibbert Journal,
Vol. 3 (1904-1905), p. 309.

226. The reasoning of mathematicians is founded on certain and infallible
principles. Every word they use conveys a determinate idea, and by
accurate definitions they excite the same ideas in the mind of the reader that
were in the mind of the writer. When they have defined the terms they
intend to make use of, they premise a few axioms, or self-evident
principles, that every one must assent to as soon as proposed. They then
take for granted certain postulates, that no one can deny them, such as, that
a right line may be drawn from any given point to another, and from these
plain, simple principles they have raised most astonishing speculations, and
proved the extent of the human mind to be more spacious and capacious
than any other science.—Adams, John.
Diary, Works (Boston, 1850), Vol. 2, p. 21.

227. It may be observed of mathematicians that they only meddle with such
things as are certain, passing by those that are doubtful and unknown. They
profess not to know all things, neither do they affect to speak of all things.
What they know to be true, and can make good by invincible arguments,
that they publish and insert among their theorems. Of other things they are
silent and pass no judgment at all, choosing rather to acknowledge their
ignorance, than affirm anything rashly. They affirm nothing among their
arguments or assertions which is not most manifestly known and examined
with utmost rigour, rejecting all probable conjectures and little witticisms.
They submit nothing to authority, indulge no affection, detest subterfuges of
words, and declare their sentiments, as in a court of justice, without passion,

without apology; knowing that their reasons, as Seneca testifies of them, are
not brought to persuade, but to compel.—Barrow, Isaac.
Mathematical Lectures (London, 1734), p.
64.

228. What is exact about mathematics but exactness? And is not this a
consequence of the inner sense of truth?—Goethe.
Sprüche in Prosa, Natur, 6, 948.

229. ... the three positive characteristics that distinguish mathematical
knowledge from other knowledge ... may be briefly expressed as follows:
first, mathematical knowledge bears more distinctly the imprint of truth on
all its results than any other kind of knowledge; secondly, it is always a sure
preliminary step to the attainment of other correct knowledge; thirdly, it has
no need of other knowledge.—Schubert, H.
Mathematical Essays and Recreations
(Chicago, 1898), p. 35.

230. It is now necessary to indicate more definitely the reason why
mathematics not only carries conviction in itself, but also transmits
conviction to the objects to which it is applied. The reason is found, first of
all, in the perfect precision with which the elementary mathematical
concepts are determined; in this respect each science must look to its own
salvation.... But this is not all. As soon as human thought attempts long
chains of conclusions, or difficult matters generally, there arises not only
the danger of error but also the suspicion of error, because since all details
cannot be surveyed with clearness at the same instant one must in the end
be satisfied with a belief that nothing has been overlooked from the
beginning. Every one knows how much this is the case even in arithmetic,
the most elementary use of mathematics. No one would imagine that the
higher parts of mathematics fare better in this respect; on the contrary, in
more complicated conclusions the uncertainty and suspicion of hidden
errors increases in rapid progression. How does mathematics manage to rid
itself of this inconvenience which attaches to it in the highest degree? By
making proofs more rigorous? By giving new rules according to which the

old rules shall be applied? Not in the least. A very great uncertainty
continues to attach to the result of each single computation. But there are
checks. In the realm of mathematics each point may be reached by a
hundred different ways; and if each of a hundred ways leads to the same
point, one may be sure that the right point has been reached. A calculation
without a check is as good as none. Just so it is with every isolated proof in
any speculative science whatever; the proof may be ever so ingenious, and
ever so perfectly true and correct, it will still fail to convince permanently.
He will therefore be much deceived, who, in metaphysics, or in psychology
which depends on metaphysics, hopes to see his greatest care in the precise
determination of the concepts and in the logical conclusions rewarded by
conviction, much less by success in transmitting conviction to others. Not
only must the conclusions support each other, without coercion or suspicion
of subreption, but in all matters originating in experience, or judging
concerning experience, the results of speculation must be verified by
experience, not only superficially, but in countless special cases.—Herbart,
J. F.
Werke [Kehrbach] (Langensalza, 1890), Bd.
5, p. 105.

231. [In mathematics] we behold the conscious logical activity of the
human mind in its purest and most perfect form. Here we learn to realize
the laborious nature of the process, the great care with which it must
proceed, the accuracy which is necessary to determine the exact extent of
the general propositions arrived at, the difficulty of forming and
comprehending abstract concepts; but here we learn also to place
confidence in the certainty, scope and fruitfulness of such intellectual
activity.—Helmholtz, H.
Ueber das Verhältniss der
Naturwissenschaften zur Gesammtheit der
Wissenschaft, Vorträge und Reden, Bd. 1
(1896), p. 176.

232. It is true that mathematics, owing to the fact that its whole content is
built up by means of purely logical deduction from a small number of
universally comprehended principles, has not unfittingly been designated as

the science of the self-evident [Selbstverständlichen]. Experience however,
shows that for the majority of the cultured, even of scientists, mathematics
remains the science of the incomprehensible [Unverständlichen].—
Pringsheim, Alfred.
Ueber Wert und angeblichen Unwert der
Mathematik, Jahresbericht der Deutschen
Mathematiker Vereinigung (1904), p. 357.

233. Mathematical reasoning is deductive in the sense that it is based upon
definitions which, as far as the validity of the reasoning is concerned (apart
from any existential import), needs only the test of self-consistency. Thus
no external verification of definitions is required in mathematics, as long as
it is considered merely as mathematics.—Whitehead, A. N.
Universal Algebra (Cambridge, 1898),
Preface, p. vi.

234. The mathematician pays not the least regard either to testimony or
conjecture, but deduces everything by demonstrative reasoning, from his
definitions and axioms. Indeed, whatever is built upon conjecture, is
improperly called science; for conjecture may beget opinion, but cannot
produce knowledge.—Reid, Thomas.
Essays on the Intellectual Powers of Man,
Essay 1, chap. 3.

235. ... for the saving the long progression of the thoughts to remote and
first principles in every case, the mind should provide itself several stages;
that is to say, intermediate principles, which it might have recourse to in the
examining those positions that come in its way. These, though they are not
self-evident principles, yet, if they have been made out from them by a
wary and unquestionable deduction, may be depended on as certain and
infallible truths, and serve as unquestionable truths to prove other points
depending upon them, by a nearer and shorter view than remote and general
maxims.... And thus mathematicians do, who do not in every new problem
run it back to the first axioms through all the whole train of intermediate
propositions. Certain theorems that they have settled to themselves upon

sure demonstration, serve to resolve to them multitudes of propositions
which depend on them, and are as firmly made out from thence as if the
mind went afresh over every link of the whole chain that tie them to first
self-evident principles.—Locke, John.
The Conduct of the Understanding, Sect. 21.

236. Those intervening ideas, which serve to show the agreement of any
two others, are called proofs; and where the agreement or disagreement is
by this means plainly and clearly perceived, it is called demonstration; it
being shown to the understanding, and the mind made to see that it is so. A
quickness in the mind to find out these intermediate ideas, (that shall
discover the agreement or disagreement of any other) and to apply them
right, is, I suppose, that which is called sagacity.—Locke, John.
An Essay concerning Human Understanding,
Bk. 6, chaps. 2, 3.

237. ... the speculative propositions of mathematics do not relate to facts; ...
all that we are convinced of by any demonstration in the science, is of a
necessary connection subsisting between certain suppositions and certain
conclusions. When we find these suppositions actually take place in a
particular instance, the demonstration forces us to apply the conclusion.
Thus, if I could form a triangle, the three sides of which were accurately
mathematical lines, I might affirm of this individual figure, that its three
angles are equal to two right angles; but, as the imperfection of my senses
puts it out of my power to be, in any case, certain of the exact
correspondence of the diagram which I delineate, with the definitions given
in the elements of geometry, I never can apply with confidence to a
particular figure, a mathematical theorem. On the other hand, it appears
from the daily testimony of our senses that the speculative truths of
geometry may be applied to material objects with a degree of accuracy
sufficient for the purposes of life; and from such applications of them,
advantages of the most important kind have been gained to society.—
Stewart, Dugald.
Elements of the Philosophy of the Human
Mind, Part 3, chap. 1, sect. 3.

238. No process of sound reasoning can establish a result not contained in
the premises.—Mellor, J. W.
Higher Mathematics for Students of
Chemistry and Physics (New York, 1902), p.
2.

239. ... we cannot get more out of the mathematical mill than we put into it,
though we may get it in a form infinitely more useful for our purpose.—
Hopkinson, John.
James Forrest Lecture, 1894.

240. The iron labor of conscious logical reasoning demands great
perseverance and great caution; it moves on but slowly, and is rarely
illuminated by brilliant flashes of genius. It knows little of that facility with
which the most varied instances come thronging into the memory of the
philologist or historian. Rather is it an essential condition of the methodical
progress of mathematical reasoning that the mind should remain
concentrated on a single point, undisturbed alike by collateral ideas on the
one hand, and by wishes and hopes on the other, and moving on steadily in
the direction it has deliberately chosen.—Helmholtz, H.
Ueber das Verhältniss der
Naturwissenschaften zur Gesammtheit der
Wissenschaft, Vorträge und Reden, Bd. 1
(1896), p. 178.

241. If it were always necessary to reduce everything to intuitive
knowledge, demonstration would often be insufferably prolix. This is why
mathematicians have had the cleverness to divide the difficulties and to
demonstrate separately the intervening propositions. And there is art also in
this; for as the mediate truths (which are called lemmas, since they appear
to be a digression) may be assigned in many ways, it is well, in order to aid
the understanding and memory, to choose of them those which greatly
shorten the process, and appear memorable and worthy in themselves of
being demonstrated. But there is another obstacle, viz.: that it is not easy to
demonstrate all the axioms, and to reduce demonstrations wholly to

intuitive knowledge. And if we had chosen to wait for that, perhaps we
should not yet have the science of geometry.—Leibnitz, G. W.
New Essay on Human Understanding
[Langley], Bk. 4, chaps. 2, 8.

242. In Pure Mathematics, where all the various truths are necessarily
connected with each other, (being all necessarily connected with those
hypotheses which are the principles of the science), an arrangement is
beautiful in proportion as the principles are few; and what we admire
perhaps chiefly in the science, is the astonishing variety of consequences
which may be demonstrably deduced from so small a number of premises.
—Stewart, Dugald.
The Elements of the Philosophy of the
Human Mind, Part 3, chap. 1, sect. 3.

243. Whenever ... a controversy arises in mathematics, the issue is not
whether a thing is true or not, but whether the proof might not be conducted
more simply in some other way, or whether the proposition demonstrated is
sufficiently important for the advancement of the science as to deserve
especial enunciation and emphasis, or finally, whether the proposition is not
a special case of some other and more general truth which is as easily
discovered.—Schubert, H.
Mathematical Essays and Recreations
(Chicago, 1898), p. 28.

244. ... just as the astronomer, the physicist, the geologist, or other student
of objective science looks about in the world of sense, so, not
metaphorically speaking but literally, the mind of the mathematician goes
forth in the universe of logic in quest of the things that are there; exploring
the heights and depths for facts—ideas, classes, relationships, implications,
and the rest; observing the minute and elusive with the powerful
microscope of his Infinitesimal Analysis; observing the elusive and vast
with the limitless telescope of his Calculus of the Infinite; making guesses
regarding the order and internal harmony of the data observed and
collocated; testing the hypotheses, not merely by the complete induction

peculiar to mathematics, but, like his colleagues of the outer world,
resorting also to experimental tests and incomplete induction; frequently
finding it necessary, in view of unforeseen disclosures, to abandon one
hopeful hypothesis or to transform it by retrenchment or by enlargement:—
thus, in his own domain, matching, point for point, the processes, methods
and experience familiar to the devotee of natural science.—Keyser, Cassius
J.
Lectures on Science, Philosophy and Art
(New York, 1908), p. 26.

245. That mathematics “do not cultivate the power of generalization,” ...
will be admitted by no person of competent knowledge, except in a very
qualified sense. The generalizations of mathematics, are, no doubt, a
different thing from the generalizations of physical science; but in the
difficulty of seizing them, and the mental tension they require, they are no
contemptible preparation for the most arduous efforts of the scientific mind.
Even the fundamental notions of the higher mathematics, from those of the
differential calculus upwards are products of a very high abstraction.... To
perceive the mathematical laws common to the results of many
mathematical operations, even in so simple a case as that of the binomial
theorem, involves a vigorous exercise of the same faculty which gave us
Kepler’s laws, and rose through those laws to the theory of universal
gravitation. Every process of what has been called Universal Geometry—
the great creation of Descartes and his successors, in which a single train of
reasoning solves whole classes of problems at once, and others common to
large groups of them—is a practical lesson in the management of wide
generalizations, and abstraction of the points of agreement from those of
difference among objects of great and confusing diversity, to which the
purely inductive sciences cannot furnish many superior. Even so elementary
an operation as that of abstracting from the particular configuration of the
triangles or other figures, and the relative situation of the particular lines or
points, in the diagram which aids the apprehension of a common
geometrical demonstration, is a very useful, and far from being always an
easy, exercise of the faculty of generalization so strangely imagined to have
no place or part in the processes of mathematics.—Mill, John Stuart.

An Examination of Sir William Hamilton’s
Philosophy (London, 1878), pp. 612, 613.

246. When the greatest of American logicians, speaking of the powers that
constitute the born geometrician, had named Conception, Imagination, and
Generalization, he paused. Thereupon from one of the audience there came
the challenge, “What of reason?” The instant response, not less just than
brilliant, was: “Ratiocination—that is but the smooth pavement on which
the chariot rolls.”—Keyser, C. J.
Lectures on Science, Philosophy and Art
(New York, 1908), p. 31.

247. ... the reasoning process [employed in mathematics] is not different
from that of any other branch of knowledge, ... but there is required, and in
a great degree, that attention of mind which is in some part necessary for
the acquisition of all knowledge, and in this branch is indispensably
necessary. This must be given in its fullest intensity; ... the other elements
especially characteristic of a mathematical mind are quickness in perceiving
logical sequence, love of order, methodical arrangement and harmony,
distinctness of conception.—Price, B.
Treatise on Infinitesimal Calculus (Oxford,
1868), Vol. 3, p. 6.

248. Histories make men wise; poets, witty; the mathematics, subtile;
natural philosophy, deep; moral, grave; logic and rhetoric, able to contend.
—Bacon, Francis.
Essays, Of Studies.

249. The Mathematician deals with two properties of objects only, number
and extension, and all the inductions he wants have been formed and
finished ages ago. He is now occupied with nothing but deduction and
verification.—Huxley, T. H.
On the Educational Value of the Natural
History Sciences; Lay Sermons, Addresses

and Reviews; (New York, 1872), p. 87.

250. [Mathematics] is that [subject] which knows nothing of observation,
nothing of experiment, nothing of induction, nothing of causation.—
Huxley, T. H.
The Scientific Aspects of Positivism,
Fortnightly Review (1898); Lay Sermons,
Addresses and Reviews, (New York, 1872), p.
169.

251. We are told that “Mathematics is that study which knows nothing of
observation, nothing of experiment, nothing of induction, nothing of
causation.” I think no statement could have been made more opposite to the
facts of the case; that mathematical analysis is constantly invoking the aid
of new principles, new ideas, and new methods, not capable of being
defined by any form of words, but springing direct from the inherent
powers and activities of the human mind, and from continually renewed
introspection of that inner world of thought of which the phenomena are as
varied and require as close attention to discern as those of the outer physical
world (to which the inner one in each individual man may, I think, be
conceived to stand somewhat in the same relation of correspondence as a
shadow to the object from which it is projected, or as the hollow palm of
one hand to the closed fist which it grasps of the other), that it is
unceasingly calling forth the faculties of observation and comparison, that
one of its principal weapons is induction, that it has frequent recourse to
experimental trial and verification, and that it affords a boundless scope for
the exercise of the highest efforts of the imagination and invention.—
Sylvester, J. J.
Presidential Address to British Association,
Exeter British Association Report (1869), pp.
1-9.; Collected Mathematical Papers, Vol. 2,
p. 654.

252. The actual evolution of mathematical theories proceeds by a process of
induction strictly analogous to the method of induction employed in

building up the physical sciences; observation, comparison, classification,
trial, and generalisation are essential in both cases. Not only are special
results, obtained independently of one another, frequently seen to be really
included in some generalisation, but branches of the subject which have
been developed quite independently of one another are sometimes found to
have connections which enable them to be synthesised in one single body of
doctrine. The essential nature of mathematical thought manifests itself in
the discernment of fundamental identity in the mathematical aspects of
what are superficially very different domains. A striking example of this
species of immanent identity of mathematical form was exhibited by the
discovery of that distinguished mathematician ... Major MacMahon, that all
possible Latin squares are capable of enumeration by the consideration of
certain differential operators. Here we have a case in which an enumeration,
which appears to be not amenable to direct treatment, can actually be
carried out in a simple manner when the underlying identity of the
operation is recognised with that involved in certain operations due to
differential operators, the calculus of which belongs superficially to a
wholly different region of thought from that relating to Latin squares.—
Hobson, E. W.
Presidential Address British Association for
the Advancement of Science (1910); Nature,
Vol. 84, p. 290.

253. It has been asserted ... that the power of observation is not developed
by mathematical studies; while the truth is, that; from the most elementary
mathematical notion that arises in the mind of a child to the farthest verge to
which mathematical investigation has been pushed and applied, this power
is in constant exercise. By observation, as here used, can only be meant the
fixing of the attention upon objects (physical or mental) so as to note
distinctive peculiarities—to recognize resemblances, differences, and other
relations. Now the first mental act of the child recognizing the distinction
between one and more than one, between one and two, two and three, etc.,
is exactly this. So, again, the first geometrical notions are as pure an
exercise of this power as can be given. To know a straight line, to
distinguish it from a curve; to recognize a triangle and distinguish the
several forms—what are these, and all perception of form, but a series of

observations? Nor is it alone in securing these fundamental conceptions of
number and form that observation plays so important a part. The very
genius of the common geometry as a method of reasoning—a system of
investigation—is, that it is but a series of observations. The figure being
before the eye in actual representation, or before the mind in conception, is
so closely scrutinized, that all its distinctive features are perceived;
auxiliary lines are drawn (the imagination leading in this), and a new series
of inspections is made; and thus, by means of direct, simple observations,
the investigation proceeds. So characteristic of common geometry is this
method of investigation, that Comte, perhaps the ablest of all writers upon
the philosophy of mathematics, is disposed to class geometry, as to its
method, with the natural sciences, being based upon observation. Moreover,
when we consider applied mathematics, we need only to notice that the
exercise of this faculty is so essential, that the basis of all such reasoning,
the very material with which we build, have received the name
observations. Thus we might proceed to consider the whole range of the
human faculties, and find for the most of them ample scope for exercise in
mathematical studies. Certainly, the memory will not be found to be
neglected. The very first steps in number—counting, the multiplication
table, etc., make heavy demands on this power; while the higher branches
require the memorizing of formulas which are simply appalling to the
uninitiated. So the imagination, the creative faculty of the mind, has
constant exercise in all original mathematical investigations, from the
solution of the simplest problems to the discovery of the most recondite
principle; for it is not by sure, consecutive steps, as many suppose, that we
advance from the known to the unknown. The imagination, not the logical
faculty, leads in this advance. In fact, practical observation is often in
advance of logical exposition. Thus, in the discovery of truth, the
imagination habitually presents hypotheses, and observation supplies facts,
which it may require ages for the tardy reason to connect logically with the
known. Of this truth, mathematics, as well as all other sciences, affords
abundant illustrations. So remarkably true is this, that today it is seriously
questioned by the majority of thinkers, whether the sublimest branch of
mathematics,—the infinitesimal calculus—has anything more than an
empirical foundation, mathematicians themselves not being agreed as to its
logical basis. That the imagination, and not the logical faculty, leads in all
original investigation, no one who has ever succeeded in producing an

original demonstration of one of the simpler propositions of geometry, can
have any doubt. Nor are induction, analogy, the scrutinization of premises
or the search for them, or the balancing of probabilities, spheres of mental
operations foreign to mathematics. No one, indeed, can claim pre-eminence
for mathematical studies in all these departments of intellectual culture, but
it may, perhaps, be claimed that scarcely any department of science affords
discipline to so great a number of faculties, and that none presents so
complete a gradation in the exercise of these faculties, from the first
principles of the science to the farthest extent of its applications, as
mathematics.—Olney, Edward.
Kiddle and Schem’s Encyclopedia of
Education, (New York, 1877), Article
“Mathematics.”

254. The opinion appears to be gaining ground that this very general
conception of functionality, born on mathematical ground, is destined to
supersede the narrower notion of causation, traditional in connection with
the natural sciences. As an abstract formulation of the idea of determination
in its most general sense, the notion of functionality includes and transcends
the more special notion of causation as a one-sided determination of future
phenomena by means of present conditions; it can be used to express the
fact of the subsumption under a general law of past, present, and future
alike, in a sequence of phenomena. From this point of view the remark of
Huxley that Mathematics “knows nothing of causation” could only be taken
to express the whole truth, if by the term “causation” is understood
“efficient causation.” The latter notion has, however, in recent times been to
an increasing extent regarded as just as irrelevant in the natural sciences as
it is in Mathematics; the idea of thorough-going determinancy, in
accordance with formal law, being thought to be alone significant in either
domain.—Hobson, E. W.
Presidential Address British Association for
the Advancement of Science (1910); Nature,
Vol. 84, p. 290.

255. Most, if not all, of the great ideas of modern mathematics have had
their origin in observation. Take, for instance, the arithmetical theory of

forms, of which the foundation was laid in the diophantine theorems of
Fermat, left without proof by their author, which resisted all efforts of the
myriad-minded Euler to reduce to demonstration, and only yielded up their
cause of being when turned over in the blow-pipe flame of Gauss’s
transcendent genius; or the doctrine of double periodicity, which resulted
from the observation of Jacobi of a purely analytical fact of transformation;
or Legendre’s law of reciprocity; or Sturm’s theorem about the roots of
equations, which, as he informed me with his own lips, stared him in the
face in the midst of some mechanical investigations connected (if my
memory serves me right) with the motion of compound pendulums; or
Huyghen’s method of continued fractions, characterized by Lagrange as one
of the principal discoveries of that great mathematician, and to which he
appears to have been led by the construction of his Planetary Automaton; or
the new algebra, speaking of which one of my predecessors (Mr.
Spottiswoode) has said, not without just reason and authority, from this
chair, “that it reaches out and indissolubly connects itself each year with
fresh branches of mathematics, that the theory of equations has become
almost new through it, algebraic geometry transfigured in its light, that the
calculus of variations, molecular physics, and mechanics” (he might, if
speaking at the present moment, go on to add the theory of elasticity and the
development of the integral calculus) “have all felt its influence.”—
Sylvester, J. J.
A Plea for the Mathematician, Nature, Vol. 1,
p. 238; Collected Mathematical Papers, Vol.
2, pp. 655, 656.

256. The ability to imagine relations is one of the most indispensable
conditions of all precise thinking. No subject can be named, in the
investigation of which it is not imperatively needed; but it can be nowhere
else so thoroughly acquired as in the study of mathematics.—Fiske, John.
Darwinism and other Essays (Boston, 1893),
p. 296.

257. The great science [mathematics] occupies itself at least just as much
with the power of imagination as with the power of logical conclusion.—
Herbart, F. J.

Pestalozzi’s Idee eines ABC der Anschauung.
Werke [Kehrbach] (Langensaltza, 1890), Bd.
1, p. 174.

258. The moving power of mathematical invention is not reasoning but
imagination.—De Morgan, A.
Quoted in Graves’ Life of Sir W. R. Hamilton,
Vol. 3 (1889), p. 219.

259. There is an astonishing imagination, even in the science of
mathematics.... We repeat, there was far more imagination in the head of
Archimedes than in that of Homer.—Voltaire.
A Philosophical Dictionary (Boston, 1881),
Vol. 3, p. 40. Article “Imagination.”

260. As the prerogative of Natural Science is to cultivate a taste for
observation, so that of Mathematics is, almost from the starting point, to
stimulate the faculty of invention.—Sylvester, J. J.
A Plea for the Mathematician, Nature, Vol. 1,
p. 261; Collected Mathematical Papers, Vol.
2 (Cambridge, 1908), p. 717.

261. A marveilous newtrality have these things mathematicall, and also a
strange participation between things supernaturall, immortall, intellectuall,
simple and indivisible, and things naturall, mortall, sensible, componded
and divisible.—Dee, John.
Euclid (1570), Preface.

262. Mathematics stands forth as that which unites, mediates between Man
and Nature, inner and outer world, thought and perception, as no other
subject does.—Froebel.
[Herford translation] (London, 1893), Vol. 1,
p. 84.

263. The intrinsic character of mathematical research and knowledge is
based essentially on three properties: first, on its conservative attitude
towards the old truths and discoveries of mathematics; secondly, on its
progressive mode of development, due to the incessant acquisition of new
knowledge on the basis of the old; and thirdly, on its self-sufficiency and its
consequent absolute independence.—Schubert, H.
Mathematical Essays and Recreations
(Chicago, 1898), p. 27.

264. Our science, in contrast with others, is not founded on a single period
of human history, but has accompanied the development of culture through
all its stages. Mathematics is as much interwoven with Greek culture as
with the most modern problems in Engineering. She not only lends a hand
to the progressive natural sciences but participates at the same time in the
abstract investigations of logicians and philosophers.—Klein, F.
Klein und Riecke: Ueber angewandte
Mathematik und Physik (1900), p. 228.

265. There is probably no other science which presents such different
appearances to one who cultivates it and to one who does not, as
mathematics. To this person it is ancient, venerable, and complete; a body
of dry, irrefutable, unambiguous reasoning. To the mathematician, on the
other hand, his science is yet in the purple bloom of vigorous youth,
everywhere stretching out after the “attainable but unattained” and full of
the excitement of nascent thoughts; its logic is beset with ambiguities, and
its analytic processes, like Bunyan’s road, have a quagmire on one side and
a deep ditch on the other and branch off into innumerable by-paths that end
in a wilderness.—Chapman, C. H.
Bulletin American Mathematical Society, Vol.
2 (First series), p. 61.

266. Mathematical science is in my opinion an indivisible whole, an
organism whose vitality is conditioned upon the connection of its parts. For
with all the variety of mathematical knowledge, we are still clearly
conscious of the similarity of the logical devices, the relationship of the

ideas in mathematics as a whole and the numerous analogies in its different
departments. We also notice that, the farther a mathematical theory is
developed, the more harmoniously and uniformly does its construction
proceed, and unsuspected relations are disclosed between hitherto separated
branches of the science. So it happens that, with the extension of
mathematics, its organic character is not lost but manifests itself the more
clearly.—Hilbert, D.
Mathematical Problems, Bulletin American
Mathematical Society, Vol. 8, p. 478.

267. The mathematics have always been the implacable enemies of
scientific romances.—Arago.
Oeuvres (1855), t. 3, p. 498.

268. Those skilled in mathematical analysis know that its object is not
simply to calculate numbers, but that it is also employed to find the
relations between magnitudes which cannot be expressed in numbers and
between functions whose law is not capable of algebraic expression.—
Cournot, Augustin.
Mathematical Theory of the Principles of
Wealth [Bacon, N. T.], (New York, 1897), p.
3.

269. Coterminous with space and coeval with time is the Kingdom of
Mathematics; within this range her dominion is supreme; otherwise than
according to her order nothing can exist; in contradiction to her laws
nothing takes place. On her mysterious scroll is to be found written for
those who can read it that which has been, that which is, and that which is
to come. Everything material which is the subject of knowledge has
number, order, or position; and these are her first outlines for a sketch of the
universe. If our feeble hands cannot follow out the details, still her part has
been drawn with an unerring pen, and her work cannot be gainsaid. So wide
is the range of mathematical sciences, so indefinitely may it extend beyond
our actual powers of manipulation that at some moments we are inclined to
fall down with even more than reverence before her majestic presence. But

so strictly limited are her promises and powers, about so much that we
might wish to know does she offer no information whatever, that at other
moments we are fain to call her results but a vain thing, and to reject them
as a stone where we had asked for bread. If one aspect of the subject
encourages our hopes, so does the other tend to chasten our desires, and he
is perhaps the wisest, and in the long run the happiest, among his fellows,
who has learned not only this science, but also the larger lesson which it
directly teaches, namely, to temper our aspirations to that which is possible,
to moderate our desires to that which is attainable, to restrict our hopes to
that of which accomplishment, if not immediately practicable, is at least
distinctly within the range of conception.—Spottiswoode, W.
Quoted in Sonnenschein’s Encyclopedia of
Education (London, 1906), p. 208.

270. But it is precisely mathematics, and the pure science generally, from
which the general educated public and independent students have been
debarred, and into which they have only rarely attained more than a very
meagre insight. The reason of this is twofold. In the first place, the
ascendant and consecutive character of mathematical knowledge renders its
results absolutely insusceptible of presentation to persons who are
unacquainted with what has gone before, and so necessitates on the part of
its devotees a thorough and patient exploration of the field from the very
beginning, as distinguished from those sciences which may, so to speak, be
begun at the end, and which are consequently cultivated with the greatest
zeal. The second reason is that, partly through the exigencies of academic
instruction, but mainly through the martinet traditions of antiquity and the
influence of mediæval logic-mongers, the great bulk of the elementary text-
books of mathematics have unconsciously assumed a very repellant form,—
something similar to what is termed in the theory of protective mimicry in
biology “the terrifying form.” And it is mainly to this formidableness and
touch-me-not character of exterior, concealing withal a harmless body, that
the undue neglect of typical mathematical studies is to be attributed.—
McCormack, T. J.
Preface to De Morgan’s Elementary
Illustrations of the Differential and Integral
Calculus (Chicago, 1899).

271. Mathematics in gross, it is plain, are a grievance in natural philosophy,
and with reason: for mathematical proofs, like diamonds, are hard as well as
clear, and will be touched with nothing but strict reasoning. Mathematical
proofs are out of the reach of topical arguments; and are not to be attacked
by the equivocal use of words or declaration, that make so great a part of
other discourses,—nay, even of controversies.—Locke, John.
Second Reply to the Bishop of Worcester.

272. The belief that mathematics, because it is abstract, because it is static
and cold and gray, is detached from life, is a mistaken belief. Mathematics,
even in its purest and most abstract estate, is not detached from life. It is
just the ideal handling of the problems of life, as sculpture may idealize a
human figure or as poetry or painting may idealize a figure or a scene.
Mathematics is precisely the ideal handling of the problems of life, and the
central ideas of the science, the great concepts about which its stately
doctrines have been built up, are precisely the chief ideas with which life
must always deal and which, as it tumbles and rolls about them through
time and space, give it its interests and problems, and its order and
rationality. That such is the case a few indications will suffice to show. The
mathematical concepts of constant and variable are represented familiarly in
life by the notions of fixedness and change. The concept of equation or that
of an equational system, imposing restriction upon variability, is matched in
life by the concept of natural and spiritual law, giving order to what were
else chaotic change and providing partial freedom in lieu of none at all.
What is known in mathematics under the name of limit is everywhere
present in life in the guise of some ideal, some excellence high-dwelling
among the rocks, an “ever flying perfect” as Emerson calls it, unto which
we may approximate nearer and nearer, but which we can never quite attain,
save in aspiration. The supreme concept of functionality finds its correlate
in life in the all-pervasive sense of interdependence and mutual
determination among the elements of the world. What is known in
mathematics as transformation—that is, lawful transfer of attention, serving
to match in orderly fashion the things of one system with those of another—
is conceived in life as a process of transmutation by which, in the flux of
the world, the content of the present has come out of the past and in its turn,
in ceasing to be, gives birth to its successor, as the boy is father to the man

and as things, in general, become what they are not. The mathematical
concept of invariance and that of infinitude, especially the imposing
doctrines that explain their meanings and bear their names—What are they
but mathematicizations of that which has ever been the chief of life’s hopes
and dreams, of that which has ever been the object of its deepest passion
and of its dominant enterprise, I mean the finding of the worth that abides,
the finding of permanence in the midst of change, and the discovery of a
presence, in what has seemed to be a finite world, of being that is infinite?
It is needless further to multiply examples of a correlation that is so
abounding and complete as indeed to suggest a doubt whether it be juster to
view mathematics as the abstract idealization of life than to regard life as
the concrete realization of mathematics.—Keyser, C. J.
The Humanization of the Teaching of
Mathematics; Science, New Series, Vol. 35,
pp. 645-646.

273. Mathematics, like dialectics, is an organ of the inner higher sense; in
its execution it is an art like eloquence. Both alike care nothing for the
content, to both nothing is of value but the form. It is immaterial to
mathematics whether it computes pennies or guineas, to rhetoric whether it
defends truth or error.—Goethe.
Wilhelm Meisters Wanderjahre, Zweites
Buch.

274. The genuine spirit of Mathesis is devout. No intellectual pursuit more
truly leads to profound impressions of the existence and attributes of a
Creator, and to a deep sense of our filial relations to him, than the study of
these abstract sciences. Who can understand so well how feeble are our
conceptions of Almighty Power, as he who has calculated the attraction of
the sun and the planets, and weighed in his balance the irresistible force of
the lightning? Who can so well understand how confused is our estimate of
the Eternal Wisdom, as he who has traced out the secret laws which guide
the hosts of heaven, and combine the atoms on earth? Who can so well
understand that man is made in the image of his Creator, as he who has
sought to frame new laws and conditions to govern imaginary worlds, and

found his own thoughts similar to those on which his Creator has acted?—
Hill, Thomas.
The Imagination in Mathematics; North
American Review, Vol. 85, p. 226.

275. ... what is physical is subject to the laws of mathematics, and what is
spiritual to the laws of God, and the laws of mathematics are but the
expression of the thoughts of God.—Hill, Thomas.
The Uses of Mathesis; Bibliotheca Sacra,
Vol. 32, p. 523.

276. It is in the inner world of pure thought, where all entia dwell, where is
every type of order and manner of correlation and variety of relationship, it
is in this infinite ensemble of eternal verities whence, if there be one
cosmos or many of them, each derives its character and mode of being,—it
is there that the spirit of mathesis has its home and its life.
Is it a restricted home, a narrow life, static and cold and grey with
logic, without artistic interest, devoid of emotion and mood and sentiment?
That world, it is true, is not a world of solar light, not clad in the colours
that liven and glorify the things of sense, but it is an illuminated world, and
over it all and everywhere throughout are hues and tints transcending sense,
painted there by radiant pencils of psychic light, the light in which it lies. It
is a silent world, and, nevertheless, in respect to the highest principle of art
—the interpenetration of content and form, the perfect fusion of mode and
meaning—it even surpasses music. In a sense, it is a static world, but so,
too, are the worlds of the sculptor and the architect. The figures, however,
which reason constructs and the mathematic vision beholds, transcend the
temple and the statue, alike in simplicity and in intricacy, in delicacy and in
grace, in symmetry and in poise. Not only are this home and this life thus
rich in æsthetic interests, really controlled and sustained by motives of a
sublimed and supersensuous art, but the religious aspiration, too, finds
there, especially in the beautiful doctrine of invariants, the most perfect
symbols of what it seeks—the changeless in the midst of change, abiding
things in a world of flux, configurations that remain the same despite the
swirl and stress of countless hosts of curious transformations. The domain

of mathematics is the sole domain of certainty. There and there alone
prevail the standards by which every hypothesis respecting the external
universe and all observation and all experiment must be finally judged. It is
the realm to which all speculation and all thought must repair for chastening
and sanitation—the court of last resort, I say it reverently, for all intellection
whatsoever, whether of demon or man or deity. It is there that mind as mind
attains its highest estate, and the condition of knowledge there is the
ultimate object, the tantalising goal of the aspiration, the Anders-Streben, of
all other knowledge of every kind.—Keyser, C. J.
The Universe and Beyond; Hibbert Journal,
Vol. 3 (1904-1905), pp. 313-314.

CHAPTER III
ESTIMATES OF MATHEMATICS

301. The world of ideas which it [mathematics] discloses or illuminates, the
contemplation of divine beauty and order which it induces, the harmonious
connection of its parts, the infinite hierarchy and absolute evidence of the
truths with which mathematical science is concerned, these, and such like,
are the surest grounds of its title of human regard, and would remain
unimpaired were the plan of the universe unrolled like a map at our feet,
and the mind of man qualified to take in the whole scheme of creation at a
glance.—Sylvester, J. J.
A Plea for the Mathematician, Nature, 1, p.
262; Collected Mathematical Papers
(Cambridge, 1908), 2, p. 659.

302. It may well be doubted whether, in all the range of Science, there is
any field so fascinating to the explorer—so rich in hidden treasures—so
fruitful in delightful surprises—as that of Pure Mathematics. The charm lies
chiefly ... in the absolute certainty of its results: for that is what, beyond all
mental treasures, the human intellect craves for. Let us only be sure of
something! More light, more light! Ἐν δὲ φάει καὶ ὀλέεσσον “And if our
fate be death, give light and let us die!” This is the cry that, through all the
ages, is going up from perplexed Humanity, and Science has little else to
offer, that will really meet the demands of its votaries, than the conclusions
of Pure Mathematics.—Dodgson, C. L.
A New Theory of Parallels (London, 1895),
Introduction.

303. In every case the awakening touch has been the mathematical spirit,
the attempt to count, to measure, or to calculate. What to the poet or the seer

may appear to be the very death of all his poetry and all his visions—the
cold touch of the calculating mind,—this has proved to be the spell by
which knowledge has been born, by which new sciences have been created,
and hundreds of definite problems put before the minds and into the hands
of diligent students. It is the geometrical figure, the dry algebraical formula,
which transforms the vague reasoning of the philosopher into a tangible and
manageable conception; which represents, though it does not fully describe,
which corresponds to, though it does not explain, the things and processes
of nature: this clothes the fruitful, but otherwise indefinite, ideas in such a
form that the strict logical methods of thought can be applied, that the
human mind can in its inner chamber evolve a train of reasoning the result
of which corresponds to the phenomena of the outer world.—Merz, J. T.
A History of European Thought in the
Nineteenth Century (Edinburgh and London,
1904), Vol. 1, p. 314.

304. Mathematics ... the ideal and norm of all careful thinking.—Hall, G.
Stanley.
Educational Problems (New York, 1911), p.
393.

305. Mathematics is the only true metaphysics.—Thomson, W. (Lord
Kelvin).
Thompson, S. P.: Life of Lord Kelvin
(London, 1910), p. 10.

306. He who knows not mathematics and the results of recent scientific
investigation dies without knowing truth.—Schellbach, C. H.
Quoted in Young’s Teaching of Mathematics
(London, 1907), p. 44.

307. The reasoning of mathematics is a type of perfect reasoning.—
Barnett, P. A.

Common Sense in Education and Teaching
(New York, 1905), p. 222.

308. Mathematics, once fairly established on the foundation of a few
axioms and definitions, as upon a rock, has grown from age to age, so as to
become the most solid fabric that human reason can boast.—Reid, Thomas.
Essays on the Intellectual Powers of Man,
4th. Ed., p. 461.

309. The analytical geometry of Descartes and the calculus of Newton and
Leibniz have expanded into the marvelous mathematical method—more
daring than anything that the history of philosophy records—of
Lobachevsky and Riemann, Gauss and Sylvester. Indeed, mathematics, the
indispensable tool of the sciences, defying the senses to follow its splendid
flights, is demonstrating today, as it never has been demonstrated before,
the supremacy of the pure reason.—Butler, Nicholas Murray.
The Meaning of Education and other Essays
and Addresses (New York, 1905), p. 45.

310. Mathematics is the gate and key of the sciences.... Neglect of
mathematics works injury to all knowledge, since he who is ignorant of it
cannot know the other sciences or the things of this world. And what is
worse, men who are thus ignorant are unable to perceive their own
ignorance and so do not seek a remedy.—Bacon, Roger.
Opus Majus, Part 4, Distinctia Prima, cap. 1.

311. Just as it will never be successfully challenged that the French
language, progressively developing and growing more perfect day by day,
has the better claim to serve as a developed court and world language, so no
one will venture to estimate lightly the debt which the world owes to
mathematicians, in that they treat in their own language matters of the
utmost importance, and govern, determine and decide whatever is subject,
using the word in the highest sense, to number and measurement.—Goethe.
Sprüche in Prosa, Natur, III, 868.

312. Do not imagine that mathematics is hard and crabbed, and repulsive to
common sense. It is merely the etherealization of common sense.—
Thomson, W. (Lord Kelvin).
Thompson, S. P.: Life of Lord Kelvin
(London, 1910), p. 1139.

313. The advancement and perfection of mathematics are intimately
connected with the prosperity of the State.—Napoleon I.
Correspondance de Napoléon, t. 24 (1868),
p. 112.

314. The love of mathematics is daily on the increase, not only with us but
in the army. The result of this was unmistakably apparent in our last
campaigns. Bonaparte himself has a mathematical head, and though all who
study this science may not become geometricians like Laplace or Lagrange,
or heroes like Bonaparte, there is yet left an influence upon the mind which
enables them to accomplish more than they could possibly have achieved
without this training.—Lalande.
Quoted in Bruhns’ Alexander von Humboldt
(1872), Bd. 1, p. 232.

315. In Pure Mathematics, where all the various truths are necessarily
connected with each other, (being all necessarily connected with those
hypotheses which are the principles of the science), an arrangement is
beautiful in proportion as the principles are few; and what we admire
perhaps chiefly in the science, is the astonishing variety of consequences
which may be demonstrably deduced from so small a number of premises.
—Stewart, Dugald.
Philosophy of the Human Mind, Part 3, chap.
1, sect. 3; Collected Works [Hamilton]
(Edinburgh, 1854), Vol. 4.

316. It is curious to observe how differently these great men [Plato and
Bacon] estimated the value of every kind of knowledge. Take Arithmetic for

example. Plato, after speaking slightly of the convenience of being able to
reckon and compute in the ordinary transactions of life, passes to what he
considers as a far more important advantage. The study of the properties of
numbers, he tells us, habituates the mind to the contemplation of pure truth,
and raises us above the material universe. He would have his disciples
apply themselves to this study, not that they may be able to buy or sell, not
that they may qualify themselves to be shop-keepers or travelling
merchants, but that they may learn to withdraw their minds from the ever-
shifting spectacle of this visible and tangible world, and to fix them on the
immutable essences of things.
Bacon, on the other hand, valued this branch of knowledge only on
account of its uses with reference to that visible and tangible world which
Plato so much despised. He speaks with scorn of the mystical arithmetic of
the later Platonists, and laments the propensity of mankind to employ, on
mere matters of curiosity, powers the whole exertion of which is required
for purposes of solid advantage. He advises arithmeticians to leave these
trifles, and employ themselves in framing convenient expressions which
may be of use in physical researches.—Macaulay.
Lord Bacon: Edinburgh Review, July, 1837.
Critical and Miscellaneous Essays (New
York, 1879), Vol. 1, p. 397.

317. Ath. There still remain three studies suitable for freemen. Calculation
in arithmetic is one of them; the measurement of length, surface, and depth
is the second; and the third has to do with the revolutions of the stars in
reference to one another ... there is in them something that is necessary and
cannot be set aside, ... if I am not mistaken, [something of] divine necessity;
for as to the human necessities of which men often speak when they talk in
this manner, nothing can be more ridiculous than such an application of the
words.
Cle. And what necessities of knowledge are there, Stranger, which are
divine and not human?
Ath. I conceive them to be those of which he who has no use nor any
knowledge at all cannot be a god, or demi-god, or hero to mankind, or able
to take any serious thought or charge of them.—Plato.

Republic, Bk. 7. Jowett’s Dialogues of Plato
(New York, 1897), Vol. 4, p. 334.

318. Those who assert that the mathematical sciences make no affirmation
about what is fair or good make a false assertion; for they do speak of these
and frame demonstrations of them in the most eminent sense of the word.
For if they do not actually employ these names, they do not exhibit even the
results and the reasons of these, and therefore can be hardly said to make
any assertion about them. Of what is fair, however, the most important
species are order and symmetry, and that which is definite, which the
mathematical sciences make manifest in a most eminent degree. And since,
at least, these appear to be the causes of many things—now, I mean, for
example, order, and that which is a definite thing, it is evident that they
would assert, also, the existence of a cause of this description, and its
subsistence after the same manner as that which is fair subsists in.—
Aristotle.
Metaphysics [MacMahon] Bk. 12, chap. 3.

319. Many arts there are which beautify the mind of man; of all other none
do more garnish and beautify it than those arts which are called
mathematical.—Billingsley, H.
The Elements of Geometrie of the most
ancient Philosopher Euclide of Megara
(London, 1570), Note to the Reader.

320. As the sun eclipses the stars by his brilliancy, so the man of knowledge
will eclipse the fame of others in assemblies of the people if he proposes
algebraic problems, and still more if he solves them.—Brahmagupta.
Quoted in Cajori’s History of Mathematics
(New York, 1897), p. 92.

321. So highly did the ancients esteem the power of figures and numbers,
that Democritus ascribed to the figures of atoms the first principles of the

variety of things; and Pythagoras asserted that the nature of things consisted
of numbers.—Bacon, Lord.
De Augmentis, Bk. 3; Advancement of
Learning, Bk. 2.

322. There has not been any science so much esteemed and honored as this
of mathematics, nor with so much industry and vigilance become the care
of great men, and labored in by the potentates of the world, viz. emperors,
kings, princes, etc.—Franklin, Benjamin.
On the Usefulness of Mathematics, Works
(Boston, 1840), Vol. 2, p. 28.

323. Whatever may have been imputed to some other studies under the
notion of insignificancy and loss of time, yet these [mathematics], I believe,
never caused repentance in any, except it was for their remissness in the
prosecution of them.—Franklin, Benjamin.
On the Usefulness of Mathematics, Works
(Boston, 1840), Vol. 2, p. 69.

324. What science can there be more noble, more excellent, more useful for
men, more admirably high and demonstrative, than this of the mathematics?
—Franklin, Benjamin.
On the Usefulness of Mathematics, Works
(Boston, 1840), Vol. 2, p. 69.

325. The great truths with which it [mathematics] deals, are clothed with
austere grandeur, far above all purposes of immediate convenience or profit.
It is in them that our limited understandings approach nearest to the
conception of that absolute and infinite, towards which in most other things
they aspire in vain. In the pure mathematics we contemplate absolute truths,
which existed in the divine mind before the morning stars sang together,
and which will continue to exist there, when the last of their radiant host
shall have fallen from heaven. They existed not merely in metaphysical
possibility, but in the actual contemplation of the supreme reason. The pen

of inspiration, ranging all nature and life for imagery to set forth the
Creator’s power and wisdom, finds them best symbolized in the skill of the
surveyor. “He meted out heaven as with a span;” and an ancient sage,
neither falsely nor irreverently, ventured to say, that “God is a geometer.”—
Everett, Edward.
Orations and Speeches (Boston, 1870), Vol.
3, p. 514.

326. There is no science which teaches the harmonies of nature more
clearly than mathematics,....—Carus, Paul.
Andrews: Magic Squares and Cubes
(Chicago, 1908), Introduction.

327. For it being the nature of the mind of man (to the extreme prejudice of
knowledge) to delight in the spacious liberty of generalities, as in a
champion region, and not in the enclosures of particularity; the
Mathematics were the goodliest fields to satisfy that appetite.—Bacon,
Lord.
De Augmentis, Bk. 3; Advancement of
Learning, Bk. 2.

328. I would have my son mind and understand business, read little history,
study the mathematics and cosmography; these are good, with
subordination to the things of God.... These fit for public services for which
man is born.—Cromwell, Oliver.
Letters and Speeches of Oliver Cromwell
(New York, 1899), Vol. 1, p. 371.

329. Mathematics is the life supreme. The life of the gods is mathematics.
All divine messengers are mathematicians. Pure mathematics is religion. Its
attainment requires a theophany.—Novalis.
Schriften (Berlin, 1901), Bd. 2, p. 223.

330. The Mathematics which effectually exercises, not vainly deludes or
vexatiously torments studious Minds with obscure Subtilties, perplexed
Difficulties, or contentious Disquisitions; which overcomes without
Opposition, triumphs without Pomp, compels without Force, and rules
absolutely without Loss of Liberty; which does not privately overreach a
weak Faith, but openly assaults an armed Reason, obtains a total Victory,
and puts on inevitable Chains; whose Words are so many Oracles, and
Works as many Miracles; which blabs out nothing rashly, nor designs
anything from the Purpose, but plainly demonstrates and readily performs
all Things within its Verge; which obtrudes no false Shadow of Science, but
the very Science itself, the Mind firmly adheres to it, as soon as possessed
of it, and can never after desert it of its own Accord, or be deprived of it by
any Force of others: Lastly the Mathematics, which depend upon Principles
clear to the Mind, and agreeable to Experience; which draws certain
Conclusions, instructs by profitable Rules, unfolds pleasant Questions; and
produces wonderful Effects; which is the fruitful Parent of, I had almost
said all, Arts, the unshaken Foundation of Sciences, and the plentiful
Fountain of Advantage to human Affairs.—Barrow, Isaac.
Oration before the University of Cambridge
on being elected Lucasian Professor of
Mathematics, Mathematical Lectures
(London, 1734), p. 28.

331. Doubtless the reasoning faculty, the mind, is the leading and
characteristic attribute of the human race. By the exercise of this, man
arrives at the properties of the natural bodies. This is science, properly and
emphatically so called. It is the science of pure mathematics; and in the
high branches of this science lies the truly sublime of human acquisition. If
any attainment deserves that epithet, it is the knowledge, which, from the
mensuration of the minutest dust of the balance, proceeds on the rising scale
of material bodies, everywhere weighing, everywhere measuring,
everywhere detecting and explaining the laws of force and motion,
penetrating into the secret principles which hold the universe of God
together, and balancing worlds against worlds, and system against system.
When we seek to accompany those who pursue studies at once so high, so
vast, and so exact; when we arrive at the discoveries of Newton, which pour

in day on the works of God, as if a second fiat had gone forth from his own
mouth; when, further, we attempt to follow those who set out where
Newton paused, making his goal their starting-place, and, proceeding with
demonstration upon demonstration, and discovery upon discovery, bring
new worlds and new systems of worlds within the limits of the known
universe, failing to learn all only because all is infinite; however we may
say of man, in admiration of his physical structure, that “in form and
moving he is express and admirable,” it is here, and here without
irreverence, we may exclaim, “In apprehension how like a god!” The study
of the pure mathematics will of course not be extensively pursued in an
institution, which, like this [Boston Mechanics’ Institute], has a direct
practical tendency and aim. But it is still to be remembered, that pure
mathematics lie at the foundation of mechanical philosophy, and that it is
ignorance only which can speak or think of that sublime science as useless
research or barren speculation.—Webster, Daniel.
Works (Boston, 1872), Vol. 1, p. 180.

332. The school of Plato has advanced the interests of the race as much
through geometry as through philosophy. The modern engineer, the
navigator, the astronomer, built on the truths which those early Greeks
discovered in their purely speculative investigations. And if the poetry,
statesmanship, oratory, and philosophy of our day owe much to Plato’s
divine Dialogues, our commerce, our manufactures, and our science are
equally indebted to his Conic Sections. Later instances may be abundantly
quoted, to show that the labors of the mathematician have outlasted those of
the statesman, and wrought mightier changes in the condition of the world.
Not that we would rank the geometer above the patriot, but we claim that he
is worthy of equal honor.—Hill,Thomas.
Imagination in Mathematics; North
American Review, Vol. 85, p. 228.

333. The discoveries of Newton have done more for England and for the
race, than has been done by whole dynasties of British monarchs; and we
doubt not that in the great mathematical birth of 1853, the Quaternions of
Hamilton, there is as much real promise of benefit to mankind as in any
event of Victoria’s reign.—Hill, Thomas.

Imagination in Mathematics; North
American Review, Vol. 85, p. 228.

334. Geometrical and Mechanical phenomena are the most general, the
most simple, the most abstract of all,—the most irreducible to others. It
follows that the study of them is an indispensable preliminary to that of all
others. Therefore must Mathematics hold the first place in the hierarchy of
the sciences, and be the point of departure of all Education, whether general
or special.—Comte, A.
Positive Philosophy [Martineau],
Introduction, chap. 2.

CHAPTER IV
THE VALUE OF MATHEMATICS

401. Mathematics because of its nature and structure is peculiarly fitted for
high school instruction [Gymnasiallehrfach]. Especially the higher
mathematics, even if presented only in its elements, combines within itself
all those qualities which are demanded of a secondary subject. It engages, it
fructifies, it quickens, compels attention, is as circumspect as inventive,
induces courage and self-confidence as well as modesty and submission to
truth. It yields the essence and kernel of all things, is brief in form and
overflows with its wealth of content. It discloses the depth and breadth of
the law and spiritual element behind the surface of phenomena; it impels
from point to point and carries within itself the incentive toward progress; it
stimulates the artistic perception, good taste in judgment and execution, as
well as the scientific comprehension of things. Mathematics, therefore,
above all other subjects, makes the student lust after knowledge, fills him,
as it were, with a longing to fathom the cause of things and to employ his
own powers independently; it collects his mental forces and concentrates
them on a single point and thus awakens the spirit of individual inquiry,
self-confidence and the joy of doing; it fascinates because of the view-
points which it offers and creates certainty and assurance, owing to the
universal validity of its methods. Thus, both what he receives and what he
himself contributes toward the proper conception and solution of a problem,
combine to mature the student and to make him skillful, to lead him away
from the surface of things and to exercise him in the perception of their
essence. A student thus prepared thirsts after knowledge and is ready for the
university and its sciences. Thus it appears, that higher mathematics is the
best guide to philosophy and to the philosophic conception of the world
(considered as a self-contained whole) and of one’s own being.—Dillmann,
E.
Die Mathematik die Fackelträgerin einer
neuen Zeit (Stuttgart, 1889), p. 40.

402. These Disciplines [mathematics] serve to inure and corroborate the
Mind to a constant Diligence in Study; to undergo the Trouble of an
attentive Meditation, and cheerfully contend with such Difficulties as lie in
the Way. They wholly deliver us from a credulous Simplicity, most strongly
fortify us against the Vanity of Scepticism, effectually restrain from a rash
Presumption, most easily incline us to a due Assent, perfectly subject us to
the Government of right Reason, and inspire us with Resolution to wrestle
against the unjust Tyranny of false Prejudices. If the Fancy be unstable and
fluctuating, it is to be poised by this Ballast, and steadied by this Anchor, if
the Wit be blunt it is sharpened upon this Whetstone; if luxuriant it is pared
by this Knife; if headstrong it is restrained by this Bridle; and if dull it is
roused by this Spur. The Steps are guided by no Lamp more clearly through
the dark Mazes of Nature, by no Thread more surely through the intricate
Labyrinths of Philosophy, nor lastly is the Bottom of Truth sounded more
happily by any other Line. I will not mention how plentiful a Stock of
Knowledge the Mind is furnished from these, with what wholesome Food it
is nourished, and what sincere Pleasure it enjoys. But if I speak farther, I
shall neither be the only Person, nor the first, who affirms it; that while the
Mind is abstracted and elevated from sensible Matter, distinctly views pure
Forms, conceives the Beauty of Ideas, and investigates the Harmony of
Proportions; the Manners themselves are sensibly corrected and improved,
the Affections composed and rectified, the Fancy calmed and settled, and
the Understanding raised and excited to more divine Contemplation. All
which I might defend by Authority, and confirm by the Suffrages of the
greatest Philosophers.—Barrow, Isaac.
Prefatory Oration: Mathematical Lectures
(London, 1734), p. 31.

403. No school subject so readily furnishes tasks whose purpose can be
made so clear, so immediate and so appealing to the sober second-thought
of the immature learner as the right sort of elementary school mathematics.
—Myers, George.
Arithmetic in Public School Education
(Chicago, 1911), p. 8.

404. Mathematics is a type of thought which seems ingrained in the human
mind, which manifests itself to some extent with even the primitive races,
and which is developed to a high degree with the growth of civilization.... A
type of thought, a body of results, so essentially characteristic of the human
mind, so little influenced by environment, so uniformly present in every
civilization, is one of which no well-informed mind today can be ignorant.
—Young, J. W. A.
The Teaching of Mathematics (London,
1907), p. 14.

405. Probably among all the pursuits of the University, mathematics pre-
eminently demand self-denial, patience, and perseverance from youth,
precisely at that period when they have liberty to act for themselves, and
when on account of obvious temptations, habits of restraint and application
are peculiarly valuable.—Todhunter, Isaac.
The Conflict of Studies and other Essays
(London, 1873), p. 12.

406. Mathematics renders its best service through the immediate furthering
of rigorous thought and the spirit of invention.—Herbart J. F.
Mathematischer Lehrplan für Realschulen:
Werke [Kehrbach] (Langensalza, 1890), Bd.
5, p. 170.

407. It seems to me that the older subjects, classics and mathematics, are
strongly to be recommended on the ground of the accuracy with which we
can compare the relative performance of the students. In fact the
definiteness of these subjects is obvious, and is commonly admitted. There
is however another advantage, which I think belongs in general to these
subjects, that the examinations can be brought to bear on what is really
most valuable in these subjects.—Todhunter, Isaac.
Conflict of Studies and other Essays
(London, 1873), pp. 6, 7.

408. It is better to teach the child arithmetic and Latin grammar than
rhetoric and moral philosophy, because they require exactitude of
performance it is made certain that the lesson is mastered, and that power of
performance is worth more than knowledge.—Emerson, R. W.
Lecture on Education.

409. Besides accustoming the student to demand complete proof, and to
know when he has not obtained it, mathematical studies are of immense
benefit to his education by habituating him to precision. It is one of the
peculiar excellencies of mathematical discipline, that the mathematician is
never satisfied with à peu près. He requires the exact truth. Hardly any of
the non-mathematical sciences, except chemistry, has this advantage. One
of the commonest modes of loose thought, and sources of error both in
opinion and in practice, is to overlook the importance of quantities.
Mathematicians and chemists are taught by the whole course of their
studies, that the most fundamental difference of quality depends on some
very slight difference in proportional quantity; and that from the qualities of
the influencing elements, without careful attention to their quantities, false
expectation would constantly be formed as to the very nature and essential
character of the result produced.—Mill, J. S.
An Examination of Sir William Hamilton’s
Philosophy (London, 1878), p. 611.

410. In mathematics I can report no deficience, except it be that men do not
sufficiently understand the excellent use of the Pure Mathematics, in that
they do remedy and cure many defects in the wit and faculties intellectual.
For if the wit be too dull, they sharpen it; if too wandering, they fix it; if too
inherent in the senses, they abstract it. So that as tennis is a game of no use
in itself, but of great use in respect it maketh a quick eye and a body ready
to put itself into all positions; so in the Mathematics, that use which is
collateral and intervenient is no less worthy than that which is principal and
intended.—Bacon, Lord.
De Augmentis, Bk. 3; Advancement of
Learning, Bk. 2.

411. If a man’s wit be wandering, let him study mathematics; for in
demonstrations, if his wit be called away never so little, he must begin
again.—Bacon, Lord.
Essays: On Studies.

412. If one be bird-witted, that is easily distracted and unable to keep his
attention as long as he should, mathematics provides a remedy; for in them
if the mind be caught away but a moment, the demonstration has to be
commenced anew.—Bacon, Lord.
De Augmentis, Bk. 6; Advancement of
Learning, Bk. 2.

413. The metaphysical philosopher from his point of view recognizes
mathematics as an instrument of education, which strengthens the power of
attention, develops the sense of order and the faculty of construction, and
enables the mind to grasp under the simple formulae the quantitative
differences of physical phenomena.—Jowett, B.
Dialogues of Plato (New York, 1897), Vol. 2,
p. 78.

414. Nor do I know any study which can compete with mathematics in
general in furnishing matter for severe and continued thought. Metaphysical
problems may be even more difficult; but then they are far less definite,
and, as they rarely lead to any precise conclusion, we miss the power of
checking our own operations, and of discovering whether we are thinking
and reasoning or merely fancying and dreaming.—Todhunter, Isaac.
Conflict of Studies (London, 1873), p. 13.

415. Another great and special excellence of mathematics is that it demands
earnest voluntary exertion. It is simply impossible for a person to become a
good mathematician by the happy accident of having been sent to a good
school; this may give him a preparation and a start, but by his own
individual efforts alone can he reach an eminent position.—Todhunter,
Isaac.

Conflict of Studies (London, 1873), p. 2.

416. The faculty of resolution is possibly much invigorated by
mathematical study, and especially by that highest branch of it which,
unjustly, merely on account of its retrograde operations, has been called, as
if par excellence, analysis.—Poe, E. A.
The Murders in Rue Morgue.

417. He who gives a portion of his time and talent to the investigation of
mathematical truth will come to all other questions with a decided
advantage over his opponents. He will be in argument what the ancient
Romans were in the field: to them the day of battle was a day of
comparative recreation, because they were ever accustomed to exercise
with arms much heavier than they fought; and reviews differed from a real
battle in two respects: they encountered more fatigue, but the victory was
bloodless.—Colton, C. C.
Lacon (New York, 1866).

418. Mathematics is the study which forms the foundation of the course
[West Point Military Academy]. This is necessary, both to impart to the
mind that combined strength and versatility, the peculiar vigor and rapidity
of comparison necessary for military action, and to pave the way for
progress in the higher military sciences.
Congressional Committee on Military
Affairs, 1834; U. S. Bureau of Education,
Bulletin 1912, No. 2, p. 10.

419. Mathematics, among all school subjects, is especially adapted to
further clearness, definite brevity and precision in expression, although it
offers no exercise in flights of rhetoric. This is due in the first place to the
logical rigour with which it develops thought, avoiding every departure
from the shortest, most direct way, never allowing empty phrases to enter.
Other subjects excel in the development of expression in other respects:
translation from foreign languages into the mother tongue gives exercise in

finding the proper word for the given foreign word and gives knowledge of
laws of syntax, the study of poetry and prose furnish fit patterns for
connected presentation and elegant form of expression, composition is to
exercise the pupil in a like presentation of his own or borrowed thoughts
and their development, the natural sciences teach description of natural
objects, apparatus and processes, as well as the statement of laws on the
grounds of immediate sense-perception. But all these aids for exercise in
the use of the mother tongue, each in its way valuable and indispensable, do
not guarantee, in the same manner as mathematical training, the exclusion
of words whose concepts, if not entirely wanting, are not sufficiently clear.
They do not furnish in the same measure that which the mathematician
demands particularly as regards precision of expression.—Reidt, F.
Anleitung zum mathematischen Unterricht in
höheren Schulen (Berlin, 1906), p. 17.

420. One rarely hears of the mathematical recitation as a preparation for
public speaking. Yet mathematics shares with these studies [foreign
languages, drawing and natural science] their advantages, and has another
in a higher degree than either of them.
Most readers will agree that a prime requisite for healthful experience
in public speaking is that the attention of the speaker and hearers alike be
drawn wholly away from the speaker and concentrated upon the thought. In
perhaps no other classroom is this so easy as in the mathematical, where the
close reasoning, the rigorous demonstration, the tracing of necessary
conclusions from given hypotheses, commands and secures the entire
mental power of the student who is explaining, and of his classmates. In
what other circumstances do students feel so instinctively that manner
counts for so little and mind for so much? In what other circumstances,
therefore, is a simple, unaffected, easy, graceful manner so naturally and so
healthfully cultivated? Mannerisms that are mere affectation or the result of
bad literary habit recede to the background and finally disappear, while
those peculiarities that are the expression of personality and are inseparable
from its activity continually develop, where the student frequently presents,
to an audience of his intellectual peers, a connected train of reasoning....

One would almost wish that our institutions of the science and art of
public speaking would put over their doors the motto that Plato had over the
entrance to his school of philosophy: “Let no one who is unacquainted with
geometry enter here.”—White, W. F.
A Scrap-book of Elementary Mathematics
(Chicago, 1908), p. 210.

421. The training which mathematics gives in working with symbols is an
excellent preparation for other sciences; ... the world’s work requires
constant mastery of symbols.—Young, J. W. A.
The Teaching of Mathematics (New York,
1907), p. 42.

422. One striking peculiarity of mathematics is its unlimited power of
evolving examples and problems. A student may read a book of Euclid, or a
few chapters of Algebra, and within that limited range of knowledge it is
possible to set him exercises as real and as interesting as the propositions
themselves which he has studied; deductions which might have pleased the
Greek geometers, and algebraic propositions which Pascal and Fermat
would not have disdained to investigate.—Todhunter, Isaac.
Private Study of Mathematics: Conflict of
Studies and other Essays (London, 1873), p.
82.

423. Would you have a man reason well, you must use him to it betimes;
exercise his mind in observing the connection between ideas, and following
them in train. Nothing does this better than mathematics, which therefore, I
think should be taught to all who have the time and opportunity, not so
much to make them mathematicians, as to make them reasonable creatures;
for though we all call ourselves so, because we are born to it if we please,
yet we may truly say that nature gives us but the seeds of it, and we are
carried no farther than industry and application have carried us.—Locke,
John.
Conduct of the Understanding, Sect. 6.

424. Secondly, the study of mathematics would show them the necessity
there is in reasoning, to separate all the distinct ideas, and to see the
habitudes that all those concerned in the present inquiry have to one
another, and to lay by those which relate not to the proposition in hand, and
wholly to leave them out of the reckoning. This is that which, in other
respects besides quantity is absolutely requisite to just reasoning, though in
them it is not so easily observed and so carefully practised. In those parts of
knowledge where it is thought demonstration has nothing to do, men reason
as it were in a lump; and if upon a summary and confused view, or upon a
partial consideration, they can raise the appearance of a probability, they
usually rest content; especially if it be in a dispute where every little straw
is laid hold on, and everything that can but be drawn in any way to give
color to the argument is advanced with ostentation. But that mind is not in a
posture to find truth that does not distinctly take all the parts asunder, and,
omitting what is not at all to the point, draws a conclusion from the result of
all the particulars which in any way influence it.—Locke, John.
Conduct of the Understanding, Sect. 7.

425. I have before mentioned mathematics, wherein algebra gives new
helps and views to the understanding. If I propose these it is not to make
every man a thorough mathematician or deep algebraist; but yet I think the
study of them is of infinite use even to grown men; first by experimentally
convincing them, that to make anyone reason well, it is not enough to have
parts wherewith he is satisfied, and that serve him well enough in his
ordinary course. A man in those studies will see, that however good he may
think his understanding, yet in many things, and those very visible, it may
fail him. This would take off that presumption that most men have of
themselves in this part; and they would not be so apt to think their minds
wanted no helps to enlarge them, that there could be nothing added to the
acuteness and penetration of their understanding.—Locke, John.
The Conduct of the Understanding, Sect. 7.

426. I have mentioned mathematics as a way to settle in the mind a habit of
reasoning closely and in train; not that I think it necessary that all men
should be deep mathematicians, but that, having got the way of reasoning
which that study necessarily brings the mind to, they might be able to

transfer it to other parts of knowledge, as they shall have occasion. For in
all sorts of reasoning, every single argument should be managed as a
mathematical demonstration; the connection and dependence of ideas
should be followed till the mind is brought to the source on which it
bottoms, and observes the coherence all along;....—Locke, John.
The Conduct of the Understanding, Sect. 7.

427. As an exercise of the reasoning faculty, pure mathematics is an
admirable exercise, because it consists of reasoning alone, and does not
encumber the student with an exercise of judgment: and it is well to begin
with learning one thing at a time, and to defer a combination of mental
exercises to a later period.—Whately, R.
Annotations to Bacon’s Essays (Boston,
1873), Essay 1, p. 493.

428. It hath been an old remark, that Geometry is an excellent Logic. And it
must be owned that when the definitions are clear; when the postulata
cannot be refused, nor the axioms denied; when from the distinct
contemplation and comparison of figures, their properties are derived, by a
perpetual well-connected chain of consequences, the objects being still kept
in view, and the attention ever fixed upon them; there is acquired a habit of
reasoning, close and exact and methodical; which habit strengthens and
sharpens the mind, and being transferred to other subjects is of general use
in the inquiry after truth.—Berkely, George.
The Analyst, 2; Works (London, 1898), Vol. 3,
p. 10.

429. Suppose then I want to give myself a little training in the art of
reasoning; suppose I want to get out of the region of conjecture and
probability, free myself from the difficult task of weighing evidence, and
putting instances together to arrive at general propositions, and simply
desire to know how to deal with my general propositions when I get them,
and how to deduce right inferences from them; it is clear that I shall obtain
this sort of discipline best in those departments of thought in which the first
principles are unquestionably true. For in all our thinking, if we come to

erroneous conclusions, we come to them either by accepting false premises
to start with—in which case our reasoning, however good, will not save us
from error; or by reasoning badly, in which case the data we start from may
be perfectly sound, and yet our conclusions may be false. But in the
mathematical or pure sciences,—geometry, arithmetic, algebra,
trigonometry, the calculus of variations or of curves,—we know at least that
there is not, and cannot be, error in our first principles, and we may
therefore fasten our whole attention upon the processes. As mere exercises
in logic, therefore, these sciences, based as they all are on primary truths
relating to space and number, have always been supposed to furnish the
most exact discipline. When Plato wrote over the portal of his school. “Let
no one ignorant of geometry enter here,” he did not mean that questions
relating to lines and surfaces would be discussed by his disciples. On the
contrary, the topics to which he directed their attention were some of the
deepest problems,—social, political, moral,—on which the mind could
exercise itself. Plato and his followers tried to think out together
conclusions respecting the being, the duty, and the destiny of man, and the
relation in which he stood to the gods and to the unseen world. What had
geometry to do with these things? Simply this: That a man whose mind has
not undergone a rigorous training in systematic thinking, and in the art of
drawing legitimate inferences from premises, was unfitted to enter on the
discussion of these high topics; and that the sort of logical discipline which
he needed was most likely to be obtained from geometry—the only
mathematical science which in Plato’s time had been formulated and
reduced to a system. And we in this country [England] have long acted on
the same principle. Our future lawyers, clergy, and statesmen are expected
at the University to learn a good deal about curves, and angles, and numbers
and proportions; not because these subjects have the smallest relation to the
needs of their lives, but because in the very act of learning them they are
likely to acquire that habit of steadfast and accurate thinking, which is
indispensable to success in all the pursuits of life.—Fitch, J. C.
Lectures on Teaching (New York, 1906), pp.
291-292.

430. It is admitted by all that a finished or even a competent reasoner is not
the work of nature alone; the experience of every day makes it evident that

education develops faculties which would otherwise never have manifested
their existence. It is, therefore, as necessary to learn to reason before we
can expect to be able to reason, as it is to learn to swim or fence, in order to
attain either of those arts. Now, something must be reasoned upon, it
matters not much what it is, provided it can be reasoned upon with
certainty. The properties of mind or matter, or the study of languages,
mathematics, or natural history, may be chosen for this purpose. Now of all
these, it is desirable to choose the one which admits of the reasoning being
verified, that is, in which we can find out by other means, such as
measurement and ocular demonstration of all sorts, whether the results are
true or not. When the guiding property of the loadstone was first
ascertained, and it was necessary to learn how to use this new discovery,
and to find out how far it might be relied on, it would have been thought
advisable to make many passages between ports that were well known
before attempting a voyage of discovery. So it is with our reasoning
faculties: it is desirable that their powers should be exerted upon objects of
such a nature, that we can tell by other means whether the results which we
obtain are true or false, and this before it is safe to trust entirely to reason.
Now the mathematics are peculiarly well adapted for this purpose, on the
following grounds:
1. Every term is distinctly explained, and has but one meaning, and it
is rarely that two words are employed to mean the same thing.
2. The first principles are self-evident, and, though derived from
observation, do not require more of it than has been made by children in
general.
3. The demonstration is strictly logical, taking nothing for granted
except self-evident first principles, resting nothing upon probability, and
entirely independent of authority and opinion.
4. When the conclusion is obtained by reasoning, its truth or falsehood
can be ascertained, in geometry by actual measurement, in algebra by
common arithmetical calculation. This gives confidence, and is absolutely
necessary, if, as was said before, reason is not to be the instructor, but the
pupil.
5. There are no words whose meanings are so much alike that the ideas
which they stand for may be confounded. Between the meaning of terms

there is no distinction, except a total distinction, and all adjectives and
adverbs expressing difference of degrees are avoided.—De Morgan,
Augustus.
On the Study and Difficulties of Mathematics
(Chicago, 1898), chap. 1.

431. The instruction of children should aim gradually to combine knowing
and doing [Wissen und Können]. Among all sciences mathematics seems to
be the only one of a kind to satisfy this aim most completely.—Kant,
Immanuel.
Werke [Rosenkranz und Schubert], Bd. 9
(Leipzig, 1838), p. 409.

432. Every discipline must be honored for reason other than its utility,
otherwise it yields no enthusiasm for industry.
For both reasons, I consider mathematics the chief subject for the
common school. No more highly honored exercise for the mind can be
found; the buoyancy [Spannkraft] which it produces is even greater than
that produced by the ancient languages, while its utility is unquestioned.—
Herbart, J. F.
Mathematischer Lehrplan für
Realgymnasien, Werke [Kehrbach],
(Langensalza, 1890), Bd. 5, p. 167.

433. The motive for the study of mathematics is insight into the nature of
the universe. Stars and strata, heat and electricity, the laws and processes of
becoming and being, incorporate mathematical truths. If language imitates
the voice of the Creator, revealing His heart, mathematics discloses His
intellect, repeating the story of how things came into being. And the value
of mathematics, appealing as it does to our energy and to our honor, to our
desire to know the truth and thereby to live as of right in the household of
God, is that it establishes us in larger and larger certainties. As literature
develops emotion, understanding, and sympathy, so mathematics develops
observation, imagination, and reason.—Chancellor, W. E.

A Theory of Motives, Ideals and Values in
Education (Boston and New York, 1907), p.
406.

434. Mathematics in its pure form, as arithmetic, algebra, geometry, and the
applications of the analytic method, as well as mathematics applied to
matter and force, or statics and dynamics, furnishes the peculiar study that
gives to us, whether as children or as men, the command of nature in this its
quantitative aspect; mathematics furnishes the instrument, the tool of
thought, which we wield in this realm.—Harris, W. T.
Psychologic Foundations of Education (New
York, 1898), p. 325.

435. Little can be understood of even the simplest phenomena of nature
without some knowledge of mathematics, and the attempt to penetrate
deeper into the mysteries of nature compels simultaneous development of
the mathematical processes.—Young, J. W. A.
The Teaching of Mathematics (New York,
1907), p. 16.

436. For many parts of nature can neither be invented with sufficient
subtility nor demonstrated with sufficient perspicuity nor accommodated
unto use with sufficient dexterity, without the aid and intervening of
mathematics.—Bacon, Lord.
De Augmentis, Bk. 2; Advancement of
Learning, Bk. 3.

437. I confess, that after I began ... to discern how useful mathematicks may
be made to physicks, I have often wished that I had employed about the
speculative part of geometry, and the cultivation of the specious Algebra I
had been taught very young, a good part of that time and industry, that I had
spent about surveying and fortification (of which I remember I once wrote
an entire treatise) and other parts of practick mathematicks.—Boyle,
Robert.

The Usefulness of Mathematicks to Natural
Philosophy; Works (London, 1772), Vol. 3, p.
426.

438. Mathematics gives the young man a clear idea of demonstration and
habituates him to form long trains of thought and reasoning methodically
connected and sustained by the final certainty of the result; and it has the
further advantage, from a purely moral point of view, of inspiring an
absolute and fanatical respect for truth. In addition to all this, mathematics,
and chiefly algebra and infinitesimal calculus, excite to a high degree the
conception of the signs and symbols—necessary instruments to extend the
power and reach of the human mind by summarizing an aggregate of
relations in a condensed form and in a kind of mechanical way. These
auxiliaries are of special value in mathematics because they are there
adequate to their definitions, a characteristic which they do not possess to
the same degree in the physical and mathematical [natural?] sciences.
There are, in fact, a mass of mental and moral faculties that can be put
in full play only by instruction in mathematics; and they would be made
still more available if the teaching was directed so as to leave free play to
the personal work of the student.—Berthelot, M. P. E. M.
Science as an Instrument of Education;
Popular Science Monthly (1897), p. 253.

439. Mathematical knowledge, therefore, appears to us of value not only in
so far as it serves as means to other ends, but for its own sake as well, and
we behold, both in its systematic external and internal development, the
most complete and purest logical mind-activity, the embodiment of the
highest intellect-esthetics.—Pringsheim, Alfred.
Ueber Wert und angeblichen Unwert der
Mathematik; Jahresbericht der Deutschen
Mathematiker Vereinigung, Bd. 13, p. 381.

440. The advantages which mathematics derives from the peculiar nature of
those relations about which it is conversant, from its simple and definite

phraseology, and from the severe logic so admirably displayed in the
concatenation of its innumerable theorems, are indeed immense, and well
entitled to separate and ample illustration.—Stewart, Dugald.
Philosophy of the Human Mind, Part 2, chap.
2, sect. 3.

441. I do not intend to go deeply into the question how far mathematical
studies, as the representatives of conscious logical reasoning, should take a
more important place in school education. But it is, in reality, one of the
questions of the day. In proportion as the range of science extends, its
system and organization must be improved, and it must inevitably come
about that individual students will find themselves compelled to go through
a stricter course of training than grammar is in a position to supply. What
strikes me in my own experience with students who pass from our classical
schools to scientific and medical studies, is first, a certain laxity in the
application of strictly universal laws. The grammatical rules, in which they
have been exercised, are for the most part followed by long lists of
exceptions; accordingly they are not in the habit of relying implicitly on the
certainty of a legitimate deduction from a strictly universal law. Secondly, I
find them for the most part too much inclined to trust to authority, even in
cases where they might form an independent judgment. In fact, in
philological studies, inasmuch as it is seldom possible to take in the whole
of the premises at a glance, and inasmuch as the decision of disputed
questions often depends on an æsthetic feeling for beauty of expression, or
for the genius of the language, attainable only by long training, it must often
happen that the student is referred to authorities even by the best teachers.
Both faults are traceable to certain indolence and vagueness of thought, the
sad effects of which are not confined to subsequent scientific studies. But
certainly the best remedy for both is to be found in mathematics, where
there is absolute certainty in the reasoning, and no authority is recognized
but that of one’s own intelligence.—Helmholtz, H.
On the Relation of Natural Science to
Science in general; Popular Lectures on
Scientific Subjects; Atkinson (New York,
1900), pp. 25-26.

442. What renders a problem definite, and what leaves it indefinite, may
best be understood from mathematics. The very important idea of solving a
problem within limits of error is an element of rational culture, coming
from the same source. The art of totalizing fluctuations by curves is capable
of being carried, in conception, far beyond the mathematical domain, where
it is first learned. The distinction between laws and coefficients applies in
every department of causation. The theory of Probable Evidence is the
mathematical contribution to Logic, and is of paramount importance.—
Bain, Alexander.
Education as a Science (New York, 1898), pp.
151-152.

443. We receive it as a fact, that some minds are so constituted as absolutely
to require for their nurture the severe logic of the abstract sciences; that
rigorous sequence of ideas which leads from the premises to the conclusion,
by a path, arduous and narrow, it may be, and which the youthful reason
may find it hard to mount, but where it cannot stray; and on which, if it
move at all, it must move onward and upward.... Even for intellects of a
different character, whose natural aptitude is for moral evidence and those
relations of ideas which are perceived and appreciated by taste, the study of
the exact sciences may be recommended as the best protection against the
errors into which they are most likely to fall. Although the study of
language is in many respects no mean exercise in logic, yet it must be
admitted that an eminently practical mind is hardly to be formed without
mathematical training.—Everett, Edward.
Orations and Speeches (Boston, 1870), Vol.
2, p. 510.

444. The value of mathematical instruction as a preparation for those more
difficult investigations, consists in the applicability not of its doctrines but
of its methods. Mathematics will ever remain the past perfect type of the
deductive method in general; and the applications of mathematics to the
simpler branches of physics furnish the only school in which philosophers
can effectually learn the most difficult and important of their art, the
employment of the laws of simpler phenomena for explaining and
predicting those of the more complex. These grounds are quite sufficient for

deeming mathematical training an indispensable basis of real scientific
education, and regarding with Plato, one who is ἀγεωμέτρητος, as wanting
in one of the most essential qualifications for the successful cultivation of
the higher branches of philosophy.—Mill, J. S.
System of Logic, Bk. 3, chap. 24, sect. 9.

445. This science, Geometry, is one of indispensable use and constant
reference, for every student of the laws of nature; for the relations of space
and number are the alphabet in which those laws are written. But besides
the interest and importance of this kind which geometry possesses, it has a
great and peculiar value for all who wish to understand the foundations of
human knowledge, and the methods by which it is acquired. For the student
of geometry acquires, with a degree of insight and clearness which the
unmathematical reader can but feebly imagine, a conviction that there are
necessary truths, many of them of a very complex and striking character;
and that a few of the most simple and self-evident truths which it is possible
for the mind of man to apprehend, may, by systematic deduction, lead to the
most remote and unexpected results.—Whewell, William.
The Philosophy of the Inductive Sciences,
Part 1, Bk. 2, chap. 4, sect. 8 (London, 1858).

446. Mathematics, while giving no quick remuneration, like the art of
stenography or the craft of bricklaying, does furnish the power for
deliberate thought and accurate statement, and to speak the truth is one of
the most social qualities a person can possess. Gossip, flattery, slander,
deceit, all spring from a slovenly mind that has not been trained in the
power of truthful statement, which is one of the highest utilities.—Dutton,
S. T.
Social Phases of Education in the School and
the Home (London, 1900), p. 30.

447. It is from this absolute indifference and tranquility of the mind, that
mathematical speculations derive some of their most considerable
advantages; because there is nothing to interest the imagination; because the
judgment sits free and unbiased to examine the point. All proportions, every

arrangement of quantity, is alike to the understanding, because the same
truths result to it from all; from greater from lesser, from equality and
inequality.—Burke, Edmund.
On the Sublime and Beautiful, Part 3, sect. 2.

448. Out of the interaction of form and content in mathematics grows an
acquaintance with methods which enable the student to produce
independently within certain though moderate limits, and to extend his
knowledge through his own reflection. The deepening of the consciousness
of the intellectual powers connected with this kind of activity, and the
gradual awakening of the feeling of intellectual self-reliance may well be
considered as the most beautiful and highest result of mathematical training.
—Pringsheim, Alfred.
Ueber Wert und angeblichen Unwert der
Mathematik; Jahresbericht der Deutschen
Mathematiker Vereinigung (1904), p. 374.

449. He who would know what geometry is, must venture boldly into its
depths and learn to think and feel as a geometer. I believe that it is
impossible to do this, and to study geometry as it admits of being studied
and am conscious it can be taught, without finding the reason invigorated,
the invention quickened, the sentiment of the orderly and beautiful
awakened and enhanced, and reverence for truth, the foundation of all
integrity of character, converted into a fixed principle of the mental and
moral constitution, according to the old and expressive adage “abeunt
studia in mores.”—Sylvester, J. J.
A probationary Lecture on Geometry;
Collected Mathematical Papers (Cambridge,
1908), Vol. 2, p. 9.

450. Mathematical knowledge adds vigour to the mind, frees it from
prejudice, credulity, and superstition.—Arbuthnot, John.
Usefulness of Mathematical Learning.

451. When the boy begins to understand that the visible point is preceded
by an invisible point, that the shortest distance between two points is
conceived as a straight line before it is ever drawn with the pencil on paper,
he experiences a feeling of pride, of satisfaction. And justly so, for the
fountain of all thought has been opened to him, the difference between the
ideal and the real, potentia et actu, has become clear to him; henceforth the
philosopher can reveal him nothing new, as a geometrician he has
discovered the basis of all thought.—Goethe.
Sprüche in Prosa, Ethisches, VI, 455.

452. In mathematics, ... and in natural philosophy since mathematics was
applied to it, we see the noblest instance of the force of the human mind,
and of the sublime heights to which it may rise by cultivation. An
acquaintance with such sciences naturally leads us to think well of our
faculties, and to indulge sanguine expectations concerning the improvement
of other parts of knowledge. To this I may add, that, as mathematical and
physical truths are perfectly uninteresting in their consequences, the
understanding readily yields its assent to the evidence which is presented to
it; and in this way may be expected to acquire the habit of trusting to its
own conclusions, which will contribute to fortify it against the weaknesses
of scepticism, in the more interesting inquiries after moral truth in which it
may afterwards engage.—Stewart, Dugald.
Philosophy of the Human Mind, Part 3, chap.
1, sect. 3.

453. Those that can readily master the difficulties of Mathematics find a
considerable charm in the study, sometimes amounting to fascination. This
is far from universal; but the subject contains elements of strong interest of
a kind that constitutes the pleasures of knowledge. The marvellous devices
for solving problems elate the mind with the feeling of intellectual power;
and the innumerable constructions of the science leave us lost in wonder.—
Bain, Alexander.
Education as a Science (New York, 1898), p.
153.

454. Thinking is merely the comparing of ideas, discerning relations of
likeness and of difference between ideas, and drawing inferences. It is
seizing general truths on the basis of clearly apprehended particulars. It is
but generalizing and particularizing. Who will deny that a child can deal
profitably with sequences of ideas like: How many marbles are 2 marbles
and 3 marbles? 2 pencils and 3 pencils? 2 balls and 3 balls? 2 children and 3
children? 2 inches and 3 inches? 2 feet and 3 feet? 2 and 3? Who has not
seen the countenance of some little learner light up at the end of such a
series of questions with the exclamation, “Why it’s always that way. Isn’t
it?” This is the glow of pleasure that the generalizing step always affords
him who takes the step himself. This is the genuine life-giving joy which
comes from feeling that one can successfully take this step. The reality of
such a discovery is as great, and the lasting effect upon the mind of him that
makes it is as sure as was that by which the great Newton hit upon the
generalization of the law of gravitation. It is through these thrills of
discovery that love to learn and intellectual pleasure are begotten and
fostered. Good arithmetic teaching abounds in such opportunities.—Myers,
George.
Arithmetic in Public Education (Chicago), p.
13.

455. A general course in mathematics should be required of all officers for
its practical value, but no less for its educational value in training the mind
to logical forms of thought, in developing the sense of absolute truthfulness,
together with a confidence in the accomplishment of definite results by
definite means.—Echols, C. P.
Mathematics at West Point and Annapolis; U.
S. Bureau of Education, Bulletin 1912, No. 2,
p. 11.

456. Exercise in the most rigorous thinking that is possible will of its own
accord strengthen the sense of truth and right, for each advance in the
ability to distinguish between correct and false thoughts, each habit making
for rigour in thought development will increase in the sound pupil the
ability and the wish to ascertain what is right in life and to defend it.—
Reidt, F.

Anleitung zum mathematischen Unterricht in
den höheren Schulen (Berlin, 1906), p. 28.

457. I do not maintain that the chief value of the study of arithmetic consists
in the lessons of morality that arise from this study. I claim only that, to be
impressed from day to day, that there is something that is right as an answer
to the questions with which one is able to grapple, and that there is a wrong
answer—that there are ways in which the right answer can be established as
right, that these ways automatically reject error and slovenliness, and that
the learner is able himself to manipulate these ways and to arrive at the
establishment of the true as opposed to the untrue, this relentless hewing to
the line and stopping at the line, must color distinctly the thought life of the
pupil with more than a tinge of morality.... To be neighborly with truth, to
feel one’s self somewhat facile in ways of recognizing and establishing
what is right, what is correct, to find the wrong persistently and unfailingly
rejected as of no value, to feel that one can apply these ways for himself,
that one can think and work independently, have a real, a positive, and a
purifying effect upon moral character. They are the quiet, steady undertones
of the work that always appeal to the learner for the sanction of his best
judgment, and these are the really significant matters in school work. It is
not the noise and bluster, not even the dramatics or the polemics from the
teacher’s desk, that abide longest and leave the deepest and stablest imprint
upon character. It is these still, small voices that speak unmistakably for the
right and against the wrong and the erroneous that really form human
character. When the school subjects are arranged on the basis of the degree
to which they contribute to the moral upbuilding of human character good
arithmetic will be well up the list.—Myers, George.
Arithmetic in Public Education (Chicago), p.
18.

458. In destroying the predisposition to anger, science of all kind is useful;
but the mathematics possess this property in the most eminent degree.—Dr.
Rush.
Quoted in Day’s Collacon (London, no date).

459. The mathematics are the friends to religion, inasmuch as they charm
the passions, restrain the impetuosity of the imagination, and purge the
mind from error and prejudice. Vice is error, confusion and false reasoning;
and all truth is more or less opposite to it. Besides, mathematical truth may
serve for a pleasant entertainment for those hours which young men are apt
to throw away upon their vices; the delightfulness of them being such as to
make solitude not only easy but desirable.—Arbuthnot, John.
Usefulness of Mathematical Learning.

460. There is no prophet which preaches the superpersonal God more
plainly than mathematics.—Carus, Paul.
Reflections on Magic Squares; Monist
(1906), p. 147.

461. Mathematics must subdue the flights of our reason; they are the staff of
the blind; no one can take a step without them; and to them and experience
is due all that is certain in physics.—Voltaire.
Oeuvres Complètes (Paris, 1880), t. 35, p.
219.

CHAPTER V
THE TEACHING OF MATHEMATICS

501. In mathematics two ends are constantly kept in view: First, stimulation
of the inventive faculty, exercise of judgment, development of logical
reasoning, and the habit of concise statement; second, the association of the
branches of pure mathematics with each other and with applied science, that
the pupil may see clearly the true relations of principles and things.
International Commission on the Teaching of
Mathematics, American Report; U. S. Bureau
of Education, Bulletin 1912, No. 4, p. 7.

502. The ends to be attained [in the teaching of mathematics in the
secondary schools] are the knowledge of a body of geometrical truths, the
power to draw correct inferences from given premises, the power to use
algebraic processes as a means of finding results in practical problems, and
the awakening of interest in the science of mathematics.
International Commission on the Teaching of
Mathematics, American Report; U. S. Bureau
of Education, Bulletin 1912, No. 4, p. 7.

503. General preparatory instruction must continue to be the aim in the
instruction at the higher institutions of learning. Exclusive selection and
treatment of subject matter with reference to specific avocations is
disadvantageous.
Resolution adopted by the German
Association for the Advancement of Scientific
and Mathematical Instruction; Jahresbericht
der Deutschen Mathematiker Vereinigung
(1896), p. 41.

504. In the secondary schools mathematics should be a part of general
culture and not contributory to technical training of any kind; it should
cultivate space intuition, logical thinking, the power to rephrase in clear
language thoughts recognized as correct, and ethical and esthetic effects; so
treated, mathematics is a quite indispensable factor of general education in
so far as the latter shows its traces in the comprehension of the development
of civilization and the ability to participate in the further tasks of
civilization.
Unterrichtsblätter für Mathematik und
Naturwissenschaft (1904), p. 128.

505. Indeed, the aim of teaching [mathematics] should be rather to
strengthen his [the pupil’s] faculties, and to supply a method of reasoning
applicable to other subjects, than to furnish him with an instrument for
solving practical problems.—Magnus, Philip.
Perry’s Teaching of Mathematics (London,
1902), p. 84.

506. The participation in the general development of the mental powers
without special reference to his future vocation must be recognized as the
essential aim of mathematical instruction.—Reidt, F.
Anleitung zum Mathematischen Unterricht
an höheren Schulen (Berlin, 1906), p. 12.

507. I am of the decided opinion, that mathematical instruction must have
for its first aim a deep penetration and complete command of abstract
mathematical theory together with a clear insight into the structure of the
system, and doubt not that the instruction which accomplishes this is
valuable and interesting even if it neglects practical applications. If the
instruction sharpens the understanding, if it arouses the scientific interest,
whether mathematical or philosophical, if finally it calls into life an esthetic
feeling for the beauty of a scientific edifice, the instruction will take on an
ethical value as well, provided that with the interest it awakens also the
impulse toward scientific activity. I contend, therefore, that even without

reference to its applications mathematics in the high schools has a value
equal to that of the other subjects of instruction.—Goetting, E.
Ueber das Lehrziel im mathematischen
Unterricht der höheren Realanstalten;
Jahresbericht der Deutschen Mathematiker
Vereinigung, Bd. 2, p. 192.

508. Mathematics will not be properly esteemed in wider circles until more
than the a b c of it is taught in the schools, and until the unfortunate
impression is gotten rid of that mathematics serves no other purpose in
instruction than the formal training of the mind. The aim of mathematics is
its content, its form is a secondary consideration and need not necessarily
be that historic form which is due to the circumstance that mathematics
took permanent shape under the influence of Greek logic.—Hankel, H.
Die Entwickelung der Mathematik in den
letzten Jahrhunderten (Tübingen, 1884), p. 6.

509. The idea that aptitude for mathematics is rarer than aptitude for other
subjects is merely an illusion which is caused by belated or neglected
beginners.—Herbart, J. F.
Umriss pädagogischer Vorlesungen; Werke
[Kehrbach] (Langensalza, 1902), Bd. 10, p.
101.

510. I believe that the useful methods of mathematics are easily to be
learned by quite young persons, just as languages are easily learned in
youth. What a wondrous philosophy and history underlie the use of almost
every word in every language—yet the child learns to use the word
unconsciously. No doubt when such a word was first invented it was studied
over and lectured upon, just as one might lecture now upon the idea of a
rate, or the use of Cartesian co-ordinates, and we may depend upon it that
children of the future will use the idea of the calculus, and use squared
paper as readily as they now cipher.... When Egyptian and Chaldean
philosophers spent years in difficult calculations, which would now be
thought easy by young children, doubtless they had the same notions of the

depth of their knowledge that Sir William Thomson might now have of his.
How is it, then, that Thomson gained his immense knowledge in the time
taken by a Chaldean philosopher to acquire a simple knowledge of
arithmetic? The reason is plain. Thomson, when a child, was taught in a few
years more than all that was known three thousand years ago of the
properties of numbers. When it is found essential to a boy’s future that
machinery should be given to his brain, it is given to him; he is taught to
use it, and his bright memory makes the use of it a second nature to him;
but it is not till after-life that he makes a close investigation of what there
actually is in his brain which has enabled him to do so much. It is taken
because the child has much faith. In after years he will accept nothing
without careful consideration. The machinery given to the brain of children
is getting more and more complicated as time goes on; but there is really no
reason why it should not be taken in as early, and used as readily, as were
the axioms of childish education in ancient Chaldea.—Perry, John.
The Teaching of Mathematics (London,
1902), p. 14.

511. The ancients devoted a lifetime to the study of arithmetic; it required
days to extract a square root or to multiply two numbers together. Is there
any harm in skipping all that, in letting the school boy learn multiplication
sums, and in starting his more abstract reasoning at a more advanced point?
Where would be the harm in letting the boy assume the truth of many
propositions of the first four books of Euclid, letting him assume their truth
partly by faith, partly by trial? Giving him the whole fifth book of Euclid by
simple algebra? Letting him assume the sixth as axiomatic? Letting him, in
fact, begin his severer studies where he is now in the habit of leaving off?
We do much less orthodox things. Every here and there in one’s
mathematical studies one makes exceedingly large assumptions, because
the methodical study would be ridiculous even in the eyes of the most
pedantic of teachers. I can imagine a whole year devoted to the
philosophical study of many things that a student now takes in his stride
without trouble. The present method of training the mind of a mathematical
teacher causes it to strain at gnats and to swallow camels. Such gnats are
most of the propositions of the sixth book of Euclid; propositions generally

about incommensurables; the use of arithmetic in geometry; the
parallelogram of forces, etc., decimals.—Perry, John.
The Teaching of Mathematics (London,
1904), p. 12.

512. The teaching of elementary mathematics should be conducted so that
the way should be prepared for the building upon them of the higher
mathematics. The teacher should always bear in mind and look forward to
what is to come after. The pupil should not be taught what may be sufficient
for the time, but will lead to difficulties in the future.... I think the fault in
teaching arithmetic is that of not attending to general principles and
teaching instead of particular rules.... I am inclined to attack the teaching of
mathematics on the grounds that it does not dwell sufficiently on a few
general axiomatic principles.—Hudson, W. H. H.
Perry’s Teaching of Mathematics (London,
1904), p. 33.

513. “Mathematics in Prussia! Ah, sir, they teach mathematics in Prussia as
you teach your boys rowing in England: they are trained by men who have
been trained by men who have themselves been trained for generations
back.”—Langley, E. M.
Perry’s Teaching of Mathematics (London,
1904), p. 43.

514. A superficial knowledge of mathematics may lead to the belief that this
subject can be taught incidentally, and that exercises akin to counting the
petals of flowers or the legs of a grasshopper are mathematical. Such work
ignores the fundamental idea out of which quantitative reasoning grows—
the equality of magnitudes. It leaves the pupil unaware of that relativity
which is the essence of mathematical science. Numerical statements are
frequently required in the study of natural history, but to repeat these as a
drill upon numbers will scarcely lend charm to these studies, and certainly
will not result in mathematical knowledge.—Speer, W. W.
Primary Arithmetic (Boston, 1897), pp. 26-
27.

515. Mathematics is no more the art of reckoning and computation than
architecture is the art of making bricks or hewing wood, no more than
painting is the art of mixing colors on a palette, no more than the science of
geology is the art of breaking rocks, or the science of anatomy the art of
butchering.—Keyser, C. J.
Lectures on Science, Philosophy and Art
(New York, 1908), p. 29.

516. The study of mathematics—from ordinary reckoning up to the higher
processes—must be connected with knowledge of nature, and at the same
time with experience, that it may enter the pupil’s circle of thought.—
Herbart, J. F.
Letters and Lectures on Education [Felkin]
(London, 1908), p. 117.

517. First, as concerns the success of teaching mathematics. No instruction
in the high schools is as difficult as that of mathematics, since the large
majority of students are at first decidedly disinclined to be harnessed into
the rigid framework of logical conclusions. The interest of young people is
won much more easily, if sense-objects are made the starting point and the
transition to abstract formulation is brought about gradually. For this reason
it is psychologically quite correct to follow this course.
Not less to be recommended is this course if we inquire into the
essential purpose of mathematical instruction. Formerly it was too
exclusively held that this purpose is to sharpen the understanding. Surely
another important end is to implant in the student the conviction that correct
thinking based on true premises secures mastery over the outer world. To
accomplish this the outer world must receive its share of attention from the
very beginning.
Doubtless this is true but there is a danger which needs pointing out. It
is as in the case of language teaching where the modern tendency is to
secure in addition to grammar also an understanding of the authors. The
danger lies in grammar being completely set aside leaving the subject
without its indispensable solid basis. Just so in the teaching of mathematics
it is possible to accumulate interesting applications to such an extent as to

stunt the essential logical development. This should in no wise be
permitted, for thus the kernel of the whole matter is lost. Therefore: We do
want throughout a quickening of mathematical instruction by the
introduction of applications, but we do not want that the pendulum, which
in former decades may have inclined too much toward the abstract side,
should now swing to the other extreme; we would rather pursue the proper
middle course.—Klein, Felix.
Ueber den Mathematischen Unterricht an
den höheren Schulen; Jahresbericht der
Deutschen Mathematiker Vereinigung, Bd.
11, p. 131.

518. It is above all the duty of the methodical text-book to adapt itself to the
pupil’s power of comprehension, only challenging his higher efforts with
the increasing development of his imagination, his logical power and the
ability of abstraction. This indeed constitutes a test of the art of teaching, it
is here where pedagogic tact becomes manifest. In reference to the axioms,
caution is necessary. It should be pointed out comparatively early, in how
far the mathematical body differs from the material body. Furthermore,
since mathematical bodies are really portions of space, this space is to be
conceived as mathematical space and to be clearly distinguished from real
or physical space. Gradually the student will become conscious that the
portion of the real space which lies beyond the visible stellar universe is not
cognizable through the senses, that we know nothing of its properties and
consequently have no basis for judgments concerning it. Mathematical
space, on the other hand, may be subjected to conditions, for instance, we
may condition its properties at infinity, and these conditions constitute the
axioms, say the Euclidean axioms. But every student will require years
before the conviction of the truth of this last statement will force itself upon
him.—Holzmüller, Gustav.
Methodisches Lehrbuch der Elementar-
Mathematik (Leipzig, 1904), Teil 1, Vorwort,
pp. 4-5.

519. Like almost every subject of human interest, this one [mathematics] is
just as easy or as difficult as we choose to make it. A lifetime may be spent

by a philosopher in discussing the truth of the simplest axiom. The simplest
fact as to our existence may fill us with such wonder that our minds will
remain overwhelmed with wonder all the time. A Scotch ploughman makes
a working religion out of a system which appalls a mental philosopher.
Some boys of ten years of age study the methods of the differential
calculus; other much cleverer boys working at mathematics to the age of
nineteen have a difficulty in comprehending the fundamental ideas of the
calculus.—Perry, John.
The Teaching of Mathematics (London,
1902), pp. 19-20.

520. Poor teaching leads to the inevitable idea that the subject
[mathematics] is only adapted to peculiar minds, when it is the one
universal science and the one whose four ground-rules are taught us almost
in infancy and reappear in the motions of the universe.—Safford, T. H.
Mathematical Teaching (Boston, 1907), p.
19.

521. The number of mathematical students ... would be much augmented if
those who hold the highest rank in science would condescend to give more
effective assistance in clearing the elements of the difficulties which they
present.—De Morgan, A.
Study and Difficulties of Mathematics
(Chicago, 1902), Preface.

522. He that could teach mathematics well, would not be a bad teacher in
any of the rest [physics, chemistry, biology, psychology] unless by the
accident of total inaptitude for experimental illustration; while the mere
experimentalist is likely to fall into the error of missing the essential
condition of science as reasoned truth; not to speak of the danger of making
the instruction an affair of sensation, glitter, or pyrotechnic show.—Bain,
Alexander.
Education as a Science (New York, 1898), p.
298.

523. I should like to draw attention to the inexhaustible variety of the
problems and exercises which it [mathematics] furnishes; these may be
graduated to precisely the amount of attainment which may be possessed,
while yet retaining an interest and value. It seems to me that no other
branch of study at all compares with mathematics in this. When we propose
a deduction to a beginner we give him an exercise in many cases that would
have been admired in the vigorous days of Greek geometry. Although
grammatical exercises are well suited to insure the great benefits connected
with the study of languages, yet these exercises seem to me stiff and
artificial in comparison with the problems of mathematics. It is not absurd
to maintain that Euclid and Apollonius would have regarded with interest
many of the elegant deductions which are invented for the use of our
students in geometry; but it seems scarcely conceivable that the great
masters in any other line of study could condescend to give a moment’s
attention to the elementary books of the beginner.—Todhunter, Isaac.
Conflict of Studies (London, 1873), pp. 10-
11.

524. The visible figures by which principles are illustrated should, so far as
possible, have no accessories. They should be magnitudes pure and simple,
so that the thought of the pupil may not be distracted, and that he may know
what features of the thing represented he is to pay attention to.
Report of the Committee of Ten on Secondary
School Subjects, (New York, 1894), p. 109.

525. Geometrical reasoning, and arithmetical process, have each its own
office: to mix the two in elementary instruction, is injurious to the proper
acquisition of both.—De Morgan, A.
Trigonometry and Double Algebra (London,
1849), p. 92.

526. Equations are Expressions of Arithmetical Computation, and properly
have no place in Geometry, except as far as Quantities truly Geometrical
(that is, Lines, Surfaces, Solids, and Proportions) may be said to be some
equal to others. Multiplications, Divisions, and such sort of Computations,

are newly received into Geometry, and that unwarily, and contrary to the
first Design of this Science. For whosoever considers the Construction of a
Problem by a right Line and a Circle, found out by the first Geometricians,
will easily perceive that Geometry was invented that we might
expeditiously avoid, by drawing Lines, the Tediousness of Computation.
Therefore these two Sciences ought not to be confounded. The Ancients did
so industriously distinguish them from one another, that they never
introduced Arithmetical Terms into Geometry. And the Moderns, by
confounding both, have lost the Simplicity in which all the Elegance of
Geometry consists. Wherefore that is Arithmetically more simple which is
determined by the more simple Equation, but that is Geometrically more
simple which is determined by the more simple drawing of Lines; and in
Geometry, that ought to be reckoned best which is geometrically most
simple.—Newton.
On the Linear Construction of Equations;
Universal Arithmetic (London, 1769), Vol. 2,
p. 470.

527. As long as algebra and geometry proceeded along separate paths, their
advance was slow and their applications limited.
But when these sciences joined company, they drew from each other
fresh vitality and thenceforward marched on at a rapid pace toward
perfection.—Lagrange.
Leçons Élémentaires sur les Mathematiques,
Leçon cinquiéme. [McCormack].

528. The greatest enemy to true arithmetic work is found in so-called
practical or illustrative problems, which are freely given to our pupils, of a
degree of difficulty and complexity altogether unsuited to their age and
mental development.... I am, myself, no bad mathematician, and all the
reasoning powers with which nature endowed me have long been as fully
developed as they are ever likely to be; but I have, not infrequently, been
puzzled, and at times foiled, by the subtle logical difficulty running through
one of these problems, given to my own children. The head-master of one
of our Boston high schools confessed to me that he had sometimes been

unable to unravel one of these tangled skeins, in trying to help his own
daughter through her evening’s work. During this summer, Dr. Fairbairn,
the distinguished head of one of the colleges of Oxford, England, told me
that not only had he himself encountered a similar difficulty, in the case of
his own children, but that, on one occasion, having as his guest one of the
first mathematicians of England, the two together had been completely
puzzled by one of these arithmetical conundrums.—Walker, F. A.
Discussions in Education (New York, 1899),
pp. 253-254.

529. It is often assumed that because the young child is not competent to
study geometry systematically he need be taught nothing geometrical; that
because it would be foolish to present to him physics and mechanics as
sciences it is useless to present to him any physical or mechanical
principles.
An error of like origin, which has wrought incalculable mischief,
denies to the scholar the use of the symbols and methods of algebra in
connection with his early essays in numbers because, forsooth, he is not as
yet capable of mastering quadratics!... The whole infant generation,
wrestling with arithmetic, seek for a sign and groan and travail together in
pain for the want of it; but no sign is given them save the sign of the
prophet Jonah, the withered gourd, fruitless endeavor, wasted strength.—
Walker, F. A.
Industrial Education; Discussions in
Education (New York, 1899), p. 132.

530. Particular and contingent inventions in the solution of problems,
which, though many times more concise than a general method would
allow, yet, in my judgment, are less proper to instruct a learner, as acrostics,
and such kind of artificial poetry, though never so excellent, would be but
improper examples to instruct one that aims at Ovidean poetry.—Newton,
Isaac.
Letter to Collins, 1670; Macclesfield,
Correspondence of Scientific Men (Oxford,
1841), Vol. 2, p. 307.

531. The logic of the subject [algebra], which, both educationally and
scientifically speaking, is the most important part of it, is wholly neglected.
The whole training consists in example grinding. What should have been
merely the help to attain the end has become the end itself. The result is that
algebra, as we teach it, is neither an art nor a science, but an ill-digested
farrago of rules, whose object is the solution of examination problems....
The result, so far as problems worked in examinations go, is, after all, very
miserable, as the reiterated complaints of examiners show; the effect on the
examinee is a well-known enervation of mind, an almost incurable
superficiality, which might be called Problematic Paralysis—a disease
which unfits a man to follow an argument extending beyond the length of a
printed octavo page.—Chrystal, George.
Presidential Address British Association for
the Advancement of Science, 1885; Nature,
Vol. 32, pp. 447-448.

532. It is a serious question whether America, following England’s lead, has
not gone into problem-solving too extensively. Certain it is that we are
producing no text-books in which the theory is presented in the delightful
style which characterizes many of the French works ..., or those of the
recent Italian school, or, indeed, those of the continental writers in general.
—Smith, D. E.
The Teaching of Elementary Mathematics
(New York, 1902), p. 219.

533. The problem for a writer of a text-book has come now, in fact, to be
this—to write a book so neatly trimmed and compacted that no coach, on
looking through it, can mark a single passage which the candidate for a
minimum pass can safely omit. Some of these text-books I have seen,
where the scientific matter has been, like the lady’s waist in the nursery
song, compressed “so gent and sma’,” that the thickness barely, if at all,
surpasses what is devoted to the publisher’s advertisements. We shall
return, I verily believe, to the Compendium of Martianus Capella. The
result of all this is that science, in the hands of specialists, soars higher and
higher into the light of day, while educators and the educated are left more
and more to wander in primeval darkness.—Chrystal, George.

Presidential Address British Association for
the Advancement of Science, 1885; Nature,
Vol. 32, p. 448.

534. Some persons have contended that mathematics ought to be taught by
making the illustrations obvious to the senses. Nothing can be more absurd
or injurious: it ought to be our never-ceasing effort to make people think,
not feel.—Coleridge, S. T.
Lectures on Shakespere (Bohn Library), p.
52.

535. I have come to the conclusion that the exertion, without which a
knowledge of mathematics cannot be acquired, is not materially increased
by logical rigor in the method of instruction.—Pringsheim, Alfred.
Jahresbericht der Deutschen Mathematiker
Vereinigung (1898), p. 143.

536. The only way in which to treat the elements of an exact and rigorous
science is to apply to them all the rigor and exactness possible.—
D’Alembert.
Quoted by De Morgan: Trigonometry and
Double Algebra (London, 1849), Title page.

537. It is an error to believe that rigor in proof is an enemy of simplicity. On
the contrary we find it confirmed by numerous examples that the rigorous
method is at the same time the simpler and the more easily comprehended.
The very effort for rigor forces us to find out simpler methods of proof.—
Hilbert, D.
Mathematical Problems; Bulletin American
Mathematical Society, Vol. 8, p. 441.

538. Few will deny that even in the first scientific instruction in
mathematics the most rigorous method is to be given preference over all

others. Especially will every teacher prefer a consistent proof to one which
is based on fallacies or proceeds in a vicious circle, indeed it will be
morally impossible for the teacher to present a proof of the latter kind
consciously and thus in a sense deceive his pupils. Notwithstanding these
objectionable so-called proofs, so far as the foundation and the development
of the system is concerned, predominate in our textbooks to the present
time. Perhaps it will be answered, that rigorous proof is found too difficult
for the pupil’s power of comprehension. Should this be anywhere the case,
—which would only indicate some defect in the plan or treatment of the
whole,—the only remedy would be to merely state the theorem in a historic
way, and forego a proof with the frank confession that no proof has been
found which could be comprehended by the pupil; a remedy which is ever
doubtful and should only be applied in the case of extreme necessity. But
this remedy is to be preferred to a proof which is no proof, and is therefore
either wholly unintelligible to the pupil, or deceives him with an appearance
of knowledge which opens the door to all superficiality and lack of
scientific method.—Grassmann, Hermann.
Stücke aus dem Lehrbuche der Arithmetik;
Werke, Bd. 2 (Leipsig, 1904), p. 296.

539. The average English author [of mathematical texts] leaves one under
the impression that he has made a bargain with his reader to put before him
the truth, the greater part of the truth, and nothing but the truth; and that if
he has put the facts of his subject into his book, however difficult it may be
to unearth them, he has fulfilled his contract with his reader. This is a very
much mistaken view, because effective teaching requires a great deal more
than a bare recitation of facts, even if these are duly set forth in logical
order—as in English books they often are not. The probable difficulties
which will occur to the student, the objections which the intelligent student
will naturally and necessarily raise to some statement of fact or theory—
these things our authors seldom or never notice, and yet a recognition and
anticipation of them by the author would be often of priceless value to the
student. Again, a touch of humour (strange as the contention may seem) in
mathematical works is not only possible with perfect propriety, but very
helpful; and I could give instances of this even from the pure mathematics
of Salmon and the physics of Clerk Maxwell.—Minchin, G. M.

Perry’s Teaching of Mathematics (London,
1902), pp. 59-61.

540. Remember this, the rule for giving an extempore lecture is—let the
mind rest from the subject entirely for an interval preceding the lecture,
after the notes are prepared; the thoughts will ferment without your
knowing it, and enter into new combinations; but if you keep the mind
active upon the subject up to the moment, the subject will not ferment but
stupefy.—De Morgan, A.
Letter to Hamilton; Graves: Life of W. R.
Hamilton (New York, 1882-1889), Vol. 3, p.
487.

CHAPTER VI
STUDY AND RESEARCH IN MATHEMATICS

601. The first thing to be attended to in reading any algebraic treatise is the
gaining a perfect understanding of the different processes there exhibited,
and of their connection with one another. This cannot be attained by the
mere reading of the book, however great the attention which may be given.
It is impossible in a mathematical work to fill up every process in the
manner in which it must be filled up in the mind of the student before he
can be said to have completely mastered it. Many results must be given of
which the details are suppressed, such are the additions, multiplications,
extractions of square roots, etc., with which the investigations abound.
These must not be taken on trust by the student, but must be worked out by
his own pen, which must never be out of his own hand while engaged in
any mathematical process.—De Morgan, A.
Study and Difficulties of Mathematics
(Chicago, 1902), chap. 12.

602. The student should not lose any opportunity of exercising himself in
numerical calculation and particularly in the use of logarithmic tables. His
power of applying mathematics to questions of practical utility is in direct
proportion to the facility which he possesses in computation.—De
Morgan,A.
Study and Difficulties of Mathematics
(Chicago, 1902), chap. 12.

603. The examples which a beginner should choose for practice should be
simple and should not contain very large numbers. The powers of the mind
cannot be directed to two things at once; if the complexity of the numbers

used requires all the student’s attention, he cannot observe the principle of
the rule which he is following.—De Morgan, A.
Study and Difficulties of Mathematics
(Chicago, 1902), chap. 3.

604. Euclid and Archimedes are allowed to be knowing, and to have
demonstrated what they say: and yet whosoever shall read over their
writings without perceiving the connection of their proofs, and seeing what
they show, though he may understand all their words, yet he is not the more
knowing. He may believe, indeed, but does not know what they say, and so
is not advanced one jot in mathematical knowledge by all his reading of
those approved mathematicians.—Locke, John.
Conduct of the Understanding, sect. 24.

605. The student should read his author with the most sustained attention, in
order to discover the meaning of every sentence. If the book is well written,
it will endure and repay his close attention: the text ought to be fairly
intelligible, even without illustrative examples. Often, far too often, a reader
hurries over the text without any sincere and vigorous effort to understand
it; and rushes to some example to clear up what ought not to have been
obscure, if it had been adequately considered. The habit of scrupulously
investigating the text seems to me important on several grounds. The close
scrutiny of language is a very valuable exercise both for studious and
practical life. In the higher departments of mathematics the habit is
indispensable: in the long investigations which occur there it would be
impossible to interpose illustrative examples at every stage, the student
must therefore encounter and master, sentence by sentence, an extensive
and complicated argument.—Todhunter, Isaac.
Private Study of Mathematics; Conflict of
Studies and other Essays (London, 1873), p.
67.

606. It must happen that in some cases the author is not understood, or is
very imperfectly understood; and the question is what is to be done. After
giving a reasonable amount of attention to the passage, let the student pass

on, reserving the obscurity for future efforts.... The natural tendency of
solitary students, I believe, is not to hurry away prematurely from a hard
passage, but to hang far too long over it; the just pride that does not like to
acknowledge defeat, and the strong will that cannot endure to be thwarted,
both urge to a continuance of effort even when success seems hopeless. It is
only by experience we gain the conviction that when the mind is thoroughly
fatigued it has neither the power to continue with advantage its course in an
assigned direction, nor elasticity to strike out a new path; but that, on the
other hand, after being withdrawn for a time from the pursuit, it may return
and gain the desired end.—Todhunter, Isaac.
Private Study of Mathematics; Conflict of
Studies and other Essays (London, 1873), p.
68.

607. Every mathematical book that is worth reading must be read
“backwards and forwards,” if I may use the expression. I would modify
Lagrange’s advice a little and say, “Go on, but often return to strengthen
your faith.” When you come on a hard or dreary passage, pass it over; and
come back to it after you have seen its importance or found the need for it
further on.—Chrystal, George.
Algebra, Part 2 (Edinburgh, 1889), Preface,
p. 8.

608. The large collection of problems which our modern Cambridge books
supply will be found to be almost an exclusive peculiarity of these books;
such collections scarcely exist in foreign treatises on mathematics, nor even
in English treatises of an earlier date. This fact shows, I think, that a
knowledge of mathematics may be gained without the perpetual working of
examples.... Do not trouble yourselves with the examples, make it your
main business, I might almost say your exclusive business, to understand
the text of your author.—Todhunter, Isaac.
Private Study of Mathematics; Conflict of
Studies and other Essays (London, 1873), p.
74.

609. In my opinion the English excel in the art of writing text-books for
mathematical teaching; as regards the clear exposition of theories and the
abundance of excellent examples, carefully selected, very few books exist
in other countries which can compete with those of Salmon and many other
distinguished English authors that could be named.—Cremona, L.
Projective Geometry [Leudesdorf] (Oxford,
1885), Preface.

610. The solution of fallacies, which give rise to absurdities, should be to
him who is not a first beginner in mathematics an excellent means of testing
for a proper intelligible insight into mathematical truth, of sharpening the
wit, and of confining the judgment and reason within strictly orderly limits.
—Viola, J.
Mathematische Sophismen (Wien, 1864),
Vorwort.

611. Success in the solution of a problem generally depends in a great
measure on the selection of the most appropriate method of approaching it;
many properties of conic sections (for instance) being demonstrable by a
few steps of pure geometry which would involve the most laborious
operations with trilinear co-ordinates, while other properties are almost self-
evident under the method of trilinear co-ordinates, which it would perhaps
be actually impossible to prove by the old geometry.—Whitworth, W. A.
Modern Analytic Geometry (Cambridge,
1866), p. 154.

612. The deep study of nature is the most fruitful source of mathematical
discoveries. By offering to research a definite end, this study has the
advantage of excluding vague questions and useless calculations; besides it
is a sure means of forming analysis itself and of discovering the elements
which it most concerns us to know, and which natural science ought always
to conserve.—Fourier, J.
Théorie Analytique de la Chaleur, Discours
Préliminaire.

613. It is certainly true that all physical phenomena are subject to strictly
mathematical conditions, and mathematical processes are unassailable in
themselves. The trouble arises from the data employed. Most phenomena
are so highly complex that one can never be quite sure that he is dealing
with all the factors until the experiment proves it. So that experiment is
rather the criterion of mathematical conclusions and must lead the way.—
Dolbear, A. E.
Matter, Ether, Motion (Boston, 1894), p. 89.

614. Students should learn to study at an early stage the great works of the
great masters instead of making their minds sterile through the everlasting
exercises of college, which are of no use whatever, except to produce a new
Arcadia where indolence is veiled under the form of useless activity.... Hard
study on the great models has ever brought out the strong; and of such must
be our new scientific generation if it is to be worthy of the era to which it is
born and of the struggles to which it is destined.—Beltrami.
Giornale di matematiche, Vol. 11, p. 153.
[Young, J. W.]

615. The history of mathematics may be instructive as well as agreeable; it
may not only remind us of what we have, but may also teach us to increase
our store. Says De Morgan, “The early history of the mind of men with
regards to mathematics leads us to point out our own errors; and in this
respect it is well to pay attention to the history of mathematics.” It warns us
against hasty conclusions; it points out the importance of a good notation
upon the progress of the science; it discourages excessive specialization on
the part of the investigator, by showing how apparently distinct branches
have been found to possess unexpected connecting links; it saves the
student from wasting time and energy upon problems which were, perhaps,
solved long since; it discourages him from attacking an unsolved problem
by the same method which has led other mathematicians to failure; it
teaches that fortifications can be taken by other ways than by direct attack,
that when repulsed from a direct assault it is well to reconnoitre and occupy
the surrounding ground and to discover the secret paths by which the
apparently unconquerable position can be taken.—Cajori, F.

History of Mathematics (New York, 1897),
pp. 1-2.

616. The history of mathematics is important also as a valuable contribution
to the history of civilization. Human progress is closely identified with
scientific thought. Mathematical and physical researches are a reliable
record of intellectual progress.—Cajori, F.
History of Mathematics (New York, 1897), p.
4.

617. It would be rash to say that nothing remains for discovery or
improvement even in elementary mathematics, but it may be safely asserted
that the ground has been so long and so thoroughly explored as to hold out
little hope of profitable return for a casual adventurer.—Todhunter, Isaac.
Private Study of Mathematics; Conflict of
Studies and other Essays (London, 1873), p.
73.

618. We do not live in a time when knowledge can be extended along a
pathway smooth and free from obstacles, as at the time of the discovery of
the infinitesimal calculus, and in a measure also when in the development
of projective geometry obstacles were suddenly removed which, having
hemmed progress for a long time, permitted a stream of investigators to
pour in upon virgin soil. There is no longer any browsing along the beaten
paths; and into the primeval forest only those may venture who are
equipped with the sharpest tools.—Burkhardt, H.
Mathematisches und wissenschaftliches
Denken; Jahresbericht der Deutschen
Mathematiker Vereinigung, Bd. 11, p. 55.

619. Though we must not without further consideration condemn a body of
reasoning merely because it is easy, nevertheless we must not allow
ourselves to be lured on merely by easiness; and we should take care that
every problem which we choose for attack, whether it be easy or difficult,

shall have a useful purpose, that it shall contribute in some measure to the
up-building of the great edifice.—Segre, Corradi.
Some Recent Tendencies in Geometric
Investigation; Rivista di Matematica (1891),
p. 63. Bulletin American Mathematical
Society, 1904, p. 465. [Young, J. W.].

620. No mathematician now-a-days sets any store on the discovery of
isolated theorems, except as affording hints of an unsuspected new sphere
of thought, like meteorites detached from some undiscovered planetary orb
of speculation.—Sylvester, J. J.
Notes to the Exeter Association Address;
Collected Mathematical Papers (Cambridge,
1908), Vol. 2, p. 715.

621. Isolated, so-called “pretty theorems” have even less value in the eyes
of a modern mathematician than the discovery of a new “pretty flower” has
to the scientific botanist, though the layman finds in these the chief charm
of the respective sciences.—Hankel, Hermann.
Die Entwickelung der Mathematik in den
letzten Jahrhunderten (Tübingen, 1884), p.
15.

622. It is, so to speak, a scientific tact, which must guide mathematicians in
their investigations, and guard them from spending their forces on
scientifically worthless problems and abstruse realms, a tact which is
closely related to esthetic tact and which is the only thing in our science
which cannot be taught or acquired, and is yet the indispensable endowment
of every mathematician.—Hankel, Hermann.
Die Entwickelung der Mathematik in den
letzten Jahrhunderten (Tübingen, 1884), p.
21.

623. The mathematician requires tact and good taste at every step of his
work, and he has to learn to trust to his own instinct to distinguish between
what is really worthy of his efforts and what is not; he must take care not to
be the slave of his symbols, but always to have before his mind the realities
which they merely serve to express. For these and other reasons it seems to
me of the highest importance that a mathematician should be trained in no
narrow school; a wide course of reading in the first few years of his
mathematical study cannot fail to influence for good the character of the
whole of his subsequent work.—Glaisher, J. W. L.
Presidential Address British Association for
the Advancement of Science, Section A,
(1890); Nature, Vol. 42, p. 467.

624. As long as a branch of science offers an abundance of problems, so
long it is alive; a lack of problems foreshadows extinction or the cessation
of independent development.—Hilbert, D.
Mathematical Problems; Bulletin American
Mathematical Society, Vol. 8, p. 438.

625. In mathematics as in other fields, to find one self lost in wonder at
some manifestation is frequently the half of a new discovery.—Dirichlet, P.
G. L.
Werke, Bd. 2 (Berlin, 1897), p. 233.

626. 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 a great measure it is really the intelligence of other
people that confronts us in science.—Mach, Ernst.
Popular Scientific Lectures (Chicago, 1910),
p. 196.

627. It is probably this fact [referring to the circumstance that the problems
of the parallel axiom, the squaring of the circle, the solution of the equation
of the fifth degree, have finally found fully satisfactory and rigorous
solutions] along with other philosophical reasons that gives rise to the
conviction (which every mathematician shares, but which no one has yet
supported by proof) that every definite mathematical problem must
necessarily be susceptible of an exact settlement, either in the form of an
actual answer to the question asked, or by the proof of the impossibility of
its solution and therewith the necessary failure of all attempts.... This
conviction of the solvability of every mathematical problem is a powerful
incentive to the worker. We hear within us the perpetual call: There is the
problem. Seek its solution. You can find it by pure reason, for in
mathematics there is no ignorabimus.—Hilbert, D.
Mathematical Problems; Bulletin American
Mathematical Society, Vol. 8, pp. 444-445.

628. He who seeks for methods without having a definite problem in mind
seeks for the most part in vain.—Hilbert, D.
Mathematical Problems; Bulletin American
Mathematical Society, Vol. 8, p. 444.

629. A mathematical problem should be difficult in order to entice us, yet
not completely inaccessible, lest it mock at our efforts. It should be to us a
guide post on the mazy paths to hidden truths, and ultimately a reminder of
our pleasure in the successful solution.—Hilbert, D.
Mathematical Problems; Bulletin American
Mathematical Society, Vol. 8, p. 438.

630. The great mathematicians have acted on the principle “Divinez avant
de demontrer,” and it is certainly true that almost all important discoveries
are made in this fashion.—Kasner, Edward.
The Present Problems in Geometry; Bulletin
American Mathematical Society, Vol. 11, p.
285.

631. “Divide et impera” is as true in algebra as in statecraft; but no less true
and even more fertile is the maxim “auge et impera.” The more to do or to
prove, the easier the doing or the proof.—Sylvester, J. J.
Proof of the Fundamental Theorem of
Invariants; Philosophic Magazine (1878), p.
186; Collected Mathematical Papers, Vol. 3,
p. 126.

632. As in the domains of practical life so likewise in science there has
come about a division of labor. The individual can no longer control the
whole field of mathematics: it is only possible for him to master separate
parts of it in such a manner as to enable him to extend the boundaries of
knowledge by creative research.—Lampe, E.
Die reine Mathematik in den Jahren 1884-
1899, p. 10.

633. With the extension of mathematical knowledge will it not finally
become impossible for the single investigator to embrace all departments of
this knowledge? In answer let me point out how thoroughly it is ingrained
in mathematical science that every real advance goes hand in hand with the
invention of sharper tools and simpler methods which at the same time
assist in understanding earlier theories and to cast aside some more
complicated developments. It is therefore possible for the individual
investigator, when he makes these sharper tools and simpler methods his
own, to find his way more easily in the various branches of mathematics
than is possible in any other science.—Hilbert, D.
Mathematical Problems; Bulletin American
Mathematical Society, Vol. 8, p. 479.

634. It would seem at first sight as if the rapid expansion of the region of
mathematics must be a source of danger to its future progress. Not only
does the area widen but the subjects of study increase rapidly in number,
and the work of the mathematician tends to become more and more
specialized. It is, of course, merely a brilliant exaggeration to say that no
mathematician is able to understand the work of any other mathematician,

but it is certainly true that it is daily becoming more and more difficult for a
mathematician to keep himself acquainted, even in a general way, with the
progress of any of the branches of mathematics except those which form the
field of his own labours. I believe, however, that the increasing extent of the
territory of mathematics will always be counteracted by increased facilities
in the means of communication. Additional knowledge opens to us new
principles and methods which may conduct us with the greatest ease to
results which previously were most difficult of access; and improvements in
notation may exercise the most powerful effects both in the simplification
and accessibility of a subject. It rests with the worker in mathematics not
only to explore new truths, but to devise the language by which they may be
discovered and expressed; and the genius of a great mathematician displays
itself no less in the notation he invents for deciphering his subject than in
the results attained.... I have great faith in the power of well-chosen notation
to simplify complicated theories and to bring remote ones near and I think it
is safe to predict that the increased knowledge of principles and the
resulting improvements in the symbolic language of mathematics will
always enable us to grapple satisfactorily with the difficulties arising from
the mere extent of the subject.—Glaisher, J. W. L.
Presidential Address British Association for
the Advancement of Science, Section A.,
(1890), Nature, Vol. 42, p. 466.

635. Quite distinct from the theoretical question of the manner in which
mathematics will rescue itself from the perils to which it is exposed by its
own prolific nature is the practical problem of finding means of rendering
available for the student the results which have been already accumulated,
and making it possible for the learner to obtain some idea of the present
state of the various departments of mathematics.... The great mass of
mathematical literature will be always contained in Journals and
Transactions, but there is no reason why it should not be rendered far more
useful and accessible than at present by means of treatises or higher text-
books. The whole science suffers from want of avenues of approach, and
many beautiful branches of mathematics are regarded as difficult and
technical merely because they are not easily accessible.... I feel very
strongly that any introduction to a new subject written by a competent

person confers a real benefit on the whole science. The number of excellent
text-books of an elementary kind that are published in this country makes it
all the more to be regretted that we have so few that are intended for the
advanced student. As an example of the higher kind of text-book, the want
of which is so badly felt in many subjects, I may mention the second part of
Prof. Chrystal’s “Algebra” published last year, which in a small compass
gives a great mass of valuable and fundamental knowledge that has hitherto
been beyond the reach of an ordinary student, though in reality lying so
close at hand. I may add that in any treatise or higher text-book it is always
desirable that references to the original memoirs should be given, and, if
possible, short historic notices also. I am sure that no subject loses more
than mathematics by any attempt to dissociate it from its history.—
Glaisher, J. W. L.
Presidential Address British Association for
the Advancement of Science, Section A
(1890); Nature, Vol. 42, p. 466.

636. The more a science advances, the more will it be possible to
understand immediately results which formerly could be demonstrated only
by means of lengthy intermediate considerations: a mathematical subject
cannot be considered as finally completed until this end has been attained.
—Gordan, Paul.
Formensystem binärer Formen (Leipzig,
1875), p. 2.

637. An old French geometer used to say that a mathematical theory was
never to be considered complete till you had made it so clear that you could
explain it to the first man you met in the street.—Smith, H. J. S.
Nature, Vol. 8 (1873), p. 452.

638. In order to comprehend and fully control arithmetical concepts and
methods of proof, a high degree of abstraction is necessary, and this
condition has at times been charged against arithmetic as a fault. I am of the
opinion that all other fields of knowledge require at least an equally high
degree of abstraction as mathematics,—provided, that in these fields the

foundations are also everywhere examined with the rigour and
completeness which is actually necessary.—Hilbert, D.
Die Theorie der algebraischen Zahlkorper,
Vorwort; Jahresbericht der Deutschen
Mathematiker Vereinigung, Bd. 4.

639. The anxious precision of modern mathematics is necessary for
accuracy, ... it is necessary for research. It makes for clearness of thought
and for fertility in trying new combinations of ideas. When the initial
statements are vague and slipshod, at every subsequent stage of thought,
common sense has to step in to limit applications and to explain meanings.
Now in creative thought common sense is a bad master. Its sole criterion for
judgment is that the new ideas shall look like the old ones, in other words it
can only act by suppressing originality.—Whitehead, A. N.
Introduction to Mathematics (New York,
1911), p. 157.

640. Mathematicians attach great importance to the elegance of their
methods and their results. This is not pure dilettantism. What is it indeed
that gives us the feeling of elegance in a solution, in a demonstration? It is
the harmony of the diverse parts, their symmetry, their happy balance; in a
word it is all that introduces order, all that gives unity, that permits us to see
clearly and to comprehend at once both the ensemble and the details. But
this is exactly what yields great results, in fact the more we see this
aggregate clearly and at a single glance, the better we perceive its analogies
with other neighboring objects, consequently the more chances we have of
divining the possible generalizations. Elegance may produce the feeling of
the unforeseen by the unexpected meeting of objects we are not accustomed
to bring together; there again it is fruitful, since it thus unveils for us
kinships before unrecognized. It is fruitful even when it results only from
the contrast between the simplicity of the means and the complexity of the
problem set; it makes us then think of the reason for this contrast and very
often makes us see that chance is not the reason; that it is to be found in
some unexpected law. In a word, the feeling of mathematical elegance is
only the satisfaction due to any adaptation of the solution to the needs of
our mind, and it is because of this very adaptation that this solution can be

for us an instrument. Consequently this esthetic satisfaction is bound up
with the economy of thought.—Poincaré, H.
The Future of Mathematics; Monist, Vol. 20,
p. 80. [Halsted].

641. The importance of a result is largely relative, is judged differently by
different men, and changes with the times and circumstances. It has often
happened that great importance has been attached to a problem merely on
account of the difficulties which it presented; and indeed if for its solution it
has been necessary to invent new methods, noteworthy artifices, etc., the
science has gained more perhaps through these than through the final result.
In general we may call important all investigations relating to things which
in themselves are important; all those which have a large degree of
generality, or which unite under a single point of view subjects apparently
distinct, simplifying and elucidating them; all those which lead to results
that promise to be the source of numerous consequences; etc.—Segre,
Corradi.
Some Recent Tendencies in Geometric
Investigations. Rivista di Matematica, Vol. 1,
p. 44. Bulletin American Mathematical
Society, 1904, p. 444. [Young, J. W.].

642. Geometric writings are not rare in which one would seek in vain for an
idea at all novel, for a result which sooner or later might be of service, for
anything in fact which might be destined to survive in the science; and one
finds instead treatises on trivial problems or investigations on special forms
which have absolutely no use, no importance, which have their origin not in
the science itself but in the caprice of the author; or one finds applications
of known methods which have already been made thousands of times; or
generalizations from known results which are so easily made that the
knowledge of the latter suffices to give at once the former. Now such work
is not merely useless; it is actually harmful because it produces a real
incumbrance in the science and an embarrassment for the more serious
investigators; and because often it crowds out certain lines of thought which
might well have deserved to be studied.—Segre, Corradi.

On some Recent Tendencies in Geometric
Investigations; Rivista di Matematica, 1891,
p. 43. Bulletin American Mathematical
Society, 1904, p. 443 [Young, J. W.].

643. A student who wishes now-a-days to study geometry by dividing it
sharply from analysis, without taking account of the progress which the
latter has made and is making, that student no matter how great his genius,
will never be a whole geometer. He will not possess those powerful
instruments of research which modern analysis puts into the hands of
modern geometry. He will remain ignorant of many geometrical results
which are to be found, perhaps implicitly, in the writings of the analyst. And
not only will he be unable to use them in his own researches, but he will
probably toil to discover them himself, and, as happens very often, he will
publish them as new, when really he has only rediscovered them.—Segre,
Corradi.
On some recent Tendencies in Geometrical
Investigations; Rivista di Matematica, 1891,
p. 43. Bulletin American Mathematical
Society, 1904, p. 443 [Young, J. W.].

644. Research may start from definite problems whose importance it
recognizes and whose solution is sought more or less directly by all forces.
But equally legitimate is the other method of research which only selects
the field of its activity and, contrary to the first method, freely reconnoitres
in the search for problems which are capable of solution. Different
individuals will hold different views as to the relative value of these two
methods. If the first method leads to greater penetration it is also easily
exposed to the danger of unproductivity. To the second method we owe the
acquisition of large and new fields, in which the details of many things
remain to be determined and explored by the first method.—Clebsch, A.
Zum Gedächtniss an Julius Plücker;
Göttinger Abhandlungen, 16, 1871,
Mathematische Classe, p. 6.

645. During a conversation with the writer in the last weeks of his life,
Sylvester remarked as curious that notwithstanding he had always
considered the bent of his mind to be rather analytical than geometrical, he
found in nearly every case that the solution of an analytical problem turned
upon some quite simple geometrical notion, and that he was never satisfied
until he could present the argument in geometrical language.—MacMahon,
P. A.
Proceedings London Royal Society, Vol. 63,
p. 17.

646. The origin of a science is usually to be sought for not in any systematic
treatise, but in the investigation and solution of some particular problem.
This is especially the case in the ordinary history of the great improvements
in any department of mathematical science. Some problem, mathematical or
physical, is proposed, which is found to be insoluble by known methods.
This condition of insolubility may arise from one of two causes: Either
there exists no machinery powerful enough to effect the required reduction,
or the workmen are not sufficiently expert to employ their tools in the
performance of an entirely new piece of work. The problem proposed is,
however, finally solved, and in its solution some new principle, or new
application of old principles, is necessarily introduced. If a principle is
brought to light it is soon found that in its application it is not necessarily
limited to the particular question which occasioned its discovery, and it is
then stated in an abstract form and applied to problems of gradually
increasing generality.
Other principles, similar in their nature, are added, and the original
principle itself receives such modifications and extensions as are from time
to time deemed necessary. The same is true of new applications of old
principles; the application is first thought to be merely confined to a
particular problem, but it is soon recognized that this problem is but one,
and generally a very simple one, out of a large class, to which the same
process of investigation and solution are applicable. The result in both of
these cases is the same. A time comes when these several problems,
solutions, and principles are grouped together and found to produce an
entirely new and consistent method; a nomenclature and uniform system of

notation is adopted, and the principles of the new method become entitled
to rank as a distinct science.—Craig, Thomas.
A Treatise on Projection, Preface. U. S.
Coast and Geodetic Survey, Treasury
Department Document, No. 61.

647. The aim of research is the discovery of the equations which subsist
between the elements of phenomena.—Mach, Ernst.
Popular Scientific Lectures (Chicago, 1910),
p. 205.

648. Let him [the author] be permitted also in all humility to add ... that in
consequence of the large arrears of algebraical and arithmetical speculations
waiting in his mind their turn to be called into outward existence, he is
driven to the alternative of leaving the fruits of his meditations to perish (as
has been the fate of too many foregone theories, the still-born progeny of
his brain, now forever resolved back again into the primordial matter of
thought), or venturing to produce from time to time such imperfect sketches
as the present, calculated to evoke the mental co-operation of his readers, in
whom the algebraical instinct has been to some extent developed, rather
than to satisfy the strict demands of rigorously systematic exposition.—
Sylvester, J. J.
Philosophic Magazine (1863), p. 460.

649. In other branches of science, where quick publication seems to be so
much desired, there may possibly be some excuse for giving to the world
slovenly or ill-digested work, but there is no such excuse in mathematics.
The form ought to be as perfect as the substance, and the demonstrations as
rigorous as those of Euclid. The mathematician has to deal with the most
exact facts of Nature, and he should spare no effort to render his
interpretation worthy of his subject, and to give to his work its highest
degree of perfection. “Pauca sed matura” was Gauss’s motto.—Glaisher, J.
W. L.
Presidential Address British Association for
the Advancement of Science, Section A,

(1890); Nature, Vol. 42, p. 467.

650. It is the man not the method that solves the problem.—Maschke, H.
Present Problems of Algebra and Analysis;
Congress of Arts and Sciences (New York and
Boston, 1905), Vol. 1, p. 530.

651. Today it is no longer questioned that the principles of the analysts are
the more far-reaching. Indeed, the synthesists lack two things in order to
engage in a general theory of algebraic configurations: these are on the one
hand a definition of imaginary elements, on the other an interpretation of
general algebraic concepts. Both of these have subsequently been
developed in synthetic form, but to do this the essential principle of
synthetic geometry had to be set aside. This principle which manifests itself
so brilliantly in the theory of linear forms and the forms of the second
degree, is the possibility of immediate proof by means of visualized
constructions.—Klein, Felix.
Riemannsche Flächen (Leipzig, 1906), Bd. 1,
p. 234.

652. Abstruse mathematical researches ... are ... often abused for having no
obvious physical application. The fact is that the most useful parts of
science have been investigated for the sake of truth, and not for their
usefulness. A new branch of mathematics, which has sprung up in the last
twenty years, was denounced by the Astronomer Royal before the
University of Cambridge as doomed to be forgotten, on account of its
uselessness. Now it turns out that the reason why we cannot go further in
our investigations of molecular action is that we do not know enough of this
branch of mathematics.—Clifford, W. K.
Conditions of Mental Development; Lectures
and Essays (London, 1901), Vol. 1, p. 115.

653. In geometry, as in most sciences, it is very rare that an isolated
proposition is of immediate utility. But the theories most powerful in

practice are formed of propositions which curiosity alone brought to light,
and which long remained useless without its being able to divine in what
way they should one day cease to be so. In this sense it may be said, that in
real science, no theory, no research, is in effect useless.—Voltaire.
A Philosophical Dictionary, Article
“Geometry”; (Boston, 1881), Vol. 1, p. 374.

654. Scientific subjects do not progress necessarily on the lines of direct
usefulness. Very many applications of the theories of pure mathematics
have come many years, sometimes centuries, after the actual discoveries
themselves. The weapons were at hand, but the men were not able to use
them.—Forsyth, A. R.
Perry’s Teaching of Mathematics (London,
1902), p. 35.

655. It is no paradox to say that in our most theoretical moods we may be
nearest to our most practical applications.—Whitehead, A. N.
Introduction to Mathematics (New York), p.
100.

656. Although with the majority of those who study and practice in these
capacities [engineers, builders, surveyors, geographers, navigators,
hydrographers, astronomers], second-hand acquirements, trite formulas, and
appropriate tables are sufficient for ordinary purposes, yet these trite
formulas and familiar rules were originally or gradually deduced from the
profound investigations of the most gifted minds, from the dawn of science
to the present day.... The further developments of the science, with its
possible applications to larger purposes of human utility and grander
theoretical generalizations, is an achievement reserved for a few of the
choicest spirits, touched from time to time by Heaven to these highest
issues. The intellectual world is filled with latent and undiscovered truth as
the material world is filled with latent electricity.—Everett, Edward.
Orations and Speeches, Vol. 3 (Boston,
1870), p. 513.

657. If we view mathematical speculations with reference to their use, it
appears that they should be divided into two classes. To the first belong
those which furnish some marked advantage either to common life or to
some art, and the value of such is usually determined by the magnitude of
this advantage. The other class embraces those speculations which, though
offering no direct advantage, are nevertheless valuable in that they extend
the boundaries of analysis and increase our resources and skill. Now since
many investigations, from which great advantage may be expected, must be
abandoned solely because of the imperfection of analysis, no small value
should be assigned to those speculations which promise to enlarge the field
of analysis.—Euler.
Novi Comm. Petr., Vol. 4, Preface.

658. The discovery of the conic sections, attributed to Plato, first threw
open the higher species of form to the contemplation of geometers. But for
this discovery, which was probably regarded in Plato’s time and long after
him, as the unprofitable amusement of a speculative brain, the whole course
of practical philosophy of the present day, of the science of astronomy, of
the theory of projectiles, of the art of navigation, might have run in a
different channel; and the greatest discovery that has ever been made in the
history of the world, the law of universal gravitation, with its innumerable
direct and indirect consequences and applications to every department of
human research and industry, might never to this hour have been elicited.—
Sylvester, J. J.
A Probationary Lecture on Geometry;
Collected Mathematical Papers, Vol. 2
(Cambridge, 1908), p. 7.

659. No more impressive warning can be given to those who would confine
knowledge and research to what is apparently useful, than the reflection that
conic sections were studied for eighteen hundred years merely as an
abstract science, without regard to any utility other than to satisfy the
craving for knowledge on the part of mathematicians, and that then at the
end of this long period of abstract study, they were found to be the
necessary key with which to attain the knowledge of the most important
laws of nature.—Whitehead, A. N.

Introduction to Mathematics (New York,
York, 1911), pp. 136-137.

660. The Greeks in the first vigour of their pursuit of mathematical truth, at
the time of Plato and soon after, had by no means confined themselves to
those propositions which had a visible bearing on the phenomena of nature;
but had followed out many beautiful trains of research concerning various
kinds of figures, for the sake of their beauty alone; as for instance in their
doctrine of Conic Sections, of which curves they had discovered all the
principal properties. But it is curious to remark, that these investigations,
thus pursued at first as mere matters of curiosity and intellectual
gratification, were destined, two thousand years later, to play a very
important part in establishing that system of celestial motions which
succeeded the Platonic scheme of cycles and epicycles. If the properties of
conic sections had not been demonstrated by the Greeks and thus rendered
familiar to the mathematicians of succeeding ages, Kepler would probably
not have been able to discover those laws respecting the orbits and motions
of planets which were the occasion of the greatest revolution that ever
happened in the history of science.—Whewell, W.
History of Scientific Ideas, Bk. 2, chap. 14,
sect. 3.

661. The greatest mathematicians, as Archimedes, Newton, and Gauss,
always united theory and applications in equal measure.—Klein, Felix.
Elementarmathematik vom höheren
Standpunkte aus (Leipzig, 1909), Bd. 2, p.
392.

662. We may see how unexpectedly recondite parts of pure mathematics
may bear upon physical science, by calling to mind the circumstance that
Fresnel obtained one of the most curious confirmations of the theory (the
laws of Circular Polarization by reflection) through an interpretation of an
algebraical expression, which, according to the original conventional
meaning of the symbols, involved an impossible quantity.—Whewell, W.

History of Scientific Ideas, Bk. 2, chap. 14,
sect. 8.

663. A great department of thought must have its own inner life, however
transcendent may be the importance of its relations to the outside. No
department of science, least of all one requiring so high a degree of mental
concentration as Mathematics, can be developed entirely, or even mainly,
with a view to applications outside its own range. The increased complexity
and specialisation of all branches of knowledge makes it true in the present,
however it may have been in former times, that important advances in such
a department as Mathematics can be expected only from men who are
interested in the subject for its own sake, and who, whilst keeping an open
mind for suggestions from outside, allow their thought to range freely in
those lines of advance which are indicated by the present state of their
subject, untrammelled by any preoccupation as to applications to other
departments of science. Even with a view to applications, if Mathematics is
to be adequately equipped for the purpose of coping with the intricate
problems which will be presented to it in the future by Physics, Chemistry
and other branches of physical science, many of these problems probably of
a character which we cannot at present forecast, it is essential that
Mathematics should be allowed to develop freely on its own lines.—
Hobson, E. W.
Presidential Address British Association for
the Advancement of Science, Section A,
(1910); Nature, Vol. 84, p. 286.

664. To emphasize this opinion that mathematicians would be unwise to
accept practical issues as the sole guide or the chief guide in the current of
their investigations, ... let me take one more instance, by choosing a subject
in which the purely mathematical interest is deemed supreme, the theory of
functions of a complex variable. That at least is a theory in pure
mathematics, initiated in that region, and developed in that region; it is built
up in scores of papers, and its plan certainly has not been, and is not now,
dominated or guided by considerations of applicability to natural
phenomena. Yet what has turned out to be its relation to practical issues?
The investigations of Lagrange and others upon the construction of maps

appear as a portion of the general property of conformal representation;
which is merely the general geometrical method of regarding functional
relations in that theory. Again, the interesting and important investigations
upon discontinuous two-dimensional fluid motion in hydrodynamics, made
in the last twenty years, can all be, and now are all, I believe, deduced from
similar considerations by interpreting functional relations between complex
variables. In the dynamics of a rotating heavy body, the only substantial
extension of our knowledge since the time of Lagrange has accrued from
associating the general properties of functions with the discussion of the
equations of motion. Further, under the title of conjugate functions, the
theory has been applied to various questions in electrostatics, particularly in
connection with condensors and electrometers. And, lastly, in the domain of
physical astronomy, some of the most conspicuous advances made in the
last few years have been achieved by introducing into the discussion the
ideas, the principles, the methods, and the results of the theory of functions
... the refined and extremely difficult work of Poincaré and others in
physical astronomy has been possible only by the use of the most elaborate
developments of some purely mathematical subjects, developments which
were made without a thought of such applications.—Forsyth, A. R.
Presidential Address British Association for
the Advancement of Science, Section A,
(1897); Nature, Vol. 56, p. 377.

CHAPTER VII
MODERN MATHEMATICS

701. Surely this is the golden age of mathematics.—Pierpont, James.
History of Mathematics in the Nineteenth
Century; Congress of Arts and Sciences
(Boston and New York, 1905), Vol. 1, p. 493.

702. The golden age of mathematics—that was not the age of Euclid, it is
ours. Ours is the age when no less than six international congresses have
been held in the course of nine years. It is in our day that more than a dozen
mathematical societies contain a growing membership of more than two
thousand men representing the centers of scientific light throughout the
great culture nations of the world. It is in our time that over five hundred
scientific journals are each devoted in part, while more than two score
others are devoted exclusively, to the publication of mathematics. It is in
our time that the Jahrbuch über die Fortschritte der Mathematik, though
admitting only condensed abstracts with titles, and not reporting on all the
journals, has, nevertheless, grown to nearly forty huge volumes in as many
years. It is in our time that as many as two thousand books and memoirs
drop from the mathematical press of the world in a single year, the
estimated number mounting up to fifty thousand in the last generation.
Finally, to adduce yet another evidence of a similar kind, it requires not less
than seven ponderous tomes of the forthcoming Encyclopaedie der
Mathematischen Wissenschaften to contain, not expositions, not
demonstrations, but merely compact reports and bibliographic notices
sketching developments that have taken place since the beginning of the
nineteenth century.—Keyser, C. J.
Lectures on Science, Philosophy and Art
(New York, 1908), p. 8.

703. I have said that mathematics is the oldest of the sciences; a glance at
its more recent history will show that it has the energy of perpetual youth.
The output of contributions to the advance of the science during the last
century and more has been so enormous that it is difficult to say whether
pride in the greatness of achievement in this subject, or despair at his
inability to cope with the multiplicity of its detailed developments, should
be the dominant feeling of the mathematician. Few people outside of the
small circle of mathematical specialists have any idea of the vast growth of
mathematical literature. The Royal Society Catalogue contains a list of
nearly thirty-nine thousand papers on subjects of Pure Mathematics alone,
which have appeared in seven hundred serials during the nineteenth century.
This represents only a portion of the total output, the very large number of
treatises, dissertations, and monographs published during the century being
omitted.—Hobson, E. W.
Presidential Address British Association for
the Advancement of Science, Section A,
(1910); Nature, Vol. 84, p. 285.

704. Mathematics is one of the oldest of the sciences; it is also one of the
most active, for its strength is the vigour of perpetual youth.—Forsyth, A.
R.
Presidential Address British Association for
the Advancement of Science, Section A,
(1897); Nature, Vol. 56, p. 378.

705. The nineteenth century which prides itself upon the invention of steam
and evolution, might have derived a more legitimate title to fame from the
discovery of pure mathematics.—Russell, Bertrand.
International Monthly, Vol. 4 (1901), p. 83.

706. One of the chiefest triumphs of modern mathematics consists in having
discovered what mathematics really is.—Russell, Bertrand.
International Monthly, Vol. 4 (1901), p. 84.

707. Modern mathematics, that most astounding of intellectual creations,
has projected the mind’s eye through infinite time and the mind’s hand into
boundless space.—Butler, N. M.
The Meaning of Education and other Essays
and Addresses (New York, 1905), p. 44.

708. The extraordinary development of mathematics in the last century is
quite unparalleled in the long history of this most ancient of sciences. Not
only have those branches of mathematics which were taken over from the
eighteenth century steadily grown, but entirely new ones have sprung up in
almost bewildering profusion, and many of them have promptly assumed
proportions of vast extent.—Pierpont, J.
The History of Mathematics in the Nineteenth
Century; Congress of Arts and Sciences
(Boston and New York, 1905), Vol. 1, p. 474.

709. The Modern Theory of Functions—that stateliest of all the pure
creations of the human intellect.—Keyser, C. J.
Lectures on Science, Philosophy and Art
(New York, 1908), p. 16.

710. If a mathematician of the past, an Archimedes or even a Descartes,
could view the field of geometry in its present condition, the first feature to
impress him would be its lack of concreteness. There are whole classes of
geometric theories which proceed not only without models and diagrams,
but without the slightest (apparent) use of spatial intuition. In the main this
is due, to the power of the analytic instruments of investigations as
compared with the purely geometric.—Kasner, Edward.
The Present Problems in Geometry; Bulletin
American Mathematical Society, 1905, p.
285.

711. In Euclid each proposition stands by itself; its connection with others is
never indicated; the leading ideas contained in its proof are not stated;

general principles do not exist. In modern methods, on the other hand, the
greatest importance is attached to the leading thoughts which pervade the
whole; and general principles, which bring whole groups of theorems under
one aspect, are given rather than separate propositions. The whole tendency
is toward generalization. A straight line is considered as given in its entirety,
extending both ways to infinity, while Euclid is very careful never to admit
anything but finite quantities. The treatment of the infinite is in fact another
fundamental difference between the two methods. Euclid avoids it, in
modern mathematics it is systematically introduced, for only thus is
generality obtained.—Cayley, Arthur.
Encyclopedia Britannica (9th edition),
Article“Geometry.”

712. This is one of the greatest advantages of modern geometry over the
ancient, to be able, through the consideration of positive and negative
quantities, to include in a single enunciation the several cases which the
same theorem may present by a change in the relative position of the
different parts of a figure. Thus in our day the nine principal problems and
the numerous particular cases, which form the object of eighty-three
theorems in the two books De sectione determinata of Appolonius
constitute only one problem which is resolved by a single equation.—
Chasles, M.
Histoire de la Géométrie, chap. 1, sect. 35.

713. Euclid always contemplates a straight line as drawn between two
definite points, and is very careful to mention when it is to be produced
beyond this segment. He never thinks of the line as an entity given once for
all as a whole. This careful definition and limitation, so as to exclude an
infinity not immediately apparent to the senses, was very characteristic of
the Greeks in all their many activities. It is enshrined in the difference
between Greek architecture and Gothic architecture, and between Greek
religion and modern religion. The spire of a Gothic cathedral and the
importance of the unbounded straight line in modern Geometry are both
emblematic of the transformation of the modern world.—Whitehead, A. N.
Introduction to Mathematics (New York,
1911), p. 119.

714. The geometrical problems and theorems of the Greeks always refer to
definite, oftentimes to rather complicated figures. Now frequently the points
and lines of such a figure may assume very many different relative
positions; each of these possible cases is then considered separately. On the
contrary, present day mathematicians generate their figures one from
another, and are accustomed to consider them subject to variation; in this
manner they unite the various cases and combine them as much as possible
by employing negative and imaginary magnitudes. For example, the
problems which Appolonius treats in his two books De sectione rationis,
are solved today by means of a single, universally applicable construction;
Apollonius, on the contrary, separates it into more than eighty different
cases varying only in position. Thus, as Hermann Hankel has fittingly
remarked, the ancient geometry sacrifices to a seeming simplicity the true
simplicity which consists in the unity of principles; it attained a trivial
sensual presentability at the cost of the recognition of the relations of
geometric forms in all their changes and in all the variations of their
sensually presentable positions.—Reye, Theodore.
Die synthetische Geometrie im Altertum und
in der Neuzeit; Jahresbericht der Deutschen
Mathematiker Vereinigung, Bd. 2, pp. 346-
347.

715. It is known that the mathematics prescribed for the high school
[Gymnasien] is essentially Euclidean, while it is modern mathematics, the
theory of functions and the infinitesimal calculus, which has secured for us
an insight into the mechanism and laws of nature. Euclidean mathematics is
indeed, a prerequisite for the theory of functions, but just as one, though he
has learned the inflections of Latin nouns and verbs, will not thereby be
enabled to read a Latin author much less to appreciate the beauties of a
Horace, so Euclidean mathematics, that is the mathematics of the high
school, is unable to unlock nature and her laws. Euclidean mathematics
assumes the completeness and invariability of mathematical forms; these
forms it describes with appropriate accuracy and enumerates their inherent
and related properties with perfect clearness, order, and completeness, that
is, Euclidean mathematics operates on forms after the manner that anatomy
operates on the dead body and its members.

On the other hand, the mathematics of variable magnitudes—function
theory or analysis—considers mathematical forms in their genesis. By
writing the equation of the parabola, we express its law of generation, the
law according to which the variable point moves. The path, produced before
the eyes of the student by a point moving in accordance to this law, is the
parabola.
If, then, Euclidean mathematics treats space and number forms after
the manner in which anatomy treats the dead body, modern mathematics
deals, as it were, with the living body, with growing and changing forms,
and thus furnishes an insight, not only into nature as she is and appears, but
also into nature as she generates and creates,—reveals her transition steps
and in so doing creates a mind for and understanding of the laws of
becoming. Thus modern mathematics bears the same relation to Euclidean
mathematics that physiology or biology ... bears to anatomy. But it is
exactly in this respect that our view of nature is so far above that of the
ancients; that we no longer look on nature as a quiescent complete whole,
which compels admiration by its sublimity and wealth of forms, but that we
conceive of her as a vigorous growing organism, unfolding according to
definite, as delicate as far-reaching, laws; that we are able to lay hold of the
permanent amidst the transitory, of law amidst fleeting phenomena, and to
be able to give these their simplest and truest expression through the
mathematical formulas.—Dillmann, E.
Die Mathematik die Fackelträgerin einer
neuen Zeit (Stuttgart, 1889), p. 37.

716. The Excellence of Modern Geometry is in nothing more evident, than
in those full and adequate Solutions it gives to Problems; representing all
possible Cases in one view, and in one general Theorem many times
comprehending whole Sciences; which deduced at length into Propositions,
and demonstrated after the manner of the Ancients, might well become the
subjects of large Treatises: For whatsoever Theorem solves the most
complicated Problem of the kind, does with a due Reduction reach all the
subordinate Cases.—Halley, E.
An Instance of the Excellence of Modern
Algebra, etc.; Philosophical Transactions,
1694, p. 960.

717. One of the most conspicuous and distinctive features of thought in the
nineteenth century is its critical spirit. Beginning with the calculus, it soon
permeates all analysis, and toward the close of the century it overhauls and
recasts the foundations of geometry and aspires to further conquests in
mechanics and in the immense domains of mathematical physics.... A
searching examination of the foundations of arithmetic and the calculus has
brought to light the insufficiency of much of the reasoning formerly
considered as conclusive.—Pierpont, J.
History of Mathematics in the Nineteenth
Century; Congress of Arts and Sciences
(Boston and New York, 1905), Vol. 1, p. 482.

718. If we compare a mathematical problem with an immense rock, whose
interior we wish to penetrate, then the work of the Greek mathematicians
appears to us like that of a robust stonecutter, who, with indefatigable
perseverance, attempts to demolish the rock gradually from the outside by
means of hammer and chisel; but the modern mathematician resembles an
expert miner, who first constructs a few passages through the rock and then
explodes it with a single blast, bringing to light its inner treasures.—
Hankel, Hermann.
Die Entwickelung der Mathematik in den
letzten Jahrhunderten (Tübingen, 1884), p. 9.

719. All the modern higher mathematics is based on a calculus of
operations, on laws of thought. All mathematics, from the first, was so in
reality; but the evolvers of the modern higher calculus have known that it is
so. Therefore elementary teachers who, at the present day, persist in
thinking about algebra and arithmetic as dealing with laws of number, and
about geometry as dealing with laws of surface and solid content, are doing
the best that in them lies to put their pupils on the wrong track for reaching
in the future any true understanding of the higher algebras. Algebras deal
not with laws of number, but with such laws of the human thinking
machinery as have been discovered in the course of investigations on
numbers. Plane geometry deals with such laws of thought as were
discovered by men intent on finding out how to measure surface; and solid

geometry with such additional laws of thought as were discovered when
men began to extend geometry into three dimensions.—Boole M. E.
Logic of Arithmetic (Oxford, 1903), Preface,
pp. 18-19.

720. It is not only a decided preference for synthesis and a complete denial
of general methods which characterizes the ancient mathematics as against
our newer science [modern mathematics]: besides this external formal
difference there is another real, more deeply seated, contrast, which arises
from the different attitudes which the two assumed relative to the use of the
concept of variability. For while the ancients, on account of considerations
which had been transmitted to them from the philosophic school of the
Eleatics, never employed the concept of motion, the spatial expression for
variability, in their rigorous system, and made incidental use of it only in
the treatment of phonoromically generated curves, modern geometry dates
from the instant that Descartes left the purely algebraic treatment of
equations and proceeded to investigate the variations which an algebraic
expression undergoes when one of its variables assumes a continuous
succession of values.—Hankel, Hermann.
Untersuchungen über die unendlich oft
oszillierenden und unstetigen Functionen;
Ostwald’s Klassiker der exacten
Wissenschaften, No. 153, pp. 44-45.

721. Without doubt one of the most characteristic features of mathematics
in the last century is the systematic and universal use of the complex
variable. Most of its great theories received invaluable aid from it, and
many owe their very existence to it.—Pierpont, J.
History of Mathematics in the Nineteenth
Century; Congress of Arts and Sciences
(Boston and New York, 1905), Vol. 1, p. 474.

722. The notion, which is really the fundamental one (and I cannot too
strongly emphasise the assertion), underlying and pervading the whole of

modern analysis and geometry, is that of imaginary magnitude in analysis
and of imaginary space in geometry.—Cayley, Arthur.
Presidential Address; Collected Works, Vol.
11, p. 434.

723. The solution of the difficulties which formerly surrounded the
mathematical infinite is probably the greatest achievement of which our age
has to boast.—Russell, Bertrand.
The Study of Mathematics; Philosophical
Essays (London, 1910), p. 77.

724. Induction and analogy are the special characteristics of modern
mathematics, in which theorems have given place to theories and no truth is
regarded otherwise than as a link in an infinite chain. “Omne exit in
infinitum” is their favorite motto and accepted axiom.—Sylvester, J. J.
A Plea for the Mathematician; Nature, Vol. 1,
p. 261.

725. The conception of correspondence plays a great part in modern
mathematics. It is the fundamental notion in the science of order as
distinguished from the science of magnitude. If the older mathematics were
mostly dominated by the needs of mensuration, modern mathematics are
dominated by the conception of order and arrangement. It may be that this
tendency of thought or direction of reasoning goes hand in hand with the
modern discovery in physics, that the changes in nature depend not only or
not so much on the quantity of mass and energy as on their distribution or
arrangement.—Merz, J. T.
History of European Thought in the
Nineteenth Century (Edinburgh and London,
1903), p. 736.

726. Now this establishment of correspondence between two aggregates
and investigation of the propositions that are carried over by the

correspondence may be called the central idea of modern mathematics.—
Clifford, W. K.
Philosophy of the Pure Sciences; Lectures
and Essays (London, 1901), Vol. 1, p. 402.

727. In our century the conceptions substitution and substitution group,
transformation and transformation group, operation and operation group,
invariant, differential invariant and differential parameter, appear more and
more clearly as the most important conceptions of mathematics.—Lie,
Sophus.
Leipziger Berichte, No. 47 (1895), p. 261.

728. Generality of points of view and of methods, precision and elegance in
presentation, have become, since Lagrange, the common property of all
who would lay claim to the rank of scientific mathematicians. And, even if
this generality leads at times to abstruseness at the expense of intuition and
applicability, so that general theorems are formulated which fail to apply to
a single special case, if furthermore precision at times degenerates into a
studied brevity which makes it more difficult to read an article than it was
to write it; if, finally, elegance of form has well-nigh become in our day the
criterion of the worth or worthlessness of a proposition,—yet are these
conditions of the highest importance to a wholesome development, in that
they keep the scientific material within the limits which are necessary both
intrinsically and extrinsically if mathematics is not to spend itself in
trivialities or smother in profusion.—Hankel, Hermann.
Die Entwickelung der Mathematik in den
letzten Jahrhunderten (Tübingen, 1884), pp.
14-15.

729. The development of abstract methods during the past few years has
given mathematics a new and vital principle which furnishes the most
powerful instrument for exhibiting the essential unity of all its branches.—
Young, J. W.
Fundamental Concepts of Algebra and
Geometry (New York, 1911), p. 225.

730. Everybody praises the incomparable power of the mathematical
method, but so is everybody aware of its incomparable unpopularity.—
Rosanes, J.
Jahresbericht der Deutschen Mathematiker
Vereinigung, Bd. 13, p. 17.

731. Indeed the modern developments of mathematics constitute not only
one of the most impressive, but one of the most characteristic, phenomena
of our age. It is a phenomenon, however, of which the boasted intelligence
of a “universalized” daily press seems strangely unaware; and there is no
other great human interest, whether of science or of art, regarding which the
mind of the educated public is permitted to hold so many fallacious
opinions and inferior estimates.—Keyser, C. J.
Lectures on Science, Philosophy and Arts
(New York, 1908), p. 8.

732. It may be asserted without exaggeration that the domain of
mathematical knowledge is the only one of which our otherwise omniscient
journalism has not yet possessed itself.—Pringsheim, Alfred.
Ueber Wert und angeblichen Unwert der
Mathematik; Jahresbericht der Deutschen
Mathematiker Vereinigung, (1904) p. 357.

733. [The] inaccessibility of special fields of mathematics, except by the
regular way of logically antecedent acquirements, renders the study
discouraging or hateful to weak or indolent minds.—Lefevre, Arthur.
Number and its Algebra (Boston, 1903), sect.
223.

734. The majority of mathematical truths now possessed by us presuppose
the intellectual toil of many centuries. A mathematician, therefore, who
wishes today to acquire a thorough understanding of modern research in
this department, must think over again in quickened tempo the
mathematical labors of several centuries. This constant dependence of new

truths on old ones stamps mathematics as a science of uncommon
exclusiveness and renders it generally impossible to lay open to uninitiated
readers a speedy path to the apprehension of the higher mathematical truths.
For this reason, too, the theories and results of mathematics are rarely
adapted for popular presentation.... This same inaccessibility of
mathematics, although it secures for it a lofty and aristocratic place among
the sciences, also renders it odious to those who have never learned it, and
who dread the great labor involved in acquiring an understanding of the
questions of modern mathematics. Neither in the languages nor in the
natural sciences are the investigations and results so closely interdependent
as to make it impossible to acquaint the uninitiated student with single
branches or with particular results of these sciences, without causing him to
go through a long course of preliminary study.—Schubert, H.
Mathematical Essays and Recreations
(Chicago, 1898), p. 32.

735. Such is the character of mathematics in its profounder depths and in its
higher and remoter zones that it is well nigh impossible to convey to one
who has not devoted years to its exploration a just impression of the scope
and magnitude of the existing body of the science. An imagination formed
by other disciplines and accustomed to the interests of another field may
scarcely receive suddenly an apocalyptic vision of that infinite interior
world. But how amazing and how edifying were such a revelation, if it only
could be made.—Keyser, C. J.
Lectures on Science, Philosophy and Art
(New York, 1908), p. 6.

736. It is not so long since, during one of the meetings of the Association,
one of the leading English newspapers briefly described a sitting of this
Section in the words, “Saturday morning was devoted to pure mathematics,
and so there was nothing of any general interest:” still, such toleration is
better than undisguised and ill-informed hostility.—Forsyth, A. R.
Report of the 67th meeting of the British
Association for the Advancement of Science.

737. The science [of mathematics] has grown to such vast proportion that
probably no living mathematician can claim to have achieved its mastery as
a whole.—Whitehead, A. N.
An Introduction to Mathematics (New York,
1911), p. 252.

738. There is perhaps no science of which the development has been carried
so far, which requires greater concentration and will power, and which by
the abstract height of the qualities required tends more to separate one from
daily life.
Provisional Report of the American
Subcommittee of the International
Commission on the Teaching of
Mathematics; Bulletin American Society
(1910), p. 97.

739. Angling may be said to be so like the mathematics, that it can never be
fully learnt.—Walton, Isaac.
The Complete Angler, Preface.

740. The flights of the imagination which occur to the pure mathematician
are in general so much better described in his formulæ than in words, that it
is not remarkable to find the subject treated by outsiders as something
essentially cold and uninteresting—... the only successful attempt to invest
mathematical reasoning with a halo of glory—that made in this section by
Prof. Sylvester—is known to a comparative few,....—Tait, P. G.
Presidential Address British Association for
the Advancement of Science (1871); Nature
Vol. 4, p. 271.

CHAPTER VIII
THE MATHEMATICIAN

801. The real mathematician is an enthusiast per se. Without enthusiasm no
mathematics.—Novalis.
Schriften (Berlin, 1901), Zweiter Teil, p. 223.

802. It is true that a mathematician, who is not somewhat of a poet, will
never be a perfect mathematician.—Weierstrass.
Quoted by Mittag-Leffler; Compte rendu du
deuxième congrês international des
mathématiciens (Paris, 1902), p. 149.

803. The mathematician is perfect only in so far as he is a perfect being, in
so far as he perceives the beauty of truth; only then will his work be
thorough, transparent, comprehensive, pure, clear, attractive and even
elegant. All this is necessary to resemble Lagrange.—Goethe.
Wilhelm Meister’s Wanderjahre, Zweites
Buch; Sprüche in Prosa; Natur, VI, 950.

804. A thorough advocate in a just cause, a penetrating mathematician
facing the starry heavens, both alike bear the semblance of divinity.—
Goethe.
Wilhelm Meister’s Wanderjahre, Zweites
Buch; Sprüche in Prosa; Natur, VI, 947.

805. Mathematicians practice absolute freedom.—Adams, Henry.
A Letter to American Teachers of History
(Washington, 1910), p. 169.

806. The mathematical method is the essence of mathematics. He who fully
comprehends the method is a mathematician.—Novalis.
Schriften (Berlin, 1901), Zweiter Teil, p. 190.

807. He who is unfamiliar with mathematics [literally, he who is a layman
in mathematics] remains more or less a stranger to our time.—Dillmann, E.
Die Mathematik die Fackelträgerin einer
neuen Zeit (Stuttgart, 1889), p. 39.

808. Enlist a great mathematician and a distinguished Grecian; your
problem will be solved. Such men can teach in a dwelling-house as well as
in a palace. Part of the apparatus they will bring; part we will furnish.
[Advice given to the Trustees of Johns Hopkins University on the choice of
a professorial staff.]—Gilman, D. C.
Report of the President of Johns Hopkins
University (1888), p. 29.

809. Persons, who have a decided mathematical talent, constitute, as it
were, a favored class. They bear the same relation to the rest of mankind
that those who are academically trained bear to those who are not.—
Moebius, P. J.
Ueber die Anlage zur Mathematik (Leipzig,
1900), p. 4.

810. One may be a mathematician of the first rank without being able to
compute. It is possible to be a great computer without having the slightest
idea of mathematics.—Novalis.
Schriften, Zweiter Teil (Berlin, 1901), p. 223.

811. It has long been a complaint against mathematicians that they are hard
to convince: but it is a far greater disqualification both for philosophy, and
for the affairs of life, to be too easily convinced; to have too low a standard
of proof. The only sound intellects are those which, in the first instance, set

their standards of proof high. Practice in concrete affairs soon teaches them
to make the necessary abatement: but they retain the consciousness, without
which there is no sound practical reasoning, that in accepting inferior
evidence because there is no better to be had, they do not by that acceptance
raise it to completeness.—Mill, J. S.
An Examination of Sir William Hamilton’s
Philosophy (London, 1878), p. 611.

812. It is easier to square the circle than to get round a mathematician.—De
Morgan, A.
Budget of Paradoxes (London, 1872), p. 90.

813. Mathematicians are like Frenchmen: whatever you say to them they
translate into their own language and forthwith it is something entirely
different.—Goethe.
Maximen und Reflexionen, Sechste
Abtheilung.

814. What I chiefly admired, and thought altogether unaccountable, was the
strong disposition I observed in them [the mathematicians of Laputa]
towards news and politics; perpetually inquiring into public affairs; giving
their judgments in matters of state; and passionately disputing every inch of
party opinion. I have indeed observed the same disposition among most of
the mathematicians I have known in Europe, although I could never
discover the least analogy between the two sciences.—Swift, Jonathan.
Gulliver’s Travels, Part 3, chap. 2.

815. The great mathematician, like the great poet or naturalist or great
administrator, is born. My contention shall be that where the mathematic
endowment is found, there will usually be found associated with it, as
essential implications in it, other endowments in generous measure, and that
the appeal of the science is to the whole mind, direct no doubt to the central
powers of thought, but indirectly through sympathy of all, rousing,
enlarging, developing, emancipating all, so that the faculties of will, of

intellect and feeling learn to respond, each in its appropriate order and
degree, like the parts of an orchestra to the “urge and ardor” of its leader
and lord.—Keyser, C. J.
Lectures on Science, Philosophy and Art
(New York, 1908), p. 22.

816. Whoever limits his exertions to the gratification of others, whether by
personal exhibition, as in the case of the actor and of the mimic, or by those
kinds of literary composition which are calculated for no end but to please
or to entertain, renders himself, in some measure, dependent on their
caprices and humours. The diversity among men, in their judgments
concerning the objects of taste, is incomparably greater than in their
speculative conclusions; and accordingly, a mathematician will publish to
the world a geometrical demonstration, or a philosopher, a process of
abstract reasoning, with a confidence very different from what a poet would
feel, in communicating one of his productions even to a friend.—Stewart,
Dugald.
Elements of the Philosophy of the Human
Mind, Part 3, chap. 1, sect. 3.

817. Considering that, among all those who up to this time made
discoveries in the sciences, it was the mathematicians alone who had been
able to arrive at demonstrations—that is to say, at proofs certain and evident
—I did not doubt that I should begin with the same truths that they have
investigated, although I had looked for no other advantage from them than
to accustom my mind to nourish itself upon truths and not to be satisfied
with false reasons.—Descartes.
Discourse upon Method, Part 2; Philosophy
of Descartes [Torrey] (New York, 1892), p.
48.

818. When the late Sophus Lie ... was asked to name the characteristic
endowment of the mathematician, his answer was the following quaternion:
Phantasie, Energie, Selbstvertrauen, Selbstkritik.—Keyser, C. J.

Lectures on Philosophy, Science and Art
(New York, 1908), p. 31.

819. The existence of an extensive Science of Mathematics, requiring the
highest scientific genius in those who contributed to its creation, and calling
for the most continued and vigorous exertion of intellect in order to
appreciate it when created, etc.—Mill, J. S.
System of Logic, Bk. 2, chap. 4, sect. 4.

820. It may be true, that men, who are mere mathematicians, have certain
specific shortcomings, but that is not the fault of mathematics, for it is
equally true of every other exclusive occupation. So there are mere
philologists, mere jurists, mere soldiers, mere merchants, etc. To such idle
talk it might further be added: that whenever a certain exclusive occupation
is coupled with specific shortcomings, it is likewise almost certainly
divorced from certain other shortcomings.—Gauss.
Gauss-Schumacher Briefwechsel, Bd. 4,
(Altona, 1862), p. 387.

821. Mathematical studies ... when combined, as they now generally are,
with a taste for physical science, enlarge infinitely our views of the wisdom
and power displayed in the universe. The very intimate connexion indeed,
which, since the date of the Newtonian philosophy, has existed between the
different branches of mathematical and physical knowledge, renders such a
character as that of a mere mathematician a very rare and scarcely possible
occurrence.—Stewart, Dugald.
Elements of the Philosophy of the Human
Mind, part 3, chap. 1, sect. 3.

822. Once when lecturing to a class he [Lord Kelvin] used the word
“mathematician,” and then interrupting himself asked his class: “Do you
know what a mathematician is?” Stepping to the blackboard he wrote upon
it:—

Then putting his finger on what he had written, he turned to his class
and said: “A mathematician is one to whom that is as obvious as that twice
two makes four is to you. Liouville was a mathematician.—Thompson, S. P.
Life of Lord Kelvin (London, 1910), p. 1139.

823. It is not surprising, in view of the polydynamic constitution of the
genuinely mathematical mind, that many of the major heroes of the science,
men like Desargues and Pascal, Descartes and Leibnitz, Newton, Gauss and
Bolzano, Helmholtz and Clifford, Riemann and Salmon and Plücker and
Poincaré, have attained to high distinction in other fields not only of science
but of philosophy and letters too. And when we reflect that the very greatest
mathematical achievements have been due, not alone to the peering,
microscopic, histologic vision of men like Weierstrass, illuminating the
hidden recesses, the minute and intimate structure of logical reality, but to
the larger vision also of men like Klein who survey the kingdoms of
geometry and analysis for the endless variety of things that nourish there, as
the eye of Darwin ranged over the flora and fauna of the world, or as a
commercial monarch contemplates its industry, or as a statesman beholds an
empire; when we reflect not only that the Calculus of Probability is a
creation of mathematics but that the master mathematician is constantly
required to exercise judgment—judgment, that is, in matters not of certainty
—balancing probabilities not yet reduced nor even reducible perhaps to
calculation; when we reflect that he is called upon to exercise a function
analogous to that of the comparative anatomist like Cuvier, comparing
theories and doctrines of every degree of similarity and dissimilarity of
structure; when, finally, we reflect that he seldom deals with a single idea at
a time, but is for the most part engaged in wielding organized hosts of them,
as a general wields at once the division of an army or as a great civil
administrator directs from his central office diverse and scattered but related
groups of interests and operations; then, I say, the current opinion that
devotion to mathematics unfits the devotee for practical affairs should be
known for false on a priori grounds. And one should be thus prepared to
find that as a fact Gaspard Monge, creator of descriptive geometry, author

of the classic “Applications de l’analyse à la géométrie;” Lazare Carnot,
author of the celebrated works, “Géométrie de position,” and “Réflections
sur la Métaphysique du Calcul infinitesimal;” Fourier, immortal creator of
the “Théorie analytique de la chaleur;” Arago, rightful inheritor of Monge’s
chair of geometry; Poncelet, creator of pure projective geometry; one
should not be surprised, I say, to find that these and other mathematicians in
a land sagacious enough to invoke their aid, rendered, alike in peace and in
war, eminent public service.—Keyser, C. J.
Lectures on Science, Philosophy and Art
(New York, 1908), pp. 32-33.

824. If in Germany the goddess Justitia had not the unfortunate habit of
depositing the ministerial portfolios only in the cradles of her own progeny,
who knows how many a German mathematician might not also have made
an excellent minister.—Pringsheim, A.
Jahresbericht der Deutschen Mathematiker
Vereinigung, Bd. 13 (1904), p. 372.

825. We pass with admiration along the great series of mathematicians, by
whom the science of theoretical mechanics has been cultivated, from the
time of Newton to our own. There is no group of men of science whose
fame is higher or brighter. The great discoveries of Copernicus, Galileo,
Newton, had fixed all eyes on those portions of human knowledge on which
their successors employed their labors. The certainty belonging to this line
of speculation seemed to elevate mathematicians above the students of
other subjects; and the beauty of mathematical relations and the subtlety of
intellect which may be shown in dealing with them, were fitted to win
unbounded applause. The successors of Newton and the Bernoullis, as
Euler, Clairaut, D’Alembert, Lagrange, Laplace, not to introduce living
names, have been some of the most remarkable men of talent which the
world has seen.—Whewell, W.
History of the Inductive Sciences, Vol. 1, Bk.
4, chap. 6, sect. 6.

826. The persons who have been employed on these problems of applying
the properties of matter and the laws of motion to the explanation of the
phenomena of the world, and who have brought to them the high and
admirable qualities which such an office requires, have justly excited in a
very eminent degree the admiration which mankind feels for great
intellectual powers. Their names occupy a distinguished place in literary
history; and probably there are no scientific reputations of the last century
higher, and none more merited, than those earned by great mathematicians
who have laboured with such wonderful success in unfolding the
mechanism of the heavens; such for instance as D’Alembert, Clairaut,
Euler, Lagrange, Laplace.—Whewell, W.
Astronomy and General Physics (London,
1833), Bk. 3, chap. 4, p. 327.

827. Two extreme views have always been held as to the use of
mathematics. To some, mathematics is only measuring and calculating
instruments, and their interest ceases as soon as discussions arise which
cannot benefit those who use the instruments for the purposes of application
in mechanics, astronomy, physics, statistics, and other sciences. At the other
extreme we have those who are animated exclusively by the love of pure
science. To them pure mathematics, with the theory of numbers at the head,
is the only real and genuine science, and the applications have only an
interest in so far as they contain or suggest problems in pure mathematics.
Of the two greatest mathematicians of modern times, Newton and
Gauss, the former can be considered as a representative of the first, the
latter of the second class; neither of them was exclusively so, and Newton’s
inventions in the science of pure mathematics were probably equal to
Gauss’s work in applied mathematics. Newton’s reluctance to publish the
method of fluxions invented and used by him may perhaps be attributed to
the fact that he was not satisfied with the logical foundations of the
Calculus; and Gauss is known to have abandoned his electro-dynamic
speculations, as he could not find a satisfying physical basis....
Newton’s greatest work, the “Principia”, laid the foundation of
mathematical physics; Gauss’s greatest work, the “Disquisitiones
Arithmeticae”, that of higher arithmetic as distinguished from algebra. Both

works, written in the synthetic style of the ancients, are difficult, if not
deterrent, in their form, neither of them leading the reader by easy steps to
the results. It took twenty or more years before either of these works
received due recognition; neither found favour at once before that great
tribunal of mathematical thought, the Paris Academy of Sciences....
The country of Newton is still pre-eminent for its culture of
mathematical physics, that of Gauss for the most abstract work in
mathematics.—Merz, J. T.
History of European Thought in the
Nineteenth Century (Edinburgh and London,
1903), p. 630.

828. As there is no study which may be so advantageously entered upon
with a less stock of preparatory knowledge than mathematics, so there is
none in which a greater number of uneducated men have raised themselves,
by their own exertions, to distinction and eminence.... Many of the
intellectual defects which, in such cases, are commonly placed to the
account of mathematical studies, ought to be ascribed to the want of a
liberal education in early youth.—Stewart, Dugald.
Elements of the Philosophy of the Human
Mind, Part 3, chap. 1, sect. 3.

829. I know, indeed, and can conceive of no pursuit so antagonistic to the
cultivation of the oratorical faculty ... as the study of Mathematics. An
eloquent mathematician must, from the nature of things, ever remain as rare
a phenomenon as a talking fish, and it is certain that the more anyone gives
himself up to the study of oratorical effect the less will he find himself in a
fit state to mathematicize. It is the constant aim of the mathematician to
reduce all his expressions to their lowest terms, to retrench every
superfluous word and phrase, and to condense the Maximum of meaning
into the Minimum of language. He has to turn his eye ever inwards, to see
everything in its dryest light, to train and inure himself to a habit of internal
and impersonal reflection and elaboration of abstract thought, which makes
it most difficult for him to touch or enlarge upon any of those themes which
appeal to the emotional nature of his fellow-men. When called upon to

speak in public he feels as a man might do who has passed all his life in
peering through a microscope, and is suddenly called upon to take charge of
a astronomical observatory. He has to get out of himself, as it were, and
change the habitual focus of his vision.—Sylvester, J. J.
Baltimore Address; Mathematical Papers,
Vol. 3, pp. 72-73.

830. An accomplished mathematician, i.e. a most wretched orator.—
Barrow, Isaac.
Mathematical Lectures (London, 1734), p.
32.

831. Nemo mathematicus genium indemnatus habebit. [No mathematician2
is esteemed a genius until condemned.]
Juvenal, Liberii, Satura VI, 562.

832. Taking ... the mathematical faculty, probably fewer than one in a
hundred really possess it, the great bulk of the population having no natural
ability for the study, or feeling the slightest interest in it.3 And if we attempt
to measure the amount of variation in the faculty itself between a first-class
mathematician and the ordinary run of people who find any kind of
calculation confusing and altogether devoid of interest, it is probable that
the former could not be estimated at less than a hundred times the latter, and
perhaps a thousand times would more nearly measure the difference
between them.—Wallace, A. R.
Darwinism, chap. 15.

833. ... the present gigantic development of the mathematical faculty is
wholly unexplained by the theory of natural selection, and must be due to
some altogether distinct cause.—Wallace, A. R.
Darwinism, chap. 15.

834. Dr. Wallace, in his “Darwinism”, declares that he can find no ground
for the existence of pure scientists, especially mathematicians, on the
hypothesis of natural selection. If we put aside the fact that great power in
theoretical science is correlated with other developments of increasing
brain-activity, we may, I think, still account for the existence of pure
scientists as Dr. Wallace would himself account for that of worker-bees.
Their function may not fit them individually to survive in the struggle for
existence, but they are a source of strength and efficiency to the society
which produces them.—Pearson, Karl.
Grammar of Science (London, 1911), Part 1,
p. 221.

835. It is only in mathematics, and to some extent in poetry, that originality
may be attained at an early age, but even then it is very rare (Newton and
Keats are examples), and it is not notable until adolescence is completed.—
Ellis, Havelock.
A Study of British Genius (London, 1904), p.
142.

836. The Anglo-Dane appears to possess an aptitude for mathematics which
is not shared by the native of any other English district as a whole, and it is
in the exact sciences that the Anglo-Dane triumphs.4—Ellis, Havelock.
A Study of British Genius (London, 1904), p.
69.

837. In the whole history of the world there was never a race with less
liking for abstract reasoning than the Anglo-Saxon.... Common-sense and
compromise are believed in, logical deductions from philosophical
principles are looked upon with suspicion, not only by legislators, but by all
our most learned professional men.—Perry, John.
The Teaching of Mathematics (London,
1902), pp. 20-21.

838. The degree of exactness of the intuition of space may be different in
different individuals, perhaps even in different races. It would seem as if a
strong naïve space-intuition were an attribute pre-eminently of the Teutonic
race, while the critical, purely logical sense is more fully developed in the
Latin and Hebrew races. A full investigation of this subject, somewhat on
the lines suggested by Francis Galton in his researches on heredity, might
be interesting.—Klein, Felix.
The Evanston Colloquium Lectures (New
York, 1894), p. 46.

839. This [the fact that the pursuit of mathematics brings into harmonious
action all the faculties of the human mind] accounts for the extraordinary
longevity of all the greatest masters of the Analytic art, the Dii Majores of
the mathematical Pantheon. Leibnitz lived to the age of 70; Euler to 76;
Lagrange to 77; Laplace to 78; Gauss to 78; Plato, the supposed inventor of
the conic sections, who made mathematics his study and delight, who called
them the handles or aids to philosophy, the medicine of the soul, and is said
never to have let a day go by without inventing some new theorems, lived
to 82; Newton, the crown and glory of his race, to 85; Archimedes, the
nearest akin, probably, to Newton in genius, was 75, and might have lived
on to be 100, for aught we can guess to the contrary, when he was slain by
the impatient and ill-mannered sergeant, sent to bring him before the
Roman general, in the full vigour of his faculties, and in the very act of
working out a problem; Pythagoras, in whose school, I believe, the word
mathematician (used, however, in a somewhat wider than its present sense)
originated, the second founder of geometry, the inventor of the matchless
theorem which goes by his name, the pre-cognizer of the undoubtedly mis-
called Copernican theory, the discoverer of the regular solids and the
musical canon who stands at the very apex of this pyramid of fame, (if we
may credit the tradition) after spending 22 years studying in Egypt, and 12
in Babylon, opened school when 56 or 57 years old in Magna Græcia,
married a young wife when past 60, and died, carrying on his work with
energy unspent to the last, at the age of 99. The mathematician lives long
and lives young; the wings of his soul do not early drop off, nor do its pores
become clogged with the earthy particles blown from the dusty highways of
vulgar life.—Sylvester, J. J.

Presidential Address to the British
Association; Collected Mathematical Papers,
Vol. 2 (1908), p. 658.

840. The game of chess has always fascinated mathematicians, and there is
reason to suppose that the possession of great powers of playing that game
is in many features very much like the possession of great mathematical
ability. There are the different pieces to learn, the pawns, the knights, the
bishops, the castles, and the queen and king. The board possesses certain
possible combinations of squares, as in rows, diagonals, etc. The pieces are
subject to certain rules by which their motions are governed, and there are
other rules governing the players.... One has only to increase the number of
pieces, to enlarge the field of the board, and to produce new rules which are
to govern either the pieces or the player, to have a pretty good idea of what
mathematics consists.—Shaw, J. B.
What is Mathematics? Bulletin American
Mathematical Society Vol. 18 (1912), pp.
386-387.

841. Every man is ready to join in the approval or condemnation of a
philosopher or a statesman, a poet or an orator, an artist or an architect. But
who can judge of a mathematician? Who will write a review of Hamilton’s
Quaternions, and show us wherein it is superior to Newton’s Fluxions?—
Hill, Thomas.
Imagination in Mathematics; North
American Review, Vol. 85, p. 224.

842. The pursuit of mathematical science makes its votary appear singularly
indifferent to the ordinary interests and cares of men. Seeking eternal truths,
and finding his pleasures in the realities of form and number, he has little
interest in the disputes and contentions of the passing hour. His views on
social and political questions partake of the grandeur of his favorite
contemplations, and, while careful to throw his mite of influence on the side
of right and truth, he is content to abide the workings of those general laws
by which he doubts not that the fluctuations of human history are as

unerringly guided as are the perturbations of the planetary hosts.—Hill,
Thomas.
Imagination in Mathematics; North
American Review, Vol. 85, p. 227.

843. There is something sublime in the secrecy in which the really great
deeds of the mathematician are done. No popular applause follows the act;
neither contemporary nor succeeding generations of the people understand
it. The geometer must be tried by his peers, and those who truly deserve the
title of geometer or analyst have usually been unable to find so many as
twelve living peers to form a jury. Archimedes so far outstripped his
competitors in the race, that more than a thousand years elapsed before any
man appeared, able to sit in judgment on his work, and to say how far he
had really gone. And in judging of those men whose names are worthy of
being mentioned in connection with his,—Galileo, Descartes, Leibnitz,
Newton, and the mathematicians created by Leibnitz and Newton’s
calculus,—we are forced to depend upon their testimony of one another.
They are too far above our reach for us to judge of them.—Hill, Thomas.
Imagination in Mathematics; North
American Review, Vol. 85, p. 223.

844. To think the thinkable—that is the mathematician’s aim.—Keyser, C.
J.
The Universe and Beyond; Hibbert Journal,
Vol. 3 (1904-1905), p. 312.

845. Every common mechanic has something to say in his craft about good
and evil, useful and useless, but these practical considerations never enter
into the purview of the mathematician.—Aristippus the Cyrenaic.
Quoted in Hicks, R. D., Stoic and Epicurean,
(New York, 1910) p. 210.

CHAPTER IX
PERSONS AND ANECDOTES
(A-M)

901. Alexander is said to have asked Menæchmus to teach him geometry
concisely, but Menæchmus replied: “O king, through the country there are
royal roads and roads for common citizens, but in geometry there is one
road for all.”
Stobœus (Edition Wachsmuth, Berlin, 1884),
Ecl. 2, p. 30

902. Alexander the king of the Macedonians, began like a wretch to learn
geometry, that he might know how little the earth was, whereof he had
possessed very little. Thus, I say, like a wretch for this, because he was to
understand that he did bear a false surname. For who can be great in so
small a thing? Those things that were delivered were subtile, and to be
learned by diligent attention: not which that mad man could perceive, who
sent his thoughts beyond the ocean sea. Teach me, saith he, easy things. To
whom his master said: These things be the same, and alike difficult unto all.
Think thou that the nature of things saith this. These things whereof thou
complainest, they are the same unto all: more easy things can be given unto
none; but whosoever will, shall make those things more easy unto himself.
How? With uprightness of mind.—Seneca.
Epistle 91 [Thomas Lodge].

903. Archimedes ... had stated that given the force, any given weight might
be moved, and even boasted, we are told, relying on the strength of
demonstration, that if there were another earth, by going into it he could
remove this. Hiero being struck with amazement at this, and entreating him
to make good this problem by actual experiment, and show some great

weight moved by a small engine, he fixed accordingly upon a ship of
burden out of the king’s arsenal, which could not be drawn out of the dock
without great labor and many men; and, loading her with many passengers
and a full freight, sitting himself the while far off with no great endeavor,
but only holding the head of the pulley in his hand and drawing the cords by
degrees, he drew the ship in a straight line, as smoothly and evenly, as if she
had been in the sea. The king, astonished at this, and convinced of the
power of the art, prevailed upon Archimedes to make him engines
accommodated to all the purposes, offensive and defensive, of a siege ... the
apparatus was, in most opportune time, ready at hand for the Syracusans,
and with it also the engineer himself.—Plutarch.
Life of Marcellus [Dryden].

904. These machines [used in the defense of the Syracusans against the
Romans under Marcellus] he [Archimedes] had designed and contrived, not
as matters of any importance, but as mere amusements in geometry; in
compliance with king Hiero’s desire and request, some time before, that he
should reduce to practice some part of his admirable speculation in science,
and by accommodating the theoretic truth to sensation and ordinary use,
bring it more within the appreciation of people in general. Eudoxus and
Archytas had been the first originators of this far-famed and highly-prized
art of mechanics, which they employed as an elegant illustration of
geometrical truths, and as means of sustaining experimentally, to the
satisfaction of the senses conclusions too intricate for proof by words and
diagrams. As, for example, to solve the problem, so often required in
constructing geometrical figures, given the two extremes, to find the two
mean lines of a proportion, both these mathematicians had recourse to the
aid of instruments, adapting to their purpose certain curves and sections of
lines. But what with Plato’s indignation at it, and his invectives against it as
the mere corruption and annihilation of the one good of geometry,—which
was thus shamefully turning its back upon the unembodied objects of pure
intelligence to recur to sensation, and to ask help (not to be obtained
without base supervisions and depravation) from matter; so it was that
mechanics came to be separated from geometry, and, repudiated and
neglected by philosophers, took its place as a military art.—Plutarch.
Life of Marcellus [Dryden].

905. Archimedes was not free from the prevailing notion that geometry was
degraded by being employed to produce anything useful. It was with
difficulty that he was induced to stoop from speculation to practice. He was
half ashamed of those inventions which were the wonder of hostile nations,
and always spoke of them slightingly as mere amusements, as trifles in
which a mathematician might be suffered to relax his mind after intense
application to the higher parts of his science.—Macaulay.
Lord Bacon; Edinburgh Review, July 1837;
Critical and Miscellaneous Essays (New
York, 1879), Vol. 1, p. 380.

906.

Call Archimedes from his buried tomb
Upon the plain of vanished Syracuse,
And feelingly the sage shall make report
How insecure, how baseless in itself,
Is the philosophy, whose sway depends
On mere material instruments—how weak
Those arts, and high inventions, if unpropped
By virtue.
—Wordsworth.
The Excursion.

907.
Zu Archimedes kam einst ein wissbegieriger Jüngling.
“Weihe mich,” sprach er zu ihm, “ein in die göttliche Kunst,
Die so herrliche Frucht dem Vaterlande getragen,
Und die Mauern der Stadt vor der Sambuca beschützt!”
“Göttlich nennst du die Kunst? Sie ists,” versetzte der Weise;
“Aber das war sie, mein Sohn, eh sie dem Staat noch gedient.
Willst du nur Früchte von ihr, die kann auch die Sterbliche zeugen;
Wer um die Göttin freit, suche in ihr nicht das Weib.”
—Schiller.
Archimedes und der Schüler.

[To Archimedes once came a youth intent upon knowledge.
Said he “Initiate me into the Science divine,
Which to our country has borne glorious fruits in abundance,
And which the walls of the town ’gainst the Sambuca protects.”
“Callst thou the science divine? It is so,” the wise man responded;
“But so it was, my son, ere the state by her service was blest.
Would’st thou have fruit of her only? Mortals with that can provide
thee,
He who the goddess would woo, seek not the woman in her.”]

908. Archimedes possessed so high a spirit, so profound a soul, and such
treasures of highly scientific knowledge, that though these inventions [used
to defend Syracuse against the Romans] had now obtained him the renown

of more than human sagacity, he yet would not deign to leave behind him
any commentary or writing on such subjects; but, repudiating as sordid and
ignoble the whole trade of engineering, and every sort of art that lends itself
to mere use and profit, he placed his whole affection and ambition in those
purer speculations where there can be no reference to the vulgar needs of
life; studies, the superiority of which to all others is unquestioned, and in
which the only doubt can be whether the beauty and grandeur of the
subjects examined, or the precision and cogency of the methods and means
of proof, most deserve our admiration.—Plutarch.
Life of Marcellus [Dryden].

909. Nothing afflicted Marcellus so much as the death of Archimedes, who
was then, as fate would have it, intent upon working out some problem by a
diagram, and having fixed his mind alike and his eyes upon the subject of
his speculation, he never noticed the incursion of the Romans, nor that the
city was taken. In this transport of study and contemplation, a soldier,
unexpectedly coming up to him, commanded him to follow to Marcellus,
which he declined to do before he had worked out his problem to a
demonstration; the soldier, enraged, drew his sword and ran him through.
Others write, that a Roman soldier, running upon him with a drawn sword,
offered to kill him; and that Archimedes, looking back, earnestly besought
him to hold his hand a little while, that he might not leave what he was at
work upon inconclusive and imperfect; but the soldier, nothing moved by
his entreaty, instantly killed him. Others again relate, that as Archimedes
was carrying to Marcellus mathematical instruments, dials, spheres, and
angles, by which the magnitude of the sun might be measured to the sight,
some soldiers seeing him, and thinking that he carried gold in a vessel, slew
him. Certain it is, that his death was very afflicting to Marcellus; and that
Marcellus ever after regarded him that killed him as a murderer; and that he
sought for his kindred and honoured them with signal favours.—Plutarch.
Life of Marcellus [Dryden].

910. [Archimedes] is said to have requested his friends and relations that
when he was dead, they would place over his tomb a sphere containing a
cylinder, inscribing it with the ratio which the containing solid bears to the
contained.—Plutarch.

Life of Marcellus [Dryden].

911. Archimedes, who combined a genius for mathematics with a physical
insight, must rank with Newton, who lived nearly two thousand years later,
as one of the founders of mathematical physics.... The day (when having
discovered his famous principle of hydrostatics he ran through the streets
shouting Eureka! Eureka!) ought to be celebrated as the birthday of
mathematical physics; the science came of age when Newton sat in his
orchard.—Whitehead, A. N.
An Introduction to Mathematics (New York,
1911), p. 38.

912. It is not possible to find in all geometry more difficult and more
intricate questions or more simple and lucid explanations [than those given
by Archimedes]. Some ascribe this to his natural genius; while others think
that incredible effort and toil produced these, to all appearance, easy and
unlaboured results. No amount of investigation of yours would succeed in
attaining the proof, and yet, once seen, you immediately believe you would
have discovered it; by so smooth and so rapid a path he leads you to the
conclusion required.—Plutarch.
Life of Marcellus [Dryden].

913. One feature which will probably most impress the mathematician
accustomed to the rapidity and directness secured by the generality of
modern methods is the deliberation with which Archimedes approaches the
solution of any one of his main problems. Yet this very characteristic, with
its incidental effects, is calculated to excite the more admiration because the
method suggests the tactics of some great strategist who foresees
everything, eliminates everything not immediately conducive to the
execution of his plan, masters every position in its order, and then suddenly
(when the very elaboration of the scheme has almost obscured, in the mind
of the spectator, its ultimate object) strikes the final blow. Thus we read in
Archimedes proposition after proposition the bearing of which is not
immediately obvious but which we find infallibly used later on; and we are
led by such easy stages that the difficulties of the original problem, as

presented at the outset, are scarcely appreciated. As Plutarch says: “It is not
possible to find in geometry more difficult and troublesome questions, or
more simple and lucid explanations.” But it is decidedly a rhetorical
exaggeration when Plutarch goes on to say that we are deceived by the
easiness of the successive steps into the belief that anyone could have
discovered them for himself. On the contrary, the studied simplicity and the
perfect finish of the treatises involve at the same time an element of
mystery. Though each step depends on the preceding ones, we are left in the
dark as to how they were suggested to Archimedes. There is, in fact, much
truth in a remark by Wallis to the effect that he seems “as it were of set
purpose to have covered up the traces of his investigation as if he had
grudged posterity the secret of his method of inquiry while he wished to
extort from them assent to his results.” Wallis adds with equal reason that
not only Archimedes but nearly all the ancients so hid away from posterity
their method of Analysis (though it is certain that they had one) that more
modern mathematicians found it easier to invent a new Analysis than to
seek out the old.—Heath, T. L.
The Works of Archimedes (Cambridge,
1897), Preface.

914. It is a great pity Aristotle had not understood mathematics as well as
Mr. Newton, and made use of it in his natural philosophy with good
success: his example had then authorized the accommodating of it to
material things.—Locke, John.
Second Reply to the Bishop of Worcester.

915. The opinion of Bacon on this subject [geometry] was diametrically
opposed to that of the ancient philosophers. He valued geometry chiefly, if
not solely, on account of those uses, which to Plato appeared so base. And it
is remarkable that the longer Bacon lived the stronger this feeling became.
When in 1605 he wrote the two books on the Advancement of Learning, he
dwelt on the advantages which mankind derived from mixed mathematics;
but he at the same time admitted that the beneficial effect produced by
mathematical study on the intellect, though a collateral advantage, was “no
less worthy than that which was principal and intended.” But it is evident
that his views underwent a change. When near twenty years later, he

published the De Augmentis, which is the Treatise on the Advancement of
Learning, greatly expanded and carefully corrected, he made important
alterations in the part which related to mathematics. He condemned with
severity the pretensions of the mathematicians, “delicias et fastum
mathematicorum.” Assuming the well-being of the human race to be the
end of knowledge, he pronounced that mathematical science could claim no
higher rank than that of an appendage or an auxiliary to other sciences.
Mathematical science, he says, is the handmaid of natural philosophy; she
ought to demean herself as such; and he declares that he cannot conceive by
what ill chance it has happened that she presumes to claim precedence over
her mistress.—Macaulay.
Lord Bacon: Edinburgh Review, July, 1837;
Critical and Miscellaneous Essays (New
York, 1879), Vol. 1, p. 380.

916. If Bacon erred here [in valuing mathematics only for its uses], we must
acknowledge that we greatly prefer his error to the opposite error of Plato.
We have no patience with a philosophy which, like those Roman matrons
who swallowed abortives in order to preserve their shapes, takes pains to be
barren for fear of being homely.—Macaulay.
Lord Bacon, Edinburgh Review, July, 1837;
Critical and Miscellaneous Essays (New
York, 1879), Vol. 2, p. 381.

917. He [Lord Bacon] appears to have been utterly ignorant of the
discoveries which had just been made by Kepler’s calculations ... he does
not say a word about Napier’s Logarithms, which had been published only
nine years before and reprinted more than once in the interval. He
complained that no considerable advance had been made in Geometry
beyond Euclid, without taking any notice of what had been done by
Archimedes and Apollonius. He saw the importance of determining
accurately the specific gravities of different substances, and himself
attempted to form a table of them by a rude process of his own, without
knowing of the more scientific though still imperfect methods previously
employed by Archimedes, Ghetaldus and Porta. He speaks of the εὔυρηκα
of Archimedes in a manner which implies that he did not clearly appreciate

either the problem to be solved or the principles upon which the solution
depended. In reviewing the progress of Mechanics, he makes no mention
either of Archimedes, or Stevinus, Galileo, Guldinus, or Ghetaldus. He
makes no allusion to the theory of Equilibrium. He observes that a ball of
one pound weight will fall nearly as fast through the air as a ball of two,
without alluding to the theory of acceleration of falling bodies, which had
been made known by Galileo more than thirty years before. He proposed an
inquiry with regard to the lever,—namely, whether in a balance with arms
of different length but equal weight the distance from the fulcrum has any
effect upon the inclination—though the theory of the lever was as well
understood in his own time as it is now.... He speaks of the poles of the
earth as fixed, in a manner which seems to imply that he was not acquainted
with the precession of the equinoxes; and in another place, of the north pole
being above and the south pole below, as a reason why in our hemisphere
the north winds predominate over the south.—Spedding, J.
Works of Francis Bacon (Boston), Preface to
De Interpretatione Naturae Prooemium.

918. Bacon himself was very ignorant of all that had been done by
mathematics; and, strange to say, he especially objected to astronomy being
handed over to the mathematicians. Leverrier and Adams, calculating an
unknown planet into a visible existence by enormous heaps of algebra,
furnish the last comment of note on this specimen of the goodness of
Bacon’s view.... Mathematics was beginning to be the great instrument of
exact inquiry: Bacon threw the science aside, from ignorance, just at the
time when his enormous sagacity, applied to knowledge, would have made
him see the part it was to play. If Newton had taken Bacon for his master,
not he, but somebody else, would have been Newton.—De Morgan, A.
Budget of Paradoxes (London, 1872), pp. 53-
54.

919. Daniel Bernoulli used to tell two little adventures, which he said had
given him more pleasure than all the other honours he had received.
Travelling with a learned stranger, who, being pleased with his
conversation, asked his name; “I am Daniel Bernoulli,” answered he with
great modesty; “and I,” said the stranger (who thought he meant to laugh at

him) “am Isaac Newton.” Another time, having to dine with the celebrated
Koenig, the mathematician, who boasted, with some degree of self-
complacency, of a difficult problem he had solved with much trouble,
Bernoulli went on doing the honours of his table, and when they went to
drink coffee he presented Koenig with a solution of the problem more
elegant than his own.—Hutton, Charles.
A Philosophical and Mathematical
Dictionary (London, 1815), Vol. 1, p. 226.

920. Following the example of Archimedes who wished his tomb decorated
with his most beautiful discovery in geometry and ordered it inscribed with
a cylinder circumscribed by a sphere, James Bernoulli requested that his
tomb be inscribed with his logarithmic spiral together with the words,
“Eadem mutata resurgo,” a happy allusion to the hope of the Christians,
which is in a way symbolized by the properties of that curve.—Fontenelle.
Eloge de M. Bernoulli; Oeuvres de
Fontenelle, t. 5 (1758), p. 112.

921. This formula [for computing Bernoulli’s numbers] was first given by
James Bernoulli. He gave no general demonstration; but was quite aware of
the importance of his theorem, for he boasts that by means of it he
calculated intra semi-quadrantem horae! the sum of the 10th powers of the
first thousand integers, and found it to be
91,409,924,241,424,243,424,241,924,242,500.
—Chrystal, G.
Algebra, Part 2 (Edinburgh, 1879), p. 209.

922. In the year 1692, James Bernoulli, discussing the logarithmic spiral [or
equiangular spiral, ρ = αθ] ... shows that it reproduces itself in its evolute, its
involute, and its caustics of both reflection and refraction, and then adds:
"But since this marvellous spiral, by such a singular and wonderful
peculiarity, pleases me so much that I can scarce be satisfied with thinking
about it, I have thought that it might not be inelegantly used for a symbolic
representation of various matters. For since it always produces a spiral
similar to itself, indeed precisely the same spiral, however it may be

involved or evolved, or reflected or refracted, it may be taken as an emblem
of a progeny always in all things like the parent, simillima filia matri. Or, if
it is not forbidden to compare a theorem of eternal truth to the mysteries of
our faith, it may be taken as an emblem of the eternal generation of the Son,
who as an image of the Father, emanating from him, as light from light,
remains ὁμοούσιος with him, howsoever overshadowed. Or, if you prefer,
since our spira mirabilis remains, amid all changes, most persistently itself,
and exactly the same as ever, it may be used as a symbol, either of fortitude
and constancy in adversity, or, of the human body, which after all its
changes, even after death, will be restored to its exact and perfect self, so
that, indeed, if the fashion of Archimedes were allowed in these days, I
should gladly have my tombstone bear this spiral, with the motto, “Though
changed, I arise again exactly the same, Eadem numero mutata resurgo.”—
Hill, Thomas.
The Uses of Mathesis; Bibliotheca Sacra,
Vol. 32, pp. 515-516.

923. Babbage was one of the founders of the Cambridge Analytical Society
whose purpose he stated was to advocate "the principles of pure d-ism as
opposed to the dot-age of the university.”—Ball, W. W. R.
History of Mathematics (London, 1901), p.
451.

924. Bolyai [Janos] when in garrison with cavalry officers, was provoked
by thirteen of them and accepted all their challenges on condition that he be
permitted after each duel to play a bit on his violin. He came out victor
from his thirteen duels, leaving his thirteen adversaries on the square.—
Halsted, G. B.
Bolyai’s Science Absolute of Space (Austin,
1896), Introduction, p. 29.

925. Bolyai [Janos] projected a universal language for speech as we have it
for music and mathematics.—Halsted, G. B.
Bolyai’s Science Absolute of Space (Austin,
1896), Introduction, p. 29.

926. [Bolyai’s Science Absolute of Space]—the most extraordinary two
dozen pages in the history of thought!—Halsted, G. B.
Bolyai’s Science Absolute of Space (Austin,
1896), Introduction, p. 18.

927. [Wolfgang Bolyai] was extremely modest. No monument, said he,
should stand over his grave, only an apple-tree, in memory of the three
apples: the two of Eve and Paris, which made hell out of earth, and that of
Newton, which elevated the earth again into the circle of the heavenly
bodies.—Cajori, F.
History of Elementary Mathematics (New
York, 1910), p. 273.

928. Bernard Bolzano dispelled the clouds that throughout all the foregone
centuries had enveloped the notion of Infinitude in darkness, completely
sheared the great term of its vagueness without shearing it of its strength,
and thus rendered it forever available for the purposes of logical discourse.
—Keyser, C. J.
Lectures on Science, Philosophy and Art
(New York, 1908), p. 42.

929. Let me tell you how at one time the famous mathematician Euclid
became a physician. It was during a vacation, which I spent in Prague as I
most always did, when I was attacked by an illness never before
experienced, which manifested itself in chilliness and painful weariness of
the whole body. In order to ease my condition I took up Euclid’s Elements
and read for the first time his doctrine of ratio, which I found treated there
in a manner entirely new to me. The ingenuity displayed in Euclid’s
presentation filled me with such vivid pleasure, that forthwith I felt as well
as ever.—Bolzano, Bernard.
Selbstbiographie (Wien, 1875), p. 20.

930. Mr. Cayley, of whom it may be so truly said, whether the matter he
takes in hand be great or small, “nihil tetigit quod non ornavit,”....—

Sylvester, J. J.
Philosophic Transactions of the Royal
Society, Vol. 17 (1864), p. 605.

931. It is not Cayley’s way to analyze concepts into their ultimate
elements.... But he is master of the empirical utilization of the material: in
the way he combines it to form a single abstract concept which he
generalizes and then subjects to computative tests, in the way the newly
acquired data are made to yield at a single stroke the general comprehensive
idea to the subsequent numerical verification of which years of labor are
devoted. Cayley is thus the natural philosopher among mathematicians.—
Noether, M.
Mathematische Annalen, Bd. 46 (1895), p.
479.

932. When Cayley had reached his most advanced generalizations he
proceeded to establish them directly by some method or other, though he
seldom gave the clue by which they had first been obtained: a proceeding
which does not tend to make his papers easy reading....
His literary style is direct, simple and clear. His legal training had an
influence, not merely upon his mode of arrangement but also upon his
expression; the result is that his papers are severe and present a curious
contrast to the luxuriant enthusiasm which pervades so many of Sylvester’s
papers. He used to prepare his work for publication as soon as he carried his
investigations in any subject far enough for his immediate purpose.... A
paper once written out was promptly sent for publication; this practice he
maintained throughout life.... The consequence is that he has left few
arrears of unfinished or unpublished papers; his work has been given by
himself to the world.—Forsyth, A. R.
Proceedings of London Royal Society, Vol. 58
(1895), pp. 23-24.

933. Cayley was singularly learned in the work of other men, and catholic
in his range of knowledge. Yet he did not read a memoir completely

through: his custom was to read only so much as would enable him to grasp
the meaning of the symbols and understand its scope. The main result
would then become to him a subject of investigation: he would establish it
(or test it) by algebraic analysis and, not infrequently, develop it so to obtain
other results. This faculty of grasping and testing rapidly the work of others,
together with his great knowledge, made him an invaluable referee; his
services in this capacity were used through a long series of years by a
number of societies to which he was almost in the position of standing
mathematical advisor.—Forsyth, A. R.
Proceedings London Royal Society, Vol. 58
(1895), pp. 11-12.

934. Bertrand, Darboux, and Glaisher have compared Cayley to Euler, alike
for his range, his analytical power, and, not least, for his prolific production
of new views and fertile theories. There is hardly a subject in the whole of
pure mathematics at which he has not worked.—Forsyth, A. R.
Proceedings London Royal Society, Vol. 58
(1895), p. 21.

935. The mathematical talent of Cayley was characterized by clearness and
extreme elegance of analytical form; it was re-enforced by an incomparable
capacity for work which has caused the distinguished scholar to be
compared with Cauchy.—Hermite, C.
Comptes Rendus, t. 120 (1895), p. 234.

936. J. J. Sylvester was an enthusiastic supporter of reform [in the teaching
of geometry]. The difference in attitude on this question between the two
foremost British mathematicians, J. J. Sylvester, the algebraist, and Arthur
Cayley, the algebraist and geometer, was grotesque. Sylvester wished to
bury Euclid “deeper than e’er plummet sounded” out of the schoolboy’s
reach; Cayley, an ardent admirer of Euclid, desired the retention of
Simson’s Euclid. When reminded that this treatise was a mixture of Euclid
and Simson, Cayley suggested striking out Simson’s additions and keeping
strictly to the original treatise.—Cajori, F.

History of Elementary Mathematics (New
York, 1910), p. 285.

937. Tait once urged the advantage of Quaternions on Cayley (who never
used them), saying: “You know Quaternions are just like a pocket-map.”
“That may be,” replied Cayley, “but you’ve got to take it out of your
pocket, and unfold it, before it’s of any use.” And he dismissed the subject
with a smile.—Thompson, S. P.
Life of Lord Kelvin (London, 1910), p. 1137.

938. As he [Clifford] spoke he appeared not to be working out a question,
but simply telling what he saw. Without any diagram or symbolic aid he
described the geometrical conditions on which the solution depended, and
they seemed to stand out visibly in space. There were no longer
consequences to be deduced, but real and evident facts which only required
to be seen.... So whole and complete was his vision that for the time the
only strange thing was that anybody should fail to see it in the same way.
When one endeavored to call it up again, and not till then, it became clear
that the magic of genius had been at work, and that the common sight had
been raised to that higher perception by the power that makes and
transforms ideas, the conquering and masterful quality of the human mind
which Goethe called in one word das Dämonische.—Pollock, F.
Clifford’s Lectures and Essays (New York,
1901), Vol. 1, Introduction, pp. 5-6.

939. Much of his [Clifford’s] best work was actually spoken before it was
written. He gave most of his public lectures with no visible preparation
beyond very short notes, and the outline seemed to be filled in without
effort or hesitation. Afterwards he would revise the lecture from a shorthand
writer’s report, or sometimes write down from memory almost exactly what
he had said. It fell out now and then, however, that neither of these things
was done; in such cases there is now no record of the lecture at all.—
Pollock, F.
Clifford’s Lectures and Essays (New York,
1901), Vol. 1, Introduction, p. 10.

940. I cannot find anything showing early aptitude for acquiring languages;
but that he [Clifford] had it and was fond of exercising it in later life is
certain. One practical reason for it was the desire of being able to read
mathematical papers in foreign journals; but this would not account for his
taking up Spanish, of which he acquired a competent knowledge in the
course of a tour to the Pyrenees. When he was at Algiers in 1876 he began
Arabic, and made progress enough to follow in a general way a course of
lessons given in that language. He read modern Greek fluently, and at one
time he was furious about Sanskrit. He even spent some time on
hieroglyphics. A new language is a riddle before it is conquered, a power in
the hand afterwards: to Clifford every riddle was a challenge, and every
chance of new power a divine opportunity to be seized. Hence he was
likewise interested in the various modes of conveying and expressing
language invented for special purposes, such as the Morse alphabet and
shorthand.... I have forgotten to mention his command of French and
German, the former of which he knew very well, and the latter quite
sufficiently;....—Pollock, F.
Clifford’s Lectures and Essays (New York,
1901), Vol. 1, Introduction, pp. 11-12.

941. The most remarkable thing was his [Clifford’s] great strength as
compared with his weight, as shown in some exercises. At one time he
could pull up on the bar with either hand, which is well known to be one of
the greatest feats of strength. His nerve at dangerous heights was
extraordinary. I am appalled now to think that he climbed up and sat on the
cross bars of the weathercock on a church tower, and when by way of doing
something worse I went up and hung by my toes to the bars he did the
same.
Quoted from a letter by one of Clifford’s
friends to Pollock, F.: Clifford’s Lectures and
Essays (New York, 1901), Vol. 1,
Introduction, p. 8.

942. [Comte] may truly be said to have created the philosophy of higher
mathematics.—Mill, J. S.
System of Logic (New York, 1846), p. 369.

943. These specimens, which I could easily multiply, may suffice to justify
a profound distrust of Auguste Comte, wherever he may venture to speak as
a mathematician. But his vast general ability, and that personal intimacy
with the great Fourier, which I most willingly take his own word for having
enjoyed, must always give an interest to his views on any subject of pure or
applied mathematics.—Hamilton, W. R.
Graves’ Life of W. R. Hamilton (New York,
1882-1889), Vol. 3, p. 475.

944. The manner of Demoivre’s death has a certain interest for
psychologists. Shortly before it, he declared that it was necessary for him to
sleep some ten minutes or a quarter of an hour longer each day than the
preceding one: the day after he had thus reached a total of something over
twenty-three hours he slept up to the limit of twenty-four hours, and then
died in his sleep.—Ball, W. W. R.
History of Mathematics (London, 1911), p.
394.

945. De Morgan was explaining to an actuary what was the chance that a
certain proportion of some group of people would at the end of a given time
be alive; and quoted the actuarial formula, involving π, which, in answer to
a question, he explained stood for the ratio of the circumference of a circle
to its diameter. His acquaintance, who had so far listened to the explanation
with interest, interrupted him and exclaimed, “My dear friend, that must be
a delusion, what can a circle have to do with the number of people alive at a
given time?”—Ball, W. W. R.
Mathematical Recreations and Problems
(London, 1896), p. 180; See also De
Morgan’s Budget of Paradoxes (London,
1872), p. 172.

946. A few days afterwards, I went to him [the same actuary referred to in
945] and very gravely told him that I had discovered the law of human
mortality in the Carlisle Table, of which he thought very highly. I told him
that the law was involved in this circumstance. Take the table of the

expectation of life, choose any age, take its expectation and make the
nearest integer a new age, do the same with that, and so on; begin at what
age you like, you are sure to end at the place where the age past is equal, or
most nearly equal, to the expectation to come. “You don’t mean that this
always happens?”—“Try it.” He did try, again and again; and found it as I
said. “This is, indeed, a curious thing; this is a discovery!” I might have sent
him about trumpeting the law of life: but I contented myself with informing
him that the same thing would happen with any table whatsoever in which
the first column goes up and the second goes down;....—De Morgan, A.
Budget of Paradoxes (London, 1872), p. 172.

947. [De Morgan relates that some person had made up 800 anagrams on
his name, of which he had seen about 650. Commenting on these he says:]
Two of these I have joined in the title-page:
[Ut agendo surgamus arguendo gustamus.]
A few of the others are personal remarks.
Great gun! do us a sum!
is a sneer at my pursuit; but,
Go! great sum!
is more dignified....
Adsum, nugator, suge!
is addressed to a student who continues talking after the lecture has
commenced: ...
Graduatus sum! nego
applies to one who declined to subscribe for an M. A. degree.—De
Morgan, Augustus.
Budget of Paradoxes (London, 1872), p. 82.

948. Descartes is the completest type which history presents of the purely
mathematical type of mind—that in which the tendencies produced by
mathematical cultivation reign unbalanced and supreme.—Mill, J. S.
An Examination of Sir W. Hamilton’s
Philosophy (London, 1878), p. 626.

949. To Descartes, the great philosopher of the 17th century, is due the
undying credit of having removed the bann which until then rested upon
geometry. The analytical geometry, as Descartes’ method was called, soon
led to an abundance of new theorems and principles, which far transcended
everything that ever could have been reached upon the path pursued by the
ancients.—Hankel, H.
Die Entwickelung der Mathematik in den
letzten Jahrhunderten (Tübingen, 1884), p.
10.

950. [The application of algebra has] far more than any of his metaphysical
speculations, immortalized the name of Descartes, and constitutes the
greatest single step ever made in the progress of the exact sciences.—Mill,
J. S.
An Examination of Sir W. Hamilton’s
Philosophy (London, 1878), p. 617.

951. ... καί φασιν ὅτι Πτολεμαῖος ἤρετό ποτε αύτόν [Εὐκλειδην], εἴ τίς
ἐστιν περὶ γεωμετρίαν ὁδὸς συντομωτέρα τῆς στοιχειώσεως· ὁδὲ
ἀπεκρὶνατο μὴ εἶναι βασιλικὴν ἀτραπὸν ἐπὶ γεωμετρίαν.
[ ... they say that Ptolemy once asked him (Euclid) whether there was
in geometry no shorter way than that of the elements, and he replied, “There
is no royal road to geometry.”]—Proclus.
(Edition Friedlein, 1873), Prol. II, 39.

952. Someone who had begun to read geometry with Euclid, when he had
learned the first proposition, asked Euclid, “But what shall I get by learning
these things?” whereupon Euclid called his slave and said, “Give him three-
pence, since he must make gain out of what he learns.”—Stobæus.
(Edition Wachsmuth, 1884), Ecl. II.

953. The sacred writings excepted, no Greek has been so much read and so
variously translated as Euclid.5—De Morgan, A.

Smith’s Dictionary of Greek and Roman
Biology and Mythology (London, 1902),
Article, “Eucleides.”

954. The thirteen books of Euclid must have been a tremendous advance,
probably even greater than that contained in the “Principia” of Newton.—
De Morgan, A.
Smith’s Dictionary of Greek and Roman
Biography and Mythology (London, 1902),
Article, “Eucleides.”

955. To suppose that so perfect a system as that of Euclid’s Elements was
produced by one man, without any preceding model or materials, would be
to suppose that Euclid was more than man. We ascribe to him as much as
the weakness of human understanding will permit, if we suppose that the
inventions in geometry, which had been made in a tract of preceding ages,
were by him not only carried much further, but digested into so admirable a
system, that his work obscured all that went before it, and made them be
forgot and lost.—Reid, Thomas.
Essay on the Powers of the Human Mind
(Edinburgh, 1812), Vol. 2, p. 368.

956. It is the invaluable merit of the great Basle mathematician Leonhard
Euler, to have freed the analytical calculus from all geometrical bonds, and
thus to have established analysis as an independent science, which from his
time on has maintained an unchallenged leadership in the field of
mathematics.—Hankel, H.
Die Entwickelung der Mathematik in den
letzten Jahrhunderten (Tübingen, 1884), p.
12.

957. We may safely say, that the whole form of modern mathematical
thinking was created by Euler. It is only with the greatest difficulty that one
is able to follow the writings of any author immediately preceding Euler,

because it was not yet known how to let the formulas speak for themselves.
This art Euler was the first one to teach.—Rudio, F.
Quoted by Ahrens W.: Scherz und Ernst in
der Mathematik (Leipzig, 1904), p. 251.

958. The general knowledge of our author [Leonhard Euler] was more
extensive than could well be expected, in one who had pursued, with such
unremitting ardor, mathematics and astronomy as his favorite studies. He
had made a very considerable progress in medical, botanical, and chemical
science. What was still more extraordinary, he was an excellent scholar, and
possessed in a high degree what is generally called erudition. He had
attentively read the most eminent writers of ancient Rome; the civil and
literary history of all ages and all nations was familiar to him; and
foreigners, who were only acquainted with his works, were astonished to
find in the conversation of a man, whose long life seemed solely occupied
in mathematical and physical researches and discoveries, such an extensive
acquaintance with the most interesting branches of literature. In this respect,
no doubt, he was much indebted to an uncommon memory, which seemed
to retain every idea that was conveyed to it, either from reading or from
meditation.—Hutton, Charles.
Philosophical and Mathematical Dictionary
(London, 1815), pp. 493-494.

959. Euler could repeat the Aeneid from the beginning to the end, and he
could even tell the first and last lines in every page of the edition which he
used. In one of his works there is a learned memoir on a question in
mechanics, of which, as he himself informs us, a verse of Aeneid6 gave him
the first idea.—Brewster, David.
Letters of Euler (New York, 1872), Vol. 1, p.
24.

960. Most of his [Euler’s] memoirs are contained in the transactions of the
Academy of Sciences at St. Petersburg, and in those of the Academy at
Berlin. From 1728 to 1783 a large portion of the Petropolitan transactions
were filled by his writings. He had engaged to furnish the Petersburg

Academy with memoirs in sufficient number to enrich its acts for twenty
years—a promise more than fulfilled, for down to 1818 [Euler died in 1793]
the volumes usually contained one or more papers of his. It has been said
that an edition of Euler’s complete works would fill 16,000 quarto pages.—
Cajori, F.
History of Mathematics (New York, 1897),
pp. 253-254.

961. Euler who could have been called almost without metaphor, and
certainly without hyperbole, analysis incarnate.—Arago.
Oeuvres, t. 2 (1854), p. 433.

962. Euler calculated without any apparent effort, just as men breathe, as
eagles sustain themselves in the air.—Arago.
Oeuvres, t. 2 (1854), p. 133.

963. Two of his [Euler’s] pupils having computed to the 17th term, a
complicated converging series, their results differed one unit in the fiftieth
cipher; and an appeal being made to Euler, he went over the calculation in
his mind, and his decision was found correct.—Brewster, David.
Letters of Euler (New York, 1872), Vol. 2, p.
22.

964. In 1735 the solving of an astronomical problem, proposed by the
Academy, for which several eminent mathematicians had demanded several
months’ time, was achieved in three days by Euler with aid of improved
methods of his own.... With still superior methods this same problem was
solved by the illustrious Gauss in one hour.—Cajori, F.
History of Mathematics (New York, 1897), p.
248.

965. Euler’s Tentamen novae theorae musicae had no great success, as it
contained too much geometry for musicians, and too much music for

geometers.—Fuss, N.
Quoted by Brewster: Letters of Euler (New
York, 1872), Vol. 1, p. 26.

966. Euler was a believer in God, downright and straight-forward. The
following story is told by Thiebault, in his Souvenirs de vingt ans de séjour
à Berlin,.... Thiebault says that he has no personal knowledge of the truth of
the story, but that it was believed throughout the whole of the north of
Europe. Diderot paid a visit to the Russian Court at the invitation of the
Empress. He conversed very freely, and gave the younger members of the
Court circle a good deal of lively atheism. The Empress was much amused,
but some of her counsellors suggested that it might be desirable to check
these expositions of doctrine. The Empress did not like to put a direct
muzzle on her guest’s tongue, so the following plot was contrived. Diderot
was informed that a learned mathematician was in possession of an
algebraical demonstration of the existence of God, and would give it him
before all the Court, if he desired to hear it. Diderot gladly consented:
though the name of the mathematician is not given, it was Euler. He
advanced toward Diderot, and said gravely, and in a tone of perfect
conviction:
Monsieur, donc Dieu existe; repondez!

Diderot, to whom algebra was Hebrew, was embarrassed and
disconcerted; while peals of laughter rose on all sides. He asked permission
to return to France at once, which was granted.—De Morgan, A.
Budget of Paradoxes (London, 1872), p. 251.

967. Fermat died with the belief that he had found along-sought-for law of
n
prime numbers in the formula 22 + 1 = a prime, but he admitted that he was
unable to prove it rigorously. The law is not true, as was pointed out by
5
Euler in the example 22 + 1 = 4,294,967,297 = 6,700,417 times 641. The
American lightning calculator Zerah Colburn, when a boy, readily found
the factors but was unable to explain the method by which he made his
marvellous mental computation.—Cajori, F.

History of Mathematics (New York, 1897), p.
180.

968. I crave the liberty to conceal my name, not to suppress it. I have
composed the letters of it written in Latin in this sentence—

In Mathesi a sole fundes.7—Flamsteed, J.

Macclesfield: Correspondence of Scientific
Men (Oxford, 1841), Vol. 2, p. 90.

969. To the Memory of Fourier
Fourier! with solemn and profound delight,
Joy born of awe, but kindling momently
To an intense and thrilling ecstacy,
I gaze upon thy glory and grow bright:
As if irradiate with beholden light;
As if the immortal that remains of thee
Attuned me to thy spirit’s harmony,
Breathing serene resolve and tranquil might.
Revealed appear thy silent thoughts of youth,
As if to consciousness, and all that view
Prophetic, of the heritage of truth
To thy majestic years of manhood due:
Darkness and error fleeing far away,
And the pure mind enthroned in perfect day.
—Hamilton, W. R.
Graves’ Life of W. R. Hamilton, (New York,
1882), Vol. 1, p. 596.

970. Astronomy and Pure Mathematics are the magnetic poles toward
which the compass of my mind ever turns.—Gauss to Bolyai.
Briefwechsel (Schmidt-Stakel), (1899), p. 55.

971. [Gauss calculated the elements of the planet Ceres] and his analysis
proved him to be the first of theoretical astronomers no less than the
greatest of “arithmeticians.”—Ball, W. W. R.
History of Mathematics (London, 1901), p.
458.

972. The mathematical giant [Gauss], who from his lofty heights embraces
in one view the stars and the abysses....—Bolyai, W.
Kurzer Grundriss eines Versuchs (Maros
Vasarhely, 1851), p. 44.

973. Almost everything, which the mathematics of our century has brought
forth in the way of original scientific ideas, attaches to the name of Gauss.
—Kronecker, L.
Zahlentheorie, Teil 1 (Leipzig, 1901), p. 43.

974. I am giving this winter two courses of lectures to three students, of
which one is only moderately prepared, the other less than moderately, and
the third lacks both preparation and ability. Such are the onera of a
mathematical profession.—Gauss to Bessel, 1810.
Gauss-Bessel Briefwechsel (1880), p. 107.

975. Gauss once said “Mathematics is the queen of the sciences and
number-theory the queen of mathematics.” If this be true we may add that
the Disquisitiones is the Magna Charta of number-theory. The advantage
which science gained by Gauss’ long-lingering method of publication is
this: What he put into print is as true and important today as when first
published; his publications are statutes, superior to other human statutes in
this, that nowhere and never has a single error been detected in them. This
justifies and makes intelligible the pride with which Gauss said in the
evening of his life of the first larger work of his youth: “The Disquisitiones
arithmeticae belong to history.”—Cantor, M.
Allgemeine Deutsche Biographie, Bd. 8
(1878), p. 435.

976. Here I am at the limit which God and nature has assigned to my
individuality. I am compelled to depend upon word, language and image in
the most precise sense, and am wholly unable to operate in any manner
whatever with symbols and numbers which are easily intelligible to the
most highly gifted minds.—Goethe.
Letter to Naumann (1826); Vogel: Goethe’s
Selbstzeugnisse (Leipzig, 1903), p. 56.

977. Dirichlet was not satisfied to study Gauss’ “Disquisitiones
arithmeticae” once or several times, but continued throughout life to keep in
close touch with the wealth of deep mathematical thoughts which it
contains by perusing it again and again. For this reason the book was never
placed on the shelf but had an abiding place on the table at which he
worked.... Dirichlet was the first one, who not only fully understood this
work, but made it also accessible to others.—Kummer, E. E.
Dirichlet: Werke, Bd. 2, p. 315.

978. [The famous attack of Sir William Hamilton on the tendency of
mathematical studies] affords the most express evidence of those fatal
lacunae in the circle of his knowledge, which unfitted him for taking a
comprehensive or even an accurate view of the processes of the human
mind in the establishment of truth. If there is any pre-requisite which all
must see to be indispensable in one who attempts to give laws to the human
intellect, it is a thorough acquaintance with the modes by which human
intellect has proceeded, in the case where, by universal acknowledgment,
grounded on subsequent direct verification, it has succeeded in ascertaining
the greatest number of important and recondite truths. This requisite Sir W.
Hamilton had not, in any tolerable degree, fulfilled. Even of pure
mathematics he apparently knew little but the rudiments. Of mathematics as
applied to investigating the laws of physical nature; of the mode in which
the properties of number, extension, and figure, are made instrumental to
the ascertainment of truths other than arithmetical or geometrical—it is too
much to say that he had even a superficial knowledge: there is not a line in
his works which shows him to have had any knowledge at all.—Mill, J. S.
Examination of Sir William Hamilton’s
Philosophy (London, 1878), p. 607.

979. Helmholtz—the physiologist who learned physics for the sake of his
physiology, and mathematics for the sake of his physics, and is now in the
first rank of all three.—Clifford, W. K.
Aims and Instruments of Scientific Thought;
Lectures and Essays, Vol. 1 (London, 1901),
p. 165.

980. It is said of Jacobi, that he attracted the particular attention and
friendship of Böckh, the director of the philological seminary at Berlin, by
the great talent he displayed for philology, and only at the end of two years’
study at the University, and after a severe mental struggle, was able to make
his final choice in favor of mathematics.—Sylvester, J. J.
Collected Mathematical Papers, Vol. 2
(Cambridge, 1908), p. 651.

981. When Dr. Johnson felt, or fancied he felt, his fancy disordered, his
constant recurrence was to the study of arithmetic.—Boswell, J.
Life of Johnson (Harper’s Edition, 1871),
Vol. 2, p. 264.

982. Endowed with two qualities, which seemed incompatible with each
other, a volcanic imagination and a pertinacity of intellect which the most
tedious numerical calculations could not daunt, Kepler conjectured that the
movements of the celestial bodies must be connected together by simple
laws, or, to use his own expression, by harmonic laws. These laws he
undertook to discover. A thousand fruitless attempts, errors of calculation
inseparable from a colossal undertaking, did not prevent him a single
instant from advancing resolutely toward the goal of which he imagined he
had obtained a glimpse. Twenty-two years were employed by him in this
investigation, and still he was not weary of it! What, in reality, are twenty-
two years of labor to him who is about to become the legislator of worlds;
who shall inscribe his name in ineffaceable characters upon the frontispiece
of an immortal code; who shall be able to exclaim in dithyrambic language,
and without incurring the reproach of anyone, “The die is cast; I have
written my book; it will be read either in the present age or by posterity, it

matters not which; it may well await a reader, since God has waited six
thousand years for an interpreter of his words.”—Arago.
Eulogy on Laplace: [Baden Powell]
Smithsonian Report, 1874, p. 132.

983. The great masters of modern analysis are Lagrange, Laplace, and
Gauss, who were contemporaries. It is interesting to note the marked
contrast in their styles. Lagrange is perfect both in form and matter, he is
careful to explain his procedure, and though his arguments are general they
are easy to follow. Laplace on the other hand explains nothing, is indifferent
to style, and, if satisfied that his results are correct, is content to leave them
either with no proof or with a faulty one. Gauss is as exact and elegant as
Lagrange, but even more difficult to follow than Laplace, for he removes
every trace of the analysis by which he reached his results, and studies to
give a proof which while rigorous shall be as concise and synthetical as
possible.—Ball, W. W. R.
History of Mathematics (London, 1901), p.
463.

984. Lagrange, in one of the later years of his life, imagined that he had
overcome the difficulty [of the parallel axiom]. He went so far as to write a
paper, which he took with him to the Institute, and began to read it. But in
the first paragraph something struck him which he had not observed: he
muttered Il faut que j’y songe encore, and put the paper in his pocket.—De
Morgan, A.
Budget of Paradoxes (London, 1872), p. 173.

985. I never come across one of Laplace’s “Thus it plainly appears”
without feeling sure that I have hours of hard work before me to fill up the
chasm and find out and show how it plainly appears.—Bowditch, N.
Quoted by Cajori: Teaching and History of
Mathematics in the U. S. (Washington, 1896),
p. 104.

986. Biot, who assisted Laplace in revising it [The Mécanique Céleste] for
the press, says that Laplace himself was frequently unable to recover the
details in the chain of reasoning, and if satisfied that the conclusions were
correct, he was content to insert the constantly recurring formula, “Il est
àisé a voir.”—Ball, W. W. R.
History of Mathematics (London, 1901), p
427.

987. It would be difficult to name a man more remarkable for the greatness
and the universality of his intellectual powers than Leibnitz.—Mill, J. S.
System of Logic, Bk. 2, chap. 5, sect. 6.

988. The influence of his [Leibnitz’s] genius in forming that peculiar taste
both in pure and in mixed mathematics which has prevailed in France, as
well as in Germany, for a century past, will be found, upon examination, to
have been incomparably greater than that of any other individual.—
Stewart, Dugald.
Philosophy of the Human Mind, Part 3, chap.
1, sect. 3.

989. Leibnitz’s discoveries lay in the direction in which all modern progress
in science lies, in establishing order, symmetry, and harmony, i.e.,
comprehensiveness and perspicuity,—rather than in dealing with single
problems, in the solution of which followers soon attained greater dexterity
than himself.—Merz, J. T.
Leibnitz, Chap. 6.

990. It was his [Leibnitz’s] love of method and order, and the conviction
that such order and harmony existed in the real world, and that our success
in understanding it depended upon the degree and order which we could
attain in our own thoughts, that originally was probably nothing more than a
habit which by degrees grew into a formal rule.8 This habit was acquired by
early occupation with legal and mathematical questions. We have seen how
the theory of combinations and arrangements of elements had a special

interest for him. We also saw how mathematical calculations served him as
a type and model of clear and orderly reasoning, and how he tried to
introduce method and system into logical discussions, by reducing to a
small number of terms the multitude of compound notions he had to deal
with. This tendency increased in strength, and even in those early years he
elaborated the idea of a general arithmetic, with a universal language of
symbols, or a characteristic which would be applicable to all reasoning
processes, and reduce philosophical investigations to that simplicity and
certainty which the use of algebraic symbols had introduced into
mathematics.
A mental attitude such as this is always highly favorable for
mathematical as well as for philosophical investigations. Wherever progress
depends upon precision and clearness of thought, and wherever such can be
gained by reducing a variety of investigations to a general method, by
bringing a multitude of notions under a common term or symbol, it proves
inestimable. It necessarily imports the special qualities of number—viz.,
their continuity, infinity and infinite divisibility—like mathematical
quantities—and destroys the notion that irreconcilable contrasts exist in
nature, or gaps which cannot be bridged over. Thus, in his letter to Arnaud,
Leibnitz expresses it as his opinion that geometry, or the philosophy of
space, forms a step to the philosophy of motion—i.e., of corporeal things—
and the philosophy of motion a step to the philosophy of mind.—Merz, J. T.
Leibnitz (Philadelphia), pp. 44-45.

991. Leibnitz believed he saw the image of creation in his binary arithmetic
in which he employed only two characters, unity and zero. Since God may
be represented by unity, and nothing by zero, he imagined that the Supreme
Being might have drawn all things from nothing, just as in the binary
arithmetic all numbers are expressed by unity with zero. This idea was so
pleasing to Leibnitz, that he communicated it to the Jesuit Grimaldi,
President of the Mathematical Board of China, with the hope that this
emblem of the creation might convert to Christianity the reigning emperor
who was particularly attached to the sciences.—Laplace.
Essai Philosophique sur les Probabilités;
Oeuvres (Paris, 1896), t. 7, p. 119.

992. Sophus Lie, great comparative anatomist of geometric theories.—
Keyser, C. J.
Lectures on Science, Philosophy and Art
(New York, 1908), p. 31.

993. It has been the final aim of Lie from the beginning to make progress in
the theory of differential equations; as subsidiary to this may be regarded
both his geometrical developments and the theory of continuous groups.—
Klein, F.
Lectures on Mathematics (New York, 1911),
p. 24.

994. To fully understand the mathematical genius of Sophus Lie, one must
not turn to books recently published by him in collaboration with Dr. Engel,
but to his earlier memoirs, written during the first years of his scientific
career. There Lie shows himself the true geometer that he is, while in his
later publications, finding that he was but imperfectly understood by the
mathematicians accustomed to the analytic point of view, he adopted a very
general analytic form of treatment that is not always easy to follow.—
Klein, F.
Lectures on Mathematics (New York, 1911),
p. 9.

995. It is said that the composing of the Lilawati was occasioned by the
following circumstance. Lilawati was the name of the author’s [Bhascara]
daughter, concerning whom it appeared, from the qualities of the ascendant
at her birth, that she was destined to pass her life unmarried, and to remain
without children. The father ascertained a lucky hour for contracting her in
marriage, that she might be firmly connected and have children. It is said
that when that hour approached, he brought his daughter and his intended
son near him. He left the hour cup on the vessel of water and kept in
attendance a time-knowing astrologer, in order that when the cup should
subside in the water, those two precious jewels should be united. But, as the
intended arrangement was not according to destiny, it happened that the
girl, from a curiosity natural to children, looked into the cup, to observe the

water coming in at the hole, when by chance a pearl separated from her
bridal dress, fell into the cup, and, rolling down to the hole, stopped the
influx of water. So the astrologer waited in expectation of the promised
hour. When the operation of the cup had thus been delayed beyond all
moderate time, the father was in consternation, and examining, he found
that a small pearl had stopped the course of the water, and that the long-
expected hour was passed. In short, the father, thus disappointed, said to his
unfortunate daughter, I will write a book of your name, which shall remain
to the latest times—for a good name is a second life, and the ground-work
of eternal existence.—Fizi.
Preface to the Lilawati. Quoted by A. Hutton:
A Philosophical and Mathematical
Dictionary, Article “Algebra” (London,
1815).

996. Is there anyone whose name cannot be twisted into either praise or
satire? I have had given to me,
Thomas Babington Macaulay
Mouths big: a Cantab anomaly.
—De Morgan, A.
Budget of Paradoxes (London, 1872), p. 83.

CHAPTER X
PERSONS AND ANECDOTES
(N-Z)

1001. When he had a few moments for diversion, he [Napoleon] not
unfrequently employed them over a book of logarithms, in which he always
found recreation.—Abbott, J. S. C.
Napoleon Bonaparte (New York, 1904), Vol.
1, chap. 10.

1002. The name of Sir Isaac Newton has by general consent been placed at
the head of those great men who have been the ornaments of their species....
The philosopher [Laplace], indeed, to whom posterity will probably assign
a place next to Newton, has characterized the Principia as pre-eminent
above all the productions of human intellect.—Brewster, D.
Life of Sir Isaac Newton (London, 1831), pp.
1, 2.

1003. Newton and Laplace need myriads of ages and thick-strewn celestial
areas. One may say a gravitating solar system is already prophesied in the
nature of Newton’s mind.—Emerson.
Essay on History.

1004. The law of gravitation is indisputably and incomparably the greatest
scientific discovery ever made, whether we look at the advance which it
involved, the extent of truth disclosed, or the fundamental and satisfactory
nature of this truth.—Whewell, W.
History of the Inductive Sciences, Bk. 7,
chap. 2, sect. 5.

1005. Newton’s theory is the circle of generalization which includes all the
others [as Kepler’s laws, Ptolemy’s theory, etc.];—the highest point of the
inductive ascent;—the catastrophe of the philosophic drama to which Plato
had prologized;—the point to which men’s minds had been journeying for
two thousand years.—Whewell, W.
History of the Inductive Sciences, Bk. 7,
chap. 2, sect. 5.

1006. The efforts of the great philosopher [Newton] were always
superhuman; the questions which he did not solve were incapable of
solution in his time.—Arago.
Eulogy on Laplace, [Baden Powell]
Smithsonian Report, 1874, p. 133.

1007.
Nature and Nature’s laws lay hid in night:
God said, “Let Newton be!” and all was light.
—Pope, A.
Epitaph intended for Sir Isaac Newton.

1008.
There Priest of Nature! dost thou shine,
Newton! a King among the Kings divine.
—Southey.
Translation of a Greek Ode on Astronomy.

1009.
O’er Nature’s laws God cast the veil of night,
Out-blaz’d a Newton’s soul—and all was light.
—Hill, Aaron.
On Sir Isaac Newton.

1010. Taking mathematics from the beginning of the world to the time
when Newton lived, what he had done was much the better half.—Leibnitz.
Quoted by F. R. Moulton: Introduction to
Astronomy (New York, 1906), p. 199.

1011. Newton was the greatest genius that ever existed, and the most
fortunate, for we cannot find more than once a system of the world to
establish.—Lagrange.
Quoted by F. R. Moulton: Introduction to
Astronomy (New York, 1906), p. 199.

1012. A monument to Newton! a monument to Shakespeare! Look up to
Heaven—look into the Human Heart. Till the planets and the passions—the
affections and the fixed stars are extinguished—their names cannot die.—
Wilson, John.
Noctes Ambrosianae.

1013. Such men as Newton and Linnaeus are incidental, but august,
teachers of religion.—Wilson, John.
Essays: Education of the People.

1014. Sir Isaac Newton, the supreme representative of Anglo-Saxon genius.
—Ellis, Havelock.
Study of British Genius (London, 1904), p.
49.

1015. Throughout his life Newton must have devoted at least as much
attention to chemistry and theology as to mathematics....—Ball, W. W. R.
History of Mathematics (London, 1901), p.
335.

1016. There was a time when he [Newton] was possessed with the old
fooleries of astrology; and another when he was so far gone in those of

chemistry, as to be upon the hunt after the philosopher’s stone.—Rev. J.
Spence.
Anecdotes, Observations, and Characters of
Books and Men (London, 1868), p. 54.

1017. For several years this great man [Newton] was intensely occupied in
endeavoring to discover a way of changing the base metals into gold....
There were periods when his furnace fires were not allowed to go out for
six weeks; he and his secretary sitting up alternate nights to replenish them.
—Parton, James.
Sir Isaac Newton.

1018. On the day of Cromwell’s death, when Newton was sixteen, a great
storm raged all over England. He used to say, in his old age, that on that day
he made his first purely scientific experiment. To ascertain the force of the
wind, he first jumped with the wind and then against it; and, by comparing
these distances with the extent of his own jump on a calm day, he was
enabled to compute the force of the storm. When the wind blew thereafter,
he used to say it was so many feet strong.—Parton, James.
Sir Isaac Newton.

1019. Newton lectured now and then to the few students who chose to hear
him; and it is recorded that very frequently he came to the lecture-room and
found it empty. On such occasions he would remain fifteen minutes, and
then, if no one came, return to his apartments.—Parton, James.
Sir Isaac Newton.

1020. Sir Isaac Newton, though so deep in algebra and fluxions, could not
readily make up a common account: and, when he was Master of the Mint,
used to get somebody else to make up his accounts for him.—Rev. J.
Spence.
Anecdotes, Observations, and Characters of
Books and Men (London, 1858), p. 132.

1021. We have one of his [Newton’s] college memorandum-books, which is
highly interesting. The following are some of the entries: “Drills, gravers, a
hone, a hammer, and a mandril, 5s.;” “a magnet, 16s.;” “compasses, 2s.;”
“glass bubbles, 4s.;” “at the tavern several other times, £1;” “spent on my
cousin, 12s.;” “on other acquaintances, 10s.;” “Philosophical Intelligences,
9s. 6d.;” “lost at cards twice, 15s.;” “at the tavern twice, 3s. 6d.;” “to three
prisms, £3;” “four ounces of putty, 1s. 4d.;” “Bacon’s Miscellanies, 1s.
6d.;” “a bible binding, 3s.;” “for oranges to my sister, 4s. 2d.;” “for
aquafortis, sublimate, oyle pink, fine silver, antimony, vinegar, spirit of
wine, white lead, salt of tartar, £2;“ “Theatrum chemicum, £1 8s”—Parton,
James.
Sir Isaac Newton.

1022. On one occasion, when he was giving a dinner to some friends at the
university, he left the table to get them a bottle of wine; but, on his way to
the cellar, he fell into reflection, forgot his errand and his company, went to
his chamber, put on his surplice, and proceeded to the chapel. Sometimes he
would go into the street half dressed, and on discovering his condition, run
back in great haste, much abashed. Often, while strolling in his garden, he
would suddenly stop, and then run rapidly to his room, and begin to write,
standing, on the first piece of paper that presented itself. Intending to dine
in the public hall, he would go out in a brown study, take the wrong turn,
walk a while, and then return to his room, having totally forgotten the
dinner. Once having dismounted from his horse to lead him up a hill, the
horse slipped his head out of the bridle; but Newton, oblivious, never
discovered it till, on reaching a tollgate at the top of the hill, he turned to
remount and perceived that the bridle which he held in his hand had no
horse attached to it. His secretary records that his forgetfulness of his dinner
was an excellent thing for his old housekeeper, who “sometimes found both
dinner and supper scarcely tasted of, which the old woman has very
pleasantly and mumpingly gone away with.” On getting out of bed in the
morning, he has been discovered to sit on his bedside for hours without
dressing himself, utterly absorbed in thought.—Parton, James.
Sir Isaac Newton.

1023. I don’t know what I may seem to the world, but, as to myself, I seem
to have been only as a boy playing on the seashore, and diverting myself in
now and then finding a smoother pebble or a prettier shell than ordinary,
whilst the great ocean of truth lay all undiscovered before me.—Newton, I.
Quoted by Rev. J. Spence: Anecdotes,
Observations, and Characters of Books and
Men (London, 1858), p. 40.

1024. If I have seen farther than Descartes, it is by standing on the
shoulders of giants.—Newton, I.
Quoted by James Parton: Sir Isaac Newton.

1025. Newton could not admit that there was any difference between him
and other men, except in the possession of such habits as ... perseverance
and vigilance. When he was asked how he made his discoveries, he
answered, “by always thinking about them;“ and at another time he
declared that if he had done anything, it was due to nothing but industry and
patient thought: “I keep the subject of my inquiry constantly before me, and
wait till the first dawning opens gradually, by little and little, into a full and
clear light”—Whewell, W.
History of the Inductive Sciences, Bk. 7,
chap. 2, sect. 5.

1026. Newton took no exercise, indulged in no amusements, and worked
incessantly, often spending eighteen or nineteen hours out of the twenty-
four in writing.—Ball, W. W. R.
History of Mathematics (London, 1901), p.
358.

1027. Foreshadowings of the principles and even of the language of [the
infinitesimal] calculus can be found in the writings of Napier, Kepler,
Cavalieri, Pascal, Fermat, Wallis, and Barrow. It was Newton’s good luck to
come at a time when everything was ripe for the discovery, and his ability

enabled him to construct almost at once a complete calculus.—Ball, W. W.
R.
History of Mathematics (London, 1901), p.
356.

1028. Kepler’s suggestion of gravitation with the inverse distance, and
Bouillaud’s proposed substitution of the inverse square of the distance, are
things which Newton knew better than his modern readers. I have
discovered two anagrams on his name, which are quite conclusive: the
notion of gravitation was not new; but Newton went on.—De Morgan, A.
Budget of Paradoxes (London, 1872), p. 82.

1029. For other great mathematicians or philosophers, he [Gauss] used the
epithets magnus, or clarus, or clarissimus; for Newton alone he kept the
prefix summus.—Ball, W. W. R.
History of Mathematics (London, 1901), p.
362.

1030. To know him [Sylvester] was to know one of the historic figures of
all time, one of the immortals; and when he was really moved to speak, his
eloquence equalled his genius.—Halsted, G. B.
F. Cajori’s Teaching and History of
Mathematics in the U. S. (Washington, 1890),
p. 265.

1031. Professor Sylvester’s first high class at the new university Johns
Hopkins consisted of only one student, G. B. Halsted, who had persisted in
urging Sylvester to lecture on the modern algebra. The attempt to lecture on
this subject led him into new investigations in quantics.—Cajori, F.
Teaching and History of Mathematics in the
U. S. (Washington, 1890), p. 264.

1032. But for the persistence of a student of this university in urging upon
me his desire to study with me the modern algebra I should never have been
led into this investigation; and the new facts and principles which I have
discovered in regard to it (important facts, I believe), would, so far as I am
concerned, have remained still hidden in the womb of time. In vain I
represented to this inquisitive student that he would do better to take up
some other subject lying less off the beaten track of study, such as the
higher parts of the calculus or elliptic functions, or the theory of
substitutions, or I wot not what besides. He stuck with perfect
respectfulness, but with invincible pertinacity, to his point. He would have
the new algebra (Heaven knows where he had heard about it, for it is almost
unknown in this continent), that or nothing. I was obliged to yield, and what
was the consequence? In trying to throw light upon an obscure explanation
in our text-book, my brain took fire, I plunged with re-quickened zeal into a
subject which I had for years abandoned, and found food for thoughts
which have engaged my attention for a considerable time past, and will
probably occupy all my powers of contemplation advantageously for
several months to come.—Sylvester, J. J.
Johns Hopkins Commemoration Day
Address; Collected Mathematical Papers,
Vol. 3, p. 76.

1033. Sylvester was incapable of reading mathematics in a purely receptive
way. Apparently a subject either fired in his brain a train of active and
restless thought, or it would not retain his attention at all. To a man of such
a temperament, it would have been peculiarly helpful to live in an
atmosphere in which his human associations would have supplied the
stimulus which he could not find in mere reading. The great modern work
in the theory of functions and in allied disciplines, he never became
acquainted with....
What would have been the effect if, in the prime of his powers, he had
been surrounded by the influences which prevail in Berlin or in Göttingen?
It may be confidently taken for granted that he would have done splendid
work in those domains of analysis, which have furnished the laurels of the
great mathematicians of Germany and France in the second half of the
present century.—Franklin, F.

Johns Hopkins University Circulars 16
(1897), p. 54.

1034. If we survey the mathematical works of Sylvester, we recognize
indeed a considerable abundance, but in contradistinction to Cayley—not a
versatility toward separate fields, but, with few exceptions—a confinement
to arithmetic-algebraic branches....
The concept of Function of a continuous variable, the fundamental
concept of modern mathematics, plays no role, is indeed scarcely
mentioned in the entire work of Sylvester—Sylvester was combinatorist
[combinatoriker].—Noether, M.
Mathematische Annalen, Bd. 50 (1898), pp.
134-135.

1035. Sylvester’s methods! He had none. “Three lectures will be delivered
on a New Universal Algebra,” he would say; then, “The course must be
extended to twelve.” It did last all the rest of that year. The following year
the course was to be Substitutions-Theorie, by Netto. We all got the text. He
lectured about three times, following the text closely and stopping sharp at
the end of the hour. Then he began to think about matrices again. “I must
give one lecture a week on those,” he said. He could not confine himself to
the hour, nor to the one lecture a week. Two weeks were passed, and Netto
was forgotten entirely and never mentioned again. Statements like the
following were not unfrequent in his lectures: “I haven’t proved this, but I
am as sure as I can be of anything that it must be so. From this it will
follow, etc.” At the next lecture it turned out that what he was so sure of
was false. Never mind, he kept on forever guessing and trying, and
presently a wonderful discovery followed, then another and another.
Afterward he would go back and work it all over again, and surprise us with
all sorts of side lights. He then made another leap in the dark, more
treasures were discovered, and so on forever.—Davis, E. W.
Cajori’s Teaching and History of
Mathematics in the U.S. (Washington, 1890),
pp. 265-266.

1036. I can see him [Sylvester] now, with his white beard and few locks of
gray hair, his forehead wrinkled o’er with thoughts, writing rapidly his
figures and formulae on the board, sometimes explaining as he wrote, while
we, his listeners, caught the reflected sounds from the board. But stop,
something is not right, he pauses, his hand goes to his forehead to help his
thought, he goes over the work again, emphasizes the leading points, and
finally discovers his difficulty. Perhaps it is some error in his figures,
perhaps an oversight in the reasoning. Sometimes, however, the difficulty is
not elucidated, and then there is not much to the rest of the lecture. But at
the next lecture we would hear of some new discovery that was the outcome
of that difficulty, and of some article for the Journal, which he had begun. If
a text-book had been taken up at the beginning, with the intention of
following it, that text-book was most likely doomed to oblivion for the rest
of the term, or until the class had been made listeners to every new thought
and principle that had sprung from the laboratory of his mind, in
consequence of that first difficulty. Other difficulties would soon appear, so
that no text-book could last more than half of the term. In this way his class
listened to almost all of the work that subsequently appeared in the Journal.
It seemed to be the quality of his mind that he must adhere to one subject.
He would think about it, talk about it to his class, and finally write about it
for the Journal. The merest accident might start him, but once started, every
moment, every thought was given to it, and, as much as possible, he read
what others had done in the same direction; but this last seemed to be his
real point; he could not read without finding difficulties in the way of
understanding the author. Thus, often his own work reproduced what had
been done by others, and he did not find it out until too late.
A notable example of this is in his theory of cyclotomic functions,
which he had reproduced in several foreign journals, only to find that he
had been greatly anticipated by foreign authors. It was manifest, one of the
critics said, that the learned professor had not read Kummer’s elementary
results in the theory of ideal primes. Yet Professor Smith’s report on the
theory of numbers, which contained a full synopsis of Kummer’s theory,
was Professor Sylvester’s constant companion.
This weakness of Professor Sylvester, in not being able to read what
others had done, is perhaps a concomitant of his peculiar genius. Other
minds could pass over little difficulties and not be troubled by them, and so

go on to a final understanding of the results of the author. But not so with
him. A difficulty, however small, worried him, and he was sure to have
difficulties until the subject had been worked over in his own way, to
correspond with his own mode of thought. To read the work of others,
meant therefore to him an almost independent development of it. Like the
man whose pleasure in life is to pioneer the way for society into the forests,
his rugged mind could derive satisfaction only in hewing out its own paths;
and only when his efforts brought him into the uncleared fields of
mathematics did he find his place in the Universe.—Hathaway, A. S.
F. Cajori’s Teaching and History of
Mathematics in the U. S. (Washington, 1890),
pp. 266-267.

1037. Professor Cayley has since informed me that the theorem about
whose origin I was in doubt, will be found in Schläfli’s “De Eliminatione.”
This is not the first unconscious plagiarism I have been guilty of towards
this eminent man whose friendship I am proud to claim. A more glaring
case occurs in a note by me in the “Comptes Rendus,” on the twenty-seven
straight lines of cubic surfaces, where I believe I have followed (like one
walking in his sleep), down to the very nomenclature and notation, the
substance of a portion of a paper inserted by Schläfli in the “Mathematical
Journal,” which bears my name as one of the editors upon the face.—
Sylvester, J. J.
Philosophical Transactions of the Royal
Society (1864), p. 642.

1038. He [Sylvester] had one remarkable peculiarity. He seldom
remembered theorems, propositions, etc., but had always to deduce them
when he wished to use them. In this he was the very antithesis of Cayley,
who was thoroughly conversant with everything that had been done in
every branch of mathematics.
I remember once submitting to Sylvester some investigations that I had
been engaged on, and he immediately denied my first statement, saying that
such a proposition had never been heard of, let alone proved. To his
astonishment, I showed him a paper of his own in which he had proved the

proposition; in fact, I believe the object of his paper had been the very proof
which was so strange to him.—Durfee, W. P.
F. Cajori’s Teaching and History of
Mathematics in the U. S. (Washington, 1890),
p. 268.

1039. A short, broad man of tremendous vitality, the physical type of
Hereward, the last of the English, and his brother-in-arms, Winter,
Sylvester’s capacious head was ever lost in the highest cloud-lands of pure
mathematics. Often in the dead of night he would get his favorite pupil, that
he might communicate the very last product of his creative thought.
Everything he saw suggested to him something new in the higher algebra.
This transmutation of everything into new mathematics was a revelation to
those who knew him intimately. They began to do it themselves. His ease
and fertility of invention proved a constant encouragement, while his
contempt for provincial stupidities, such as the American hieroglyphics for
π and e, which have even found their way into Webster’s Dictionary, made
each young worker apply to himself the strictest tests.—Halsted, G. B.
F. Cajori’s Teaching and History of
Mathematics in the U. S. (Washington, 1890),
p. 265.

1040. Sylvester’s writings are flowery and eloquent. He was able to make
the dullest subject bright, fresh and interesting. His enthusiasm is evident in
every line. He would get quite close up to his subject, so that everything
else looked small in comparison, and for the time would think and make
others think that the world contained no finer matter for contemplation. His
handwriting was bad, and a trouble to his printers. His papers were finished
with difficulty. No sooner was the manuscript in the editor’s hands than
alterations, corrections, ameliorations and generalizations would suggest
themselves to his mind, and every post would carry further directions to the
editors and printers.—MacMahon. P. A.
Nature, Vol. 55 (1897), p. 494.

1041. The enthusiasm of Sylvester for his own work, which manifests itself
here as always, indicates one of his characteristic qualities: a high degree of
subjectivity in his productions and publications. Sylvester was so fully
possessed by the matter which for the time being engaged his attention, that
it appeared to him and was designated by him as the summit of all that is
important, remarkable and full of future promise. It would excite his
phantasy and power of imagination in even a greater measure than his
power of reflection, so much so that he could never marshal the ability to
master his subject-matter, much less to present it in an orderly manner.
Considering that he was also somewhat of a poet, it will be easier to
overlook the poetic flights which pervade his writing, often bombastic,
sometimes furnishing apt illustrations; more damaging is the complete lack
of form and orderliness of his publications and their sketchlike character,....
which must be accredited at least as much to lack of objectivity as to a
superfluity of ideas. Again, the text is permeated with associated emotional
expressions, bizarre utterances and paradoxes and is everywhere
accompanied by notes, which constitute an essential part of Sylvester’s
method of presentation, embodying relations, whether proximate or remote,
which momentarily suggested themselves. These notes, full of inspiration
and occasional flashes of genius, are the more stimulating owing to their
incompleteness. But none of his works manifest a desire to penetrate the
subject from all sides and to allow it to mature; each mere surmise,
conceptions which arose during publication, immature thoughts and even
errors were ushered into publicity at the moment of their inception, with
utmost carelessness, and always with complete unfamiliarity of the
literature of the subject. Nowhere is there the least trace of self-criticism.
No one can be expected to read the treatises entire, for in the form in which
they are available they fail to give a clear view of the matter under
contemplation.
Sylvester’s was not a harmoniously gifted or well-balanced mind, but
rather an instinctively active and creative mind, free from egotism. His
reasoning moved in generalizations, was frequently influenced by analysis
and at times was guided even by mystical numerical relations. His
reasoning consists less frequently of pure intelligible conclusions than of
inductions, or rather conjectures incited by individual observations and
verifications. In this he was guided by an algebraic sense, developed

through long occupation with processes of forms, and this led him luckily to
general fundamental truths which in some instances remain veiled. His lack
of system is here offset by the advantage of freedom from purely
mechanical logical activity.
The exponents of his essential characteristics are an intuitive talent and
a faculty of invention to which we owe a series of ideas of lasting value and
bearing the germs of fruitful methods. To no one more fittingly than to
Sylvester can be applied one of the mottos of the Philosophic Magazine:
“Admiratio generat quaestionem, quaestio investigationem investigatio
inventionem”—Noether, M.
Mathematische Annalen, Bd. 50 (1898), pp.
155-160.

1042. Perhaps I may without immodesty lay claim to the appellation of
Mathematical Adam, as I believe that I have given more names (passed into
general circulation) of the creatures of the mathematical reason than all the
other mathematicians of the age combined.—Sylvester, J. J.
Nature, Vol. 37 (1887-1888), p. 162.

1043. Tait dubbed Maxwell dp/dt, for according to thermodynamics dp/dt =
JCM (where C denotes Carnot’s function) the initials of (J. C.) Maxwell’s
name. On the other hand Maxwell denoted Thomson by T and Tait by T´; so
that it became customary to quote Thomson and Tait’s Treatise on Natural
Philosophy as T and T´.—Macfarlane, A.
Bibliotheca Mathematica, Bd. 3 (1903), p.
189.

1044. In future times Tait will be best known for his work in the quaternion
analysis. Had it not been for his expositions, developments and
applications, Hamilton’s invention would be today, in all probability, a
mathematical curiosity.—Macfarlane, A.
Bibliotheca Mathematica, Bd. 3 (1903), p.
189.

1045. Not seldom did he [Sir William Thomson], in his writings, set down
some mathematical statement with the prefacing remark “it is obvious that”
to the perplexity of mathematical readers, to whom the statement was
anything but obvious from such mathematics as preceded it on the page. To
him it was obvious for physical reasons that might not suggest themselves
at all to the mathematician, however competent.—Thompson, S. P.
Life of Lord Kelvin (London, 1910), p. 1136.

1046. The following is one of the many stories told of “old Donald
McFarlane” the faithful assistant of Sir William Thomson.
The father of a new student when bringing him to the University, after
calling to see the Professor [Thomson] drew his assistant to one side and
besought him to tell him what his son must do that he might stand well with
the Professor. “You want your son to stand weel with the Profeessorr?”
asked McFarlane. “Yes.” “Weel, then, he must just have a guid bellyful o’
mathematics!“—Thompson, S. P.
Life of Lord Kelvin (London, 1910), p. 420.

1047. The following story (here a little softened from the vernacular) was
narrated by Lord Kelvin himself when dining at Trinity Hall:—
A certain rough Highland lad at the university had done exceedingly
well, and at the close of the session gained prizes both in mathematics and
in metaphysics. His old father came up from the farm to see his son receive
the prizes, and visited the College. Thomson was deputed to show him
round the place. “Weel, Mr. Thomson,” asked the old man, “and what may
these mathematics be, for which my son has getten a prize?” “I told him,”
replied Thomson, “that mathematics meant reckoning with figures, and
calculating.” “Oo ay,” said the old man, “he’ll ha’ getten that fra’ me: I were
ever a braw hand at the countin’.” After a pause he resumed: “And what,
Mr. Thomson, might these metapheesics be?” “I endeavoured,” replied
Thomson, “to explain how metaphysics was the attempt to express in
language the indefinite.” The old Highlander stood still and scratched his
head. “Oo ay: may be he’ll ha’ getten that fra’ his mither. She were aye a
bletherin’ body”—Thompson, S. P.
Life of Lord Kelvin (London, 1910), p. 1124.

1048. Lord Kelvin, unable to meet his classes one day, posted the following
notice on the door of his lecture room,—
“Professor Thomson will not meet his classes today.”
The disappointed class decided to play a joke on the professor. Erasing the
“c” they left the legend to read,—
“Professor Thomson will not meet his lasses today.”
When the class assembled the next day in anticipation of the effect of their
joke, they were astonished and chagrined to find that the professor had
outwitted them. The legend of yesterday was now found to read,—
“Professor Thomson will not meet his asses today.” 9
—Northrup, Cyrus.
University of Washington Address, November
2, 1908.

1049. One morning a great noise proceeded from one of the classrooms [of
the Braunsberger gymnasium] and on investigation it was found that
Weierstrass, who was to give the recitation, had not appeared. The director
went in person to Weierstrass’ dwelling and on knocking was told to come
in. There sat Weierstrass by a glimmering lamp in a darkened room though
it was daylight outside. He had worked the night through and had not
noticed the approach of daylight. When the director reminded him of the
noisy throng of students who were waiting for him, his only reply was that
he could impossibly interrupt his work; that he was about to make an
important discovery which would attract attention in scientific circles.—
Lampe, E.
Karl Weierstrass: Jahrbuch der Deutschen
Mathematiker Vereinigung, Bd. 6 (1897), pp.
38-39.

1050. Weierstrass related ... that he followed Sylvester’s papers on the
theory of algebraic forms very attentively until Sylvester began to employ
Hebrew characters. That was more than he could stand and after that he quit
him.—Lampe, E.
Naturwissenschaftliche Rundschau, Bd. 12
(1897), p. 361.

CHAPTER XI
MATHEMATICS AS A FINE ART

1101. The world of idea which it discloses or illuminates, the contemplation
of divine beauty and order which it induces, the harmonious connexion of
its parts, the infinite hierarchy and absolute evidence of the truths with
which it is concerned, these, and such like, are the surest grounds of the title
of mathematics to human regard, and would remain unimpeached and
unimpaired were the plan of the universe unrolled like a map at our feet,
and the mind of man qualified to take in the whole scheme of creation at a
glance.—Sylvester, J. J.
Presidential Address, British Association
Report (1869); Collected Mathematical
Papers, Vol. 2, p. 659.

1102. Mathematics has a triple end. It should furnish an instrument for the
study of nature. Furthermore it has a philosophic end, and, I venture to say,
an end esthetic. It ought to incite the philosopher to search into the notions
of number, space, and time; and, above all, adepts find in mathematics
delights analogous to those that painting and music give. They admire the
delicate harmony of number and of forms; they are amazed when a new
discovery discloses for them an unlooked for perspective; and the joy they
thus experience, has it not the esthetic character although the senses take no
part in it? Only the privileged few are called to enjoy it fully, it is true; but
is it not the same with all the noblest arts? Hence I do not hesitate to say
that mathematics deserves to be cultivated for its own sake, and that the
theories not admitting of application to physics deserve to be studied as
well as others. Poincaré, Henri.
The Relation of Analysis and Mathematical
Physics; Bulletin American Mathematical
Society, Vol. 4 (1899), p. 248.

1103. I like to look at mathematics almost more as an art than as a science;
for the activity of the mathematician, constantly creating as he is, guided
though not controlled by the external world of the senses, bears a
resemblance, not fanciful I believe but real, to the activity of an artist, of a
painter let us say. Rigorous deductive reasoning on the part of the
mathematician may be likened here to technical skill in drawing on the part
of the painter. Just as no one can become a good painter without a certain
amount of skill, so no one can become a mathematician without the power
to reason accurately up to a certain point. Yet these qualities, fundamental
though they are, do not make a painter or mathematician worthy of the
name, nor indeed are they the most important factors in the case. Other
qualities of a far more subtle sort, chief among which in both cases is
imagination, go to the making of a good artist or good mathematician.—
Bôcher, Maxime.
Fundamental Conceptions and Methods in
Mathematics; Bulletin American
Mathematical Society, Vol. 9 (1904), p. 133.

1104. Mathematics, rightly viewed, possesses not only truth, but supreme
beauty—a beauty cold and austere, like that of sculpture, without appeal to
any part of our weaker nature, without the gorgeous trappings of painting or
music, yet sublimely pure, and capable of a stern perfection such as only the
greatest art can show. The true spirit of delight, the exaltation, the sense of
being more than man, which is the touchstone of the highest excellence, is
to be found in mathematics as surely as in poetry. What is best in
mathematics deserves not merely to be learned as a task, but to be
assimilated as a part of daily thought, and brought again and again before
the mind with ever-renewed encouragement. Real life is, to most men, a
long second-best, a perpetual compromise between the real and the
possible; but the world of pure reason knows no compromise, no practical
limitations, no barrier to the creative activity embodying in splendid
edifices the passionate aspiration after the perfect from which all great work
springs. Remote from human passions, remote even from the pitiful facts of
nature, the generations have gradually created an ordered cosmos, where
pure thought can dwell as in its natural home, and where one, at least, of

our nobler impulses can escape from the dreary exile of the natural world.
—Russell, Bertrand.
The Study of Mathematics: Philosophical
Essays (London, 1910), p. 73.

1105. It was not alone the striving for universal culture which attracted the
great masters of the Renaissance, such as Brunellesco, Leonardo de Vinci,
Raphael, Michael Angelo and especially Albrecht Dürer, with irresistible
power to the mathematical sciences. They were conscious that, with all the
freedom of the individual phantasy, art is subject to necessary laws, and
conversely, with all its rigor of logical structure, mathematics follows
esthetic laws.—Rudio, F.
Virchow-Holtzendorf: Sammlung
gemeinverständliche wissenschaftliche
Vorträge, Heft 142, p. 19.

1106. Surely the claim of mathematics to take a place among the liberal arts
must now be admitted as fully made good. Whether we look at the advances
made in modern geometry, in modern integral calculus, or in modern
algebra, in each of these three a free handling of the material employed is
now possible, and an almost unlimited scope is left to the regulated play of
fancy. It seems to me that the whole of aesthetic (so far as at present
revealed) may be regarded as a scheme having four centres, which may be
treated as the four apices of a tetrahedron, namely Epic, Music, Plastic, and
Mathematic. There will be found a common plane to every three of these,
outside of which lies the fourth; and through every two may be drawn a
common axis opposite to the axis passing through the other two. So far is
certain and demonstrable. I think it also possible that there is a centre of
gravity to each set of three, and that the line joining each such centre with
the outside apex will intersect in a common point—the centre of gravity of
the whole body of aesthetic; but what that centre is or must be I have not
had time to think out.—Sylvester, J. J.
Proof of the hitherto undemonstrated
Fundamental Theorem of Invariants:
Collected Mathematical Papers, Vol. 3, p.
123.

1107. It is with mathematics not otherwise than it is with music, painting or
poetry. Anyone can become a lawyer, doctor or chemist, and as such may
succeed well, provided he is clever and industrious, but not every one can
become a painter, or a musician, or a mathematician: general cleverness and
industry alone count here for nothing.—Moebius, P. J.
Ueber die Anlage zur Mathematik (Leipzig,
1900), p. 5.

1108. The true mathematician is always a good deal of an artist, an
architect, yes, of a poet. Beyond the real world, though perceptibly
connected with it, mathematicians have intellectually created an ideal
world, which they attempt to develop into the most perfect of all worlds,
and which is being explored in every direction. None has the faintest
conception of this world, except he who knows it.—Pringsheim, A.
Jahresbericht der Deutschen Mathematiker
Vereinigung, Bd. 32, p. 381.

1109. Who has studied the works of such men as Euler, Lagrange, Cauchy,
Riemann, Sophus Lie, and Weierstrass, can doubt that a great
mathematician is a great artist? The faculties possessed by such men,
varying greatly in kind and degree with the individual, are analogous with
those requisite for constructive art. Not every mathematician possesses in a
specially high degree that critical faculty which finds its employment in the
perfection of form, in conformity with the ideal of logical completeness; but
every great mathematician possesses the rarer faculty of constructive
imagination.—Hobson, E. W.
Presidential Address British Association for
the Advancement of Science (1910) Nature,
Vol. 84, p. 290.

1110. Mathematics has beauties of its own—a symmetry and proportion in
its results, a lack of superfluity, an exact adaptation of means to ends, which
is exceedingly remarkable and to be found elsewhere only in the works of
the greatest beauty. It was a felicitous expression of Goethe’s to call a noble
cathedral “frozen music,” but it might even better be called “petrified

mathematics.” The beauties of mathematics—of simplicity, of symmetry, of
completeness—can and should be exemplified even to young children.
When this subject is properly and concretely presented, the mental emotion
should be that of enjoyment of beauty, not that of repulsion from the ugly
and the unpleasant.—Young, J. W. A.
The Teaching of Mathematics (New York,
1907), p. 44.

1111. A peculiar beauty reigns in the realm of mathematics, a beauty which
resembles not so much the beauty of art as the beauty of nature and which
affects the reflective mind, which has acquired an appreciation of it, very
much like the latter.—Kummer, E. E.
Berliner Monatsberichte (1867), p. 395.

1112. Mathematics make the mind attentive to the objects which it
considers. This they do by entertaining it with a great variety of truths,
which are delightful and evident, but not obvious. Truth is the same thing to
the understanding as music to the ear and beauty to the eye. The pursuit of it
does really as much gratify a natural faculty implanted in us by our wise
Creator as the pleasing of our senses: only in the former case, as the object
and faculty are more spiritual, the delight is more pure, free from regret,
turpitude, lassitude, and intemperance that commonly attend sensual
pleasures.—Arbuthnot, John.
Usefulness of Mathematical Learning.

1113. However far the calculating reason of the mathematician may seem
separated from the bold flight of the artist’s phantasy, it must be
remembered that these expressions are but momentary images snatched
arbitrarily from among the activities of both. In the projection of new
theories the mathematician needs as bold and creative a phantasy as the
productive artist, and in the execution of the details of a composition the
artist too must calculate dispassionately the means which are necessary for
the successful consummation of the parts. Common to both is the creation,
the generation, of forms out of mind.—Lampe, E.

Die Entwickelung der Mathematik, etc.
(Berlin, 1893), p. 4.

1114. As pure truth is the polar star of our science [mathematics], so it is the
great advantage of our science over others that it awakens more easily the
love of truth in our pupils.... If Hegel justly said, “Whoever does not know
the works of the ancients, has lived without knowing beauty,” Schellbach
responds with equal right, “Who does not know mathematics, and the
results of recent scientific investigation, dies without knowing truth”—
Simon, Max.
Quoted in J. W. A. Young: Teaching of
Mathematics (New York, 1907), p. 44.

1115. Büchsel in his reminiscences from the life of a country parson relates
that he sought his recreation in Lacroix’s Differential Calculus and thus
found intellectual refreshment for his calling. Instances like this make
manifest the great advantage which occupation with mathematics affords to
one who lives remote from the city and is compelled to forego the pleasures
of art. The entrancing charm of mathematics, which captivates every one
who devotes himself to it, and which is comparable to the fine frenzy under
whose ban the poet completes his work, has ever been incomprehensible to
the spectator and has often caused the enthusiastic mathematician to be held
in derision. A classic illustration is the example of Archimedes,....—Lampe,
E.
Die Entwickelung der Mathematik, etc.
(Berlin 1893), p. 22.

1116. Among the memoirs of Kirchhoff are some of uncommon beauty.
Beauty, I hear you ask, do not the Graces flee where integrals stretch forth
their necks? Can anything be beautiful, where the author has no time for the
slightest external embellishment?... Yet it is this very simplicity, the
indispensableness of each word, each letter, each little dash, that among all
artists raises the mathematician nearest to the World-creator; it establishes a
sublimity which is equalled in no other art,—something like it exists at
most in symphonic music. The Pythagoreans recognized already the

similarity between the most subjective and the most objective of the arts....
Ultima se tangunt. How expressive, how nicely characterizing withal is
mathematics! As the musician recognizes Mozart, Beethoven, Schubert in
the first chords, so the mathematician would distinguish his Cauchy, Gauss,
Jacobi, Helmholtz in a few pages. Extreme external elegance, sometimes a
somewhat weak skeleton of conclusions characterizes the French; the
English, above all Maxwell, are distinguished by the greatest dramatic bulk.
Who does not know Maxwell’s dynamic theory of gases? At first there is
the majestic development of the variations of velocities, then enter from one
side the equations of condition and from the other the equations of central
motions,—higher and higher surges the chaos of formulas,—suddenly four
words burst forth: “Put n = 5.” The evil demon V disappears like the sudden
ceasing of the basso parts in music, which hitherto wildly permeated the
piece; what before seemed beyond control is now ordered as by magic.
There is no time to state why this or that substitution was made, he who
cannot feel the reason may as well lay the book aside; Maxwell is no
program-musician who explains the notes of his composition. Forthwith the
formulas yield obediently result after result, until the temperature-
equilibrium of a heavy gas is reached as a surprising final climax and the
curtain drops....
Kirchhoff’s whole tendency, and its true counterpart, the form of his
presentation, was different.... He is characterized by the extreme precision
of his hypotheses, minute execution, a quiet rather than epic development
with utmost rigor, never concealing a difficulty, always dispelling the
faintest obscurity. To return once more to my allegory, he resembled
Beethoven, the thinker in tones.—He who doubts that mathematical
compositions can be beautiful, let him read his memoir on Absorption and
Emission (Gesammelte Abhandlungen, Leipzig, 1882, p. 571-598) or the
chapter of his mechanics devoted to Hydrodynamics.—Boltzmann, L.
Gustav Robert Kirchhoff (Leipzig 1888), pp.
28-30.

1117.
On poetry and geometric truth,
And their high privilege of lasting life,
From all internal injury exempt,

I mused; upon these chiefly: and at length,
My senses yielding to the sultry air,
Sleep seized me, and I passed into a dream.
—Wordsworth.
The Prelude, Bk. 5.

1118. Geometry seems to stand for all that is practical, poetry for all that is
visionary, but in the kingdom of the imagination you will find them close
akin, and they should go together as a precious heritage to every youth.—
Milner, Florence.
School Review, 1898, p. 114.

1119. The beautiful has its place in mathematics as elsewhere. The prose of
ordinary intercourse and of business correspondence might be held to be the
most practical use to which language is put, but we should be poor indeed
without the literature of imagination. Mathematics too has its triumphs of
the creative imagination, its beautiful theorems, its proofs and processes
whose perfection of form has made them classic. He must be a “practical”
man who can see no poetry in mathematics.—White, W. F.
A Scrap-book of Elementary Mathematics
(Chicago, 1908), p. 208.

1120. I venture to assert that the feelings one has when the beautiful
symbolism of the infinitesimal calculus first gets a meaning, or when the
delicate analysis of Fourier has been mastered, or while one follows Clerk
Maxwell or Thomson into the strange world of electricity, now growing so
rapidly in form and being, or can almost feel with Stokes the pulsations of
light that gives nature to our eyes, or track with Clausius the courses of
molecules we can measure, even if we know with certainty that we can
never see them—I venture to assert that these feelings are altogether
comparable to those aroused in us by an exquisite poem or a lofty thought.
—Workman, W. P.
F. Spencer: Aim and Practice of Teaching
(New York, 1897), p. 194.

1121. It is an open secret to the few who know it, but a mystery and
stumbling block to the many, that Science and Poetry are own sisters;
insomuch that in those branches of scientific inquiry which are most
abstract, most formal, and most remote from the grasp of the ordinary
sensible imagination, a higher power of imagination akin to the creative
insight of the poet is most needed and most fruitful of lasting work.—
Pollock, F.
Clifford’s Lectures and Essays (New York,
1901), Vol. 1, Introduction, p. 1.

1122. It is as great a mistake to maintain that a high development of the
imagination is not essential to progress in mathematical studies as to hold
with Ruskin and others that science and poetry are antagonistic pursuits.—
Hoffman, F. S.
Sphere of Science (London, 1898), p. 107.

1123. We have heard much about the poetry of mathematics, but very little
of it has as yet been sung. The ancients had a juster notion of their poetic
value than we. The most distinct and beautiful statements of any truth must
take at last the mathematical form. We might so simplify the rules of moral
philosophy, as well as of arithmetic, that one formula would express them
both.—Thoreau, H. D.
A Week on the Concord and Merrimac Rivers
(Boston, 1893), p. 477.

1124. We do not listen with the best regard to the verses of a man who is
only a poet, nor to his problems if he is only an algebraist; but if a man is at
once acquainted with the geometric foundation of things and with their
festal splendor, his poetry is exact and his arithmetic musical.—Emerson,
R. W.
Society and Solitude, Chap. 7, Works and
Days.

1125. Mathesis and Poetry are ... the utterance of the same power of
imagination, only that in the one case it is addressed to the head, and in the
other, to the heart.—Hill, Thomas.
North American Review, Vol. 85, p. 230.

1126. The Mathematics are usually considered as being the very antipodes
of Poesy. Yet Mathesis and Poesy are of the closest kindred, for they are
both works of the imagination. Poesy is a creation, a making, a fiction; and
the Mathematics have been called, by an admirer of them, the sublimest and
most stupendous of fictions. It is true, they are not only μάθησις, learning,
but ποίησις, a creation.—Hill, Thomas.
North American Review, Vol. 85, p. 229.

1127.

Music and poesy used to quicken you:
The mathematics, and the metaphysics,
Fall to them as you find your stomach serves you.
No profit grows, where is no pleasure ta’en:—
In brief, sir, study what you most affect.
—Shakespeare.
Taming of the Shrew, Act 1, Scene 1.

1128. Music has much resemblance to algebra.—Novalis.
Schriften, Teil 2 (Berlin, 1901), p. 549.

1129.
I do present you with a man of mine,
Cunning in music and in mathematics,
To instruct her fully in those sciences,
Whereof, I know, she is not ignorant.
—Shakespeare.
Taming of the Shrew, Act 2, Scene 1.

1130. Saturated with that speculative spirit then pervading the Greek mind,
he [Pythagoras] endeavoured to discover some principle of homogeneity in
the universe. Before him, the philosophers of the Ionic school had sought it
in the matter of things; Pythagoras looked for it in the structure of things.
He observed the various numerical relations or analogies between numbers
and the phenomena of the universe. Being convinced that it was in numbers
and their relations that he was to find the foundation to true philosophy, he
proceeded to trace the origin of all things to numbers. Thus he observed that
musical strings of equal lengths stretched by weights having the proportion
of ½, ⅔, ¾, produced intervals which were an octave, a fifth and a fourth.
Harmony, therefore, depends on musical proportion; it is nothing but a
mysterious numerical relation. Where harmony is, there are numbers. Hence
the order and beauty of the universe have their origin in numbers. There are
seven intervals in the musical scale, and also seven planets crossing the
heavens. The same numerical relations which underlie the former must
underlie the latter. But where number is, there is harmony. Hence his

spiritual ear discerned in the planetary motions a wonderful “Harmony of
spheres”—Cajori, F.
History of Mathematics (New York, 1897), p.
67.

1131. May not Music be described as the Mathematic of sense, Mathematic
as Music of the reason? the soul of each the same! Thus the musician feels
Mathematic, the mathematician thinks Music,—Music the dream,
Mathematic the working life—each to receive its consummation from the
other when the human intelligence, elevated to its perfect type, shall shine
forth glorified in some future Mozart-Dirichlet or Beethoven-Gauss—a
union already not indistinctly foreshadowed in the genius and labours of a
Helmholtz!—Sylvester, J. J.
On Newton’s Rule for the Discovery of
Imaginary Roots; Collected Mathematical
Papers, Vol. 2, p. 419.

1132. Just as the musician is able to form an acoustic image of a
composition which he has never heard played by merely looking at its
score, so the equation of a curve, which he has never seen, furnishes the
mathematician with a complete picture of its course. Yea, even more: as the
score frequently reveals to the musician niceties which would escape his ear
because of the complication and rapid change of the auditory impressions,
so the insight which the mathematician gains from the equation of a curve is
much deeper than that which is brought about by a mere inspection of the
curve.—Pringsheim, A.
Jahresbericht der Deutschen Mathematiker
Vereinigung. Bd. 13, p. 364.

1133. Mathematics and music, the most sharply contrasted fields of
scientific activity which can be found, and yet related, supporting each
other, as if to show forth the secret connection which ties together all the
activities of our mind, and which leads us to surmise that the manifestations
of the artist’s genius are but the unconscious expressions of a mysteriously
acting rationality.—Helmholtz, H.

Vorträge und Reden, Bd. 1 (Braunschweig,
1884), p. 82.

1134. Among all highly civilized peoples the golden age of art has always
been closely coincident with the golden age of the pure sciences,
particularly with mathematics, the most ancient among them.
This coincidence must not be looked upon as accidental, but as natural,
due to an inner necessity. Just as art can thrive only when the artist, relieved
of the anxieties of existence, can listen to the inspirations of his spirit and
follow in their lead, so mathematics, the most ideal of the sciences, will
yield its choicest blossoms only when life’s dismal phantom dissolves and
fades away, when the striving after naked truth alone predominates,
conditions which prevail only in nations while in the prime of their
development.—Lampe, E.
Die Entwickelung der Mathematik etc.
(Berlin, 1893), p. 4.

1135. Till the fifteenth century little progress appears to have been made in
the science or practice of music; but since that era it has advanced with
marvelous rapidity, its progress being curiously parallel with that of
mathematics, inasmuch as great musical geniuses appeared suddenly among
different nations, equal in their possession of this special faculty to any that
have since arisen. As with the mathematical so with the musical faculty—it
is impossible to trace any connection between its possession and survival in
the struggle for existence.—Wallace, A. R.
Darwinism, Chap. 15.

1136. In my opinion, there is absolutely no trustworthy proof that talents
have been improved by their exercise through the course of a long series of
generations. The Bach family shows that musical talent, and the Bernoulli
family that mathematical power, can be transmitted from generation to
generation, but this teaches us nothing as to the origin of such talents. In
both families the high-watermark of talent lies, not at the end of the series
of generations, as it should do if the results of practice are transmitted, but

in the middle. Again, talents frequently appear in some member of a family
which has not been previously distinguished.
Gauss was not the son of a mathematician; Handel’s father was a
surgeon, of whose musical powers nothing is known; Titian was the son and
also the nephew of a lawyer, while he and his brother, Francesco Vecellio,
were the first painters in a family which produced a succession of seven
other artists with diminishing talents. These facts do not, however, prove
that the condition of the nerve-tracts and centres of the brain, which
determine the specific talent, appeared for the first time in these men: the
appropriate condition surely existed previously in their parents, although it
did not achieve expression. They prove, as it seems to me, that a high
degree of endowment in a special direction, which we call talent, cannot
have arisen from the experience of previous generations, that is, by the
exercise of the brain in the same specific direction.—Weismann, August.
Essays upon Heredity [A. E. Shipley],
(Oxford, 1891), Vol. 1, p. 97.

CHAPTER XII
MATHEMATICS AS A LANGUAGE

1201. The new mathematics is a sort of supplement to language, affording a
means of thought about form and quantity and a means of expression, more
exact, compact, and ready than ordinary language. The great body of
physical science, a great deal of the essential facts of financial science, and
endless social and political problems are only accessible and only thinkable
to those who have had a sound training in mathematical analysis, and the
time may not be very remote when it will be understood that for complete
initiation as an efficient citizen of one of the new great complex world wide
states that are now developing, it is as necessary to be able to compute, to
think in averages and maxima and minima, as it is now to be able to read
and to write.—Wells, H. G.
Mankind in the Making (London, 1904), pp.
191-192.

1202. Mathematical language is not only the simplest and most easily
understood of any, but the shortest also.—Brougham, H. L.
Works (Edinburgh, 1872), Vol. 7, p. 317.

1203. Mathematics is the science of definiteness, the necessary vocabulary
of those who know.—White, W. F.
A Scrap-book of Elementary Mathematics
(Chicago, 1908), p. 7.

1204. Mathematics, too, is a language, and as concerns its structure and
content it is the most perfect language which exists, superior to any
vernacular; indeed, since it is understood by every people, mathematics

may be called the language of languages. Through it, as it were, nature
herself speaks; through it the Creator of the world has spoken, and through
it the Preserver of the world continues to speak.—Dillmann, C.
Die Mathematik die Fackelträgerin einer
neuen Zeit (Stuttgart, 1889), p. 5.

1205. Would it sound too presumptuous to speak of perception as a
quintessence of sensation, language (that is, communicable thought) of
perception, mathematics of language? We should then have four terms
differentiating from inorganic matter and from each other the Vegetable,
Animal, Rational, and Super-sensual modes of existence.—Sylvester, J. J.
Presidential Address, British Association;
Collected Mathematical Papers, Vol. 2, p.
652.

1206. Little could Plato have imagined, when, indulging his instinctive love
of the true and beautiful for their own sakes, he entered upon these refined
speculations and revelled in a world of his own creation, that he was writing
the grammar of the language in which it would be demonstrated in after
ages that the pages of the universe are written.—Sylvester, J. J.
A Probationary Lecture on Geometry;
Collected Mathematical Papers, Vol. 2, p. 7.

1207. It is the symbolic language of mathematics only which has yet proved
sufficiently accurate and comprehensive to demand familiarity with this
conception of an inverse process.—Venn, John.
Symbolic Logic (London and New York,
1894), p. 74.

1208. Without this language [mathematics] most of the intimate analogies
of things would have remained forever unknown to us; and we should
forever have been ignorant of the internal harmony of the world, which is
the only true objective reality....

This harmony ... is the sole objective reality, the only truth we can
attain; and when I add that the universal harmony of the world is the source
of all beauty, it will be understood what price we should attach to the slow
and difficult progress which little by little enables us to know it better.—
Poincaré, H.
The Value of Science [Halsted] Popular
Science Monthly, 1906, pp. 195-196.

1209. The most striking characteristic of the written language of algebra
and of the higher forms of the calculus is the sharpness of definition, by
which we are enabled to reason upon the symbols by the mere laws of
verbal logic, discharging our minds entirely of the meaning of the symbols,
until we have reached a stage of the process where we desire to interpret
our results. The ability to attend to the symbols, and to perform the verbal,
visible changes in the position of them permitted by the logical rules of the
science, without allowing the mind to be perplexed with the meaning of the
symbols until the result is reached which you wish to interpret, is a
fundamental part of what is called analytical power. Many students find
themselves perplexed by a perpetual attempt to interpret not only the result,
but each step of the process. They thus lose much of the benefit of the
labor-saving machinery of the calculus and are, indeed, frequently
incapacitated for using it.—Hill, Thomas.
Uses of Mathesis; Bibliotheca Sacra, Vol. 32,
p. 505.

1210. The prominent reason why a mathematician can be judged by none
but mathematicians, is that he uses a peculiar language. The language of
mathesis is special and untranslatable. In its simplest forms it can be
translated, as, for instance, we say a right angle to mean a square corner.
But you go a little higher in the science of mathematics, and it is impossible
to dispense with a peculiar language. It would defy all the power of
Mercury himself to explain to a person ignorant of the science what is
meant by the single phrase “functional exponent.” How much more
impossible, if we may say so, would it be to explain a whole treatise like
Hamilton’s Quaternions, in such a wise as to make it possible to judge of its
value! But to one who has learned this language, it is the most precise and

clear of all modes of expression. It discloses the thought exactly as
conceived by the writer, with more or less beauty of form, but never with
obscurity. It may be prolix, as it often is among French writers; may delight
in mere verbal metamorphoses, as in the Cambridge University of England;
or adopt the briefest and clearest forms, as under the pens of the geometers
of our Cambridge; but it always reveals to us precisely the writer’s thought.
—Hill, Thomas.
North American Review, Vol. 85, pp. 224-
225.

1211. The domain, over which the language of analysis extends its sway, is,
indeed, relatively limited, but within this domain it so infinitely excels
ordinary language that its attempt to follow the former must be given up
after a few steps. The mathematician, who knows how to think in this
marvelously condensed language, is as different from the mechanical
computer as heaven from earth.—Pringsheim, A.
Jahresberichte der Deutschen Mathematiker
Vereinigung, Bd. 13, p. 367.

1212. The results of systematic symbolical reasoning must always express
general truths, by their nature; and do not, for their justification, require
each of the steps of the process to represent some definite operation upon
quantity. The absolute universality of the interpretation of symbols is the
fundamental principle of their use.—Whewell, William.
The Philosophy of the Inductive Sciences,
Part I, Bk. 2, chap. 12, sect. 2 (London,
1858).

1213. Anyone who understands algebraic notation, reads at a glance in an
equation results reached arithmetically only with great labour and pains.—
Cournot, A.
Theory of Wealth [N. T. Bacon], (New York,
1897), p. 4.

1214. As arithmetic and algebra are sciences of great clearness, certainty,
and extent, which are immediately conversant about signs, upon the skilful
use whereof they entirely depend, so a little attention to them may possibly
help us to judge of the progress of the mind in other sciences, which,
though differing in nature, design, and object, may yet agree in the general
methods of proof and inquiry.—Berkeley, George.
Alciphron, or the Minute Philosopher,
Dialogue 7, sect. 12.

1215. In general the position as regards all such new calculi is this—That
one cannot accomplish by them anything that could not be accomplished
without them. However, the advantage is, that, provided such a calculus
corresponds to the inmost nature of frequent needs, anyone who masters it
thoroughly is able—without the unconscious inspiration of genius which no
one can command—to solve the respective problems, yea, to solve them
mechanically in complicated cases in which, without such aid, even genius
becomes powerless. Such is the case with the invention of general algebra,
with the differential calculus, and in a more limited region with Lagrange’s
calculus of variations, with my calculus of congruences, and with Möbius’s
calculus. Such conceptions unite, as it were, into an organic whole countless
problems which otherwise would remain isolated and require for their
separate solution more or less application of inventive genius.—Gauss, C.
J.
Werke, Bd. 8, p. 298.

1216. The invention of what we may call primary or fundamental notation
has been but little indebted to analogy, evidently owing to the small extent
of ideas in which comparison can be made useful. But at the same time
analogy should be attended to, even if for no other reason than that, by
making the invention of notation an art, the exertion of individual caprice
ceases to be allowable. Nothing is more easy than the invention of notation,
and nothing of worse example and consequence than the confusion of
mathematical expressions by unknown symbols. If new notation be
advisable, permanently or temporarily, it should carry with it some mark of
distinction from that which is already in use, unless it be a demonstrable
extension of the latter.—De Morgan, A.

Calculus of Functions; Encyclopedia
Metropolitana, Addition to Article 26.

1217. Before the introduction of the Arabic notation, multiplication was
difficult, and the division even of integers called into play the highest
mathematical faculties. Probably nothing in the modern world could have
more astonished a Greek mathematician than to learn that, under the
influence of compulsory education, the whole population of Western
Europe, from the highest to the lowest, could perform the operation of
division for the largest numbers. This fact would have seemed to him a
sheer impossibility.... Our modern power of easy reckoning with decimal
fractions is the most miraculous result of a perfect notation.—Whitehead,
A. N.
Introduction to Mathematics (New York,
1911), p. 59.

1218. Mathematics is often considered a difficult and mysterious science,
because of the numerous symbols which it employs. Of course, nothing is
more incomprehensible than a symbolism which we do not understand.
Also a symbolism, which we only partially understand and are
unaccustomed to use, is difficult to follow. In exactly the same way the
technical terms of any profession or trade are incomprehensible to those
who have never been trained to use them. But this is not because they are
difficult in themselves. On the contrary they have invariably been
introduced to make things easy. So in mathematics, granted that we are
giving any serious attention to mathematical ideas, the symbolism is
invariably an immense simplification.—Whitehead, A. N.
Introduction to Mathematics (New York,
1911), pp. 59-60.

1219. Symbolism is useful because it makes things difficult. Now in the
beginning everything is self-evident, and it is hard to see whether one self-
evident proposition follows from another or not. Obviousness is always the
enemy to correctness. Hence we must invent a new and difficult symbolism
in which nothing is obvious.... Thus the whole of Arithmetic and Algebra

has been shown to require three indefinable notions and five
indemonstrable propositions.—Russell, Bertrand.
International Monthly, 1901, p. 85.

1220. The employment of mathematical symbols is perfectly natural when
the relations between magnitudes are under discussion; and even if they are
not rigorously necessary, it would hardly be reasonable to reject them,
because they are not equally familiar to all readers and because they have
sometimes been wrongly used, if they are able to facilitate the exposition of
problems, to render it more concise, to open the way to more extended
developments, and to avoid the digressions of vague argumentation.—
Cournot, A.
Theory of Wealth [N. T. Bacon], (New York,
1897), pp. 3-4.

1221. An all-inclusive geometrical symbolism, such as Hamilton and
Grassmann conceived of, is impossible.—Burkhardt, H.
Jahresbericht der Deutschen Mathematiker
Vereinigung, Bd. 5, p. 52.

1222. The language of analysis, most perfect of all, being in itself a
powerful instrument of discoveries, its notations, especially when they are
necessary and happily conceived, are so many germs of new calculi.—
Laplace.
Oeuvres, t. 7 (Paris, 1896), p. xl.

CHAPTER XIII
MATHEMATICS AND LOGIC

1301. Mathematics belongs to every inquiry, moral as well as physical.
Even the rules of logic, by which it is rigidly bound, could not be deduced
without its aid. The laws of argument admit of simple statement, but they
must be curiously transposed before they can be applied to the living speech
and verified by observation. In its pure and simple form the syllogism
cannot be directly compared with all experience, or it would not have
required an Aristotle to discover it. It must be transmuted into all the
possible shapes in which reasoning loves to clothe itself. The transmutation
is the mathematical process in the establishment of the law.—Peirce,
Benjamin.
Linear Associative Algebra; American
Journal of Mathematics, Vol. 4 (1881), p. 97.

1302. In mathematics we see the conscious logical activity of our mind in
its purest and most perfect form; here is made manifest to us all the labor
and the great care with which it progresses, the precision which is necessary
to determine exactly the source of the established general theorems, and the
difficulty with which we form and comprehend abstract conceptions; but we
also learn here to have confidence in the certainty, breadth, and fruitfulness
of such intellectual labor.—Helmholtz, H.
Vorträge und Reden, Bd. 1 (Braunschweig,
1896), p. 176.

1303. Mathematical demonstrations are a logic of as much or more use,
than that commonly learned at schools, serving to a just formation of the
mind, enlarging its capacity, and strengthening it so as to render the same
capable of exact reasoning, and discerning truth from falsehood in all

occurrences, even in subjects not mathematical. For which reason it is said,
the Egyptians, Persians, and Lacedaemonians seldom elected any new
kings, but such as had some knowledge in the mathematics, imagining
those, who had not, men of imperfect judgments, and unfit to rule and
govern.—Franklin, Benjamin.
Usefulness of Mathematics; Works (Boston,
1840), Vol. 2, p. 68.

1304. The mathematical conception is, from its very nature, abstract; indeed
its abstractness is usually of a higher order than the abstractness of the
logician.—Chrystal, George.
Encyclopedia Britannica (Ninth Edition),
Article “Mathematics”

1305. Mathematics, that giant pincers of scientific logic....—Halsted, G. B.
Science (1905), p. 161.

1306. Logic has borrowed the rules of geometry without understanding its
power.... I am far from placing logicians by the side of geometers who teach
the true way to guide the reason.... The method of avoiding error is sought
by every one. The logicians profess to lead the way, the geometers alone
reach it, and aside from their science there is no true demonstration.—
Pascal.
Quoted by A. Rebière: Mathématiques et
Mathématiciens (Paris, 1898), pp. 162-163.

1307. Mathematics, like dialectics, is an organ of the higher sense, in its
execution it is an art like eloquence. To both nothing but the form is of
value; neither cares anything for content. Whether mathematics considers
pennies or guineas, whether rhetoric defends truth or error, is perfectly
immaterial to either.—Goethe.
Sprüche in Prosa, Natur IV, 946.

1308. Confined to its true domain, mathematical reasoning is admirably
adapted to perform the universal office of sound logic: to induce in order to
deduce, in order to construct.... It contents itself to furnish, in the most
favorable domain, a model of clearness, of precision, and consistency, the
close contemplation of which is alone able to prepare the mind to render
other conceptions also as perfect as their nature permits. Its general
reaction, more negative than positive, must consist, above all, in inspiring
us everywhere with an invincible aversion for vagueness, inconsistency, and
obscurity, which may always be really avoided in any reasoning
whatsoever, if we make sufficient effort.—Comte, A.
Subjective Synthesis.

1309. Formal thought, consciously recognized as such, is the means of all
exact knowledge; and a correct understanding of the main formal sciences,
Logic and Mathematics, is the proper and only safe foundation for a
scientific education.—Lefevre, Arthur.
Number and its Algebra (Boston, Sect. 222.)

1310. It has come to pass, I know not how, that Mathematics and Logic,
which ought to be but the handmaids of Physic, nevertheless presume on
the strength of the certainty which they possess to exercise dominion over
it.—Bacon, Francis.
De Augmentis, Bk. 3.

1311. We may regard geometry as a practical logic, for the truths which it
considers, being the most simple and most sensible of all, are, for this
reason, the most susceptible to easy and ready application of the rules of
reasoning.—D’Alembert.
Quoted in A. Rebière: Mathématiques et
Mathématiciens (Paris, 1898), pp. 151-152.

1312. There are notable examples enough of demonstration outside of
mathematics, and it may be said that Aristotle has already given some in his
“Prior Analytics.” In fact logic is as susceptible of demonstration as

geometry,.... Archimedes is the first, whose works we have, who has
practised the art of demonstration upon an occasion where he is treating of
physics, as he has done in his book on Equilibrium. Furthermore, jurists
may be said to have many good demonstrations; especially the ancient
Roman jurists, whose fragments have been preserved to us in the Pandects.
—Leibnitz, G. W.
New Essay on Human Understanding
[Langley], Bk. 4, chap. 2, sect. 12.

1313. It is commonly considered that mathematics owes its certainty to its
reliance on the immutable principles of formal logic. This ... is only half the
truth imperfectly expressed. The other half would be that the principles of
formal logic owe such a degree of permanence as they have largely to the
fact that they have been tempered by long and varied use by
mathematicians. “A vicious circle!” you will perhaps say. I should rather
describe it as an example of the process known by mathematicians as the
method of successive approximation.—Bôcher, Maxime.
Bulletin of the American Mathematical
Society, Vol. 11, p. 120.

1314. Whatever advantage can be attributed to logic in directing and
strengthening the action of the understanding is found in a higher degree in
mathematical study, with the immense added advantage of a determinate
subject, distinctly circumscribed, admitting of the utmost precision, and free
from the danger which is inherent in all abstract logic,—of leading to
useless and puerile rules, or to vain ontological speculations. The positive
method, being everywhere identical, is as much at home in the art of
reasoning as anywhere else: and this is why no science, whether biology or
any other, can offer any kind of reasoning, of which mathematics does not
supply a simpler and purer counterpart. Thus, we are enabled to eliminate
the only remaining portion of the old philosophy which could even appear
to offer any real utility; the logical part, the value of which is irrevocably
absorbed by mathematical science.—Comte, A.
Positive Philosophy [Martineau], (London,
1875), Vol. 1, pp. 321-322.

1315. We know that mathematicians care no more for logic than logicians
for mathematics. The two eyes of exact science are mathematics and logic:
the mathematical sect puts out the logical eye, the logical sect puts out the
mathematical eye; each believing that it can see better with one eye than
with two.—De Morgan, A.
Quoted in F. Cajori: History of Mathematics
(New York, 1897), p. 316.

1316. The progress of the art of rational discovery depends in a great part
upon the art of characteristic (ars characteristica). The reason why people
usually seek demonstrations only in numbers and lines and things
represented by these is none other than that there are not, outside of
numbers, convenient characters corresponding to the notions.—Leibnitz, G.
W.
Philosophische Schriften [Gerhardt] Bd. 8, p.
198.

1317. The influence of the mathematics of Leibnitz upon his philosophy
appears chiefly in connection with his law of continuity and his prolonged
efforts to establish a Logical Calculus.... To find a Logical Calculus
(implying a universal philosophical language or system of signs) is an
attempt to apply in theological and philosophical investigations an analytic
method analogous to that which had proved so successful in Geometry and
Physics. It seemed to Leibnitz that if all the complex and apparently
disconnected ideas which make up our knowledge could be analysed into
their simple elements, and if these elements could each be represented by a
definite sign, we should have a kind of “alphabet of human thoughts.” By
the combination of these signs (letters of the alphabet of thought) a system
of true knowledge would be built up, in which reality would be more and
more adequately represented or symbolized.... In many cases the analysis
may result in an infinite series of elements; but the principles of the
Infinitesimal Calculus in mathematics have shown that this does not
necessarily render calculation impossible or inaccurate. Thus it seemed to
Leibnitz that a synthetic calculus, based upon a thorough analysis, would be
the most effective instrument of knowledge that could be devised. “I feel,”
he says, “that controversies can never be finished, nor silence imposed upon

the Sects, unless we give up complicated reasonings in favor of simple
calculations, words of vague and uncertain meaning in favor of fixed
symbols.” Thus it will appear that “every paralogism is nothing but an error
of calculation.” “When controversies arise, there will be no more necessity
of disputation between two philosophers than between two accountants.
Nothing will be needed but that they should take pen in hand, sit down with
their counting-tables, and (having summoned a friend, if they like) say to
one another: Let us calculate”—Latta, Robert.
Leibnitz, The Monadology, etc. (Oxford,
1898), p. 85.

1318. Pure mathematics was discovered by Boole in a work which he called
“The Laws of Thought“.... His work was concerned with formal logic, and
this is the same thing as mathematics.—Russell, Bertrand.
International Monthly, 1901, p. 83.

1319. Mathematics is but the higher development of Symbolic Logic.—
Whetham, W. C. D.
Recent Development of Physical Science
(Philadelphia, 1904), p. 34.

1320. Symbolic Logic has been disowned by many logicians on the plea
that its interest is mathematical, and by many mathematicians on the plea
that its interest is logical.—Whitehead, A. N.
Universal Algebra (Cambridge, 1898),
Preface, p. 6.

1321. ... the two great components of the critical movement, though distinct
in origin and following separate paths, are found to converge at last in the
thesis: Symbolic Logic is Mathematics, Mathematics is Symbolic Logic, the
twain are one.—Keyser, C. J.
Lectures on Science, Philosophy and Art
(New York, 1908), p. 19.

1322. The emancipation of logic from the yoke of Aristotle very much
resembles the emancipation of geometry from the bondage of Euclid; and,
by its subsequent growth and diversification, logic, less abundantly perhaps
but not less certainly than geometry, has illustrated the blessings of
freedom.—Keyser, C. J.
Science, Vol. 35 (1912), p. 108.

1323. I would express it as my personal view, which is probably not yet
shared generally, that pure mathematics seems to me merely a branch of
general logic; that branch which is based on the concept of numbers, to
whose economic advantages is to be attributed the tremendous development
which this particular branch has undergone as compared with the remaining
branches of logic, which until the most recent times have remained almost
stationary.—Schröder, E.
Ueber Pasigraphie etc.; Verhandlungen des
1. Internationalen Mathematiker-Kongresses
(Leipzig, 1898), p. 149.

1324. If logical training is to consist, not in repeating barbarous scholastic
formulas or mechanically tacking together empty majors and minors, but in
acquiring dexterity in the use of trustworthy methods of advancing from the
known to the unknown, then mathematical investigation must ever remain
one of its most indispensable instruments. Once inured to the habit of
accurately imagining abstract relations, recognizing the true value of
symbolic conceptions, and familiarized with a fixed standard of proof, the
mind is equipped for the consideration of quite other objects than lines and
angles. The twin treatises of Adam Smith on social science, wherein, by
deducing all human phenomena first from the unchecked action of
selfishness and then from the unchecked action of sympathy, he arrives at
mutually-limiting conclusions of transcendent practical importance, furnish
for all time a brilliant illustration of the value of mathematical methods and
mathematical discipline.—Fiske, John.
Darwinism and other Essays (Boston, 1893),
pp. 297-298.

1325. No irrational exaggeration of the claims of Mathematics can ever
deprive that part of philosophy of the property of being the natural basis of
all logical education, through its simplicity, abstractness, generality, and
freedom from disturbance by human passion. There, and there alone, we
find in full development the art of reasoning, all the resources of which,
from the most spontaneous to the most sublime, are continually applied
with far more variety and fruitfulness than elsewhere;.... The more abstract
portion of mathematics may in fact be regarded as an immense repository of
logical resources, ready for use in scientific deduction and co-ordination.—
Comte, A.
Positive Philosophy [Martineau], (London,
1875), Vol. 2, p. 439.

1326. Logic it is called [referring to Whitehead and Russell’s Principia
Mathematica] and logic it is, the logic of propositions and functions and
classes and relations, by far the greatest (not merely the biggest) logic that
our planet has produced, so much that is new in matter and in manner; but it
is also mathematics, a prolegomenon to the science, yet itself mathematics
in its most genuine sense, differing from other parts of the science only in
the respects that it surpasses these in fundamentality, generality and
precision, and lacks traditionality. Few will read it, but all will feel its
effect, for behind it is the urgence and push of a magnificent past: two
thousand five hundred years of record and yet longer tradition of human
endeavor to think aright.—Keyser, C. J.
Science, Vol. 35 (1912), p. 110.

CHAPTER XIV
MATHEMATICS AND PHILOSOPHY

1401. Socrates is praised by all the centuries for having called philosophy
from heaven to men on earth; but if, knowing the condition of our science,
he should come again and should look once more to heaven for a means of
curing men, he would there find that to mathematics, rather than to the
philosophy of today, had been given the crown because of its industry and
its most happy and brilliant successes.—Herbart, J. F.
Werke [Kehrbach], (Langensalza, 1890), Bd.
5, p. 95.

1402. It is the embarrassment of metaphysics that it is able to accomplish so
little with the many things that mathematics offers her.—Kant, E.
Metaphysische Anfangsgründe der
Naturwissenschaft, Vorrede.

1403. Philosophers, when they have possessed a thorough knowledge of
mathematics, have been among those who have enriched the science with
some of its best ideas. On the other hand it must be said that, with hardly an
exception, all the remarks on mathematics made by those philosophers who
have possessed but a slight or hasty or late-acquired knowledge of it are
entirely worthless, being either trivial or wrong.—Whitehead, A. N.
Introduction to Mathematics (New York,
1911), p. 113.

1404. The union of philosophical and mathematical productivity, which
besides in Plato we find only in Pythagoras, Descartes and Leibnitz, has
always yielded the choicest fruits to mathematics: To the first we owe

scientific mathematics in general, Plato discovered the analytic method, by
means of which mathematics was elevated above the view-point of the
elements, Descartes created the analytical geometry, our own illustrious
countryman discovered the infinitesimal calculus—and just these are the
four greatest steps in the development of mathematics.—Hankel, Hermann.
Geschichte der Mathematik im Altertum und
im Mittelalter (Leipzig, 1874), pp. 149-150.

1405. Without mathematics one cannot fathom the depths of philosophy;
without philosophy one cannot fathom the depths of mathematics; without
the two one cannot fathom anything.—Bordas-Demoulins.
Quoted in A. Rebière: Mathématiques et
Mathématiciens (Paris, 1898), p. 147.

1406. In the end mathematics is but simple philosophy, and philosophy,
higher mathematics in general.—Novalis.
Schriften (Berlin, 1901), Teil 2, p. 443.

1407. It is a safe rule to apply that, when a mathematical or philosophical
author writes with a misty profundity, he is talking nonsense.—Whitehead,
A. N.
Introduction to Mathematics (New York,
1911), p. 227.

1408. The real finisher of our education is philosophy, but it is the office of
mathematics to ward off the dangers of philosophy.—Herbart, J. F.
Pestalozzi’s Idee eines ABC der Anschauung;
Werke [Kehrbach], (Langensalza, 1890), Bd.
1, p. 168.

1409. Since antiquity mathematics has been regarded as the most
indispensable school for philosophic thought and in its highest spheres the
research of the mathematician is indeed most closely related to pure

speculation. Mathematics is the most perfect union between exact
knowledge and theoretical thought.—Curtius, E.
Berliner Monatsberichte (1873), p. 517.

1410. Geometry has been, throughout, of supreme importance in the history
of knowledge.—Russell, Bertrand.
Foundations of Geometry (Cambridge,
1897), p. 54.

1411. He is unworthy of the name of man who is ignorant of the fact that
the diagonal of a square is incommensurable with its side.—Plato.
Quoted by Sophie Germain: Mémoire sur les
surfaces élastiques.

1412. Mathematics, considered as a science, owes its origin to the idealistic
needs of the Greek philosophers, and not as fable has it, to the practical
demands of Egyptian economics.... Adam was no zoölogist when he gave
names to the beasts of the field, nor were the Egyptian surveyors
mathematicians.—Hankel, H.
Die Entwickelung der Mathematik in den
letzten Jahrhunderten (Tübingen, 1884), p. 7.

1413. There are only two ways open to man for attaining a certain
knowledge of truth: clear intuition and necessary deduction.—Descartes.
Rules for the Direction of the Mind; Torrey’s
The Philosophy of Descartes (New York,
1892), p. 104.

1414. Mathematicians have, in many cases, proved some things to be
possible and others to be impossible, which, without demonstration, would
not have been believed.... Mathematics afford many instances of
impossibilities in the nature of things, which no man would have believed,
if they had not been strictly demonstrated. Perhaps, if we were able to

reason demonstratively in other subjects, to as great extent as in
mathematics, we might find many things to be impossible, which we
conclude, without hesitation, to be possible.—Reid, Thomas.
Essay on the Intellectual Powers of Man,
Essay 4, chap. 3.

1415. If philosophers understood mathematics, they would know that
indefinite speech, which permits each one to think what he pleases and
produces a constantly increasing difference of opinion, is utterly unable, in
spite of all fine words and even in spite of the magnitude of the objects
which are under contemplation, to maintain a balance against a science
which instructs and advances through every word which it utters and which
at the same time wins for itself endless astonishment, not through its survey
of immense spaces, but through the exhibition of the most prodigious
human ingenuity which surpasses all power of description.—Herbart, J. F.
Werke Kehrbach (Langensalza, 1890), Bd. 5,
p. 105.

1416. German intellect is an excellent thing, but when a German product is
presented it must be analysed. Most probably it is a combination of intellect
(I) and tobacco-smoke (T). Certainly I3T1, and I2T1, occur; but I1T3 is more
common, and I2T15 and I1T20 occur. In many cases metaphysics (M) occurs
and I hold that IaTbMc never occurs without b + c > 2a.
N. B.—Be careful, in analysing the compounds of the three, not to
confound T and M, which are strongly suspected to be isomorphic. Thus,
I1T3M3 may easily be confounded with I1T6. As far as I dare say anything,
those who have placed Hegel, Fichte, etc., in the rank of the extenders of
Kant have imagined T and M to be identical.—De Morgan, A.
Graves’ Life of W. R. Hamilton (New York,
1882-1889), Vol. 13, p. 446.

1417. The discovery [of Ceres] was made by G. Piazzi of Palermo; and it
was the more interesting as its announcement occurred simultaneously with
a publication by Hegel in which he severely criticized astronomers for not

paying more attention to philosophy, a science, said he, which would at
once have shown them that there could not possibly be more than seven
planets, and a study of which would therefore have prevented an absurd
waste of time in looking for what in the nature of things could never be
found.—Ball, W. W. R.
History of Mathematics (London, 1901), p.
458.

1418.
But who shall parcel out
His intellect by geometric rules,
Split like a province into round and square?
—Wordsworth.
The Prelude, Bk. 2.

1419.
And Proposition, gentle maid,
Who soothly ask’d stern Demonstration’s aid,....
—Coleridge, S. T.
A Mathematical Problem.

1420. Mathematics connect themselves on the one side with common life
and physical science; on the other side with philosophy in regard to our
notions of space and time, and in the questions which have arisen as to the
universality and necessity of the truths of mathematics and the foundation
of our knowledge of them.—Cayley, Arthur.
British Association Address (1888);
Collected Mathematical Papers, Vol. 11, p.
430.

1421. Mathematical teaching ... trains the mind to capacities, which ... are
of the closest kin to those of the greatest metaphysician and philosopher.
There is some color of truth for the opposite doctrine in the case of
elementary algebra. The resolution of a common equation can be reduced to

almost as mechanical a process as the working of a sum in arithmetic. The
reduction of the question to an equation, however, is no mechanical
operation, but one which, according to the degree of its difficulty, requires
nearly every possible grade of ingenuity: not to speak of the new, and in the
present state of the science insoluble, equations, which start up at every
fresh step attempted in the application of mathematics to other branches of
knowledge.—Mill, J. S.
An Examination of Sir William Hamilton’s
Philosophy (London, 1878), p. 615.

1422. The value of mathematical instruction as a preparation for those more
difficult investigations, consists in the applicability not of its doctrines, but
of its methods. Mathematics will ever remain the most perfect type of the
Deductive Method in general; and the applications of mathematics to the
simpler branches of physics, furnish the only school in which philosophers
can effectually learn the most difficult and important portion of their art, the
employment of the laws of the simpler phenomena for explaining and
predicting those of the more complex. These grounds are quite sufficient for
deeming mathematical training an indispensable basis of real scientific
education, and regarding, with Plato, one who is ἀγεωμέτρητος, as wanting
in one of the most essential qualifications for the successful cultivation of
the higher branches of philosophy.—Mill, J. S.
System of Logic, Bk. 3, chap. 24, sect. 9.

1423. In metaphysical reasoning, the process is always short. The
conclusion is but a step or two, seldom more, from the first principles or
axioms on which it is grounded, and the different conclusions depend not
one upon another.
It is otherwise in mathematical reasoning. Here the field has no limits.
One proposition leads on to another, that to a third, and so on without end.
If it should be asked, why demonstrative reasoning has so wide a field in
mathematics, while, in other abstract subjects, it is confined within very
narrow limits, I conceive this is chiefly owing to the nature of quantity, ...
mathematical quantities being made up of parts without number, can touch