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Title: Memorabilia Mathematica - or the Philomath's Quotation-Book
Author: Moritz, Robert Edouard
Language: English
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  MEMORABILIA MATHEMATICA



  THE MACMILLAN COMPANY

  NEW YORK. BOSTON. CHICAGO. DALLAS
  ATLANTA. SAN FRANCISCO

  MACMILLAN & CO., LIMITED

  LONDON. BOMBAY. CALCUTTA
  MELBOURNE

  THE MACMILLAN CO. OF CANADA, LTD.

  TORONTO



  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 SPACE   345

     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._

    [1]  i.e., in terms of the absolutely clear and
         _in_definable.


=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
mathematical 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:--

                    /+∞
                    |  -x²
                    | e  dx = √π
                    |
                    /-∞

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 admitting 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
mathematician[2] is esteemed a genius until condemned.]

                         _Juvenal, Liberii, Satura VI, 562._

    [2]  Used here in the sense of astrologer, or
         soothsayer.


=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._

    [3]  This is the estimate furnished me by two
         mathematical masters in one of our great
         public schools of the proportion of boys who
         have any special taste or capacity for
         mathematical studies. Many more, of course,
         can be drilled into a fair knowledge of
         elementary mathematics, but only this small
         proportion possess the natural faculty which
         renders it possible for them ever to rank
         high as mathematicians, to take any pleasure
         in it, or to do any original mathematical work.


=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._

    [4]  The mathematical tendencies of Cambridge are
         due to the fact that Cambridge drains the
         ability of nearly the whole Anglo-Danish
         district.


=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 = α^θ] ... 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,

                                    /   n
                                    |  u
                     Go! great sum! | a  du
                                    /

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.”_

    [5]  Riccardi’s Bibliografia Euclidea (Bologna,
         1887), lists nearly two thousand editions.


=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 Aeneid[6] gave him the first idea.

                                               --BREWSTER, DAVID.

                         _Letters of Euler (New York, 1872), Vol.
                         1, p. 24._

    [6]  The line referred to is:
           “The anchor drops, the rushing keel is staid.”


=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:

                  a + b^n
      _Monsieur_, ------- = x, _donc Dieu existe; repondez!_
                     n

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 a
long-sought-for law of prime numbers in the formula 2^2^n + 1 = a
prime, but he admitted that he was unable to prove it rigorously.
The law is not true, as was pointed out by Euler in the example
2^2^5 + 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._

    [7]  Johannes Flamsteedius.


=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 Mecánique
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.

    [8]  This sentence has been reworded for the
         purpose of this quotation.

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._

    [9]  Author’s note. My colleague, Dr. E. T. Bell,
         informs me that this same anecdote is
         associated with the name of J. S. Blackie,
         Professor of Greek at Aberdeen and Edinburgh.


=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
1/2, 2/3, 3/4, 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 I_{a}T_{b}M_{c}
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 in innumerable points, and be compared in innumerable
different ways.--REID, THOMAS.

                         _Essays on the Powers of the Human Mind
                         (Edinburgh, 1812), Vol. 2, pp. 422-423._


=1424.= The power of Reason ... is unquestionably the most
important by far of those which are comprehended under the
general title of Intellectual. It is on the right use of this
power that our success in the pursuit of both knowledge and of
happiness depends; and it is by the exclusive possession of it
that man is distinguished, in the most essential respects, from
the lower animals. It is, indeed, from their subserviency to its
operations, that the other faculties ... derive their chief
value.--STEWART, DUGALD.

                         _Philosophy of the Human Mind; Collected
                         Works (Edinburgh, 1854), Vol. 8, p. 5._


=1425.= When ... I asked myself why was it then that the earliest
philosophers would admit to the study of wisdom only those who
had studied mathematics, as if this science was the easiest of
all and the one most necessary for preparing and disciplining the
mind to comprehend the more advanced, I suspected that they had
knowledge of a mathematical science different from that of our
time....

I believe I find some traces of these true mathematics in Pappus
and Diophantus, who, although they were not of extreme antiquity,
lived nevertheless in times long preceding ours. But I willingly
believe that these writers themselves, by a culpable ruse,
suppressed the knowledge of them; like some artisans who conceal
their secret, they feared, perhaps, that the ease and simplicity
of their method, if become popular, would diminish its importance,
and they preferred to make themselves admired by leaving to us, as
the product of their art, certain barren truths deduced with
subtlety, rather than to teach us that art itself, the knowledge
of which would end our admiration.--DESCARTES.

                         _Rules for the Direction of the Mind;
                         Philosophy of Descartes [Torrey], (New
                         York, 1892), pp. 70-71._


=1426.= If we rightly adhere to our rule [that is, that we should
occupy ourselves only with those subjects in reference to which
the mind is capable of acquiring certain and indubitable
knowledge] there will remain but few things to the study of which
we can devote ourselves. There exists in the sciences hardly a
single question upon which men of intellectual ability have not
held different opinions. But whenever two men pass contrary
judgment on the same thing, it is certain that one of the two is
wrong. More than that, neither of them has the truth; for if one
of them had a clear and precise insight into it, he could so
exhibit it to his opponent as to end the discussion by compelling
his conviction.... It follows from this, if we reckon rightly,
that among existing sciences there remain only geometry and
arithmetic, to which the observance of our rule would bring us.

                                                     --DESCARTES.

                         _Rules for the Direction of the Mind;
                         Philosophy of Descartes [Torrey], (New
                         York, 1892), p. 62._


=1427.= The same reason which led Plato to recommend the study of
arithmetic led him to recommend also the study of geometry. The
vulgar crowd of geometricians, he says, will not understand him.
They have practice always in view. They do not know that the real
use of the science is to lead men to the knowledge of abstract,
essential, eternal truth. (Plato’s Republic, Book 7). Indeed if
we are to believe Plutarch, Plato carried his feeling so far that
he considered geometry as degraded by being applied to any
purpose of vulgar utility. Archytas, it seems, had framed
machines of extraordinary power on mathematical principles.
(Plutarch, Sympos., VIII., and Life of Marcellus. The machines of
Archytas are also mentioned by Aulus Gellius and Diogenes
Laertius). Plato remonstrated with his friend, and declared that
this was to degrade a noble intellectual exercise into a low
craft, fit only for carpenters and wheelwrights. The office of
geometry, he said, was to discipline the mind, not to minister to
the base wants of the body. His interference was successful; and
from that time according to Plutarch, the science of mechanics
was considered unworthy of the attention of a philosopher.

                                                      --MACAULAY.

                         _Lord Bacon; Edinburgh Review, July,
                         1837._


=1428.= The intellectual habits of the Mathematicians are, in
some respects, the same with those [of the Metaphysicians] we
have been now considering; but, in other respects, they differ
widely. Both are favourable to the improvement of the power of
_attention_, but not in the same manner, nor in the same degree.

Those of the metaphysician give capacity of fixing the attention
on the subjects of our consciousness, without being distracted by
things external; but they afford little or no exercise to that
species of attention which enables us to follow long processes of
reasoning, and to keep in view all the various steps of an
investigation till we arrive at the conclusion. In mathematics,
such processes are much longer than in any other science; and
hence the study of it is peculiarly calculated to strengthen the
power of steady and concatenated thinking,--a power which, in all
the pursuits of life, whether speculative or active, is one of
the most valuable endowments we can possess. This command of
attention, however, it may be proper to add, is to be acquired,
not by the practice of modern methods, but by the study of Greek
geometry, more particularly, by accustoming ourselves to pursue
long trains of demonstration, without availing ourselves of the
aid of any sensible diagrams; the thoughts being directed solely
by those ideal delineations which the powers of conception and of
memory enable us to form.--STEWART, DUGALD.

                         _Philosophy of the Human Mind, Part 3,
                         chap. 1, sect. 3._


=1429.= They [the Greeks] speculated and theorized under a lively
persuasion that a Science of every part of nature was possible,
and was a fit object for the exercise of a man’s best faculties;
and they were speedily led to the conviction that such a science
must clothe its conclusions in the language of mathematics. This
conviction is eminently conspicuous in the writings of Plato....
Probably no succeeding step in the discovery of the Laws of
Nature was of so much importance as the full adoption of this
pervading conviction, that there must be Mathematical Laws of
Nature, and that it is the business of Philosophy to discover
these Laws. This conviction continues, through all the succeeding
ages of the history of the science, to be the animating and
supporting principle of scientific investigation and discovery.

                                                    --WHEWELL, W.

                         _History of the Inductive Sciences, Vol.
                         1, bk. 2, chap. 3._


=1430.= For to pass by those Ancients, the wonderful _Pythagoras_,
the sagacious _Democritus_, the divine _Plato_, the most subtle
and very learned _Aristotle_, Men whom every Age has hitherto
acknowledged as deservedly honored, as the greatest Philosophers,
the Ring-leaders of Arts; in whose Judgments how much these
Studies [mathematics] were esteemed, is abundantly proclaimed
in History and confirmed by their famous Monuments, which
are everywhere interspersed and bespangled with Mathematical
Reasonings and Examples, as with so many Stars; and consequently
anyone not in some Degree conversant in these Studies will in vain
expect to understand, or unlock their hidden Meanings, without the
Help of a Mathematical Key: For who can play well on _Aristotle’s_
Instrument but with a Mathematical Quill; or not be altogether
deaf to the Lessons of natural _Philosophy_, while ignorant of
_Geometry?_ Who void of (_Geometry_ shall I say, or) _Arithmetic_
can comprehend _Plato’s_ _Socrates_ lisping with Children
concerning Square Numbers; or can conceive _Plato_ himself
treating not only of the Universe, but the Polity of Commonwealths
regulated by the Laws of Geometry, and formed according to a
Mathematical Plan?--BARROW, ISAAC.

                         _Mathematical Lectures (London, 1734),
                         pp. 26-27._


=1431.=

        And Reason now through number, time, and space
        Darts the keen lustre of her serious eye;
        And learns from facts compar’d the laws to trace
        Whose long procession leads to Deity
                                           --BEATTIE, JAMES.

                         _The Minstrel, Bk. 2, stanza 47._


=1432.= That Egyptian and Chaldean wisdom mathematical wherewith
Moses and Daniel were furnished, ....--HOOKER, RICHARD.

                         _Ecclesiastical Polity, Bk. 3, sect. 8._


=1433.= General and certain truths are only founded in the
habitudes and relations of _abstract ideas_. A sagacious and
methodical application of our thoughts, for the finding out of
these relations, is the only way to discover all that can be
put with truth and certainty concerning them into general
propositions. By what steps we are to proceed in these, is to be
learned in the schools of mathematicians, who, from very plain and
easy beginnings, by gentle degrees, and a continued chain of
reasonings, proceed to the discovery and demonstration of truths
that appear at first sight beyond human capacity. The art of
finding proofs, and the admirable method they have invented for
the singling out and laying in order those intermediate ideas that
demonstratively show the equality or inequality of unapplicable
quantities, is that which has carried them so far and produced
such wonderful and unexpected discoveries; but whether something
like this, in respect of other ideas, as well as those of
magnitude, may not in time be found out, I will not determine.
This, I think, I may say, that if other ideas that are the real as
well as the nominal essences of their species, were pursued in the
way familiar to mathematicians, they would carry our thoughts
further, and with greater evidence and clearness than possibly we
are apt to imagine.--LOCKE, JOHN.

                         _An Essay concerning Human
                         Understanding, Bk. 4, chap. 12, sect.
                         7._


=1434.= Those long chains of reasoning, quite simple and easy,
which geometers are wont to employ in the accomplishment of their
most difficult demonstrations, led me to think that everything
which might fall under the cognizance of the human mind might be
connected together in a similar manner, and that, provided only
that one should take care not to receive anything as true which
was not so, and if one were always careful to preserve the order
necessary for deducing one truth from another, there would be
none so remote at which he might not at last arrive, nor so
concealed which he might not discover.--DESCARTES.

                         _Discourse upon Method, part 2; The
                         Philosophy of Descartes [Torrey], (New
                         York, 1892), p. 47._


=1435.= If anyone wished to write in mathematical fashion in
metaphysics or ethics, nothing would prevent him from so doing
with vigor. Some have professed to do this, and we have a promise
of mathematical demonstrations outside of mathematics; but it is
very rare that they have been successful. This is, I believe,
because they are disgusted with the trouble it is necessary to
take for a small number of readers where they would ask as in
Persius: _Quis leget haec_, and reply: _Vel duo vel nemo._

                                                      --LEIBNITZ.

                         _New Essay concerning Human
                         Understanding, Langley, Bk 2, chap. 29,
                         sect. 12._


=1436.= It is commonly asserted that mathematics and philosophy
differ from one another according to their _objects_, the former
treating of _quantity_, the latter of _quality_. All this is
false. The difference between these sciences cannot depend on
their object; for philosophy applies to everything, hence also to
_quanta_, and so does mathematics in part, inasmuch as everything
has magnitude. It is only the _different kind of rational
knowledge or application_ of reason in mathematics and philosophy
which constitutes the specific difference between these two
sciences. For philosophy is _rational knowledge from mere
concepts_, mathematics, on the contrary, is _rational knowledge
from the construction of concepts_.

We construct concepts when we represent them in intuition _a
priori_, without experience, or when we represent in intuition
the object which corresponds to our concept of it.--The
mathematician can never apply his reason to mere concepts, nor
the philosopher to the construction of concepts.--In mathematics
the reason is employed _in concreto_, however, the intuition is
not empirical, but the object of contemplation is something _a
priori_.

In this, as we see, mathematics has an advantage over philosophy,
the knowledge in the former being intuitive, in the latter, on the
contrary, only _discursive_. But the reason why in mathematics we
deal more with quantity lies in this, that magnitudes can be
constructed in intuition _a priori_, while qualities, on the
contrary, do not permit of being represented in intuition.--KANT, E.

                         _Logik; Werke [Hartenstein], (Leipzig,
                         1868), Bd. 8, pp. 23-24._


=1437.= Kant has divided human ideas into the two categories of
quantity and quality, which, if true, would destroy the
universality of Mathematics; but Descartes’ fundamental
conception of the relation of the concrete to the abstract in
Mathematics abolishes this division, and proves that all ideas of
quality are reducible to ideas of quantity. He had in view
geometrical phenomena only; but his successors have included in
this generalization, first, mechanical phenomena, and, more
recently, those of heat. There are now no geometers who do not
consider it of universal application, and admit that every
phenomenon may be as logically capable of being represented by an
equation as a curve or a motion, if only we were always capable
(which we are very far from being) of first discovering, and then
resolving it.

The limitations of Mathematical science are not, then, in its
nature. The limitations are in our intelligence: and by these we
find the domain of the science remarkably restricted, in
proportion as phenomena, in becoming special, become complex.

                                                      --COMTE, A.

                         _Positive Philosophy [Martineau], Bk. 1,
                         chap. 1._


=1438.= The great advantage of the mathematical sciences over the
moral consists in this, that the ideas of the former, being
sensible, are always clear and determinate, the smallest
distinction between them being immediately perceptible, and the
same terms are still expressive of the same ideas, without
ambiguity or variation. An oval is never mistaken for a circle,
nor an hyperbola for an ellipsis. The isosceles and scalenum are
distinguished by boundaries more exact than vice and virtue,
right or wrong. If any term be defined in geometry, the mind
readily, of itself, substitutes on all occasions, the definition
for the thing defined: Or even when no definition is employed,
the object itself may be represented to the senses, and by that
means be steadily and clearly apprehended. But the finer
sentiments of the mind, the operations of the understanding, the
various agitations of the passions, though really in themselves
distinct, easily escape us, when surveyed by reflection; nor is
it in our power to recall the original object, so often as we
have occasion to contemplate it. Ambiguity, by this means, is
gradually introduced into our reasonings: Similar objects are
readily taken to be the same: And the conclusion becomes at last
very wide off the premises.--HUME, DAVID.

                         _An Inquiry concerning Human
                         Understanding, sect. 7, part 1._


=1439.= One part of these disadvantages in moral ideas which has
made them be thought not capable of demonstration, may in a good
measure be remedied by definitions, setting down that collection
of simple ideas, which every term shall stand for; and then using
the terms steadily and constantly for that precise collection.
And what methods algebra, or something of that kind, may
hereafter suggest, to remove the other difficulties, it is not
easy to foretell. Confident, I am, that if men would in the same
method, and with the same indifferency, search after moral as
they do mathematical truths, they would find them have a stronger
connexion one with another, and a more necessary consequence from
our clear and distinct ideas, and to come nearer perfect
demonstration than is commonly imagined.--LOCKE, JOHN.

                         _An Essay concerning Human
                         Understanding, Bk. 4, chap. 3, sect.
                         20._


=1440.= That which in this respect has given the advantage to the
ideas of quantity, and made them thought more capable of
certainty and demonstration [than moral ideas], is,

First, That they can be set down and represented by sensible
marks, which have a greater and nearer correspondence with them
than any words or sounds whatsoever. Diagrams drawn on paper
are copies of the ideas in the mind, and not liable to the
uncertainty that words carry in their signification. An angle,
circle, or square, drawn in lines, lies open to the view, and
cannot be mistaken: it remains unchangeable, and may at leisure
be considered and examined, and the demonstration be revised, and
all the parts of it may be gone over more than once, without any
danger of the least change in the ideas. This cannot be done in
moral ideas: we have no sensible marks that resemble them,
whereby we can set them down; we have nothing but words to
express them by; which, though when written they remain the same,
yet the ideas they stand for may change in the same man; and it
is seldom that they are not different in different persons.

Secondly, Another thing that makes the greater difficulty in
ethics is, That moral ideas are commonly more complex than those
of the figures ordinarily considered in mathematics. From whence
these two inconveniences follow:--First, that their names are of
more uncertain signification, the precise collection of simple
ideas they stand for not being so easily agreed on; and so the
sign that is used for them in communication always, and in
thinking often, does not steadily carry with it the same idea.
Upon which the same disorder, confusion, and error follow, as
would if a man, going to demonstrate something of an heptagon,
should, in the diagram he took to do it, leave out one of the
angles, or by oversight make the figure with an angle more than
the name ordinarily imported, or he intended it should when at
first he thought of his demonstration. This often happens, and is
hardly avoidable in very complex moral ideas, where the same name
being retained, an angle, i.e. one simple idea is left out, or
put in the complex one (still called by the same name) more at
one time than another. Secondly, From the complexedness of these
moral ideas there follows another inconvenience, viz., that the
mind cannot easily retain those precise combinations so exactly
and perfectly as is necessary in the examination of the habitudes
and correspondences, agreements or disagreements, of several of
them one with another; especially where it is to be judged of by
long deductions and the intervention of several other complex
ideas to show the agreement or disagreement of two remote ones.

                                                   --LOCKE, JOHN.

                         _An Essay concerning Human
                         Understanding, Bk. 4, chap. 3, sect.
                         19._


=1441.= It has been generally taken for granted, that mathematics
alone are capable of demonstrative certainty: but to have such an
agreement or disagreement as may be intuitively perceived,
being, as I imagine, not the privileges of the ideas of number,
extension, and figure alone, it may possibly be the want of due
method and application in us, and not of sufficient evidence in
things, that demonstration has been thought to have so little to
do in other parts of knowledge, and been scarce so much as aimed
at by any but mathematicians. For whatever ideas we have wherein
the mind can perceive the immediate agreement or disagreement that
is between them, there the mind is capable of intuitive knowledge,
and where it can perceive the agreement or disagreement of any two
ideas, by an intuitive perception of the agreement or disagreement
they have with any intermediate ideas, there the mind is capable
of demonstration: which is not limited to the idea of extension,
figure, number, and their modes.--LOCKE, JOHN.

                         _An Essay concerning Human
                         Understanding, Bk. 4, chap. 2, sect. 9._


=1442.= Now I shall remark again what I have already touched
upon more than once, that it is a common opinion that only
mathematical sciences are capable of a demonstrative certainty;
but as the agreement and disagreement which may be known
intuitively is not a privilege belonging only to the ideas of
numbers and figures, it is perhaps for want of application on our
part that mathematics alone have attained to demonstrations.

                                                      --LEIBNITZ.

                         _New Essay concerning Human
                         Understanding, Bk. 4, chap. 2, sect. 9
                         [Langley]._



  CHAPTER XV

  MATHEMATICS AND SCIENCE


=1501.= How comes it about that the knowledge of other sciences,
which depend upon this [mathematics], is painfully sought, and
that no one puts himself to the trouble of studying this science
itself? I should certainly be surprised, if I did not know that
everybody regarded it as being very easy, and if I had not long
ago observed that the human mind, neglecting what it believes to
be easy, is always in haste to run after what is novel and
advanced.--DESCARTES.

                         _Rules for the Direction of the Mind;
                         Philosophy of Descartes [Torrey], (New
                         York, 1892), p. 72._


=1502.= All quantitative determinations are in the hands
of mathematics, and it at once follows from this that all
speculation which is heedless of mathematics, which does not
enter into partnership with it, which does not seek its aid in
distinguishing between the manifold modifications that must of
necessity arise by a change of quantitative determinations, is
either an empty play of thoughts, or at most a fruitless effort.
In the field of speculation many things grow which do not start
from mathematics nor give it any care, and I am far from
asserting that all that thus grow are useless weeds, among them
may be many noble plants, but without mathematics none will
develop to complete maturity.--HERBART, J. F.

                         _Werke (Kehrbach), (Langensalza, 1890),
                         Bd. 5, p. 106._


=1503.= There are few things which we know, which are not capable
of being reduc’d to a Mathematical Reasoning, and when they
cannot, it’s a sign our knowledge of them is very small and
confus’d; and where a mathematical reasoning can be had, it’s as
great folly to make use of any other, as to grope for a thing in
the dark, when you have a candle standing by you.--ARBUTHNOT.

                         _Quoted in Todhunter’s History of the
                         Theory of Probability (Cambridge and
                         London, 1865), p. 51._


=1504.= Mathematical Analysis is ... the true rational basis of
the whole system of our positive knowledge.--COMTE, A.

                         _Positive Philosophy [Martineau], Bk. 1,
                         chap. 1._


=1505.= It is only through Mathematics that we can thoroughly
understand what true science is. Here alone we can find in the
highest degree simplicity and severity of scientific law, and
such abstraction as the human mind can attain. Any scientific
education setting forth from any other point, is faulty in its
basis.--COMTE, A.

                         _Positive Philosophy [Martineau], Bk. 1,
                         chap. 1._


=1506.= In the present state of our knowledge we must regard
Mathematics less as a constituent part of natural philosophy than
as having been, since the time of Descartes and Newton, the true
basis of the whole of natural philosophy; though it is, exactly
speaking, both the one and the other. To us it is of less use for
the knowledge of which it consists, substantial and valuable as
that knowledge is, than as being the most powerful instrument
that the human mind can employ in the investigation of the laws
of natural phenomena.--COMTE, A.

                         _Positive Philosophy [Martineau],
                         Introduction, chap. 2._


=1507.= The concept of mathematics is the concept of science in
general.--NOVALIS.

                         _Schriften (Berlin, 1901), Teil 2, p.
                         222._


=1508.= I contend, that each natural science is real science only
in so far as it is mathematical.... It may be that a pure
philosophy of nature in general (that is, a philosophy which
concerns itself only with the general concepts of nature) is
possible without mathematics, but a pure science of nature
dealing with definite objects (physics or psychology), is
possible only by means of mathematics, and since each natural
science contains only as much real science as it contains _a
priori_ knowledge, each natural science becomes real science only
to the extent that it permits the application of mathematics.

                                                       --KANT, E.

                         _Metaphysische Anfangsgründe der
                         Naturwissenschaft, Vorrede._


=1509.= The theory most prevalent among teachers is that
mathematics affords the best training for the reasoning
powers;... The modern, and to my mind true, theory is that
mathematics is the abstract form of the natural sciences; and
that it is valuable as a training of the reasoning powers, not
because it is abstract, but because it is a representation of
actual things.--SAFFORD, T. H.

                         _Mathematical Teaching etc. (Boston,
                         1886), p. 9._


=1510.= It seems to me that no one science can so well serve to
co-ordinate and, as it were, bind together all of the sciences as
the queen of them all, mathematics.--DAVIS, E. W.

                         _Proceedings Nebraska Academy of
                         Sciences for 1896 (Lincoln, 1897), p.
                         282._


=1511.= And as for Mixed Mathematics, I may only make this
prediction, that there cannot fail to be more kinds of them, as
nature grows further disclosed.--BACON, FRANCIS.

                         _Advancement of Learning, Bk. 2; De
                         Augmentis, Bk. 3._


=1512.= Besides the exercise in keen comprehension and the
certain discovery of truth, mathematics has another formative
function, that of equipping the mind for the survey of a
scientific system.--GRASSMANN, H.

                         _Stücke aus dem Lehrbuche der
                         Arithmetik; Werke (Leipzig, 1904), Bd.
                         2, p. 298._


=1513.= Mathematicks may help the naturalists, both to frame
hypotheses, and to judge of those that are proposed to them,
especially such as relate to mathematical subjects in conjunction
with others.--BOYLE, ROBERT.

                         _Works (London, 1772), Vol. 3, p. 429._


=1514.= The more progress physical sciences make, the more they
tend to enter the domain of mathematics, which is a kind of
centre to which they all converge. We may even judge of the
degree of perfection to which a science has arrived by the
facility with which it may be submitted to calculation.--QUETELET.

                         _Quoted in E. Mailly’s Eulogy on
                         Quetelet; Smithsonian Report, 1874, p.
                         173._


=1515.= The mathematical formula is the point through which all
the light gained by science passes in order to be of use to
practice; it is also the point in which all knowledge gained by
practice, experiment, and observation must be concentrated before
it can be scientifically grasped. The more distant and marked the
point, the more concentrated will be the light coming from it,
the more unmistakable the insight conveyed. All scientific
thought, from the simple gravitation formula of Newton, through
the more complicated formulae of physics and chemistry, the
vaguer so called laws of organic and animated nature, down to the
uncertain statements of psychology and the data of our social and
historical knowledge, alike partakes of this characteristic, that
it is an attempt to gather up the scattered rays of light, the
different parts of knowledge, in a focus, from whence it can be
again spread out and analyzed, according to the abstract
processes of the thinking mind. But only when this can be done
with a mathematical precision and accuracy is the image sharp and
well-defined, and the deductions clear and unmistakable. As we
descend from the mechanical, through the physical, chemical, and
biological, to the mental, moral, and social sciences, the
process of focalization becomes less and less perfect,--the sharp
point, the focus, is replaced by a larger or smaller circle, the
contours of the image become less and less distinct, and with the
possible light which we gain there is mingled much darkness, the
sources of many mistakes and errors. But the tendency of all
scientific thought is toward clearer and clearer definition; it
lies in the direction of a more and more extended use of
mathematical measurements, of mathematical formulae.--MERZ, J. T.

                         _History of European Thought in the 19th
                         Century (Edinburgh and London, 1904),
                         Vol. 1, p. 333._


=1516.= From the very outset of his investigations the physicist
has to rely constantly on the aid of the mathematician, for even
in the simplest cases, the direct results of his measuring
operations are entirely without meaning until they have been
submitted to more or less of mathematical discussion. And when in
this way some interpretation of the experimental results has been
arrived at, and it has been proved that two or more physical
quantities stand in a definite relation to each other, the
mathematician is very often able to infer, from the existence of
this relation, that the quantities in question also fulfill some
other relation, that was previously unsuspected. Thus when
Coulomb, combining the functions of experimentalist and
mathematician, had discovered the law of the force exerted
between two particles of electricity, it became a purely
mathematical problem, not requiring any further experiment, to
ascertain how electricity is distributed upon a charged conductor
and this problem has been solved by mathematicians in several
cases.--FOSTER, G. C.

                         _Presidential Address British
                         Association for the Advancement of
                         Science, Section A (1877); Nature, Vol.
                         16, p. 312-313._


=1517.= Without consummate mathematical skill, on the part of
some investigators at any rate, all the higher physical problems
would be sealed to us; and without competent skill on the part of
the ordinary student no idea can be formed of the nature and
cogency of the evidence on which the solutions rest. Mathematics
are not merely a gate through which we may approach if we please,
but they are the only mode of approach to large and important
districts of thought.--VENN, JOHN.

                         _Symbolic Logic (London and New York,
                         1894), Introduction, p. xix._


=1518.= Much of the skill of the true mathematical physicist and
of the mathematical astronomer consists in the power of adapting
methods and results carried out on an exact mathematical basis to
obtain approximations sufficient for the purposes of physical
measurements. It might perhaps be thought that a scheme of
Mathematics on a frankly approximative basis would be sufficient
for all the practical purposes of application in Physics,
Engineering Science, and Astronomy, and no doubt it would be
possible to develop, to some extent at least, a species of
Mathematics on these lines. Such a system would, however, involve
an intolerable awkwardness and prolixity in the statements of
results, especially in view of the fact that the degree of
approximation necessary for various purposes is very different,
and thus that unassigned grades of approximation would have to
be provided for. Moreover, the mathematician working on these
lines would be cut off from the chief sources of inspiration, the
ideals of exactitude and logical rigour, as well as from one of
his most indispensable guides to discovery, symmetry, and
permanence of mathematical form. The history of the actual
movements of mathematical thought through the centuries shows
that these ideals are the very life-blood of the science, and
warrants the conclusion that a constant striving toward their
attainment is an absolutely essential condition of vigorous
growth. These ideals have their roots in irresistible impulses
and deep-seated needs of the human mind, manifested in its
efforts to introduce intelligibility in certain great domains of
the world of thought.--HOBSON, E. W.

                         _Presidential Address British
                         Association for the Advancement of
                         Science, Section A (1910); Nature, Vol.
                         84, pp. 285-286._


=1519.= The immense part which those laws [laws of number and
extension] take in giving a deductive character to the other
departments of physical science, is well known; and is not
surprising, when we consider that all causes operate according to
mathematical laws. The effect is always dependent upon, or in
mathematical language, is a function of, the quantity of the
agent; and generally of its position also. We cannot, therefore,
reason respecting causation, without introducing considerations
of quantity and extension at every step; and if the nature of the
phenomena admits of our obtaining numerical data of sufficient
accuracy, the laws of quantity become the grand instruments for
calculating forward to an effect, or backward to a cause.

                                                    --MILL, J. S.

                         _System of Logic, Bk. 3, chap. 24, sect.
                         9._


=1520.= The ordinary mathematical treatment of any applied
science substitutes exact axioms for the approximate results of
experience, and deduces from these axioms the rigid mathematical
conclusions. In applying this method it must not be forgotten
that the mathematical developments transcending the limits of
exactness of the science are of no practical value. It follows
that a large portion of abstract mathematics remains without
finding any practical application, the amount of mathematics
that can be usefully employed in any science being in proportion
to the degree of accuracy attained in the science. Thus, while
the astronomer can put to use a wide range of mathematical
theory, the chemist is only just beginning to apply the first
derivative, i.e. the rate of change at which certain processes
are going on; for second derivatives he does not seem to have
found any use as yet.--KLEIN, F.

                         _Lectures on Mathematics (New York,
                         1911), p. 47._


=1521.= The bond of union among the physical sciences is the
mathematical spirit and the mathematical method which pervades
them.... Our knowledge of nature, as it advances, continuously
resolves differences of quality into differences of quantity. All
exact reasoning--indeed all reasoning--about quantity is
mathematical reasoning; and thus as our knowledge increases, that
portion of it which becomes mathematical increases at a still
more rapid rate.--SMITH, H. J. S.

                         _Presidential Address British
                         Association for the Advancement of
                         Science, Section A (1873); Nature, Vol.
                         8, p. 449._


=1522.= Another way of convincing ourselves how largely this
process [of assimilation of mathematics by physics] has gone on
would be to try to conceive the effect of some intellectual
catastrophe, supposing such a thing possible, whereby all
knowledge of mathematics should be swept away from men’s minds.
Would it not be that the departure of mathematics would be the
destruction of physics? Objective physical phenomena would,
indeed, remain as they are now, but physical science would cease
to exist. We should no doubt see the same colours on looking into
a spectroscope or polariscope, vibrating strings would produce
the same sounds, electrical machines would give sparks, and
galvanometer needles would be deflected; but all these things
would have lost their meaning; they would be but as the dry
bones--the _disjecta membra_--of what is now a living and growing
science. To follow this conception further, and to try to image to
ourselves in some detail what would be the kind of knowledge of
physics which would remain possible, supposing all mathematical
ideas to be blotted out, would be extremely interesting, but it
would lead us directly into a dim and entangled region where
the subjective seems to be always passing itself off for the
objective, and where I at least could not attempt to lead the way,
gladly as I would follow any one who could show where a firm
footing is to be found. But without venturing to do more than to
look from a safe distance over this puzzling ground, we may see
clearly enough that mathematics is the connective tissue of
physics, binding what would else be merely a list of detached
observations into an organized body of science.--FOSTER, G. C.

                         _Presidential Address British
                         Association for the Advancement of
                         Science, Section A (1877); Nature, Vol.
                         16, p. 313._


=1523.= In _Plato’s_ time mathematics was purely a play of the
free intellect; the mathematic-mystical reveries of a Pythagoras
foreshadowed a far-reaching significance, but such a significance
(except in the case of music) was as yet entirely a matter of
fancy; yet even in that time mathematics was the prerequisite to
all other studies! But today, when mathematics furnishes the
_only_ language by means of which we may formulate the most
comprehensive laws of nature, laws which the ancients scarcely
dreamed of, when moreover mathematics is the _only_ means by
which these laws may be understood,--how few learn today anything
of the real essence of our mathematics!... In the schools of
today mathematics serves only as a disciplinary study, a mental
gymnastic; that it includes the highest ideal value for the
comprehension of the universe, one dares scarcely to think of in
view of our present day instruction.--LINDEMAN, F.

                         _Lehren und Lernen in der Mathematik
                         (München, 1904), p. 14._


=1524.= All applications of mathematics consist in extending the
empirical knowledge which we possess of a limited number or
region of accessible phenomena into the region of the unknown and
inaccessible; and much of the progress of pure analysis consists
in inventing definite conceptions, marked by symbols, of
complicated operations; in ascertaining their properties as
independent objects of research; and in extending their meaning
beyond the limits they were originally invented for,--thus
opening out new and larger regions of thought.--MERZ, J. T.

                         _History of European Thought in the 19th
                         Century (Edinburgh and London, 1903),
                         Vol. 1, p. 698._


=1525.= All the effects of nature are only mathematical results
of a small number of immutable laws.--LAPLACE.

                         _A Philosophical Essay on Probabilities
                         [Truscott and Emory] (New York, 1902),
                         p. 177; Oeuvres, t. 7, p. 139._


=1526.= What logarithms are to mathematics that mathematics are
to the other sciences.--NOVALIS.

                         _Schriften (Berlin, 1901), Teil 2, p.
                         222._


=1527.= Any intelligent man may now, by resolutely applying
himself for a few years to mathematics, learn more than the great
Newton knew after half a century of study and meditation.

                                                      --MACAULAY.

                         _Milton; Critical and Miscellaneous
                         Essays (New York, 1879), Vol. 1, p. 13._


=1528.= In questions of science the authority of a thousand is
not worth the humble reasoning of a single individual.--GALILEO.

                         _Quoted in Arago’s Eulogy on Laplace;
                         Smithsonian Report, 1874, p. 164._


=1529.= Behind the artisan is the chemist, behind the chemist a
physicist, behind the physicist a mathematician.--WHITE, W. F.

                         _Scrap-book of Elementary Mathematics
                         (Chicago, 1908), p. 217._


=1530.= The advance in our knowledge of physics is largely due to
the application to it of mathematics, and every year it becomes
more difficult for an experimenter to make any mark in the
subject unless he is also a mathematician.--BALL, W. W. R.

                         _History of Mathematics (London, 1901),
                         p. 503._


=1531.= In very many cases the most obvious and direct
experimental method of investigating a given problem is extremely
difficult, or for some reason or other untrustworthy. In such
cases the mathematician can often point out some other problem
more accessible to experimental treatment, the solution of which
involves the solution of the former one. For example, if we try
to deduce from direct experiments the law according to which one
pole of a magnet attracts or repels a pole of another magnet, the
observed action is so much complicated with the effects of the
mutual induction of the magnets and of the forces due to the
second pole of each magnet, that it is next to impossible to
obtain results of any great accuracy. Gauss, however, showed how
the law which applied in the case mentioned can be deduced from
the deflections undergone by a small suspended magnetic needle
when it is acted upon by a small fixed magnet placed successively
in two determinate positions relatively to the needle; and being
an experimentalist as well as a mathematician, he showed likewise
how these deflections can be measured very easily and with great
precision.--FOSTER, G. C.

                         _Presidential Address British
                         Association for the Advancement of
                         Science, Section A (1877); Nature, Vol.
                         16, p. 313._


=1532.=

        Give me to learn each secret cause;
        Let Number’s, Figure’s, Motion’s laws
        Reveal’d before me stand;
        These to great Nature’s scenes apply,
        And round the globe, and through the sky,
        Disclose her working hand.
                                              --AKENSIDE, M.

                         _Hymn to Science._


=1533.= Now there are several scores, upon which skill in
mathematicks may be useful to the experimental philosopher. For
there are some general advantages, which mathematicks may bring
to the minds of men, to whatever study they apply themselves, and
consequently to the student of natural philosophy; namely, that
these disciplines are wont to make men accurate, and very
attentive to the employment that they are about, keeping their
thoughts from wandering, and inuring them to patience in going
through with tedious and intricate demonstrations; besides, that
they much improve reason, by accustoming the mind to deduce
successive consequences, and judge of them without easily
acquiescing in anything but demonstration.--BOYLE, ROBERT.

                         _Works (London, 1772), Vol. 3, p. 426._


=1534.= It is not easy to anatomize the constitution and the
operations of a mind [like Newton’s] which makes such an advance
in knowledge. Yet we may observe that there must exist in it, in
an eminent degree, the elements which compose the mathematical
talent. It must possess distinctness of intuition, tenacity and
facility in tracing logical connection, fertility of invention,
and a strong tendency to generalization.--WHEWELL, W.

                         _History of the Inductive Sciences (New
                         York, 1894), Vol. 1, p. 416._


=1535.= The domain of physics is no proper field for mathematical
pastimes. The best security would be in giving a geometrical
training to physicists, who need not then have recourse to
mathematicians, whose tendency is to despise experimental science.
By this method will that union between the abstract and the
concrete be effected which will perfect the uses of mathematical,
while extending the positive value of physical science. Meantime,
the use of analysis in physics is clear enough. Without it we
should have no precision, and no co-ordination; and what account
could we give of our study of heat, weight, light, etc.? We should
have merely series of unconnected facts, in which we could foresee
nothing but by constant recourse to experiment; whereas, they now
have a character of rationality which fits them for purposes of
prevision.--COMTE, A.

                         _Positive Philosophy [Martineau], Bk. 3,
                         chap. 1._


=1536.= It must ever be remembered that the true positive spirit
first came forth from the pure sources of mathematical science;
and it is only the mind that has imbibed it there, and which has
been face to face with the lucid truths of geometry and
mechanics, that can bring into full action its natural
positivity, and apply it in bringing the most complex studies
into the reality of demonstration. No other discipline can fitly
prepare the intellectual organ.--COMTE, A.

                         _Positive Philosophy [Martineau], Bk. 3,
                         chap. 1._


=1537.= During the last two centuries and a half, physical
knowledge has been gradually made to rest upon a basis which it
had not before. It has become _mathematical_. The question now
is, not whether this or that hypothesis is better or worse to the
pure thought, but whether it accords with observed phenomena in
those consequences which can be shown necessarily to follow from
it, if it be true. Even in those sciences which are not yet under
the dominion of mathematics, and perhaps never will be, a working
copy of the mathematical process has been made. This is not known
to the followers of those sciences who are not themselves
mathematicians, and who very often exalt their horns against the
mathematics in consequence. They might as well be squaring the
circle, for any sense they show in this particular.--DE MORGAN, A.

                         _A Budget of Paradoxes (London, 1872),
                         p. 2._


=1538.= Among the mere talkers so far as mathematics are
concerned, are to be ranked three out of four of those who apply
mathematics to physics, who, wanting a tool only, are very
impatient of everything which is not of direct aid to the actual
methods which are in their hands.--DE MORGAN, A.

                         _Graves’ Life of Sir William Rowan
                         Hamilton (New York, 1882-1889), Vol. 3,
                         p. 348._


=1539.= Something has been said about the use of mathematics in
physical science, the mathematics being regarded as a weapon
forged by others, and the study of the weapon being completely
set aside. I can only say that there is danger of obtaining
untrustworthy results in physical science, if only the results of
mathematics are used; for the person so using the weapon can
remain unacquainted with the conditions under which it can be
rightly applied.... The results are often correct, sometimes are
incorrect; the consequence of the latter class of cases is to
throw doubt upon all the applications of such a worker until a
result has been otherwise tested. Moreover, such a practice in
the use of mathematics leads a worker to a mere repetition in the
use of familiar weapons; he is unable to adapt them with any
confidence when some new set of conditions arise with a demand
for a new method: for want of adequate instruction in the
forging of the weapon, he may find himself, sooner or later in
the progress of his subject, without any weapon worth having.

                                                 --FORSYTH, A. R.

                         _Perry’s Teaching of Mathematics
                         (London, 1902), p. 36._


=1540.= If in the range of human endeavor after sound knowledge
there is one subject that needs to be practical, it surely is
Medicine. Yet in the field of Medicine it has been found that
branches such as biology and pathology must be studied for
themselves and be developed by themselves with the single aim of
increasing knowledge; and it is then that they can be best
applied to the conduct of living processes. So also in the
pursuit of mathematics, the path of practical utility is too
narrow and irregular, not always leading far. The witness of
history shows that, in the field of natural philosophy,
mathematics will furnish the more effective assistance if, in its
systematic development, its course can freely pass beyond the
ever-shifting domain of use and application.--FORSYTH, A. R.

                         _Presidential Address British
                         Association for the Advancement of
                         Science, Section A; Nature, Vol. 56
                         (1897), p. 377._


=1541.= If the Greeks had not cultivated Conic Sections, Kepler
could not have superseded Ptolemy; if the Greeks had cultivated
Dynamics, Kepler might have anticipated Newton.--WHEWELL, W.

                         _History of the Inductive Science (New
                         York, 1894), Vol. 1, p. 311._


=1542.= If we may use the great names of Kepler and Newton to
signify stages in the progress of human discovery, it is not too
much to say that without the treatises of the Greek geometers on
the conic sections there could have been no Kepler, without
Kepler no Newton, and without Newton no science in the modern
sense of the term, or at least no such conception of nature as
now lies at the basis of all our science, of nature as subject in
the smallest as well as in its greatest phenomena, to exact
quantitative relations, and to definite numerical laws.

                                                --SMITH, H. J. S.

                         _Presidential Address British
                         Association for the Advancement of
                         Science, Section A; Nature, Vol. 8
                         (1873), p. 450._


=1543.= The silent work of the great Regiomontanus in his chamber
at Nuremberg computed the Ephemerides which made possible the
discovery of America by Columbus.--RUDIO, F.

                         _Quoted in Max Simon’s Geschichte der
                         Mathematik im Altertum (Berlin, 1909),
                         Einleitung, p. xi._


=1544.= The calculation of the eclipses of Jupiter’s satellites,
many a man might have been disposed, originally, to regard as a
most unprofitable study. But the utility of it to navigation (in
the determination of longitudes) is now well known.--WHATELY, R.

                         _Annotations to Bacon’s Essays (Boston,
                         1783), p. 492._


=1545.= Who could have imagined, when Galvani observed the
twitching of the frog muscles as he brought various metals in
contact with them, that eighty years later Europe would be
overspun with wires which transmit messages from Madrid to St.
Petersburg with the rapidity of lightning, by means of the same
principle whose first manifestations this anatomist then
observed!...

He who seeks for immediate practical use in the pursuit of
science, may be reasonably sure, that he will seek in vain.
Complete knowledge and complete understanding of the action of
forces of nature and of the mind, is the only thing that science
can aim at. The individual investigator must find his reward in
the joy of new discoveries, as new victories of thought over
resisting matter, in the esthetic beauty which a well-ordered
domain of knowledge affords, where all parts are intellectually
related, where one thing evolves from another, and all show the
marks of the mind’s supremacy; he must find his reward in the
consciousness of having contributed to the growing capital of
knowledge on which depends the supremacy of man over the forces
hostile to the spirit.--HELMHOLTZ, H.

                         _Vorträge und Reden (Braunschweig,
                         1884), Bd. 1, p. 142._


=1546.= When the time comes that knowledge will not be sought for
its own sake, and men will not press forward simply in a desire
of achievement, without hope of gain, to extend the limits of
human knowledge and information, then, indeed, will the race
enter upon its decadence.--HUGHES, C. E.

                         _Quoted in D. E. Smith’s Teaching of
                         Geometry (Boston, 1911), p. 9._


=1547.= [In the Opus Majus of Roger Bacon] there is a chapter, in
which it is proved by reason, that all sciences require
mathematics. And the arguments which are used to establish this
doctrine, show a most just appreciation of the office of
mathematics in science. They are such as follows: That other
sciences use examples taken from mathematics as the most
evident:--That mathematical knowledge is, as it were, innate to
us, on which point he refers to the well-known dialogue of Plato,
as quoted by Cicero:--That this science, being the easiest,
offers the best introduction to the more difficult:--That in
mathematics, things as known to us are identical with things as
known to nature:--That we can here entirely avoid doubt and
error, and obtain certainty and truth:--That mathematics is prior
to other sciences in nature, because it takes cognizance
of quantity, which is apprehended by intuition (_intuitu
intellectus_). “Moreover,” he adds, “there have been found famous
men, as Robert, bishop of Lincoln, and Brother Adam Marshman (de
Marisco), and many others, who by the power of mathematics have
been able to explain the causes of things; as may be seen in the
writings of these men, for instance, concerning the Rainbow and
Comets, and the generation of heat, and climates, and the
celestial bodies.”--WHEWELL, W.

                         _History of the Inductive Sciences (New
                         York, 1894), Vol. 1, p. 519. Bacon,
                         Roger: Opus Majus, Part 4, Distinctia
                         Prima, cap. 3._


=1548.= The analysis which is based upon the conception of
function discloses to the astronomer and physicist not merely the
formulae for the computation of whatever desired distances,
times, velocities, physical constants; it moreover gives him
insight into the laws of the processes of motion, teaches him to
predict future occurrences from past experiences and supplies him
with means to a scientific knowledge of nature, i.e. it enables
him to trace back whole groups of various, sometimes extremely
heterogeneous, phenomena to a minimum of simple fundamental laws.

                                                 --PRINGSHEIM, A.

                         _Jahresbericht der Deutschen
                         Mathematiker Vereinigung, Bd. 13, p.
                         366._


=1549.= “As is known, scientific physics dates its existence from
the discovery of the differential calculus. Only when it was
learned how to follow continuously the course of natural events,
attempts, to construct by means of abstract conceptions the
connection between phenomena, met with success. To do this two
things are necessary: First, simple fundamental concepts with
which to construct; second, some method by which to deduce, from
the simple fundamental laws of the construction which relate to
instants of time and points in space, laws for finite intervals
and distances, which alone are accessible to observation (can be
compared with experience).” [Riemann.]

The first of the two problems here indicated by Riemann consists
in setting up the differential equation, based upon physical
facts and hypotheses. The second is the integration of this
differential equation and its application to each separate
concrete case, this is the task of mathematics.--WEBER, HEINRICH.

                         _Die partiellen Differentialgleichungen
                         der mathematischen Physik (Braunschweig,
                         1882), Bd. 1, Vorrede._


=1550.= Mathematics is the most powerful instrument which we
possess for this purpose [to trace into their farthest results
those general laws which an inductive philosophy has supplied]:
in many sciences a profound knowledge of mathematics is
indispensable for a successful investigation. In the most
delicate researches into the theories of light, heat, and sound
it is the only instrument; they have properties which no other
language can express; and their argumentative processes are
beyond the reach of other symbols.--PRICE, B.

                         _Treatise on Infinitesimal Calculus
                         (Oxford, 1858), Vol. 3, p. 5._


=1551.= Notwithstanding the eminent difficulties of the
mathematical theory of sonorous vibrations, we owe to it such
progress as has yet been made in acoustics. The formation of the
differential equations proper to the phenomena is, independent of
their integration, a very important acquisition, on account of
the approximations which mathematical analysis allows between
questions, otherwise heterogeneous, which lead to similar
equations. This fundamental property, whose value we have so
often to recognize, applies remarkably in the present case; and
especially since the creation of mathematical thermology, whose
principal equations are strongly analogous to those of vibratory
motion.--This means of investigation is all the more valuable on
account of the difficulties in the way of direct inquiry into
the phenomena of sound. We may decide the necessity of the
atmospheric medium for the transmission of sonorous vibrations;
and we may conceive of the possibility of determining by
experiment the duration of the propagation, in the air, and then
through other media; but the general laws of the vibrations of
sonorous bodies escape immediate observation. We should know
almost nothing of the whole case if the mathematical theory did
not come in to connect the different phenomena of sound, enabling
us to substitute for direct observation an equivalent examination
of more favorable cases subjected to the same law. For instance,
when the analysis of the problem of vibrating chords has shown us
that, other things being equal, the number of oscillations is in
inverse proportion to the length of the chord, we see that the
most rapid vibrations of a very short chord may be counted, since
the law enables us to direct our attention to very slow
vibrations. The same substitution is at our command in many cases
in which it is less direct.--COMTE, A.

                         _Positive Philosophy [Martineau], Bk. 3,
                         chap. 4._


=1552.= Problems relative to the uniform propagation, or to the
varied movements of heat in the interior of solids, are reduced
... to problems of pure analysis, and the progress of this part
of physics will depend in consequence upon the advance which may
be made in the art of analysis. The differential equations ...
contain the chief results of the theory; they express, in the
most general and concise manner, the necessary relations of
numerical analysis to a very extensive class of phenomena; and
they connect forever with mathematical science one of the most
important branches of natural philosophy.--FOURIER, J.

                         _Theory of Heat [Freeman], (Cambridge,
                         1878), Chap. 3, p. 131._


=1553.= The effects of heat are subject to constant laws which
cannot be discovered without the aid of mathematical analysis.
The object of the theory is to demonstrate these laws; it reduces
all physical researches on the propagation of heat, to problems
of the integral calculus, whose elements are given by experiment.
No subject has more extensive relations with the progress of
industry and the natural sciences; for the action of heat is
always present, it influences the processes of the arts, and
occurs in all the phenomena of the universe.--FOURIER, J.

                         _Theory of Heat [Freeman], (Cambridge,
                         1878), Chap. 1, p. 12._


=1554.= Dealing with any and every amount of static electricity,
the mathematical mind has balanced and adjusted them with
wonderful advantage, and has foretold results which the
experimentalist can do no more than verify.... So in respect of
the force of gravitation, it has calculated the results of the
power in such a wonderful manner as to trace the known planets
through their courses and perturbations, and in so doing has
_discovered_ a planet before unknown.--FARADAY.

                         _Some Thoughts on the Conservation of
                         Force._


=1555.= Certain branches of natural philosophy (such as physical
astronomy and optics), ... are, in a great measure, inaccessible
to those who have not received a regular mathematical education....

                                                --STEWART, DUGALD.

                         _Philosophy of the Human Mind, Part 3,
                         chap. 1, sect. 3._


=1556.= So intimate is the union between mathematics and physics
that probably by far the larger part of the accessions to our
mathematical knowledge have been obtained by the efforts of
mathematicians to solve the problems set to them by experiment,
and to create “for each successive class of phenomena, a new
calculus or a new geometry, as the case might be, which might
prove not wholly inadequate to the subtlety of nature.”
Sometimes, indeed, the mathematician has been before the
physicist, and it has happened that when some great and new
question has occurred to the experimentalist or the observer, he
has found in the armoury of the mathematician the weapons which
he has needed ready made to his hand. But, much oftener, the
questions proposed by the physicist have transcended the utmost
powers of the mathematics of the time, and a fresh mathematical
creation has been needed to supply the logical instrument
requisite to interpret the new enigma.--SMITH, H. J. S.

                         _Presidential Address British
                         Association for the Advancement of
                         Science, Section A; Nature, Vol. 8
                         (1873), p. 450._


=1557.= Of all the great subjects which belong to the province of
his section, take that which at first sight is the least within
the domain of mathematics--I mean meteorology. Yet the part which
mathematics plays in meteorology increases every year, and seems
destined to increase. Not only is the theory of the simplest
instruments essentially mathematical, but the discussions of the
observations--upon which, be it remembered, depend the hopes
which are already entertained with increasing confidence, of
reducing the most variable and complex of all known phenomena to
exact laws--is a problem which not only belongs wholly to
mathematics, but which taxes to the utmost the resources of the
mathematics which we now possess.--SMITH, H. J. S.

                         _Presidential Address British
                         Association for the Advancement of
                         Science, Section A; Nature, Vol. 8
                         (1873), p. 449._


=1558.= You know that if you make a dot on a piece of paper, and
then hold a piece of Iceland spar over it, you will see not one
dot but two. A mineralogist, by measuring the angles of a
crystal, can tell you whether or no it possesses this property
without looking through it. He requires no scientific thought to
do that. But Sir William Roman Hamilton ... knowing these facts
and also the explanation of them which Fresnel had given,
thought about the subject, and he predicted that by looking
through certain crystals in a particular direction we should see
not two dots but a continuous circle. Mr. Lloyd made the
experiment, and saw the circle, a result which had never been
even suspected. This has always been considered one of the most
signal instances of scientific thought in the domain of physics.

                                                --CLIFFORD, W. K.

                         _Lectures and Essays (New York, 1901),
                         Vol. 1, p. 144._


=1559.= The discovery of this planet [Neptune] is justly reckoned
as the greatest triumph of mathematical astronomy. Uranus failed
to move precisely in the path which the computers predicted for
it, and was misguided by some unknown influence to an extent
which a keen eye might almost see without telescopic aid....
These minute discrepancies constituted the data which were found
sufficient for calculating the position of a hitherto unknown
planet, and bringing it to light. Leverrier wrote to Galle, in
substance: “_Direct your telescope to a point on the ecliptic in
the constellation of Aquarius, in longitude 326°, and you will
find within a degree of that place a new planet, looking like a
star of about the ninth magnitude, and having a perceptible
disc._” The planet was found at Berlin on the night of Sept. 26,
1846, in exact accordance with this prediction, within half an
hour after the astronomers began looking for it, and only about
52′ distant from the precise point that Leverrier had indicated.

                                                   --YOUNG, C. A.

                         _General Astronomy (Boston, 1891), Art.
                         653._


=1560.= I am convinced that the future progress of chemistry as
an exact science depends very much indeed upon the alliance with
mathematics.--FRANKLAND, A.

                         _American Journal of Mathematics, Vol.
                         1, p. 349._


=1561.= It is almost impossible to follow the later developments
of physical or general chemistry without a working knowledge of
higher mathematics.--MELLOR, J. W.

                         _Higher Mathematics (New York, 1902),
                         Preface._


=1562.=

        ... Mount where science guides;
        Go measure earth, weigh air, and state the tides;
        Instruct the planets in what orb to run,
        Correct old time, and regulate the sun.
                                               --THOMSON, W.

                         _On the Figure of the Earth, Title
                         page._


=1563.= Admission to its sanctuary [referring to astronomy] and
to the privileges and feelings of a votary, is only to be gained
by one means,--_sound and sufficient knowledge of mathematics,
the great instrument of all exact inquiry, without which no man
can ever make such advances in this or any other of the higher
departments of science as can entitle him to form an independent
opinion on any subject of discussion within their range._

                                                   --HERSCHEL, J.

                         _Outlines of Astronomy, Introduction,
                         sect. 7._


=1564.= The long series of connected truths which compose the
science of astronomy, have been evolved from the appearances and
observations by calculation, and a process of reasoning entirely
geometrical. It was not without reason that Plato called geometry
and arithmetic the wings of astronomy; for it is only by means of
these two sciences that we can give a rational account of any of
the appearances, or connect any fact with theory, or even render
a single observation available to the most common astronomical
purpose. It is by geometry that we are enabled to reason our
way up through the apparent motions to the real orbits of
the planets, and to assign their positions, magnitudes and
eccentricities. And it is by application of geometry--a sublime
geometry, indeed, invented for the purpose--to the general laws
of mechanics, that we demonstrate the law of gravitation, trace
it through its remotest effects on the different planets, and,
comparing these effects with what we observe, determine the
densities and weights of the minutest bodies belonging to the
system. The whole science of astronomy is in fact a tissue of
geometrical reasoning, applied to the data of observation; and it
is from this circumstance that it derives its peculiar character
of precision and certainty. To disconnect it from geometry,
therefore, and to substitute familiar illustrations and vague
description for close and logical reasoning, is to deprive it of
its principal advantages, and to reduce it to the condition of
an ordinary province of natural history.

                         _Edinburgh Review, Vol. 58 (1833-1834),
                         p. 168._


=1565.= But geometry is not only the instrument of astronomical
investigation, and the bond by which the truths are enchained
together,--it is also the instrument of explanation, affording,
by the peculiar brevity and perspicuity of its technical
processes, not only aid to the learner, but also such facilities
to the teacher as he will find it very difficult to supply,
if he voluntarily undertakes to forego its assistance. Few
undertakings, indeed, are attended with greater difficulty than
that of attempting to exhibit the connecting links of a chain of
mathematical reasoning, when we lay aside the technical symbols
and notation which relieve the memory, and speak at once to the
eyes and the understanding:....

                         _Edinburgh Review, Vol. 58 (1833-1834),
                         p. 169._


=1566.= With an ordinary acquaintance of trigonometry, and the
simplest elements of algebra, one may take up any well-written
treatise on plane astronomy, and work his way through it, from
beginning to end, with perfect ease; and he will acquire, in the
course of his progress, from the mere examples put before him, an
infinitely more correct and precise idea of astronomical methods
and theories, than he could obtain in a lifetime from the most
eloquent general descriptions that ever were written. At the same
time he will be strengthening himself for farther advances, and
accustoming his mind to habits of close comparison and rigid
demonstration, which are of infinitely more importance than the
acquisition of stores of undigested facts.

                         _Edinburgh Review, Vol. 58 (1833-1834),
                         p. 170._


=1567.= While the telescope serves as a means of penetrating
space, and of bringing its remotest regions nearer us, mathematics,
by inductive reasoning, have led us onwards to the remotest
regions of heaven, and brought a portion of them within the range
of our possibilities; nay, in our own times--so propitious to the
extension of knowledge--the application of all the elements
yielded by the present conditions of astronomy has even revealed
to the intellectual eyes a heavenly body, and assigned to it its
place, orbit, mass, before a single telescope has been directed
towards it.--HUMBOLDT, A.

                         _Cosmos [Otte], Vol. 2, part 2, sect.
                         3._


=1568.= Mighty are numbers, joined with art resistless.--EURIPIDES.

                         _Hecuba, Line 884._


=1569.= No single instrument of youthful education has such
mighty power, both as regards domestic economy and politics, and
in the arts, as the study of arithmetic. Above all, arithmetic
stirs up him who is by nature sleepy and dull, and makes him
quick to learn, retentive, shrewd, and aided by art divine he
makes progress quite beyond his natural powers.--PLATO.

                         _Laws [Jowett,] Bk. 5, p. 747._


=1570.= For all the higher arts of construction some acquaintance
with mathematics is indispensable. The village carpenter, who,
lacking rational instruction, lays out his work by empirical
rules learned in his apprenticeship, equally with the builder of
a Britannia Bridge, makes hourly reference to the laws of
quantitative relations. The surveyor on whose survey the land is
purchased; the architect in designing a mansion to be built on
it; the builder in preparing his estimates; his foreman in laying
out the foundations; the masons in cutting the stones; and the
various artisans who put up the fittings; are all guided by
geometrical truths. Railway-making is regulated from beginning to
end by mathematics: alike in the preparation of plans and
sections; in staking out the lines; in the mensuration of
cuttings and embankments; in the designing, estimating, and
building of bridges, culverts, viaducts, tunnels, stations. And
similarly with the harbors, docks, piers, and various engineering
and architectural works that fringe the coasts and overspread the
face of the country, as well as the mines that run underneath it.
Out of geometry, too, as applied to astronomy, the art of
navigation has grown; and so, by this science, has been made
possible that enormous foreign commerce which supports a large
part of our population, and supplies us with many necessaries
and most of our luxuries. And nowadays even the farmer, for the
correct laying out of his drains, has recourse to the level--that
is, to geometrical principles.--SPENCER, HERBERT.

                         _Education, chap. 1._


=1571.= [Arithmetic] is another of the great master-keys of life.
With it the astronomer opens the depths of the heavens; the
engineer, the gates of the mountains; the navigator, the pathways
of the deep. The skillful arrangement, the rapid handling of
figures, is a perfect magician’s wand. The mighty commerce of the
United States, foreign and domestic, passes through the books
kept by some thousands of diligent and faithful clerks. Eight
hundred bookkeepers, in the Bank of England, strike the monetary
balance of half the civilized world. Their skill and accuracy in
applying the common rules of arithmetic are as important as the
enterprise and capital of the merchant, or the industry and
courage of the navigator. I look upon a well-kept ledger with
something of the pleasure with which I gaze on a picture or a
statue. It is a beautiful work of art.--EVERETT, EDWARD.

                         _Orations and Speeches (Boston, 1870),
                         Vol. 3, p. 47._


=1572.= [Mathematics] 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. In which last
Respect, we may be said to receive from the _Mathematics_, the
principal Delights of Life, Securities of Health, Increase of
Fortune, and Conveniences of Labour: That we dwell elegantly and
commodiously, build decent Houses for ourselves, erect stately
Temples to God, and leave wonderful Monuments to Posterity: That
we are protected by those Rampires from the Incursions of the
Enemy; rightly use Arms, skillfully range an Army, and manage War
by Art, and not by the Madness of wild Beasts: That we have safe
Traffick through the deceitful Billows, pass in a direct Road
through the tractless Ways of the Sea, and come to the designed
Ports by the uncertain Impulse of the Winds: That we rightly cast
up our Accounts, do Business expeditiously, dispose, tabulate,
and calculate scattered Ranks of Numbers, and easily compute
them, though expressive of huge Heaps of Sand, nay immense Hills
of Atoms: That we make pacifick Separations of the Bounds of
Lands, examine the Moments of Weights in an equal Balance, and
distribute every one his own by a just Measure: That with a light
Touch we thrust forward vast Bodies which way we will, and stop a
huge Resistance with a very small Force: That we accurately
delineate the Face of this Earthly Orb, and subject the Oeconomy
of the Universe to our Sight: That we aptly digest the flowing
Series of Time, distinguish what is acted by due Intervals,
rightly account and discern the various Returns of the Seasons,
the stated Periods of Years and Months, the alternate Increments
of Days and Nights, the doubtful Limits of Light and Shadow, and
the exact Differences of Hours and Minutes: That we derive the
subtle Virtue of the Solar Rays to our Uses, infinitely extend
the Sphere of Sight, enlarge the near Appearances of Things,
bring to Hand Things remote, discover Things hidden, search
Nature out of her Concealments, and unfold her dark Mysteries:
That we delight our Eyes with beautiful Images, cunningly imitate
the Devices and portray the Works of Nature; imitate did I say?
nay excel, while we form to ourselves Things not in being,
exhibit Things absent, and represent Things past: That we
recreate our Minds and delight our Ears with melodious Sounds,
attemperate the inconstant Undulations of the Air to musical
Tunes, add a pleasant Voice to a sapless Log and draw a sweet
Eloquence from a rigid Metal; celebrate our Maker with an
harmonious Praise, and not unaptly imitate the blessed Choirs of
Heaven: That we approach and examine the inaccessible Seats of
the Clouds, the distant Tracts of Land, unfrequented Paths of the
Sea; lofty Tops of the Mountains, low Bottoms of the Valleys, and
deep Gulphs of the Ocean: That in Heart we advance to the Saints
themselves above, yea draw them to us, scale the etherial
Towers, freely range through the celestial Fields, measure the
Magnitudes, and determine the Interstices of the Stars, prescribe
inviolable Laws to the Heavens themselves, and confine the
wandering Circuits of the Stars within fixed Bounds: Lastly, that
we comprehend the vast Fabrick of the Universe, admire and
contemplate the wonderful Beauty of the Divine Workmanship, and
to learn the incredible Force and Sagacity of our own Minds, by
certain Experiments, and to acknowledge the Blessings of Heaven
with pious Affection.--BARROW, ISAAC.

                         _Mathematical Lectures (London, 1734),
                         pp. 27-30._


=1573.= Analytical and graphical treatment of statistics is
employed by the economist, the philanthropist, the business
expert, the actuary, and even the physician, with the most
surprisingly valuable results; while symbolic language involving
mathematical methods has become a part of wellnigh every large
business. The handling of pig-iron does not seem to offer any
opportunity for mathematical application. Yet graphical and
analytical treatment of the data from long-continued experiments
with this material at Bethlehem, Pennsylvania, resulted in the
discovery of the law that fatigue varied in proportion to a
certain relation between the load and the periods of rest.
Practical application of this law increased the amount handled by
each man from twelve and a half to forty-seven tons per day. Such
study would have been impossible without preliminary acquaintance
with the simple invariable elements of mathematics.--KARPINSKY, L.

                         _High School Education (New York, 1912),
                         chap. 6, p. 134._


=1574.= They [computation and arithmetic] belong then, it seems,
to the branches of learning which we are now investigating;--for
a military man must necessarily learn them with a view to the
marshalling of his troops, and so must a philosopher with the
view of understanding real being, after having emerged from the
unstable condition of becoming, or else he can never become an
apt reasoner.

That is the fact he replied.

But the guardian of ours happens to be both a military man and a
philosopher.

Unquestionably so.

It would be proper then, Glaucon, to lay down laws for this
branch of science and persuade those about to engage in the most
important state-matters to apply themselves to computation, and
study it, not in the common vulgar fashion, but with the view of
arriving at the contemplation of the nature of numbers by the
intellect itself,--not for the sake of buying and selling as
anxious merchants and retailers, but for war also, and that the
soul may acquire a facility in turning itself from what is in the
course of generation to truth and real being.--PLATO.

                         _Republic [Davis], Bk. 7, p. 525._


=1575.= The scientific part of Arithmetic and Geometry would be
of more use for regulating the thoughts and opinions of men than
all the great advantage which Society receives from the general
application of them: and this use cannot be spread through the
Society by the practice; for the Practitioners, however dextrous,
have no more knowledge of the Science than the very instruments
with which they work. They have taken up the Rules as they found
them delivered down to them by scientific men, without the least
inquiry after the Principles from which they are derived: and the
more accurate the Rules, the less occasion there is for inquiring
after the Principles, and consequently, the more difficult it is
to make them turn their attention to the First Principles; and,
therefore, a Nation ought to have both Scientific and Practical
Mathematicians.--WILLIAMSON, JAMES.

                         _Elements of Euclid with Dissertations
                         (Oxford, 1781)._


=1576.= _Where there is nothing to measure there is nothing to
calculate_, hence it is impossible to employ mathematics in
psychological investigations. Thus runs the syllogism compounded
of an adherence to usage and an apparent truth. As to the latter,
it is wholly untrue that we may calculate only where we have
measured. Exactly the opposite is true. Every hypothetically
assumed law of quantitative combination, even such as is
recognized as invalid, is subject to calculation; and in case of
deeply hidden but important matters it is imperative to try on
hypotheses and to subject the consequences which flow from them
to precise computation until it is found which one of the
various hypotheses coincides with experience. Thus the ancient
astronomers _tried_ eccentric circles, and Kepler _tried_ the
ellipse to account for the motion of the planets, the latter also
compared the squares of the times of revolution with the cubes of
the mean distances before he discovered their agreement. In like
manner Newton _tried_ whether a gravitation, varying inversely as
the square of the distance, sufficed to keep the moon in its
orbit about the earth; if this supposition had failed him, he
would have tried some other power of the distance, as the fourth
or fifth, and deduced the corresponding consequences to compare
them with the observations. Just this is the greatest benefit of
mathematics, that it enables us to survey the possibilities whose
range includes the actual, long before we have adequate definite
experience; this makes it possible to employ very incomplete
indications of experience to avoid at least the crudest
errors. Long before the transit of Venus was employed in the
determination of the sun’s parallax, it was attempted to
determine the instant at which the sun illumines exactly one-half
of the moon’s disk, in order to compute the sun’s distance from
the known distance of the moon from the earth. This was not
possible, for, owing to psychological reasons, our method of
measuring time is too crude to give us the desired instant with
sufficient accuracy; yet the attempt gave us the knowledge that
the sun’s distance from us is at least several hundred times as
great as that of the moon. This illustration shows clearly that
even a very imperfect estimate of a magnitude in a case where no
precise observation is possible, may become very instructive, if
we know how to exploit it. Was it necessary to know the scale of
our solar system in order to learn of its order in general? Or,
taking an illustration from another field, was it impossible to
investigate the laws of motion until it was known exactly how far
a body falls in a second at some definite place? Not at all. Such
determinations of _fundamental measures_ are in themselves
exceedingly difficult, but fortunately, such investigations form
a class of their own; our knowledge of _fundamental laws_ does
not need to wait on these. To be sure, computation invites
measurement, and every easily observed regularity of certain
magnitudes is an incentive to mathematical investigation.

                                                 --HERBART, J. F.

                         _Werke [Kehrbach], (Langensalza, 1890),
                         Bd. 5, p. 97._


=1577.= Those who pass for naturalists, have, for the most part,
been very little, or not at all, versed in mathematicks, if not
also jealous of them.--BOYLE, ROBERT.

                         _Works (London, 1772), Vol. 3, p. 426._


=1578.= However hurtful may have been the incursions of the
geometers, direct and indirect, into a domain which it is not for
them to cultivate, the physiologists are not the less wrong in
turning away from mathematics altogether. It is not only that
without mathematics they could not receive their due preliminary
training in the intervening sciences: it is further necessary for
them to have geometrical and mechanical knowledge, to understand
the structure and the play of the complex apparatus of the living,
and especially the animal organism. Animal mechanics, statical and
dynamical, must be unintelligible to those who are ignorant of
the general laws of rational mechanics. The laws of equilibrium
and motion are ... absolutely universal in their action, depending
wholly on the energy, and not at all on the nature of the forces
considered: and the only difficulty is in their numerical
application in cases of complexity. Thus, discarding all idea of a
numerical application in biology, we perceive that the general
theorems of statics and dynamics must be steadily verified in the
mechanism of living bodies, on the rational study of which they
cast an indispensable light. The highest orders of animals act in
repose and motion, like any other mechanical apparatus of a
similar complexity, with the one difference of the mover, which
has no power to alter the laws of motion and equilibrium. The
participation of rational mechanics in positive biology is thus
evident. Mechanics cannot dispense with geometry; and beside, we
see how anatomical and physiological speculations involve
considerations of form and position, and require a familiar
knowledge of the principal geometrical laws which may cast light
upon these complex relations.--COMTE, A.

                         _Positive Philosophy [Martineau], Bk. 5,
                         chap. 1._


=1579.= In mathematics we find the primitive source of
rationality; and to mathematics must the biologists resort for
means to carry on their researches.--COMTE, A.

                         _Positive Philosophy [Martineau], Bk. 5,
                         chap. 1._


=1580.= In this school [of mathematics] must they [biologists]
learn familiarly the real characters and conditions of scientific
evidence, in order to transfer it afterwards to the province of
their own theories. The study of it here, in the most simple and
perfect cases, is the only sound preparation for its recognition
in the most complex.

The study is equally necessary for the formation of intellectual
habits; for obtaining an aptitude in forming and sustaining
positive abstractions, without which the comparative method
cannot be used in either anatomy or physiology. The abstraction
which is to be the standard of comparison must be first clearly
formed, and then steadily maintained in its integrity, or the
analysis becomes abortive: and this is so completely in the
spirit of mathematical combinations, that practice in them is the
best preparation for it. A student who cannot accomplish the
process in the more simple case may be assured that he is not
qualified for the higher order of biological researches, and must
be satisfied with the humbler office of collecting materials for
the use of minds of another order. Hence arises another use of
mathematical training;--that of testing and classifying minds, as
well as preparing and guiding them. Probably as much good would
be done by excluding the students who only encumber the science
by aimless and desultory inquiries, as by fitly instituting those
who can better fulfill its conditions.--COMTE, A.

                         _Positive Philosophy [Martineau], Bk. 5,
                         chap. 1._


=1581.= There seems no sufficient reason why the use of
scientific fictions, so common in the hands of geometers, should
not be introduced into biology, if systematically employed, and
adopted with sufficient sobriety. In mathematical studies, great
advantages have arisen from imagining a series of hypothetical
cases, the consideration of which, though artificial, may aid the
clearing up of the real subject, or its fundamental elaboration.
This art is usually confounded with that of hypotheses; but it is
entirely different; inasmuch as in the latter case the solution
alone is imaginary; whereas in the former, the problem itself is
radically ideal. Its use can never be in biology comparable to
what it is in mathematics: but it seems to me that the abstract
character of the higher conceptions of comparative biology
renders them susceptible of such treatment. The process will be
to intercalate, among different known organisms, certain purely
fictitious organisms, so imagined as to facilitate their
comparison, by rendering the biological series more homogeneous
and continuous: and it might be that several might hereafter meet
with more or less of a realization among organisms hitherto
unexplored. It may be possible, in the present state of our
knowledge of living bodies, to conceive of a new organism capable
of fulfilling certain given conditions of existence. However that
may be, the collocation of real cases with well-imagined ones,
after the manner of geometers, will doubtless be practised
hereafter, to complete the general laws of comparative anatomy
and physiology, and possibly to anticipate occasionally the
direct exploration. Even now, the rational use of such an
artifice might greatly simplify and clear up the ordinary system
of biological instruction. But it is only the highest order of
investigators who can be trusted with it. Whenever it is adopted,
it will constitute another ground of relation between biology and
mathematics.--COMTE, A.

                         _Positive Philosophy [Martineau], Bk. 5,
                         chap. 1._


=1582.= I think it may safely enough be affirmed, that he, that
is not so much as indifferently skilled in mathematicks, can
hardly be more than indifferently skilled in the fundamental
principles of physiology.--BOYLE, ROBERT.

                         _Works (London, 1772), Vol. 3, p. 430._


=1583.= It is not only possible but necessary that mathematics be
applied to psychology; the reason for this necessity lies briefly
in this: that by no other means can be reached that which is the
ultimate aim of all speculation, namely conviction.--HERBART, J. F.

                         _Werke [Kehrbach], (Langensalza, 1890),
                         Bd. 5, p. 104._


=1584.= All more definite knowledge must start with computation;
and this is of most important consequences not only for the
theory of memory, of imagination, of understanding, but as well
for the doctrine of sensations, of desires, and affections.

                                                 --HERBART, J. F.

                         _Werke [Kehrbach], (Langensalza, 1890),
                         Bd. 5, p. 103._


=1585.= In the near future mathematics will play an important
part in medicine: already there are increasing indications that
physiology, descriptive anatomy, pathology and therapeutics
cannot escape mathematical legitimation.--DESSOIR, MAX.

                         _Westermann’s Monatsberichte, Bd. 77, p.
                         380; Ahrens: Scherz und Ernst in der
                         Mathematik (Leipzig, 1904), p. 395._


=1586.= The social sciences mathematically developed are to be
the controlling factors in civilization.--WHITE, W. F.

                         _A Scrap-book of Elementary Mathematics
                         (Chicago, 1908), p. 208._


=1587.= It is clear that this education [referring to education
preparatory to the science of sociology] must rest on a basis of
mathematical philosophy, even apart from the necessity of
mathematics to the study of inorganic philosophy. It is only in
the region of mathematics that sociologists, or anybody else, can
obtain a true sense of scientific evidence, and form the habit of
rational and decisive argumentation; can, in short, learn to
fulfill the logical conditions of all positive speculation, by
studying universal positivism at its source. This training,
obtained and employed with the more care on account of the
eminent difficulty of social science, is what sociologists have
to seek in mathematics.--COMTE, A.

                         _Positive Philosophy [Martineau], Bk. 6,
                         chap. 4._


=1588.= It is clear that the individual as a social unit and the
state as a social aggregate require a certain modicum of
mathematics, some arithmetic and algebra, to conduct their
affairs. Under this head would fall the theory of interest,
simple and compound, matters of discount and amortization, and,
if lotteries hold a prominent place in raising moneys, as in some
states, questions of probability must be added. As the state
becomes more highly organized and more interested in the
scientific analysis of its life, there appears an urgent
necessity for various statistical information, and this can be
properly obtained, reduced, correlated, and interpreted only when
the guiding spirit in the work have the necessary mathematical
training in the theory of statistics. (Figures may not lie, but
statistics compiled unscientifically and analyzed incompetently
are almost sure to be misleading, and when this condition is
unnecessarily chronic the so-called statisticians may well be
called liars.) The dependence of insurance of various kinds on
statistical information and the very great place which insurance
occupies in the modern state, albeit often controlled by private
corporations instead of by the government, makes the theories of
paramount importance to our social life.--WILSON, E. B.

                         _Bulletin American Mathematical Society,
                         Vol. 18 (1912), p. 463._


=1589.= The theory of probabilities and the theory of errors now
constitute a formidable body of knowledge of great mathematical
interest and of great practical importance. Though developed
largely through the applications to the more precise sciences of
astronomy, geodesy, and physics, their range of applicability
extends to all the sciences; and they are plainly destined to
play an increasingly important rôle in the development and in the
applications of the sciences of the future. Hence their study is
not only a commendable element in a liberal education, but some
knowledge of them is essential to a correct understanding of
daily events.--WOODWARD, R. S.

                         _Probability and Theory of Errors (New
                         York, 1906), Preface._


=1590.= It was not to be anticipated that a new science [the
science of probabilities] which took its rise in games of chance,
and which had long to encounter an obloquy, hardly yet extinct,
due to the prevailing idea that its only end was to facilitate
and encourage the calculations of gamblers, could ever have
attained its present status--that its aid should be called for in
every department of natural science, both to assist in discovery,
which it has repeatedly done (even in pure mathematics), to
minimize the unavoidable errors of observation, and to detect the
presence of causes as revealed by observed events. Nor are
commercial and other practical interests of life less indebted to
it: wherever the future has to be forecasted, risk to be provided
against, or the true lessons to be deduced from statistics, it
corrects for us the rough conjectures of common sense, and
decides which course is really, according to the lights of which
we are in possession, the wisest for us to pursue.--CROFTON, M. W.

                         _Encyclopedia Britannica, 9th Edition;
                         Article “Probability.”_


=1591.= The calculus of probabilities, when confined within just
limits, ought to interest, in an equal degree, the mathematician,
the experimentalist, and the statesman. From the time when Pascal
and Fermat established its first principles, it has rendered, and
continues daily to render, services of the most eminent kind. It
is the calculus of probabilities, which, after having suggested
the best arrangements of the tables of population and mortality,
teaches us to deduce from those numbers, in general so erroneously
interpreted, conclusions of a precise and useful character; it is
the calculus of probabilities which alone can regulate justly the
premiums to be paid for assurances; the reserve funds for the
disbursements of pensions, annuities, discounts, etc. It is under
its influence that lotteries and other shameful snares cunningly
laid for avarice and ignorance have definitely disappeared.--ARAGO.

                         _Eulogy on Laplace [Baden-Powell],
                         Smithsonian Report, 1874, p. 164._


=1592.= Men were surprised to hear that not only births, deaths,
and marriages, but the decisions of tribunals, the results of
popular elections, the influence of punishments in checking
crime, the comparative values of medical remedies, the probable
limits of error in numerical results in every department of
physical inquiry, the detection of causes, physical, social, and
moral, nay, even the weight of evidence and the validity of
logical argument, might come to be surveyed with the lynx-eyed
scrutiny of a dispassionate analysis.--HERSCHEL, J.

                         _Quoted in Encyclopedia Britannica, 9th
                         Edition; Article “Probability.”_


=1593.= If economists expect of the application of the
mathematical method any extensive concrete numerical results, and
it is to be feared that like other non-mathematicians all too
many of them think of mathematics as merely an arithmetical
science, they are bound to be disappointed and to find a paucity
of results in the works of the few of their colleagues who use
that method. But they should rather learn, as the mathematicians
among them know full well, that mathematics is much broader, that
it has an abstract quantitative (or even qualitative) side, that
it deals with relations as well as numbers, ....--WILSON, E. B.

                         _Bulletin American Mathematical Society,
                         Vol. 18 (1912), p. 464._


=1594.= The effort of the economist is to _see_, to picture the
inter-play of economic elements. The more clearly cut these
elements appear in his vision, the better; the more elements he
can grasp and hold in his mind at once, the better. The economic
world is a misty region. The first explorers used unaided vision.
Mathematics is the lantern by which what before was dimly visible
now looms up in firm, bold outlines. The old phantasmagoria
disappear. We see better. We also see further.--FISHER, IRVING.

                         _Transactions of Connecticut Academy,
                         Vol. 9 (1892), p. 119._


=1595.= In the great inquiries of the moral and social sciences
... mathematics (I always mean Applied Mathematics) affords the
only sufficient type of deductive art. Up to this time, I may
venture to say that no one ever knew what deduction is, as a
means of investigating the laws of nature, who had not learned it
from mathematics, nor can any one hope to understand it
thoroughly, who has not, at some time in his life, known enough
of mathematics to be familiar with the instrument at work.

                                                    --MILL, J. S.

                         _An Examination of Sir William
                         Hamilton’s Philosophy (London, 1878), p.
                         622._


=1596.= Let me pass on to say a word or two about the teaching of
mathematics as an academic training for general professional
life. It has immense capabilities in that respect. If you
consider how much of the effectiveness of an administrator
depends upon the capacity for co-ordinating appropriately a
number of different ideas, precise accuracy of definition,
rigidity of proof, and sustained reasoning, strict in every step,
and when you consider what substitutes for these things nine men
out of every ten without special training have to put up with, it
is clear that a man with a mathematical training has incalculable
advantages.--SHAW, W. H.

                         _Perry’s Teaching of Mathematics
                         (London, 1902), p. 73._


=1597.= Before you enter on the study of law a sufficient ground
work must be laid.... Mathematics and natural philosophy are so
useful in the most familiar occurrences of life and are so
peculiarly engaging and delightful as would induce everyone to
wish an acquaintance with them. Besides this, the faculties of
the mind, like the members of a body, are strengthened and
improved by exercise. Mathematical reasoning and deductions are,
therefore, a fine preparation for investigating the abstruse
speculations of the law.--JEFFERSON, THOMAS.

                         _Quoted in Cajori’s Teaching and History
                         of Mathematics in the U. S. (Washington,
                         1890), p. 35._


=1598.= It has been observed in England of the study of
law,--though the acquisition of the most difficult parts of its
learning, the interpretation of laws, the comparison of
authorities, and the construction of instruments, would seem to
require philological and critical training; though the weighing
of evidence and the investigation of probable truth belong to the
province of the moral sciences, and the peculiar duties of the
advocate require rhetorical skill,--yet that a large proportion
of the most distinguished members of the profession has proceeded
from the university (that of Cambridge) most celebrated for the
cultivation of mathematical studies.--EVERETT, EDWARD.

                         _Orations and Speeches (Boston, 1870),
                         Vol. 2, p. 511._


=1599.= All historic science tends to become mathematical.
Mathematical power is classifying power.--NOVALIS.

                         _Schriften (Berlin, 1901), Teil 2, p.
                         192._


=1599a.= History has never regarded itself as a science of
statistics. It was the Science of Vital Energy in relation with
time; and of late this radiating centre of its life has been
steadily tending,--together with every form of physical and
mechanical energy,--toward mathematical expression.--ADAM, HENRY.

                         _A Letter to American Teachers of
                         History (Washington, 1910), p. 115._


=1599b.= Mathematics can be shown to sustain a certain relation
to rhetoric and may aid in determining its laws.--SHERMAN L. A.

                         _University [of Nebraska] Studies, Vol.
                         1, p. 130._



  CHAPTER XVI

  ARITHMETIC


=1601.= There is no problem in all mathematics that cannot be
solved by direct counting. But with the present implements of
mathematics many operations can be performed in a few minutes
which without mathematical methods would take a lifetime.

                                                   --MACH, ERNST.

                         _Popular Scientific Lectures [McCormack]
                         (Chicago, 1898), p. 197._


=1602.= There is no inquiry which is not finally reducible to a
question of Numbers; for there is none which may not be conceived
of as consisting in the determination of quantities by each
other, according to certain relations.--COMTE, A.

                         _Positive Philosophy [Martineau], Bk. 1,
                         chap. 1._


=1603.= Pythagoras says that number is the origin of all things,
and certainly the law of number is the key that unlocks the
secrets of the universe. But the law of number possesses an
immanent order, which is at first sight mystifying, but on a more
intimate acquaintance we easily understand it to be intrinsically
necessary; and this law of number explains the wondrous
consistency of the laws of nature.--CARUS, PAUL.

                         _Reflections on Magic Squares; Monist,
                         Vol. 16 (1906), p. 139._


=1604.= An ancient writer said that arithmetic and geometry are
the _wings of mathematics_; I believe one can say without
speaking metaphorically that these two sciences are the
foundation and essence of all the sciences which deal with
quantity. Not only are they the foundation, they are also, as it
were, the capstones; for, whenever a result has been arrived at,
in order to use that result, it is necessary to translate it into
numbers or into lines; to translate it into numbers requires the
aid of arithmetic, to translate it into lines necessitates the
use of geometry.--LAGRANGE.

                         _Leçons Elémentaires sur les
                         Mathématiques, Leçon seconde._


=1605.= It is number which regulates everything and it is measure
which establishes universal order.... A quiet peace, an
inviolable order, an inflexible security amidst all change and
turmoil characterize the world which mathematics discloses and
whose depths it unlocks.--DILLMANN, E.

                         _Die Mathematik die Fackelträgerin einer
                         neuen Zeit (Stuttgart, 1889), p. 12._


=1606.=

        Number, the inducer of philosophies,
        The synthesis of letters, ....
                                                --AESCHYLUS.

                         _Quoted in, Thomson, J. A., Introduction
                         to Science, chap. 1 (London)._


=1607.= Amongst all the ideas we have, as there is none suggested
to the mind by more ways, so there is none more simple, than that
of _unity_, or one: it has no shadow of variety or composition in
it; every object our senses are employed about; every idea in our
understanding; every thought of our minds, brings this idea along
with it. And therefore it is the most intimate to our thoughts,
as well as it is, in its agreement to all other things, _the most
universal idea we have_.--LOCKE, JOHN.

                         _An Essay concerning Human
                         Understanding, Bk. 2, chap. 16, sect.
                         1._


=1608.= The _simple modes_ of _number_ are of all other the most
distinct; every the least variation, which is an unit, making
each combination as clearly different from that which approacheth
nearest to it, as the most remote; two being as distinct from
one, as two hundred; and the idea of two as distinct from the
idea of three, as the magnitude of the whole earth is from that
of a mite.--LOCKE, JOHN.

                         _An Essay concerning Human
                         Understanding, Bk. 2, chap. 16, sect.
                         3._


=1609.= The number of a class is the class of all classes similar
to the given class.--RUSSELL, BERTRAND.

                         _Principles of Mathematics (Cambridge,
                         1903), p. 115._


=1610.= Number is that property of a group of distinct things
which remains unchanged during any change to which the group may
be subjected which does not destroy the distinctness of the
individual things.--FINE, H. B.

                         _Number-system of Algebra (Boston and
                         New York, 1890), p. 3._


=1611.= The science of arithmetic may be called the science of
exact limitation of matter and things in space, force, and time.

                                                  --PARKER, F. W.

                         _Talks on Pedagogics (New York, 1894),
                         p. 64._


=1612.=

        Arithmetic is the science of the Evaluation
            of Functions,
        Algebra is the science of the Transformation
            of Functions.
                                            --HOWISON, G. H.

                         _Journal of Speculative Philosophy, Vol.
                         5, p. 175._


=1613.= That _arithmetic_ rests on pure intuition of _time_ is not
so obvious as that geometry is based on pure intuition of space,
but it may be readily proved as follows. All counting consists in
the repeated positing of unity; only in order to know how often it
has been posited, we mark it each time with a different word:
these are the numerals. Now repetition is possible only through
succession: but succession rests on the immediate intuition of
_time_, it is intelligible only by means of this latter concept:
hence counting is possible only by means of time.--This dependence
of counting on _time_ is evidenced by the fact that in all
languages multiplication is expressed by “times” [mal], that
is, by a concept of time; sexies, ἑξακις, six fois, six times.

                                                --SCHOPENHAUER, A.

                         _Die Welt als Vorstellung und Wille;
                         Werke (Frauenstaedt) (Leipzig, 1877),
                         Bd. 3, p. 39._


=1614.= The miraculous powers of modern calculation are due to
three inventions: the Arabic Notation, Decimal Fractions and
Logarithms.--CAJORI, F.

                         _History of Mathematics (New York,
                         1897), p. 161._


=1615.= The grandest achievement of the Hindoos and the one
which, of all mathematical investigations, has contributed most
to the general progress of intelligence, is the invention of the
principle of position in writing numbers.--CAJORI, F.

                         _History of Mathematics (New York,
                         1897), p. 87._


=1616.= The invention of logarithms and the calculation of the
earlier tables form a very striking episode in the history of
exact science, and, with the exception of the _Principia_ of
Newton, there is no mathematical work published in the country
which has produced such important consequences, or to which so
much interest attaches as to Napier’s Descriptio.

                                             --GLAISHER, J. W. L.

                         _Encyclopedia Britannica, 9th Edition;
                         Article “Logarithms.”_


=1617.= All minds are equally capable of attaining the science of
numbers: yet we find a prodigious difference in the powers of
different men, in that respect, after they have grown up, because
their minds have been more or less exercised in it.

                                               --JOHNSON, SAMUEL.

                         _Boswell’s Life of Johnson, Harper’s
                         Edition (1871), Vol. 2, p. 33._


=1618.= The method of arithmetical teaching is perhaps the best
understood of any of the methods concerned with elementary
studies.--BAIN, ALEXANDER.

                         _Education as a Science (New York,
                         1898), p. 288._


=1619.= What a benefite that onely thyng is, to haue the witte
whetted and sharpened, I neade not trauell to declare, sith all
men confesse it to be as greate as maie be. Excepte any witlesse
persone thinke he maie bee to wise. But he that most feareth
that, is leaste in daunger of it. Wherefore to conclude, I see
moare menne to acknowledge the benefite of nomber, than I can
espie willying to studie, to attaine the benefites of it. Many
praise it, but fewe dooe greatly practise it: onlesse it bee for
the vulgare practice, concernying Merchaundes trade. Wherein the
desire and hope of gain, maketh many willying to sustaine some
trauell. For aide of whom, I did sette forth the first parte of
_Arithmetike_. But if thei knewe how faree this seconde parte,
doeeth excell the firste parte, thei would not accoumpte any
tyme loste, that were emploied in it. Yea thei would not thinke
any tyme well bestowed till thei had gotten soche habilitie by
it, that it might be their aide in al other studies.

                                               --RECORDE, ROBERT.

                         _Whetstone of Witte (London, 1557)._


=1620.= You see then, my friend, I observed, that our real need
of this branch of science [arithmetic] is probably because it
seems to compel the soul to use our intelligence in the search
after pure truth.

Aye, remarked he, it does this to a remarkable extent.

Have you ever noticed that those who have a turn for arithmetic
are, with scarcely an exception, naturally quick in all sciences;
and that men of slow intellect, if they be trained and exercised
in this study ... become invariably quicker than they were
before?

Exactly so, he replied.

And, moreover, I think you will not easily find that many things
give the learner and student more trouble than this.

Of course not.

On all these accounts, then, we must not omit this branch of
science, but those with the best of talents should be instructed
therein.--PLATO.

                         _Republic [Davis], Bk. 7, chap. 8._


=1621.= Arithmetic has a very great and elevating effect,
compelling the soul to reason about abstract number, and if
visible or tangible objects are obtruding upon the argument,
refusing to be satisfied.--PLATO.

                         _Republic [Jowett], Bk. 7, p. 525._


=1622.= Good arithmetic contributes powerfully to purposive
effort, to concentration, to tenacity of purpose, to generalship,
to faith in right, and to the joy of achievement, which are the
elements that make up efficient citizenship.... Good arithmetic
exalts thinking, furnishes intellectual pleasure, adds appreciably
to love of right, and subordinates pure memory.--MYERS, GEORGE.

                         _Monograph on Arithmetic in Public
                         Education (Chicago), p. 21._


=1623.= On the one side we may say that the purpose of number
work is to put a child in possession of the machinery of
calculation; on the other side it is to give him a better mastery
of the world through a clear (mathematical) insight into the
varied physical objects and activities. The whole world, from one
point of view, can be definitely interpreted and appreciated by
mathematical measurements and estimates. Arithmetic in the common
school should give a child this point of view, the ability to see
and estimate things with a mathematical eye.--MCMURRAY, C. A.

                         _Special Method in Arithmetic_ (_New
                         York, 1906_), _p. 18._


=1624.= We are so accustomed to hear arithmetic spoken of as
one of the three fundamental ingredients in all schemes of
instruction, that it seems like inquiring too curiously to ask
why this should be. Reading, Writing, and Arithmetic--these
three are assumed to be of co-ordinate rank. Are they indeed
co-ordinate, and if so on what grounds?

In this modern “trivium” the art of reading is put first. Well,
there is no doubt as to its right to the foremost place. For
reading is the instrument of all our acquisition. It is
indispensable. There is not an hour in our lives in which it does
not make a great difference to us whether we can read or not.
And the art of Writing, too; that is the instrument of all
communication, and it becomes, in one form or other, useful to us
every day. But Counting--doing sums,--how often in life does this
accomplishment come into exercise? Beyond the simplest additions,
and the power to check the items of a bill, the arithmetical
knowledge required of any well-informed person in private life is
very limited. For all practical purposes, whatever I may have
learned at school of fractions, or proportion, or decimals, is,
unless I happen to be in business, far less available to me in
life than a knowledge, say, of history of my own country, or the
elementary truths of physics. The truth is, that regarded as
practical _arts_, reading, writing, and arithmetic have no right
to be classed together as co-ordinate elements of education; for
the last of these is considerably less useful to the average man
or woman not only than the other two, but than many others that
might be named. But reading, writing, and such mathematical or
logical exercise as may be gained in connection with the
manifestation of numbers, _have_ a right to constitute the
primary elements of instruction. And I believe that arithmetic,
if it deserves the high place that it conventionally holds in our
educational system, deserves it mainly on the ground that it is
to be treated as a logical exercise. It is the only branch of
mathematics which has found its way into primary and early
education; other departments of pure science being reserved for
what is called higher or university instruction. But all the
arguments in favor of teaching algebra and trigonometry to
advanced students, apply equally to the teaching of the
principles or theory of arithmetic to schoolboys. It is
calculated to do for them exactly the same kind of service, to
educate one side of their minds, to bring into play one set of
faculties which cannot be so severely or properly exercised in
any other department of learning. In short, relatively to the
needs of a beginner, Arithmetic, as a science, is just as
valuable--it is certainly quite as intelligible--as the higher
mathematics to a university student.--FITCH, J. G.

                         _Lectures on Teaching (New York, 1906),
                         pp. 267-268._


=1625.= What mathematics, therefore are expected to do for the
advanced student at the university, Arithmetic, if taught
demonstratively, is capable of doing for the children even of the
humblest school. It furnishes training in reasoning, and
particularly in deductive reasoning. It is a discipline in
closeness and continuity of thought. It reveals the nature of
fallacies, and refuses to avail itself of unverified assumptions.
It is the one department of school-study in which the sceptical
and inquisitive spirit has the most legitimate scope; in which
authority goes for nothing. In other departments of instruction
you have a right to ask for the scholar’s confidence, and to
expect many things to be received on your testimony with
the understanding that they will be explained and verified
afterwards. But here you are justified in saying to your pupil
“Believe nothing which you cannot understand. Take nothing for
granted.” In short, the proper office of arithmetic is to serve
as elementary training in logic. All through your work as
teachers you will bear in mind the fundamental difference between
knowing and thinking; and will feel how much more important
relatively to the health of the intellectual life the habit of
thinking is than the power of knowing, or even facility of
achieving visible results. But here this principle has special
significance. It is by Arithmetic more than by any other subject
in the school course that the art of thinking--consecutively,
closely, logically--can be effectually taught.--FITCH, J. G.

                         _Lectures on Teaching (New York, 1906),
                         pp. 292-293._


=1626.= Arithmetic and geometry, those wings on which the
astronomer soars as high as heaven.--BOYLE, ROBERT.

                         _Usefulness of Mathematics to Natural
                         Philosophy; Works (London, 1772), Vol.
                         3, p. 429._


=1627.= Arithmetical symbols are written diagrams and geometrical
figures are graphic formulas.--HILBERT, D.

                         _Mathematical Problems; Bulletin
                         American Mathematical Society, Vol. 8
                         (1902), p. 443._


=1628.= Arithmetic and geometry are much more certain than the
other sciences, because the objects of them are in themselves so
simple and so clear that they need not suppose anything which
experience can call in question, and both proceed by a chain of
consequences which reason deduces one from another. They are also
the easiest and clearest of all the sciences, and their object is
such as we desire; for, except for want of attention, it is
hardly supposable that a man should go astray in them. We must
not be surprised, however, that many minds apply themselves by
preference to other studies, or to philosophy. Indeed everyone
allows himself more freely the right to make his guess if the
matter be dark than if it be clear, and it is much easier to have
on any question some vague ideas than to arrive at the truth
itself on the simplest of all.--DESCARTES.

                         _Rules for the Direction of the Mind;
                         Torrey’s Philosophy of Descartes (New
                         York, 1892), p. 63._


=1629.=

        Why are _wise_ few, _fools_ numerous in the
            excesse?
        ’Cause, wanting _number_, they are
            _numberlesse_.
                                                 --LOVELACE.

                         _Noah Bridges: Vulgar Arithmetike
                         (London, 1659), p. 127._


=1630.= The clearness and distinctness of each mode of number
from all others, even those that approach nearest, makes me apt
to think that demonstrations in numbers, if they are not more
evident and exact than in extension, yet they are more general in
their use, and more determinate in their application. Because the
ideas of numbers are more precise and distinguishable than in
extension; where every equality and excess are not so easy to be
observed or measured; because our thoughts cannot in space arrive
at any determined smallness beyond which it cannot go, as an
unit; and therefore the quantity or proportion of any the least
excess cannot be discovered.--LOCKE, JOHN.

                         _An Essay concerning Human
                         Understanding, Bk. 2, chap. 16, sect.
                         4._


=1631.= Battalions of figures are like battalions of men, not
always as strong as is supposed.--SAGE, M.

                         _Mrs. Piper and the Society for
                         Psychical Research [Robertson] (New
                         York, 1909), p. 151._


=1632.= Number was born in superstition and reared in mystery,...
numbers were once made the foundation of religion and philosophy, and
the tricks of figures have had a marvellous effect on a credulous
people.--PARKER, F. W.

                         _Talks on Pedagogics (New York, 1894),
                         P. 64._


=1633.= A rule to trick th’ arithmetic.--KIPLING, R.

                         _To the True Romance._


=1634.= God made integers, all else is the work of man.

                                                  --KRONECKER, L.

                         _Jahresberichte der Deutschen
                         Mathematiker Vereinigung, Bd. 2, p. 19._


=1635.= Plato said “ἀεὶ ὁ θεὸς γεωμέτρε.” Jacobi changed this to
“ἀεὶ ὁ θεὸς ἀριθμητίζει.” Then came Kronecker and created the
memorable expression “Die ganzen Zahlen hat Gott gemacht, alles
andere ist Menschenwerk.”--KLEIN, F.

                         _Jahresbericht der Deutschen
                         Mathematiker Vereinigung, Bd. 6, p.
                         136._


=1636.= Integral numbers are the fountainhead of all mathematics.

                                                  --MINKOWSKI, H.

                         _Diophantische Approximationen (Leipzig,
                         1907), Vorrede._


=1637.= The “Disquisitiones Arithmeticae” that great book with
seven seals.--MERZ, J. T.

                         _A History of European Thought in the
                         Nineteenth Century (Edinburgh and
                         London, 1908), p. 721._


=1638.= It may fairly be said that the germs of the modern
algebra of linear substitutions and concomitants are to be found
in the fifth section of the _Disquisitiones Arithmeticae_; and
inversely, every advance in the algebraic theory of forms is an
acquisition to the arithmetical theory.--MATHEWS, G. B.

                         _Theory of Numbers (Cambridge, 1892),
                         Part 1, sect. 48._


=1639.= Strictly speaking, the theory of numbers has nothing to
do with negative, or fractional, or irrational quantities, _as
such._ No theorem which cannot be expressed without reference to
these notions is purely arithmetical: and no proof of an
arithmetical theorem, can be considered finally satisfactory if
it intrinsically depends upon extraneous analytical theories.

                                                 --MATHEWS, G. B.

                         _Theory of Numbers (Cambridge, 1892),
                         Part 1, sect. 1._


=1640.= Many of the greatest masters of the mathematical sciences
were first attracted to mathematical inquiry by problems relating
to numbers, and no one can glance at the periodicals of the
present day which contain questions for solution without noticing
how singular a charm such problems still continue to exert. The
interest in numbers seems implanted in the human mind, and it is
a pity that it should not have freer scope in this country. The
methods of the theory of numbers are peculiar to itself, and are
not readily acquired by a student whose mind has for years been
familiarized with the very different treatment which is
appropriate to the theory of continuous magnitude; it is
therefore extremely desirable that some portion of the theory
should be included in the ordinary course of mathematical
instruction at our University. From the moment that Gauss, in his
wonderful treatise of 1801, laid down the true lines of the
theory, it entered upon a new day, and no one is likely to be
able to do useful work in any part of the subject who is
unacquainted with the principles and conceptions with which he
endowed it.--GLAISHER, J. W. L.

                         _Presidential Address British
                         Association for the Advancement of
                         Science (1890); Nature, Vol. 42, p.
                         467._


=1641.= Let us look for a moment at the general significance of
the fact that calculating machines actually exist, which relieve
mathematicians of the purely mechanical part of numerical
computations, and which accomplish the work more quickly and with
a greater degree of accuracy; for the machine is not subject to
the slips of the human calculator. The existence of such a
machine proves that computation is not concerned with the
significance of numbers, but that it is concerned essentially
only with the formal laws of operation; for it is only these that
the machine can obey--having been thus constructed--an intuitive
perception of the significance of numbers being out of the
question.--KLEIN, F.

                         _Elementarmathematik vom höheren
                         Standpunkte aus. (Leipzig, 1908), Bd. 1,
                         p. 53._


=1642.= Mathematics is the queen of the sciences and arithmetic
the queen of mathematics. She often condescends to render service
to astronomy and other natural sciences, but in all relations she
is entitled to the first rank.--GAUSS.

                         _Sartorius von Waltershausen: Gauss zum
                         Gedächtniss. (Leipzig, 1866), p. 79._


=1643.=

        Zu Archimedes kam ein wissbegieriger Jüngling,
        Weihe mich, sprach er zu ihm, ein in die
            göttliche Kunst,
        Die so herrliche Dienste der Sternenkunde
            geleistet,
        Hinter dem Uranos noch einen Planeten entdeckt.
        Göttlich nennst Du die Kunst, sie ist’s,
            versetzte der Weise,
        Aber sie war es, bevor noch sie den Kosmos
            erforscht,
        Ehe sie herrliche Dienste der Sternenkunde
            geleistet,
        Hinter dem Uranos noch einen Planeten entdeckt.
        Was Du im Kosmos erblickst, ist nur der
            Göttlichen Abglanz,
        In der Olympier Schaar thronet die ewige
            Zahl.
                                          --JACOBI, C. G. J.

                         _Journal für Mathematik, Bd. 101 (1887),
                         p. 338._

        To Archimedes came a youth intent upon
            knowledge,
        Quoth he, “Initiate me into the science divine
        Which to astronomy, lo! such excellent service
            has rendered,
        And beyond Uranus’ orb a hidden planet
            revealed.”
        “Call’st thou the science divine? So it is,”
            the wise man responded,
        “But so it was long before its light on the
            Cosmos it shed,
        Ere in astronomy’s realm such excellent service
            it rendered,
        And beyond Uranus’ orb a hidden planet
            revealed.
        Only reflection divine is that which Cosmos
            discloses,
        Number herself sits enthroned among Olympia’s
            hosts.”


=1644.= The higher arithmetic presents us with an inexhaustible
store of interesting truths,--of truths too, which are not
isolated, but stand in a close internal connexion, and between
which, as our knowledge increases, we are continually discovering
new and sometimes wholly unexpected ties. A great part of its
theories derives an additional charm from the peculiarity that
important propositions, with the impress of simplicity upon them,
are often easily discoverable by induction, and yet are of so
profound a character that we cannot find their demonstration
till after many vain attempts; and even then, when we do succeed,
it is often by some tedious and artificial process, while the
simpler methods may long remain concealed.--GAUSS, C. F.

                         _Preface to Eisenstein’s Mathematische
                         Abhandlungen (Berlin, 1847), [H. J. S.
                         Smith]._


=1645.= The Theory of Numbers has acquired a great and increasing
claim to the attention of mathematicians. It is equally
remarkable for the number and importance of its results, for the
precision and rigorousness of its demonstrations, for the variety
of its methods, for the intimate relations between truths
apparently isolated which it sometimes discloses, and for the
numerous applications of which it is susceptible in other parts
of analysis.--SMITH, H. J. S.

                         _Report on the Theory of Numbers,
                         British Association, 1859; Collected
                         Mathematical Papers, Vol. 1, p. 38._


=1646.= The invention of the symbol ≡ by Gauss affords a striking
example of the advantage which may be derived from an appropriate
notation, and marks an epoch in the development of the science of
arithmetic.--MATHEWS, G. B.

                         _Theory of Numbers (Cambridge, 1892),
                         Part 1, sect. 29._


=1647.= As Gauss first pointed out, the problem of cyclotomy, or
division of the circle into a number of equal parts, depends in a
very remarkable way upon arithmetical considerations. We have
here the earliest and simplest example of those relations of the
theory of numbers to transcendental analysis, and even to pure
geometry, which so often unexpectedly present themselves, and
which, at first sight, are so mysterious.--MATHEWS, G. B.

                         _Theory of Numbers (Cambridge, 1892),
                         Part 1, sect. 167._


=1648.= I have sometimes thought that the profound mystery which
envelops our conceptions relative to prime numbers depends upon
the limitations of our faculties in regard to time, which like
space may be in its essence poly-dimensional, and that this and
such sort of truths would become self-evident to a being
whose mode of perception is according to _superficially_ as
distinguished from our own limitation to _linearly_ extended
time.--SYLVESTER, J. J.

                         _Collected Mathematical Papers, Vol. 4,
                         p. 600, footnote._



  CHAPTER XVII

  ALGEBRA


=1701.= The science of algebra, independently of any of its uses,
has all the advantages which belong to mathematics in general as
an object of study, and which it is not necessary to enumerate.
Viewed either as a science of quantity, or as a language of
symbols, it may be made of the greatest service to those who are
sufficiently acquainted with arithmetic, and who have sufficient
power of comprehension to enter fairly upon its difficulties.

                                                  --DE MORGAN, A.

                    _Elements of Algebra (London, 1837),
                    Preface._


=1702.= Algebra is generous, she often gives more than is asked
of her.--D’ALEMBERT.

                    _Quoted in Bulletin American Mathematical
                    Society, Vol. 2 (1905), p. 285._


=1703.= The operations of symbolic arithmetick seem to me to
afford men one of the clearest exercises of reason that I ever
yet met with, nothing being there to be performed without strict
and watchful ratiocination, and the whole method and progress of
that appearing at once upon the paper, when the operation is
finished, and affording the analyst a lasting, and, as it were,
visible ratiocination.--BOYLE, ROBERT.

                    _Works (London, 1772), Vol. 3, p. 426._


=1704.= The human mind has never invented a labor-saving machine
equal to algebra.--

                    _The Nation, Vol. 33, p. 237._


=1705.= They that are ignorant of Algebra cannot imagine the
wonders in this kind are to be done by it: and what further
improvements and helps advantageous to other parts of knowledge
the sagacious mind of man may yet find out, it is not easy to
determine. This at least I believe, that the _ideas of quantity_
are not those alone that are capable of demonstration and
knowledge; and that other, and perhaps more useful, parts of
contemplation, would afford us certainty, if vices, passions, and
domineering interest did not oppose and menace such endeavours.

                                                   --LOCKE, JOHN.

                    _An Essay concerning Human Understanding, Bk.
                    4, chap. 3, sect. 18._


=1706.= Algebra is but written geometry and geometry is but
figured algebra.--GERMAIN, SOPHIE.

                    _Mémoire sur la surfaces élastiques._


=1707.= So long as algebra and geometry proceeded separately
their progress was slow and their application limited, but when
these two sciences were united, they mutually strengthened each
other, and marched together at a rapid pace toward perfection.

                                                      --LAGRANGE.

                    _Leçons élémentaires sur les Mathématiques,
                    Leçon Cinquième._


=1708.= The laws of algebra, though suggested by arithmetic, do
not depend on it. They depend entirely on the conventions by
which it is stated that certain modes of grouping the symbols are
to be considered as identical. This assigns certain properties to
the marks which form the symbols of algebra. The laws regulating
the manipulation of algebraic symbols are identical with those of
arithmetic. It follows that no algebraic theorem can ever
contradict any result which could be arrived at by arithmetic;
for the reasoning in both cases merely applies the same general
laws to different classes of things. If an algebraic theorem can
be interpreted in arithmetic, the corresponding arithmetical
theorem is therefore true.--WHITEHEAD, A. N.

                    _Universal Algebra (Cambridge, 1898), p. 2._


=1709.= That a formal science like algebra, the creation of our
abstract thought, should thus, in a sense, dictate the laws of
its own being, is very remarkable. It has required the experience
of centuries for us to realize the full force of this appeal.

                                                 --MATHEWS, G. B.

                    _F. Spencer: Chapters on Aims and Practice of
                    Teaching (London, 1899), p. 184._


=1710.= The rules of algebra may be investigated by its own
principles, without any aid from geometry; and although in many
cases the two sciences may serve to illustrate each other, there
is not now the least necessity in the more elementary parts to
call in the aid of the latter in expounding the former.

                                              --CHRYSTAL, GEORGE.

                    _Encyclopedia Britannica, 9th Edition;
                    Article “Algebra.”_


=1711.= Algebra, as an art, can be of no use to any one in the
business of life; certainly not as taught in the schools. I
appeal to every man who has been through the school routine
whether this be not the case. Taught as an art it is of little
use in the higher mathematics, as those are made to feel who
attempt to study the differential calculus without knowing more
of the principles than is contained in books of rules.

                                                  --DE MORGAN, A.

                    _Elements of Algebra (London, 1837),
                    Preface._


=1712.= We may always depend upon it that algebra, which cannot
be translated into good English and sound common sense, is bad
algebra.--CLIFFORD, W. K.

                    _Common Sense in the Exact Sciences (London,
                    1885), chap. 1, sect. 7._


=1713.= The best review of arithmetic consists in the study of
algebra.--CAJORI, F.

                    _Teaching and History of Mathematics in U. S.
                    (Washington, 1896), p. 110._


=1714.= [Algebra] has for its object the resolution of equations;
taking this expression in its full logical meaning, which
signifies the transformation of implicit functions into
equivalent explicit ones. In the same way arithmetic may be
defined as destined to the determination of the values of
functions.... We will briefly say that _Algebra is the Calculus
of Functions_, and _Arithmetic the Calculus of Values_.--COMTE, A.

                    _Philosophy of Mathematics [Gillespie] (New
                    York, 1851), p. 55._


=1715.= ... the subject matter of algebraic science is the
abstract notion of time; divested of, or not yet clothed with,
any actual knowledge which we may possess of the real Events of
History, or any conception which we may frame of Cause and Effect
in Nature; but involving, what indeed it _cannot_ be divested of,
the thought of _possible_ Succession, or of pure, _ideal_
Progression.--HAMILTON, W. R.

                    _Graves’ Life of Hamilton (New York,
                    1882-1889), Vol. 3, p. 633._


=1716.= ... instead of seeking to attain consistency and
uniformity of system, as some modern writers have attempted, by
banishing this thought of time from the _higher_ Algebra, I seek
to _attain_ the same object, by systematically introducing it
into the _lower_ or earlier parts of the science.--HAMILTON, W. R.

                    _Graves’ Life of Hamilton (New York,
                    1882-1889), Vol. 3, p. 634._


=1717.= The circumstances that algebra has its origin in
arithmetic, however widely it may in the end differ from that
science, led Sir Isaac Newton to designate it “Universal
Arithmetic,” a designation which, vague as it is, indicates its
character better than any other by which it has been attempted to
express its functions--better certainly, to ordinary minds, than
the designation which has been applied to it by Sir William Rowan
Hamilton, one of the greatest mathematicians the world has seen
since the days of Newton--“the Science of Pure Time;” or even
than the title by which De Morgan would paraphrase Hamilton’s
words--“the Calculus of Succession.”--CHRYSTAL, GEORGE.

                    _Encyclopedia Britannica, 9th Edition;
                    Article “Algebra.”_


=1718.= Time is said to have only _one dimension_, and space to
have _three dimensions_.... The mathematical _quaternion_
partakes of _both_ these elements; in technical language it may
be said to be “time plus space,” or “space plus time:” and in
this sense it has, or at least involves a reference to, _four
dimensions_....

        And how the One of Time, of Space the Three,
        Might in the Chain of Symbols girdled be.
                                           --HAMILTON, W. R.

                         _Graves’ Life of Hamilton (New York,
                         1882-1889), Vol. 3, p. 635._


=1719.= It is confidently predicted, by those best qualified to
judge, that in the coming centuries Hamilton’s Quaternions will
stand out as the great discovery of our nineteenth century. Yet
how silently has the book taken its place upon the shelves of the
mathematician’s library! Perhaps not fifty men on this side of
the Atlantic have seen it, certainly not five have read it.

                                                  --HILL, THOMAS.

                         _North American Review, Vol. 85, p.
                         223._


=1720.= I think the time may come when double algebra will be the
beginner’s tool; and quaternions will be where double algebra is
now. The Lord only knows what will come above the quaternions.

                                                  --DE MORGAN, A.

                         _Graves’ Life of Hamilton (New York,
                         1882-1889), Vol. 3, p. 493._


=1721.= Quaternions came from Hamilton after his really good work
had been done; and though beautifully ingenious, have been an
unmixed evil to those who have touched them in any way, including
Clerk Maxwell.--THOMSON, WILLIAM.

                         _Thompson, S. P.: Life of Lord Kelvin
                         (London, 1910), p. 1138._


=1722.= The whole affair [quaternions] has in respect to
mathematics a value not inferior to that of “Volapuk” in respect
to language.--THOMSON, WILLIAM.

                         _Thompson, S. P.: Life of Lord Kelvin
                         (London, 1910), p. 1138._


=1723.= A quaternion of maladies! Do send me some formula by help
of which I may so doctor them that they may all become imaginary
or positively equal to nothing.--SEDGWICK.

                         _Graves’ Life of Hamilton (New York,
                         1882-1889), Vol. 3, p. 2._


=1724.= If nothing more could be said of Quaternions than that
they enable us to exhibit in a singularly compact and elegant
form, whose meaning is obvious at a glance on account of the
utter inartificiality of the method, results which in the
ordinary Cartesian co-ordinates are of the utmost complexity, a
very powerful argument for their use would be furnished. But it
would be unjust to Quaternions to be content with such a
statement; for we are fully entitled to say that in _all_ cases,
even in those to which the Cartesian methods seem specially
adapted, they give as simple an expression as any other method;
while in the great majority of cases they give a vastly simpler
one. In the common methods a judicious choice of co-ordinates is
often of immense importance in simplifying an investigation; in
Quaternions there is usually _no choice_, for (except when they
degrade to mere scalars) they are in general utterly independent
of any particular directions in space, and select of themselves
the most natural reference lines for each particular problem.

                                                    --TAIT, P. G.

                         _Presidential Address British
                         Association for the Advancement of
                         Science (1871); Nature, Vol. 4, p. 270._


=1725.= Comparing a Quaternion investigation, no matter in what
department, with the equivalent Cartesian one, even when the
latter has availed itself to the utmost of the improvements
suggested by Higher Algebra, one can hardly help making the remark
that they contrast even more strongly than the decimal notation
with the binary scale, or with the old Greek arithmetic--or than
the well-ordered subdivisions of the metrical system with the
preposterous no-systems of Great Britain, a mere fragment of which
(in the form of Table of Weights and Measures) form, perhaps the
most effective, if not the most ingenious, of the many instruments
of torture employed in our elementary teaching.--TAIT, P. G.

                         _Presidential Address British
                         Association for the Advancement of
                         Science (1871); Nature, Vol. 4, p. 271._


=1726.= It is true that, in the eyes of the pure mathematician,
Quaternions have one grand and fatal defect. They cannot be
applied to space of _n_ dimensions, they are contented to deal
with those poor three dimensions in which mere mortals are doomed
to dwell, but which cannot bound the limitless aspirations of a
Cayley or a Sylvester. From the physical point of view this,
instead of a defect, is to be regarded as the greatest possible
recommendation. It shows, in fact, Quaternions to be the special
instrument so constructed for application to the _Actual_ as to
have thrown overboard everything which is not absolutely
necessary, without the slightest consideration whether or no it
was thereby being rendered useless for application to the
_Inconceivable_.--TAIT, P. G.

                         _Presidential Address British
                         Association for the Advancement of
                         Science (1871); Nature, Vol. 4, p. 271._


=1727.= There is an old epigram which assigns the empire of the
sea to the English, of the land to the French, and of the clouds
to the Germans. Surely it was from the clouds that the Germans
fetched + and −; the ideas which these symbols have generated are
much too important to the welfare of humanity to have come from
the sea or from the land.--WHITEHEAD, A. N.

                         _An Introduction to Mathematics (New
                         York, 1911), p. 86._


=1728.= Now as to what pertains to these Surd numbers (which, as
it were by way of reproach and calumny, having no merit of their
own are also styled Irrational, Irregular, and Inexplicable) they
are by many denied to be numbers properly speaking, and are wont
to be banished from arithmetic to another Science, (which yet is
no science) viz. algebra.--BARROW, ISAAC.

                         _Mathematical Lectures (London, 1734),
                         p. 44._


=1729.= If it is true as Whewell says, that the essence of the
triumphs of science and its progress consists in that it enables
us to consider evident and necessary, views which our ancestors
held to be unintelligible and were unable to comprehend, then the
extension of the number concept to include the irrational, and we
will at once add, the imaginary, is the greatest forward step
which pure mathematics has ever taken.--HANKEL, HERMANN.

                         _Theorie der Complexen Zahlen (Leipzig,
                         1867), p. 60._


=1730.= That this subject [of imaginary magnitudes] has hitherto
been considered from the wrong point of view and surrounded by a
mysterious obscurity, is to be attributed largely to an ill-adapted
notation. If for instance, +1,−1, √−1 had been called direct,
inverse, and lateral units, instead of positive, negative, and
imaginary (or even impossible) such an obscurity would have been
out of question.--GAUSS, C. F.

                         _Theoria residiorum biquadraticorum,
                         Commentatio secunda; Werke, Bd. 2
                         (Goettingen, 1863), p. 177._


=1731.= ... the imaginary, this bosom-child of complex mysticism.

                                                --DÜHRING, EUGEN.

                         _Kritische Geschichte der allgemeinen
                         Principien der Mechanik (Leipzig, 1877),
                         p. 517._


=1732.= Judged by the only standards which are admissible in a
pure doctrine of numbers _i_ is imaginary in the same sense as
the negative, the fraction, and the irrational, but in no other
sense; all are alike mere symbols devised for the sake of
representing the results of operations even when these results
are not numbers (positive integers).--FINE, H. B.

                         _The Number-System of Algebra (Boston,
                         1890), p. 36._


=1733.= This symbol [√-1] is restricted to a precise signification
as the representative of perpendicularity in quaternions, and this
wonderful algebra of space is intimately dependent upon the
special use of the symbol for its symmetry, elegance, and power.
The immortal author of quaternions has shown that there are other
significations which may attach to the symbol in other cases. But
the strongest use of the symbol is to be found in its magical
power of doubling the actual universe, and placing by its side an
ideal universe, its exact counterpart, with which it can be
compared and contrasted, and, by means of curiously connecting
fibres, form with it an organic whole, from which modern analysis
has developed her surpassing geometry.--PEIRCE, BENJAMIN.

                         _On the Uses and Transformations of
                         Linear Algebras; American Journal of
                         Mathematics, Vol. 4 (1881), p. 216._


=1734.= The conception of the inconceivable [imaginary], this
measurement of what not only does not, but cannot exist, is one
of the finest achievements of the human intellect. No one can
deny that such imaginings are indeed imaginary. But they lead to
results grander than any which flow from the imagination of the
poet. The imaginary calculus is one of the masterkeys to physical
science. These realms of the inconceivable afford in many places
our only mode of passage to the domains of positive knowledge.
Light itself lay in darkness until this imaginary calculus threw
light upon light. And in all modern researches into electricity,
magnetism, and heat, and other subtile physical inquiries, these
are the most powerful instruments.--HILL, THOMAS.

                         _North American Review, Vol. 85, p.
                         235._


=1735.= All the fruitful uses of imaginaries, in Geometry, are those
which begin and end with real quantities, and use imaginaries only
for the intermediate steps. Now in all such cases, we have a real
spatial interpretation at the beginning and end of our argument,
where alone the spatial interpretation is important; in the
intermediate links, we are dealing in purely algebraic manner with
purely algebraic quantities, and may perform any operations which
are algebraically permissible. If the quantities with which we end
are capable of spatial interpretation, then, and only then, our
results may be regarded as geometrical. To use geometrical
language, in any other case, is only a convenient help to the
imagination. To speak, for example, of projective properties which
refer to the circular points, is a mere _memoria technica_ for
purely algebraical properties; the circular points are not to be
found in space, but only in the auxiliary quantities by which
geometrical equations are transformed. That no contradictions
arise from the geometrical interpretation of imaginaries is not
wonderful; for they are interpreted solely by the rules of
Algebra, which we may admit as valid in their interpretation to
imaginaries. The perception of space being wholly absent, Algebra
rules supreme, and no inconsistency can arise.--RUSSELL, BERTRAND.

                         _Foundations of Geometry (Cambridge,
                         1897), p. 45._


=1736.= Indeed, if one understands by algebra the application of
arithmetic operations to composite magnitudes of all kinds, whether
they be rational or irrational number or space magnitudes, then
the learned Brahmins of Hindostan are the true inventors of
algebra.--HANKEL, HERMANN.

                         _Geschichte der Mathematik im Altertum
                         und Mittelalter (Leipzig, 1874), p.
                         195._


=1737.= It is remarkable to what extent Indian mathematics enters
into the science of our time. Both the form and the spirit of the
arithmetic and algebra of modern times are essentially Indian and
not Grecian.--CAJORI, F.

                         _History of Mathematics (New York,
                         1897), p. 100._


=1738.= There are many questions in this science [algebra] which
learned men have to this time in vain attempted to solve; and
they have stated some of these questions in their writings, to
prove that this science contains difficulties, to silence those
who pretend they find nothing in it above their ability, to warn
mathematicians against undertaking to answer every question that
may be proposed, and to excite men of genius to attempt their
solution. Of these I have selected seven.

1. To divide 10 into two parts, such, that when each part is
added to its square-root and the sums multiplied together, the
product is equal to the supposed number.

2. What square is that, which being increased or diminished by
10, the sum and remainder are both square numbers?

3. A person said he owed to Zaid 10 all but the square-root of
what he owed to Amir, and that he owed Amir 5 all but the
square-root of what he owed Zaid.

4. To divide a cube number into two cube numbers.

5. To divide 10 into two parts such, that if each is divided by
the other, and the two quotients are added together, the sum is
equal to one of the parts.

6. There are three square numbers in continued geometric
proportion, such, that the sum of the three is a square number.

7. There is a square, such, that when it is increased and
diminished by its root and 2, the sum and the difference are
squares.--KHULASAT-AL-HISAB.

                         _Algebra; quoted in Hutton: A
                         Philosophical and Mathematical
                         Dictionary (London, 1815), Vol. 1, p.
                         70._


=1739.= The solution of such questions as these [referring to the
solution of cubic equations] depends on correct judgment, aided
by the assistance of God.--BIJA GANITA.

                         _Quoted in Hutton: A Philosophical and
                         Mathematical Dictionary (London, 1815),
                         Vol. 1, p. 65._


=1740.= For what is the theory of determinants? It is an algebra
upon algebra; a calculus which enables us to combine and foretell
the results of algebraical operations, in the same way as algebra
itself enables us to dispense with the performance of the special
operations of arithmetic. All analysis must ultimately clothe
itself under this form.--SYLVESTER, J. J.

                         _Philosophical Magazine, Vol. 1, (1851),
                         p. 300; Collected Mathematical Papers,
                         Vol. 1, p. 247._


=1741.=

    Fuchs. Fast möcht’ ich nun _moderne Algebra_ studieren.

    Meph.  Ich wünschte nicht euch irre zu führen.
           Was diese Wissenschaft betrifft,
           Es ist so schwer, die leere Form zu meiden,
           Und wenn ihr es nicht recht begrifft,
           Vermögt die Indices ihr kaum zu unterscheiden.
           Am Besten ist’s, wenn ihr nur _Einem_ traut
           Und auf des Meister’s Formeln baut.
           Im Ganzen--haltet euch an die _Symbole_.
           Dann geht ihr zu der Forschung Wohle
           Ins sichre Reich der Formeln ein.

    Fuchs. Ein Resultat muss beim Symbole sein?

    Meph.  Schon gut! Nur muss man sich nicht alzu ängstlich
             quälen.
           Denn eben, wo die Resultate fehlen,
           Stellt ein Symbol zur rechten Zeit sich ein.
           Symbolisch lässt sich alles schreiben,
           Müsst nur im Allgemeinen bleiben.
           Wenn man der Gleichung Lösung nicht erkannte,
           Schreibt man sie als Determinante.
           Schreib’ was du willst, nur rechne _nie_ was aus.
           Symbole lassen trefflich sich traktieren,
           Mit einem Strich ist alles auszuführen,
           Und mit Symbolen kommt man immer aus.
                                                --LASSWITZ, KURD.

                         _Der Faust-Tragödie (-n)ter Teil;
                         Zeitschrift für mathematischen und
                         naturwissenschaftlichen Unterricht, Bd.
                         14, p. 317._

    Fuchs. To study _modern algebra_ I’m most persuaded.

    Meph.  ’Twas not my wish to lead thee astray.
           But as concerns this science, truly
           ’Tis difficult to avoid the empty form,
           And should’st thou lack clear comprehension,
           Scarcely the indices thou’ll know apart.
           ’Tis safest far to trust but _one_
           And built upon your master’s formulas.
           On the whole--cling closely to your _symbols_.
           Then, for the weal of research you may gain
           An entrance to the formula’s sure domain.

    Fuchs. The symbol, it must lead to some result?

    Meph.  Granted. But never worry about results,
           For, mind you, just where the results are wanting
           A symbol at the nick of time appears.
           To symbolic treatment all things yield,
           Provided we stay in the general field.
           Should a solution prove elusive,
           Write the equation in determinant form.
           Write what you please, but _never_ calculate.
           Symbols are patient and long suffering,
           A single stroke completes the whole affair.
           Symbols for every purpose do suffice.


=1742.= As all roads are said to lead to Rome, so I find, in my
own case at least, that all algebraic inquiries sooner or later
end at the Capitol of Modern Algebra over whose shining portal
is inscribed “Theory of Invariants.”--SYLVESTER, J. J.

                         _On Newton’s Rule for the Discovery of
                         Imaginary Roots; Collected Mathematical
                         Papers, Vol. 2, p. 380._


=1743.= If we consider the beauty of the theorem [Sylvester’s
Theorem on Newton’s Rule for the Discovery of Imaginary Roots]
which has now been expounded, the interest which belongs to the
rule associated with the great name of Newton, and the long lapse
of years during which the reason and extent of that rule remained
undiscovered by mathematicians, among whom Maclaurin, Waring and
Euler are explicitly included, we must regard Professor
Sylvester’s investigations made to the Theory of Equations in
modern times, justly to be ranked with those of Fourier, Sturm
and Cauchy.--TODHUNTER, I.

                         _Theory of Equations (London, 1904), p.
                         250._


=1744.= Considering the remarkable elegance, generality, and
simplicity of the method [Homer’s Method of finding the numerical
values of the roots of an equation], it is not a little
surprising that it has not taken a more prominent place in
current mathematical textbooks.... As a matter of fact, its
spirit is purely arithmetical; and its beauty, which can only be
appreciated after one has used it in particular cases, is of that
indescribably simple kind, which distinguishes the use of
position in the decimal notation and the arrangement of the
simple rules of arithmetic. It is, in short, one of those things
whose invention was the creation of a commonplace.

                                              --CHRYSTAL, GEORGE.

                         _Algebra (London and Edinburgh, 1893),
                         Vol. 1, chap. 15, sect. 25._


=1745.= _To a missing member of a family group of terms in an
algebraical formula._

          Lone and discarded one! divorced by fate,
        Far from thy wished-for fellows--whither art
            flown?
        Where lingerest thou in thy bereaved estate,
        Like some lost star, or buried meteor stone?
        Thou mindst me much of that presumptuous one
        Who loth, aught less than greatest, to be
            great,
        From Heaven’s immensity fell headlong down
        To live forlorn, self-centred, desolate:
        Or who, like Heraclid, hard exile bore,
        Now buoyed by hope, now stretched on rack of
            fear,
        Till throned Astæa, wafting to his ear
        Words of dim portent through the Atlantic roar,
        Bade him “the sanctuary of the Muse revere
        And strew with flame the dust of Isis’
            shore.”
                                          --SYLVESTER, J. J.

                         _Inaugural Lecture, Oxford, 1885;
                         Nature, Vol. 33, p. 228._


=1746.= In every subject of inquiry there are certain entities,
the mutual relations of which, under various conditions, it is
desirable to ascertain. A certain combination of these entities
are submitted to certain processes or are made the subjects of
certain operations. The theory of invariants in its widest
scientific meaning determines these combinations, elucidates
their properties, and expresses results when possible in terms of
them. Many of the general principles of political science and
economics can be represented by means of invariantive relations
connecting the factors which enter as entities into the special
problems. The great principle of chemical science which asserts
that when elementary or compound bodies combine with one another
the total weight of the materials is unchanged, is another case
in point. Again, in physics, a given mass of gas under the
operation of varying pressure and temperature has the well-known
invariant, pressure multiplied by volume and divided by absolute
temperature.... In mathematics the entities under examination may
be arithmetical, algebraical, or geometrical; the processes to
which they are subjected may be any of those which are met with
in mathematical work.... It is the _principle_ which is so
valuable. It is the _idea_ of invariance that pervades today all
branches of mathematics.--MACMAHON, P. A.

                         _Presidential Address British
                         Association for the Advancement of
                         Science (1901); Nature, Vol. 64, p.
                         481._


=1747.= [The theory of invariants] has invaded the domain of
geometry, and has almost re-created the analytical theory; but it
has done more than this for the investigations of Cayley have
required a full reconsideration of the very foundations of
geometry. It has exercised a profound influence upon the theory
of algebraic equations; it has made its way into the theory of
differential equations; and the generalisation of its ideas is
opening out new regions of most advanced and profound functional
analysis. And so far from its course being completed, its
questions fully answered, or its interest extinct, there is no
reason to suppose that a term can be assigned to its growth and
its influence.--FORSYTH, A. R.

                         _Presidential Address British
                         Association for the Advancement of
                         Science (1897); Nature, Vol. 56, p.
                         378._


=1748.= ... the doctrine of Invariants, a theory filling the heavens
like a light-bearing ether, penetrating all the branches of
geometry and analysis, revealing everywhere abiding configurations
in the midst of change, everywhere disclosing the eternal reign of
the law of form.--KEYSER, C. J.

                         _Lectures on Science, Philosophy and Art
                         (New York, 1908), p. 28._


=1749.= It is in the mathematical doctrine of Invariance, the
realm wherein are sought and found configurations and types of
being that, amidst the swirl and stress of countless hosts of
transformations remain immutable, and the spirit dwells in
contemplation of the serene and eternal reign of the subtile laws
of Form, it is there that Theology may find, if she will, the
clearest conceptions, the noblest symbols, the most inspiring
intimations, the most illuminating illustrations, and the surest
guarantees of the object of her teaching and her quest, an
Eternal Being, unchanging in the midst of the universal flux.

                                                  --KEYSER, C. J.

                         _Lectures on Science, Philosophy and Art
                         (New York, 1908), p. 42._


=1750.= I think that young chemists desirous of raising their
science to its proper rank would act wisely in making themselves
master betimes of the theory of algebraic forms. What mechanics
is to physics, that I think is algebraic morphology, founded at
option on the theory of partitions or ideal elements, or both, is
destined to be to the chemistry of the future ... invariants and
isomerism are sister theories.--SYLVESTER, J. J.

                         _American Journal of Mathematics, Vol. 1
                         (1878), p. 126._


=1751.= The great notion of Group, ... though it had barely merged
into consciousness a hundred years ago, has meanwhile become a
concept of fundamental importance and prodigious fertility, not
only affording the basis of an imposing doctrine--the Theory of
Groups--but therewith serving also as a bond of union, a kind of
connective tissue, or rather as an immense cerebro-spinal system,
uniting together a large number of widely dissimilar doctrines as
organs of a single body.--KEYSER, C. J.

                         _Lectures on Science, Philosophy and Art
                         (New York, 1908), p. 12._


=1752.= In recent times the view becomes more and more prevalent
that many branches of mathematics are nothing but the theory of
invariants of special groups.--LIE, SOPHUS.

                         _Continuierliche Gruppen--Scheffers
                         (Leipzig, 1893), p. 665._


=1753.= Universal Algebra has been looked on with some suspicion
by many mathematicians, as being without intrinsic mathematical
interest and as being comparatively useless as an engine of
investigation.... But it may be shown that Universal Algebra has
the same claim to be a serious subject of mathematical study as
any other branch of mathematics.--WHITEHEAD, A. N.

                         _Universal Algebra (Cambridge, 1898),
                         Preface, p. vi._


=1754.= [Function] theory was, in effect, founded by Cauchy; but,
outside his own investigations, it at first made slow and
hesitating progress. At the present day, its fundamental ideas
may be said almost to govern most departments of the analysis of
continuous quantity. On many of them, it has shed a completely
new light; it has educed relations between them before unknown.
It may be doubted whether any subject is at the present day so
richly endowed with variety of method and fertility of resource;
its activity is prodigious, and no less remarkable than its
activity is its freshness.--FORSYTH, A. R.

                         _Presidential Address British
                         Association for the Advancement of
                         Science (1897); Nature, Vol. 56, p.
                         378._


=1755.= Let me mention one other contribution which this theory
[Theory of functions of a complex variable] has made to knowledge
lying somewhat outside our track. During the rigorous revision to
which the foundations of the theory have been subjected in its
re-establishment by Weierstrass, new ideas as regards number and
continuity have been introduced. With him and with others
influenced by him, there has thence sprung a new theory of higher
arithmetic; and with its growth, much has concurrently been
effected in the elucidation of the general notions of number and
quantity.... It thus appears to be the fact that, as with Plato,
or Descartes, or Leibnitz, or Kant, the activity of pure
mathematics is again lending some assistance to the better
comprehension of those notions of time, space, number, quantity,
which underlie a philosophical conception of the universe.

                                                 --FORSYTH, A. R.

                         _Presidential Address British
                         Association for the Advancement of
                         Science (1897); Nature, Vol. 56, p.
                         378._



  CHAPTER XVIII

  GEOMETRY


=1801.= The science of figures is most glorious and beautiful.
But how inaptly it has received the name geometry!--FRISCHLINUS, N.

                         _Dialog 1._


=1802.= Plato said that God geometrizes continually.--PLUTARCH.

                         _Convivialium disputationum, liber 8,
                         2._


=1803.= μηδεὶς ἐγεωμέτρητος εἰσίτω μοῦ τὴν στέγην. [Let no one
ignorant of geometry enter my door.]--PLATO.

                         _Tzetzes, Chiliad, 8, 972._


=1804.= All the authorities agree that he [Plato] made a study of
geometry or some exact science an indispensable preliminary to
that of philosophy. The inscription over the entrance to his
school ran “Let none ignorant of geometry enter my door,” and on
one occasion an applicant who knew no geometry is said to have
been refused admission as a student.--BALL, W. W. R.

                         _History of Mathematics (London, 1901),
                         p. 45._


=1805.= Form and size constitute the foundation of all search for
truth.--PARKER, F. W.

                         _Talks on Pedagogics (New York, 1894),
                         p. 72._


=1806.= At present the science [of geometry] is in flat
contradiction to the language which geometricians use, as will
hardly be denied by those who have any acquaintance with the
study: for they speak of finding the side of a square, and
applying and adding, and so on, as if they were engaged in some
business, and as if all their propositions had a practical end in
view: whereas in reality the science is pursued wholly for the
sake of knowledge.

Certainly, he said.

Then must not a further admission be made?

What admission?

The admission that this knowledge at which geometry aims is of
the eternal, and not of the perishing and transient.

That may be easily allowed. Geometry, no doubt, is the knowledge
of what eternally exists.

Then, my noble friend, geometry will draw the soul towards truth,
and create the mind of philosophy, and raise up that which is now
unhappily allowed to fall down.--PLATO.

                         _Republic [Jowett-Davies], Bk. 7, p.
                         527._


=1807.= Among them [the Greeks] geometry was held in highest
honor: nothing was more glorious than mathematics. But we have
limited the usefulness of this art to measuring and calculating.

                                                        --CICERO.

                         _Tusculanae Disputationes, 1, 2, 5._


=1808.=

                              Geometria,
        Through which a man hath the sleight
        Of length, and brede, of depth, of height.
                                              --GOWER, JOHN.

                         _Confessio Amantis, Bk. 7._


=1809.= Geometrical truths are in a way asymptotes to physical truths,
that is to say, the latter approach the former indefinitely near
without ever reaching them exactly.--D’ALEMBERT.

                         _Quoted in Rebière: Mathématiques et
                         Mathématiciens (Paris, 1898), p. 10._


=1810.= Geometry exhibits the most perfect example of logical
stratagem.--BUCKLE, H. T.

                         _History of Civilization in England (New
                         York, 1891), Vol. 2, p. 342._


=1811.= It is the glory of geometry that from so few principles,
fetched from without, it is able to accomplish so much.--NEWTON.

                         _Philosophiae Naturalis Principia
                         Mathematica, Praefat._


=1812.= Geometry is the application of strict logic to those
properties of space and figure which are self-evident, and which
therefore cannot be disputed. But the rigor of this science is
carried one step further; for no property, however evident it may
be, is allowed to pass without demonstration, if that can be
given. The question is therefore to demonstrate all geometrical
truths with the smallest possible number of assumptions.

                                                  --DE MORGAN, A.

                         _On the Study and Difficulties of
                         Mathematics (Chicago, 1902), p. 231._


=1813.= Geometry is a true natural science:--only more simple,
and therefore more perfect than any other. We must not suppose
that, because it admits the application of mathematical analysis,
it is therefore a purely logical science, independent of
observation. Every body studied by geometers presents some
primitive phenomena which, not being discoverable by reasoning,
must be due to observation alone.--COMTE, A.

                         _Positive Philosophy [Martineau], Bk. 1,
                         chap. 3._


=1814.= Geometry in every proposition speaks a language which
experience never dares to utter; and indeed of which she but half
comprehends the meaning. Experience sees that the assertions are
true, but she sees not how profound and absolute is their truth.
She unhesitatingly assents to the laws which geometry delivers,
but she does not pretend to see the origin of their obligation.
She is always ready to acknowledge the sway of pure scientific
principles as a matter of fact, but she does not dream of
offering her opinion on their authority as a matter of right;
still less can she justly claim to herself the source of that
authority.--WHEWELL, WILLIAM.

                         _The Philosophy of the Inductive
                         Sciences, Part 1, Bk. 1, chap. 6, sect.
                         1 (London, 1858)._


=1815.= Geometry is the science created to give understanding
and mastery of the external relations of things; to make easy
the explanation and description of such relations and the
transmission of this mastery.--HALSTED, G. B.

                         _Proceedings of the American Association
                         for the Advancement of Science (1904),
                         p. 359._


=1816.= A mathematical point is the most indivisible and unique
thing which art can present.--DONNE, JOHN.

                         _Letters, 21._


=1817.= It is certain that from its completeness, uniformity and
faultlessness, from its arrangement and progressive character,
and from the universal adoption of the completest and best line
of argument, Euclid’s “Elements” stand pre-eminently at the head
of all human productions. In no science, in no department of
knowledge, has anything appeared like this work: for upward of
2000 years it has commanded the admiration of mankind, and that
period has suggested little toward its improvement.--KELLAND, P.

                         _Lectures on the Principles of
                         Demonstrative Mathematics (London,
                         1843), p. 17._


=1818.= In comparing the performance in Euclid with that in
Arithmetic and Algebra there could be no doubt that Euclid had
made the deepest and most beneficial impression: in fact it might
be asserted that this constituted by far the most valuable part
of the whole training to which such persons [students, the
majority of which were not distinguished for mathematical taste
and power] were subjected.--TODHUNTER, I.

                         _Essay on Elementary Geometry; Conflict
                         of Studies and other Essays (London,
                         1873), p. 167._


=1819.= In England the geometry studied is that of Euclid, and I
hope it never will be any other; for this reason, that so much
has been written on Euclid, and all the difficulties of geometry
have so uniformly been considered with reference to the form in
which they appear in Euclid, that the study of that author is a
better key to a great quantity of useful reading than any other.

                                                  --DE MORGAN, A.

                         _Elements of Algebra (London, 1837),
                         Introduction._


=1820.= This book [Euclid] has been for nearly twenty-two
centuries the encouragement and guide of that scientific thought
which is one thing with the progress of man from a worse to a
better state. The encouragement; for it contained a body of
knowledge that was really known and could be relied on, and that
moreover was growing in extent and application. For even at the
time this book was written--shortly after the foundation of the
Alexandrian Museum--Mathematics was no longer the merely ideal
science of the Platonic school, but had started on her career of
conquest over the whole world of Phenomena. The guide; for the
aim of every scientific student of every subject was to bring his
knowledge of that subject into a form as perfect as that which
geometry had attained. Far up on the great mountain of Truth,
which all the sciences hope to scale, the foremost of that sacred
sisterhood was seen, beckoning for the rest to follow her. And
hence she was called, in the dialect of the Phythagoreans, “the
purifier of the reasonable soul.”--CLIFFORD, W. K.

                         _Lectures and Essays (London, 1901),
                         Vol. 1, p. 354._


=1821.= [Euclid] at once the inspiration and aspiration of
scientific thought.--CLIFFORD, W. K.

                         _Lectures and Essays (London, 1901), Vol
                         1, p. 355._


=1822.= The “elements” of the Great Alexandrian remain for all
time the first, and one may venture to assert, the _only_ perfect
model of logical exactness of principles, and of rigorous
development of theorems. If one would see how a science can be
constructed and developed to its minutest details from a very
small number of intuitively perceived axioms, postulates, and
plain definitions, by means of rigorous, one would almost say
chaste, syllogism, which nowhere makes use of surreptitious or
foreign aids, if one would see how a science may thus be
constructed one must turn to the elements of Euclid.--HANKEL, H.

                         _Die Entwickelung der Mathematik in den
                         letzten Jahrhunderten (Tübingen, 1884),
                         p. 7._


=1823.= If we consider him [Euclid] as meaning to be what
his commentators have taken him to be, a model of the most
unscrupulous formal rigour, we can deny that he has altogether
succeeded, though we admit that he made the nearest approach.

                                                  --DE MORGAN, A.

                         _Smith’s Dictionary of Greek and Roman
                         Biography and Mythology (London, 1902);
                         Article “Eucleides.”_


=1824.= The Elements of Euclid is as small a part of mathematics
as the Iliad is of literature; or as the sculpture of Phidias is
of the world’s total art.--KEYSER, C. J.

                         _Lectures on Science, Philosophy and Art
                         (New York, 1908), p. 8._


=1825.= I should rejoice to see ... Euclid honourably shelved
or buried “deeper than did ever plummet sound” out of the
schoolboys’ reach; morphology introduced into the elements of
algebra; projection, correlation, and motion accepted as aids to
geometry; the mind of the student quickened and elevated and his
faith awakened by early initiation into the ruling ideas of
polarity, continuity, infinity, and familiarization with the
doctrines of the imaginary and inconceivable.--SYLVESTER, J. J.

                         _A Plea for the Mathematician; Nature,
                         Vol. 1, p. 261._


=1826.= The early study of Euclid made me a hater of geometry,
... and yet, in spite of this repugnance, which had become a
second nature in me, whenever I went far enough into any
mathematical question, I found I touched, at last, a geometrical
bottom.--SYLVESTER, J. J.

                         _A Plea for the Mathematician; Nature,
                         Vol. 1, p. 262._


=1827.= Newton had so remarkable a talent for mathematics that
Euclid’s Geometry seemed to him “a trifling book,” and he wondered
that any man should have taken the trouble to demonstrate
propositions, the truth of which was so obvious to him at the
first glance. But, on attempting to read the more abstruse
geometry of Descartes, without having mastered the elements of the
science, he was baffled, and was glad to come back again to his
Euclid.--PARTON, JAMES.

                         _Sir Isaac Newton._


=1828.= As to the need of improvement there can be no question
whilst the reign of Euclid continues. My own idea of a useful
course is to begin with arithmetic, and then not Euclid but
algebra. Next, not Euclid, but practical geometry, solid as well
as plane; not demonstration, but to make acquaintance. Then not
Euclid, but elementary vectors, conjoined with algebra, and
applied to geometry. Addition first; then the scalar product.
Elementary calculus should go on simultaneously, and come into
the vector algebraic geometry after a bit. Euclid might be an
extra course for learned men, like Homer. But Euclid for children
is barbarous.--HEAVISIDE, OLIVER.

                         _Electro-Magnetic Theory (London, 1893),
                         Vol. 1, p. 148._


=1829.= Geometry is nothing if it be not rigorous, and the whole
educational value of the study is lost, if strictness of
demonstration be trifled with. The methods of Euclid are, by
almost universal consent, unexceptionable in point of rigour.

                                                --SMITH, H. J. S.

                         _Nature, Vol. 8, p. 450._


=1830.= To seek for proof of geometrical propositions by an
appeal to observation proves nothing in reality, except that the
person who has recourse to such grounds has no due apprehension
of the nature of geometrical demonstration. We have heard of
persons who convince themselves by measurement that the
geometrical rule respecting the squares on the sides of a
right-angles triangle was true: but these were persons whose
minds had been engrossed by practical habits, and in whom
speculative development of the idea of space had been stifled by
other employments.--WHEWELL, WILLIAM.

                         _The Philosophy of the Inductive
                         Sciences, (London, 1858), Part 1, Bk. 2,
                         chap. 1, sect. 4._


=1831.= No one has ever given so easy and natural a chain of
geometrical consequences [as Euclid]. There is a never-erring
truth in the results.--DE MORGAN, A.

                         _Smith’s Dictionary of Greek and Roman
                         Biography and Mythology (London, 1902);
                         Article “Eucleides.”_


=1832.= Beyond question, Egyptian geometry, such as it was, was
eagerly studied by the early Greek philosophers, and was the germ
from which in their hands grew that magnificent science to which
every Englishman is indebted for his first lessons in right
seeing and thinking.--GOW, JAMES.

                         _A Short History of Greek Mathematics
                         (Cambridge, 1884), p. 131._


=1833.=

        A figure and a step onward:
        Not a figure and a florin.
                     --MOTTO OF THE PYTHAGOREAN BROTHERHOOD.

                         _W. B. Frankland: Story of Euclid
                         (London, 1902), p. 33._


=1834.= The doctrine of proportion, as laid down in the fifth
book of Euclid, is, probably, still unsurpassed as a masterpiece
of exact reasoning; although the cumbrousness of the forms of
expression which were adopted in the old geometry has led to the
total exclusion of this part of the elements from the ordinary
course of geometrical education. A zealous defender of Euclid
might add with truth that the gap thus created in the elementary
teaching of mathematics has never been adequately supplied.

                                                --SMITH, H. J. S.

                         _Presidential Address British
                         Association for the Advancement of
                         Science (1873); Nature, Vol. 8, p. 451._


=1835.= The Definition in the Elements, according to Clavius, is
this: Magnitudes are said to be in the same Reason [ratio], a
first to a second, and a third to a fourth, when the Equimultiples
of the first and third according to any Multiplication whatsoever
are both together either short of, equal to, or exceed the
Equimultiples of the second and fourth, if those be taken,
which answer one another.... Such is Euclid’s Definition of
Proportions; that _scare_-Crow at which the over modest or
slothful Dispositions of Men are generally affrighted: they are
modest, who distrust their own Ability, as soon as a Difficulty
appears, but they are slothful that will not give some Attention
for the learning of Sciences; as if while we are involved in
Obscurity we could clear ourselves without Labour. Both of which
Sorts of Persons are to be admonished, that the former be not
discouraged, nor the latter refuse a little Care and Diligence
when a Thing requires some Study.--BARROW, ISAAC.

                         _Mathematical Lectures (London, 1734),
                         p. 388._


=1836.= Of all branches of human knowledge, there is none which,
like it [geometry] has sprung a completely armed Minerva from the
head of Jupiter; none before whose death-dealing Aegis doubt
and inconsistency have so little dared to raise their eyes.
It escapes the tedious and troublesome task of collecting
experimental facts, which is the province of the natural sciences
in the strict sense of the word: the sole form of its scientific
method is deduction. Conclusion is deduced from conclusion, and
yet no one of common sense doubts but that these geometrical
principles must find their practical application in the real
world about us. Land surveying, as well as architecture, the
construction of machinery no less than mathematical physics, are
continually calculating relations of space of the most varied
kinds by geometrical principles; they expect that the success of
their constructions and experiments shall agree with their
calculations; and no case is known in which this expectation has
been falsified, provided the calculations were made correctly and
with sufficient data.--HELMHOLTZ, H.

                         _The Origin and Significance of
                         Geometrical Axioms; Popular Scientific
                         Lectures [Atkinson], Second Series (New
                         York, 1881), p. 27._


=1837.= The amazing triumphs of this branch of mathematics
[geometry] show how powerful a weapon that form of deduction is
which proceeds by an artificial reparation of facts, in
themselves inseparable.--BUCKLE, H. T.

                         _History of Civilization in England (New
                         York, 1891), Vol. 2, p. 343._


=1838.= Every theorem in geometry is a law of external nature,
and might have been ascertained by generalizing from observation
and experiment, which in this case resolve themselves into
comparisons and measurements. But it was found practicable, and
being practicable was desirable, to deduce these truths by
ratiocination from a small number of general laws of nature, the
certainty and universality of which was obvious to the most
careless observer, and which compose the first principles and
ultimate premises of the science.--MILL, J. S.

                         _System of Logic, Bk. 3, chap. 24, sect.
                         7._


=1839.= All such reasonings [natural philosophy, chemistry,
agriculture, political economy, etc.] are, in comparison with
mathematics, very complex; requiring so much _more_ than that
does, beyond the process of merely deducing the conclusion
logically from the premises: so that it is no wonder that the
longest mathematical demonstration should be much more easily
constructed and understood, than a much shorter train of just
reasoning concerning real facts. The former has been aptly
compared to a long and steep, but even and regular, flight of
steps, which tries the breath, and the strength, and the
perseverance only; while the latter resembles a short, but rugged
and uneven, ascent up a precipice, which requires a quick eye,
agile limbs, and a firm step; and in which we have to tread now
on this side, now on that--ever considering as we proceed,
whether this or that projection will afford room for our foot, or
whether some loose stone may not slide from under us. There are
probably as many steps of pure reasoning in one of the longer of
Euclid’s demonstrations, as in the whole of an argumentative
treatise on some other subject, occupying perhaps a considerable
volume.--WHATELY, R.

                         _Elements of Logic, Bk. 4, chap. 2,
                         sect. 5._


=1840.=

        [Geometry] that held acquaintance with the stars,
        And wedded soul to soul in purest bond
        Of reason, undisturbed by space or time.
                                               --WORDSWORTH.

                         _The Prelude, Bk. 5._


=1841.= The statement that a given individual has received a
sound geometrical training implies that he has segregated from
the whole of his sense impressions a certain set of these
impressions, that he has eliminated from their consideration all
irrelevant impressions (in other words, acquired a subjective
command of these impressions), that he has developed on the basis
of these impressions an ordered and continuous system of logical
deduction, and finally that he is capable of expressing the
nature of these impressions and his deductions therefrom in terms
simple and free from ambiguity. Now the slightest consideration
will convince any one not already conversant with the idea, that
the same sequence of mental processes underlies the whole career
of any individual in any walk of life if only he is not concerned
entirely with manual labor; consequently a full training in the
performance of such sequences must be regarded as forming an
essential part of any education worthy of the name. Moreover the
full appreciation of such processes has a higher value than is
contained in the mental training involved, great though this be,
for it induces an appreciation of intellectual unity and beauty
which plays for the mind that part which the appreciation of
schemes of shape and color plays for the artistic faculties; or,
again, that part which the appreciation of a body of religious
doctrine plays for the ethical aspirations. Now geometry is not
the sole possible basis for inculcating this appreciation. Logic
is an alternative for adults, provided that the individual is
possessed of sufficient wide, though rough, experience on which
to base his reasoning. Geometry is, however, highly desirable in
that the objective bases are so simple and precise that they can
be grasped at an early age, that the amount of training for the
imagination is very large, that the deductive processes are not
beyond the scope of ordinary boys, and finally that it affords a
better basis for exercise in the art of simple and exact
expression than any other possible subject of a school course.

                                               --CARSON, G. W. L.

                         _The Functions of Geometry as a Subject
                         of Education (Tonbridge, 1910), p. 3._


=1842.= It seems to me that the thing that is wanting in the
education of women is not the acquaintance with any facts, but
accurate and scientific habits of thought, and the courage to
think that true which appears unlikely. And for supplying this
want there is a special advantage in geometry, namely that it
does not require study of a physically laborious kind, but rather
that rapid intuition which women certainly possess; so that it is
fit to become a scientific pursuit for them.--CLIFFORD, W. K.

                         _Quoted by Pollock in Clifford’s
                         Lectures and Essays (London, 1901), Vol.
                         1, Introduction, p. 43._


=1843.=

              On the lecture slate
        The circle rounded under female hands
        With flawless demonstration.
                                                 --TENNYSON.

                         _The Princess, II, l. 493._


=1844.= It is plain that that part of geometry which bears upon
strategy does concern us. For in pitching camps, or in occupying
positions, or in closing or extending the lines of an army, and
in all the other manœuvres of an army whether in battle or on
the march, it will make a great difference to a general, whether
he is a geometrician or not.--PLATO.

                         _Republic, Bk. 7, p. 526._


=1845.= Then nothing should be more effectually enacted, than
that the inhabitants of your fair city should learn geometry.
Moreover the science has indirect effects, which are not small.

Of what kind are they? he said.

There are the military advantages of which you spoke, I said; and
in all departments of study, as experience proves, any one who
has studied geometry is infinitely quicker of apprehension.--PLATO.

                         _Republic [Jowett], Bk. 7, p. 527._


=1846.= It is doubtful if we have any other subject that does so
much to bring to the front the danger of carelessness, of
slovenly reasoning, of inaccuracy, and of forgetfulness as this
science of geometry, which has been so polished and perfected as
the centuries have gone on.--SMITH, D. E.

                         _The Teaching of Geometry (Boston,
                         1911), p. 12._


=1847.= The culture of the geometric imagination, tending to
produce precision in remembrance and invention of visible forms
will, therefore, tend directly to increase the appreciation of
works of belles-letters.--HILL, THOMAS.

                         _The Uses of Mathesis; Bibliotheca
                         Sacra, Vol. 32, p. 504._


=1848.=

        Yet may we not entirely overlook
        The pleasures gathered from the rudiments
        Of geometric science. Though advanced
        In these inquiries, with regret I speak,
        No farther than the threshold, there I found
        Both elevation and composed delight:
        With Indian awe and wonder, ignorance pleased
        With its own struggles, did I meditate
        On the relations those abstractions bear
        To Nature’s laws.

               *       *       *       *       *

        More frequently from the same source I drew
        A pleasure quiet and profound, a sense
        Of permanent and universal sway,
        And paramount belief; there, recognized
        A type, for finite natures, of the one
        Supreme Existence, the surpassing life
        Which to the boundaries of space and time,
        Of melancholy space and doleful time,
        Superior and incapable of change,
        Nor touched by welterings of passion--is,
        And hath the name of God. Transcendent peace
        And silence did wait upon these thoughts
        That were a frequent comfort to my youth.

               *       *       *       *       *

                   Mighty is the charm
        Of those abstractions to a mind beset
        With images and haunted by himself,
        And specially delightful unto me
        Was that clear synthesis built up aloft
        So gracefully; even then when it appeared
        Not more than a mere plaything, or a toy
        To sense embodied: not the thing it is
        In verity, an independent world,
        Created out of pure intelligence.
                                               --WORDSWORTH.

                         _The Prelude, Bk. 6._


=1849.=

        ’Tis told by one whom stormy waters threw,
        With fellow-sufferers by the shipwreck spared,
        Upon a desert coast, that having brought
        To land a single volume, saved by chance,
        A treatise of Geometry, he wont,
        Although of food and clothing destitute,
        And beyond common wretchedness depressed,
        To part from company, and take this book
        (Then first a self taught pupil in its truths)
        To spots remote, and draw his diagrams
        With a long staff upon the sand, and thus
        Did oft beguile his sorrow, and almost
        Forget his feeling:
                                               --WORDSWORTH.

                         _The Prelude, Bk. 6._


=1850.= We study art because we receive pleasure from the great
works of the masters, and probably we appreciate them the more
because we have dabbled a little in pigments or in clay. We do
not expect to be composers, or poets, or sculptors, but we wish
to appreciate music and letters and the fine arts, and to derive
pleasure from them and be uplifted by them....

So it is with geometry. We study it because we derive pleasure
from contact with a great and ancient body of learning that has
occupied the attention of master minds during the thousands of
years in which it has been perfected, and we are uplifted by it.
To deny that our pupils derive this pleasure from the study is to
confess ourselves poor teachers, for most pupils do have positive
enjoyment in the pursuit of geometry, in spite of the tradition
that leads them to proclaim a general dislike for all study. This
enjoyment is partly that of the game,--the playing of a game that
can always be won, but that cannot be won too easily. It is
partly that of the aesthetic, the pleasure of symmetry of form,
the delight of fitting things together. But probably it lies
chiefly in the mental uplift that geometry brings, the contact
with absolute truth, and the approach that one makes to the
Infinite. We are not quite sure of any one thing in biology; our
knowledge of geology is relatively very slight, and the economic
laws of society are uncertain to every one except some individual
who attempts to set them forth; but before the world was
fashioned the square on the hypotenuse was equal to the sum of
the squares on the other two sides of a right triangle, and it
will be so after this world is dead; and the inhabitant of Mars,
if he exists, probably knows its truth as we know it. The uplift
of this contact with absolute truth, with truth eternal, gives
pleasure to humanity to a greater or less degree, depending upon
the mental equipment of the particular individual; but it
probably gives an appreciable amount of pleasure to every student
of geometry who has a teacher worthy of the name.--SMITH, D. E.

                         _The Teaching of Geometry (Boston,
                         1911), p. 16._


=1851.= No other person can judge better of either [the merits of
a writer and the merits of his works] than himself; for none have
had access to a closer or more deliberate examination of them. It
is for this reason, that in proportion that the value of a work
is intrinsic, and independent of opinion, the less eagerness will
the author feel to conciliate the suffrages of the public. Hence
that inward satisfaction, so pure and so complete, which the
study of geometry yields. The progress which an individual makes
in this science, the degree of eminence which he attains in it,
all this may be measured with the same rigorous accuracy as the
methods about which his thoughts are employed. It is only when we
entertain some doubts about the justness of our own standard,
that we become anxious to relieve ourselves from our uncertainty,
by comparing it with the standard of another. Now, in all matters
which fall under the cognizance of taste, this standard is
necessarily somewhat variable; depending on a sort of gross
estimate, always a little arbitrary, either in whole or in part;
and liable to continual alteration in its dimensions, from
negligence, temper, or caprice. In consequence of these
circumstances I have no doubt, that if men lived separate from
each other, and could in such a situation occupy themselves about
anything but self-preservation, they would prefer the study of
the exact sciences to the cultivation of the agreeable arts. It
is chiefly on account of others, that a man aims at excellence in
the latter, it is on his own account that he devotes himself to
the former. In a desert island, accordingly, I should think that
a poet could scarcely be vain; whereas a geometrician might still
enjoy the pride of discovery.--D’ALEMBERT.

                         _Essai sur les Gens Lettres; Melages
                         (Amsterdam 1764), t. 1, p. 334._


=1852.= If it were required to determine inclined planes of
varying inclinations of such lengths that a free rolling body
would descend on them in equal times, any one who understands the
mechanical laws involved would admit that this would necessitate
sundry preparations. But in the circle the proper arrangement
takes place of its own accord for an infinite variety of
positions yet with the greatest accuracy in each individual case.
For all chords which meet the vertical diameter whether at its
highest or lowest point, and whatever their inclinations, have
this in common: that the free descent along them takes place in
equal times. I remember, one bright pupil, who, after I had
stated and demonstrated this theorem to him, and he had caught
the full import of it, was moved as by a miracle. And, indeed,
there is just cause for astonishment and wonder when one beholds
such a strange union of manifold things in accordance with such
fruitful rules in so plain and simple an object as the circle.
Moreover, there is no miracle in nature, which because of its
pervading beauty or order, gives greater cause for astonishment,
unless it be, for the reason that its causes are not so clearly
comprehended, marvel being a daughter of ignorance.--KANT.

                         _Der einzig mögliche Beweisgrund zu
                         einer Demonstration des Daseins Gottes;
                         Werke (Hartenstein), Bd. 2, p. 137._


=1853.= These examples [taken from the geometry of the circle]
indicate what a countless number of other such harmonic relations
obtain in the properties of space, many of which are manifested
in the relations of the various classes of curves in higher
geometry, all of which, besides exercising the understanding
through intellectual insight, affect the emotion in a similar or
even greater degree than the occasional beauties of nature.--KANT.

                         _Der einzig mögliche Beweisgrund zu
                         einer Demonstration des Daseins Gottes;
                         Werke (Hartenstein), Bd. 2, p. 138._


=1854.= But neither thirty years, nor thirty centuries, affect
the clearness, or the charm, of Geometrical truths. Such a
theorem as “the square of the hypotenuse of a right-angled
triangle is equal to the sum of the squares of the sides” is as
dazzlingly beautiful now as it was in the day when Pythagoras
first discovered it, and celebrated its advent, it is said, by
sacrificing a hecatomb of oxen--a method of doing honor to
Science that has always seemed to me _slightly_ exaggerated and
uncalled-for. One can imagine oneself, even in these degenerate
days, marking the epoch of some brilliant scientific discovery by
inviting a convivial friend or two, to join one in a beefsteak
and a bottle of wine. But a _hecatomb_ of oxen! It would produce
a quite inconvenient supply of beef.--DODGSON, C. L.

                         _A New Theory of Parallels (London,
                         1895), Introduction, p. 16._


=1855.= After Pythagoras discovered his fundamental theorem he
sacrificed a hecatomb of oxen. Since that time all dunces[10]
[Ochsen] tremble whenever a new truth is discovered.--BOERNE.

                         _Quoted in Moszkowski: Die unsterbliche
                         Kiste (Berlin, 1908), p. 18._

    [10]  In the German vernacular a dunce or blockhead
          is called an ox.


=1856.=

              _Vom Pythagorieschen Lehrsatz._

          Die Wahrheit, sie besteht in Ewigkeit,
        Wenn erst die blöde Welt ihr Licht erkannt:
        Der Lehrsatz, nach Pythagoras benannt,
        Gilt heute, wie er galt in seiner Zeit.

          Ein Opfer hat Pythagoras geweiht
        Den Göttern, die den Lichtstrahl ihm gesandt;
        Es thaten kund, geschlachtet und verbrannt,
        Ein hundert Ochsen seine Dankbarkeit.

          Die Ochsen seit den Tage, wenn sie wittern,
        Dass eine neue Wahrheit sich enthülle,
        Erheben ein unmenschliches Gebrülle;

          Pythagoras erfüllt sie mit Entsetzen;
        Und machtlos, sich dem Licht zu wiedersetzen,
        Verschiessen sie die Augen und erzittern.
                                   --CHAMISSO, ADELBERT VON.

                         _Gedichte, 1835 (Haushenbusch), (Berlin,
                         1889), p. 302._

          Truth lasts throughout eternity,
        When once the stupid world its light discerns:
        The theorem, coupled with Pythagoras’ name,
        Holds true today, as’t did in olden times.

          A splendid sacrifice Pythagoras brought
        The gods, who blessed him with this ray divine;
        A great burnt offering of a hundred kine,
        Proclaimed afar the sage’s gratitude.

          Now since that day, all cattle [blockheads] when they
            scent
        New truth about to see the light of day,
        In frightful bellowings manifest their dismay;

          Pythagoras fills them all with terror;
        And powerless to shut out light by error,
        In sheer despair they shut their eyes and tremble.


=1857.= To the question “Which is the signally most beautiful of
geometrical truths?” Frankland replies: “One star excels another
in brightness, but the very sun will be, by common consent, a
property of the circle [Euclid, Book 3, Proposition 31] selected
for particular mention by Dante, that greatest of all exponents
of the beautiful.”--FRANKLAND, W. B.

                         _The Story of Euclid (London, 1902), p.
                         70._


=1858.=

                                       As one
        Who vers’d in geometric lore, would fain
        Measure the circle; and, though pondering long
        And deeply, that beginning, which he needs,
        Finds not; e’en such was I, intent to scan
        The novel wonder, and trace out the form,
        How to the circle fitted, and therein
        How plac’d: but the flight was not for my wing;
                                                    --DANTE.

                         _Paradise [Carey] Canto 33, lines
                         122-129._


=1859.= If geometry were as much opposed to our passions and
present interests as is ethics, we should contest it and violate
it but little less, notwithstanding all the demonstrations of
Euclid and of Archimedes, which you would call dreams and believe
full of paralogisms; and Joseph Scaliger, Hobbes, and others, who
have written against Euclid and Archimedes, would not find
themselves in such a small company as at present.--LEIBNITZ.

                         _New Essays concerning Human
                         Understanding [Langley], Bk. 1, chap. 2,
                         sect. 12._


=1860.= I have no fault to find with those who teach geometry.
That science is the only one which has not produced sects; it is
founded on analysis and on synthesis and on the calculus; it does
not occupy itself with probable truth; moreover it has the same
method in every country.--FREDERICK THE GREAT.

                         _Oeuvres (Decker), t. 7, p. 100._


=1861.= There are, undoubtedly, the most ample reasons for
stating both the principles and theorems [of geometry] in their
general form, .... But, that an unpractised learner, even in
making use of one theorem to demonstrate another, reasons rather
from particular to particular than from the general proposition,
is manifest from the difficulty he finds in applying a theorem to
a case in which the configuration of the diagram is extremely
unlike that of the diagram by which the original theorem was
demonstrated. A difficulty which, except in cases of unusual
mental powers, long practice can alone remove, and removes
chiefly by rendering us familiar with all the configurations
consistent with the general conditions of the theorem.--MILL, J. S.

                         _System of Logic, Bk. 2, chap. 3, sect.
                         3._


=1862.= The reason why I impute any defect to geometry, is,
because its original and fundamental principles are deriv’d
merely from appearances; and it may perhaps be imagin’d, that
this defect must always attend it, and keep it from ever reaching
a greater exactness in the comparison of objects or ideas, than
what our eye or imagination alone is able to attain. I own that
this defect so far attends it, as to keep it from ever aspiring
to a full certainty. But since these fundamental principles
depend on the easiest and least deceitful appearances, they
bestow on their consequences a degree of exactness, of which
these consequences are singly incapable.--HUME, D.

                         _A Treatise of Human Nature, Part 3,
                         sect. 1._


=1863.= I have already observed, that geometry, or the art, by
which we fix the proportions of figures, tho’ it much excels both
in universality and exactness, the loose judgments of the senses
and imagination; yet never attains a perfect precision and
exactness. Its first principles are still drawn from the general
appearance of the objects; and that appearance can never afford
us any security, when we examine the prodigious minuteness of
which nature is susceptible....

There remain, therefore, algebra and arithmetic as the only
sciences, in which we can carry on a chain of reasoning to any
degree of intricacy, and yet preserve a perfect exactness and
certainty.--HUME, D.

                         _A Treatise of Human Nature, Part 3,
                         sect. 1._


=1864.= All geometrical reasoning is, in the last resort,
circular: if we start by assuming points, they can only be
defined by the lines or planes which relate them; and if we start
by assuming lines or planes, they can only be defined by the
points through which they pass.--RUSSELL, BERTRAND.

                         _Foundations of Geometry (Cambridge,
                         1897), p. 120._


=1865.= The description of right lines and circles, upon which
Geometry is founded, belongs to Mechanics. Geometry does not
teach us to draw these lines, but requires them to be drawn....
it requires that the learner should first be taught to describe
these accurately, before he enters upon Geometry; then it shows
how by these operations problems may be solved. To describe right
lines and circles are problems, but not geometrical problems. The
solution of these problems is required from Mechanics; by
Geometry the use of them, when solved, is shown.... Therefore
Geometry is founded in mechanical practice, and is nothing but
that part of universal Mechanics which accurately proposes and
demonstrates the art of measuring. But since the manual arts are
chiefly conversant in the moving of bodies, it comes to pass
that Geometry is commonly referred to their magnitudes, and
Mechanics to their motion.--NEWTON.

                         _Philosophiae Naturalis Principia
                         Mathematica, Praefat._


=1866.= We must, then, admit ... that there is an independent
science of geometry just as there is an independent science of
physics, and that either of these may be treated by mathematical
methods. Thus geometry becomes the simplest of the natural
sciences, and its axioms are of the nature of physical laws, to
be tested by experience and to be regarded as true only within
the limits of error of observation--BÔCHER, MAXIME.

                         _Bulletin American Mathematical Society,
                         Vol. 2 (1904), p. 124._


=1867.= Geometry is not an experimental science; experience forms
merely the occasion for our reflecting upon the geometrical ideas
which pre-exist in us. But the occasion is necessary, if it did
not exist we should not reflect, and if our experiences were
different, doubtless our reflections would also be different.
Space is not a form of sensibility; it is an instrument which
serves us not to represent things to ourselves, but to reason
upon things.--POINCARÉ, H.

                         _On the Foundations of Geometry; Monist,
                         Vol. 9 (1898-1899), p. 41._


=1868.= It has been said that geometry is an instrument. The
comparison may be admitted, provided it is granted at the same
time that this instrument, like Proteus in the fable, ought
constantly to change its form.--ARAGO.

                         _Oeuvres, t. 2 (1854), p. 694._


=1869.= It is essential that the treatment [of geometry] should
be rid of everything superfluous, for the superfluous is an
obstacle to the acquisition of knowledge; it should select
everything that embraces the subject and brings it to a focus,
for this is of the highest service to science; it must have great
regard both to clearness and to conciseness, for their opposites
trouble our understanding; it must aim to generalize its
theorems, for the division of knowledge into small elements
renders it difficult of comprehension.--PROCLUS.

                         _Quoted in D. E. Smith: The Teaching of
                         Geometry (Boston, 1911), p. 71._


=1870.= Many are acquainted with mathematics, but mathesis few
know. For it is one thing to know a number of propositions and to
make some obvious deductions from them, by accident rather than
by any sure method of procedure, another thing to know clearly
the nature and character of the science itself, to penetrate into
its inmost recesses, and to be instructed by its universal
principles, by which facility in working out countless problems
and their proofs is secured. For as the majority of artists, by
copying the same model again and again, gain certain technical
skill in painting, but no other knowledge of the art of painting
than what their eyes suggest, so many, having read the books of
Euclid and other geometricians, are wont to devise, in imitation
of them and to prove some propositions, but the most profound
method of solving more difficult demonstrations and problems they
are utterly ignorant of.--LAFAILLE, J. C.

                         _Theoremata de Centro Gravitatis
                         (Anvers, 1632), Praefat._


=1871.= The elements of plane geometry should precede algebra for
every reason known to sound educational theory. It is more
fundamental, more concrete, and it deals with things and their
relations rather than with symbols.--BUTLER, N. M.

                         _The Meaning of Education etc. (New
                         York, 1905), p. 171._


=1872.= The reason why geometry is not so difficult as algebra,
is to be found in the less general nature of the symbols
employed. In algebra a general proposition respecting numbers is
to be proved. Letters are taken which may represent any of the
numbers in question, and the course of the demonstration, far
from making use of a particular case, does not even allow that
any reasoning, however general in its nature, is conclusive,
unless the symbols are as general as the arguments.... In
geometry on the contrary, at least in the elementary parts, any
proposition may be safely demonstrated on reasonings on any one
particular example.... It also affords some facility that the
results of elementary geometry are in many cases sufficiently
evident of themselves to the eye; for instance, that two sides of
a triangle are greater than the third, whereas in algebra many
rudimentary propositions derive no evidence from the senses; for
example, that a³−b³ is always divisible without a remainder by
a−b.--DE MORGAN, A.

                         _On the Study and Difficulties of
                         Mathematics (Chicago, 1902), chap. 13._


=1873.= The principal characteristics of the ancient geometry
are:--

(1) A wonderful clearness and definiteness of its concepts and an
almost perfect logical rigour of its conclusions.

(2) A complete want of general principles and methods.... In the
demonstration of a theorem, there were, for the ancient
geometers, as many different cases requiring separate proof as
there were different positions of the lines. The greatest
geometers considered it necessary to treat all possible cases
independently of each other, and to prove each with equal
fulness. To devise methods by which all the various cases could
all be disposed of with one stroke, was beyond the power of the
ancients.--CAJORI, F.

                         _History of Mathematics (New York,
                         1897), p. 62._


=1874.= It has been observed that the ancient geometers made use
of a kind of analysis, which they employed in the solution of
problems, although they begrudged to posterity the knowledge of
it.--DESCARTES.

                         _Rules for the Direction of the Mind;
                         The Philosophy of Descartes [Torrey]
                         (New York, 1892), p. 68._


=1875.= The ancients studied geometry with reference to the
_bodies_ under notice, or specially: the moderns study it with
reference to the _phenomena_ to be considered, or generally. The
ancients extracted all they could out of one line or surface,
before passing to another; and each inquiry gave little or no
assistance in the next. The moderns, since Descartes, employ
themselves on questions which relate to any figure whatever. They
abstract, to treat by itself, every question relating to the
same geometrical phenomenon, in whatever bodies it may be
considered. Geometers can thus rise to the study of new
geometrical conceptions, which, applied to the curves
investigated by the ancients, have brought out new properties
never suspected by them.--COMTE.

                         _Positive Philosophy [Martineau], Bk. 1,
                         chap. 3._


=1876.= It is astonishing that this subject [projective geometry]
should be so generally ignored, for mathematics offers nothing
more attractive. It possesses the concreteness of the ancient
geometry without the tedious particularity, and the power of the
analytical geometry without the reckoning, and by the beauty of
its ideas and methods illustrates the esthetic generality which
is the charm of higher mathematics, but which the elementary
mathematics generally lacks.

                         _Report of the Committee of Ten on
                         Secondary School Studies (Chicago,
                         1894), p. 116._


=1877.= There exist a small number of very simple fundamental
relations which contain the scheme, according to which the
remaining mass of theorems [in projective geometry] permit of
orderly and easy development.

By a proper appropriation of a few fundamental relations one
becomes master of the whole subject; order takes the place of
chaos, one beholds how all parts fit naturally into each other,
and arrange themselves serially in the most beautiful order, and
how related parts combine into well-defined groups. In this
manner one arrives, as it were, at the elements, which nature
herself employs in order to endow figures with numberless
properties with the utmost economy and simplicity.--STEINER, J.

                         _Werke, Bd. 1 (1881), p. 233._


=1878.= Euclid once said to his king Ptolemy, who, as is easily
understood, found the painstaking study of the “Elements”
repellant, “There exists no royal road to mathematics.” But we
may add: Modern geometry is a royal road. It has disclosed “the
organism, by means of which the most heterogeneous phenomena in
the world of space are united one with another” (Steiner), and
has, as we may say without exaggeration, almost attained to the
scientific ideal.--HANKEL, H.

                         _Die Entwickelung der Mathematik in den
                         letzten Jahrhunderten (Tübingen, 1869)._


=1879.= The two mathematically fundamental things in projective
geometry are anharmonic ratio, and the quadrilateral construction.
Everything else follows mathematically from these two.

                                             --RUSSELL, BERTRAND.

                         _Foundations of Geometry (Cambridge,
                         1897), p. 122._


=1880.= ... Projective Geometry: a boundless domain of countless
fields where reals and imaginaries, finites and infinites, enter
on equal terms, where the spirit delights in the artistic balance
and symmetric interplay of a kind of conceptual and logical
counterpoint,--an enchanted realm where thought is double and
flows throughout in parallel streams.--KEYSER, C. J.

                         _Lectures on Science, Philosophy and
                         Arts (New York, 1908), p. 2._


=1881.= The ancients, in the early days of the science,
made great use of the graphic method, even in the form of
construction; as when Aristarchus of Samos estimated the distance
of the sun and moon from the earth on a triangle constructed as
nearly as possible in resemblance to the right-angled triangle
formed by the three bodies at the instant when the moon is in
quadrature, and when therefore an observation of the angle at the
earth would define the triangle. Archimedes himself, though he
was the first to introduce calculated determinations into
geometry, frequently used the same means. The introduction of
trigonometry lessened the practice; but did not abolish it. The
Greeks and Arabians employed it still for a great number of
investigations for which we now consider the use of the Calculus
indispensable.--COMTE, A.

                         _Positive Philosophy [Martineau], Bk. 1,
                         chap. 3._


=1882.= A mathematical problem may usually be attacked by what is
termed in military parlance the method of “systematic approach;”
that is to say, its solution may be gradually felt for, even
though the successive steps leading to that solution cannot be
clearly foreseen. But a Descriptive Geometry problem must be seen
through and through before it can be attempted. The entire scope
of its conditions, as well as each step toward its solution, must
be grasped by the imagination. It must be “taken by assault.”

                                                  --CLARKE, G. S.

                         _Quoted in W. S. Hall: Descriptive
                         Geometry (New York, 1902), chap. 1._


=1883.= The grand use [of Descriptive Geometry] is in its
application to the industrial arts;--its few abstract problems,
capable of invariable solution, relating essentially to the
contacts and intersections of surfaces; so that all the
geometrical questions which may arise in any of the various arts
of construction,--as stone-cutting, carpentry, perspective,
dialing, fortification, etc.,--can always be treated as simple
individual cases of a single theory, the solution being certainly
obtainable through the particular circumstances of each case.
This creation must be very important in the eyes of philosophers
who think that all human achievement, thus far, is only a first
step toward a philosophical renovation of the labours of mankind;
towards that precision and logical character which can alone
ensure the future progression of all arts.... Of Descriptive
Geometry, it may further be said that it usefully exercises the
student’s faculty of Imagination,--of conceiving of complicated
geometrical combinations in space; and that, while it belongs to
the geometry of the ancients by the character of its solutions,
it approaches to the geometry of the moderns by the nature of the
questions which compose it.--COMTE, A.

                         _Positive Philosophy [Martineau], Bk. 1,
                         chap. 3._


=1884.= There is perhaps nothing which so occupies, as it were,
the middle position of mathematics, as trigonometry.

                                                 --HERBART, J. F.

                         _Idee eines ABC der Anschauung; Werke
                         (Kehrbach) (Langensalza, 1890), Bd. 1,
                         p. 174._


=1885.= Trigonometry contains the science of continually
undulating magnitude: meaning magnitude which becomes alternately
greater and less, without any termination to succession of
increase and decrease.... All trigonometric functions are not
undulating: but it may be stated that in common algebra nothing
but infinite series undulate: in trigonometry nothing but
infinite series do not undulate.--DE MORGAN, A.

                         _Trigonometry and Double Algebra
                         (London, 1849), Bk. 1, chap. 1._


=1886.= Sin²φ is odious to me, even though Laplace made use of it;
should it be feared that sinφ² might become ambiguous, which would
perhaps never occur, or at most very rarely when speaking of sin
(φ²), well then, let us write (sinφ)², but not sin²φ, which by
analogy should signify sin(sinφ).--GAUSS.

                         _Gauss-Schumacher Briefwechsel, Bd. 3,
                         p. 292; Bd. 4, p. 63._


=1887.= Perhaps to the student there is no part of elementary
mathematics so repulsive as is spherical trigonometry.--TAIT, P. G.

                         _Encyclopedia Britannica, 9th Edition;
                         Article “Quaternions.”_


=1888.= “Napier’s Rule of circular parts” is perhaps the happiest
example of artificial memory that is known.--CAJORI, F.

                         _History of Mathematics (New York,
                         1897), p. 165._


=1889.= The analytical equations, unknown to the ancients, which
Descartes first introduced into the study of curves and surfaces,
are not restricted to the properties of figures, and to those
properties which are the object of rational mechanics; they apply
to all phenomena in general. There cannot be a language more
universal and more simple, more free from errors and obscurities,
that is to say, better adapted to express the invariable
relations of nature.--FOURIER.

                         _Théorie Analytique de la Chaleur,
                         Discours Préliminaire._


=1890.= It is impossible not to feel stirred at the thought of
the emotions of men at certain historic moments of adventure and
discovery--Columbus when he first saw the Western shore, Pizarro
when he stared at the Pacific Ocean, Franklin when the electric
spark came from the string of his kite, Galileo when he first
turned his telescope to the heavens. Such moments are also
granted to students in the abstract regions of thought, and high
among them must be placed the morning when Descartes lay in bed
and invented the method of co-ordinate geometry.--WHITEHEAD, A. N.

                         _An Introduction to Mathematics (New
                         York, 1911), p. 122._


=1891.= It is often said that an equation contains only what has
been put into it. It is easy to reply that the new form under
which things are found often constitutes by itself an important
discovery. But there is something more: analysis, by the simple
play of its symbols, may suggest generalizations far beyond the
original limits.--PICARD, E.

                         _Bulletin American Mathematical Society,
                         Vol. 2 (1905), p. 409._


=1892.= It is not the Simplicity of the Equation, but the
Easiness of the Description, which is to determine the Choice of
our Lines for the Constructions of Problems. For the Equation
that expresses a Parabola is more simple than that that expresses
the Circle, and yet the Circle, by its more simple Construction,
is admitted before it.--NEWTON.

                         _The Linear Constructions of Equations;
                         Universal Arithmetic (London, 1769),
                         Vol. 2, p. 468._


=1893.= The pursuit of mathematics unfolds its formative power
completely only with the transition from the elementary subjects
to analytical geometry. Unquestionably the simplest geometry and
algebra already accustom the mind to sharp quantitative thinking,
as also to assume as true only axioms and what has been proven.
But the representation of functions by curves or surfaces reveals
a new world of concepts and teaches the use of one of the most
fruitful methods, which the human mind ever employed to increase
its own effectiveness. What the discovery of this method by Vieta
and Descartes brought to humanity, that it brings today to
every one who is in any measure endowed for such things: a
life-epoch-making beam of light [Lichtblick]. This method has its
roots in the farthest depths of human cognition and so has an
entirely different significance, than the most ingenious artifice
which serves a special purpose.--BOIS-REYMOND, EMIL DU.

                         _Reden, Bd. 1 (Leipzig, 1885), p. 287._


=1894.=

             _Song of the Screw._

        A moving form or rigid mass,
          Under whate’er conditions
        Along successive screws must pass
          Between each two positions.
        It turns around and slides along--
        This is the burden of my song.

        The pitch of screw, if multiplied
          By angle of rotation,
        Will give the distance it must glide
          In motion of translation.
        Infinite pitch means pure translation,
        And zero pitch means pure rotation.

        Two motions on two given screws,
          With amplitudes at pleasure,
        Into a third screw-motion fuse,
          Whose amplitude we measure
        By parallelogram construction
        (A very obvious deduction).

        Its axis cuts the nodal line
          Which to both screws is normal,
        And generates a form divine,
          Whose name, in language formal,
        Is “surface-ruled of third degree.”
        Cylindroid is the name for me.

        Rotation round a given line
          Is like a force along,
        If to say couple you decline,
          You’re clearly in the wrong;--
        ’Tis obvious, upon reflection,
        A line is not a mere direction.

        So couples with translations too
          In all respects agree;
        And thus there centres in the screw
          A wondrous harmony
        Of Kinematics and of Statics,--
        The sweetest thing in mathematics.

        The forces on one given screw,
          With motion on a second,
        In general some work will do,
          Whose magnitude is reckoned
        By angle, force, and what we call
        The coefficient virtual.

        Rotation now to force convert,
          And force into rotation;
        Unchanged the work, we can assert,
          In spite of transformation.
        And if two screws no work can claim,
        Reciprocal will be their name.

        Five numbers will a screw define,
          A screwing motion, six;
        For four will give the axial line,
          One more the pitch will fix;
        And hence we always can contrive
        One screw reciprocal to five.

        Screws--two, three, four or five, combined
          (No question here of six),
        Yield other screws which are confined
          Within one screw complex.
        Thus we obtain the clearest notion
        Of freedom and constraint of motion.

        In complex III, three several screws
          At every point you find,
        Or if you one direction choose,
          One screw is to your mind;
        And complexes of order III.
        Their own reciprocals may be.

        In IV, wherever you arrive,
          You find of screws a cone,
        On every line of complex V.
          There is precisely one;
        At each point of this complex rich,
        A plane of screws have given pitch.

        But time would fail me to discourse
          Of Order and Degree;
        Of Impulse, Energy and Force,
          And Reciprocity.
        All these and more, for motions small,
        Have been discussed by Dr. Ball.
                                                --ANONYMOUS.



  CHAPTER XIX

  THE CALCULUS AND ALLIED TOPICS


=1901.= It may be said that the conceptions of differential
quotient and integral, which in their origin certainly go back to
Archimedes, were introduced into science by the investigations of
Kepler, Descartes, Cavalieri, Fermat and Wallis.... The capital
discovery that differentiation and integration are _inverse_
operations belongs to Newton and Leibnitz.--LIE, SOPHUS.

                         _Leipziger Berichte, 47 (1895),
                         Math.-phys. Classe, p. 53._


=1902.= It appears that Fermat, the true inventor of the
differential calculus, considered that calculus as derived from
the calculus of finite differences by neglecting infinitesimals
of higher orders as compared with those of a lower order....
Newton, through his method of fluxions, has since rendered the
calculus more analytical, he also simplified and generalized the
method by the invention of his binomial theorem. Leibnitz has
enriched the differential calculus by a very happy notation.

                                                       --LAPLACE.

                         _Lés Intégrales Définies, etc.; Oeuvres,
                         t. 12 (Paris, 1898), p. 359._


=1903.= Professor Peacock’s Algebra, and Mr. Whewell’s Doctrine
of Limits should be studied by every one who desires to
comprehend the evidence of mathematical truths, and the meaning
of the obscure processes of the calculus; while, even after
mastering these treatises, the student will have much to learn on
the subject from M. Comte, of whose admirable work one of the
most admirable portions is that in which he may truly be said to
have created the philosophy of the higher mathematics.

                                                    --MILL, J. S.

                         _System of Logic, Bk. 3, chap. 24, sect.
                         6._


=1904.= If we must confine ourselves to one system of notation
then there can be no doubt that that which was invented by
Leibnitz is better fitted for most of the purposes to which the
infinitesimal calculus is applied than that of fluxions, and for
some (such as the calculus of variations) it is indeed almost
essential.--BALL, W. W. R.

                         _History of Mathematics (London, 1901),
                         p. 371._


=1905.= The difference between the method of infinitesimals and
that of limits (when exclusively adopted) is, that in the latter
it is usual to retain evanescent quantities of higher orders
until the end of the calculation and then neglect them. On the
other hand, such quantities are neglected from the commencement
in the infinitesimal method, from the conviction that they cannot
affect the final result, as they must disappear when we proceed
to the limit.--WILLIAMSON, B.

                         _Encyclopedia Britannica, 9th Edition;
                         Article “Infinitesimal Calculus,” sect.
                         14._


=1906.= When we have grasped the spirit of the infinitesimal
method, and have verified the exactness of its results either by
the geometrical method of prime and ultimate ratios, or by the
analytical method of derived functions, we may employ infinitely
small quantities as a sure and valuable means of shortening and
simplifying our proofs.--LAGRANGE.

                         _Méchanique Analytique, Preface;
                         Oeuvres, t. 2 (Paris, 1888), p. 14._


=1907.= The essential merit, the sublimity, of the infinitesimal
method lies in the fact that it is as easily performed as the
simplest method of approximation, and that it is as accurate as
the results of an ordinary calculation. This advantage would be
lost, or at least greatly impaired, if, under the pretense of
securing greater accuracy throughout the whole process, we were
to substitute for the simpler method given by Leibnitz, one less
convenient and less in harmony with the probable course of
natural events....

The objections which have been raised against the infinitesimal
method are based on the false supposition that the errors due to
neglecting infinitely small quantities during the actual
calculation will continue to exist in the result of the
calculation.--CARNOT, L.

                         _Réflections sur la Métaphysique du
                         Calcul Infinitésimal (Paris, 1813), p.
                         215._


=1908.= A limiting ratio is neither more nor less difficult to
define than an infinitely small quantity.--CARNOT, L.

                         _Réflections sur la Métaphysique du
                         Calcul Infinitésimal (Paris, 1813), p.
                         210._


=1909.= A limit is a peculiar and fundamental conception, the use
of which in proving the propositions of Higher Geometry cannot
be superseded by any combination of other hypotheses and
definitions. The axiom just noted that what is true up to the
limit is true at the limit, is involved in the very conception of
a limit: and this principle, with its consequences, leads to all
the results which form the subject of the higher mathematics,
whether proved by the consideration of evanescent triangles, by
the processes of the Differential Calculus, or in any other way.

                                                    --WHEWELL, W.

                         _The Philosophy of the Inductive
                         Sciences, Part 1, bk. 2, chap. 12, sect.
                         1, (London, 1858)._


=1910.= The differential calculus has all the exactitude of other
algebraic operations.--LAPLACE.

                         _Théorie Analytique des Probabilités,
                         Introduction; Oeuvres, t. 7 (Paris,
                         1886), p. 37._


=1911.= The method of fluxions is probably one of the greatest,
most subtle, and sublime discoveries of any age: it opens a new
world to our view, and extends our knowledge, as it were, to
infinity; carrying us beyond the bounds that seemed to have been
prescribed to the human mind, at least infinitely beyond those to
which the ancient geometry was confined.--HUTTON, CHARLES.

                         _A Philosophical and Mathematical
                         Dictionary (London, 1815), Vol. 1, p.
                         525._


=1912.= The states and conditions of matter, as they occur in
nature, are in a state of perpetual flux, and these qualities
may be effectively studied by the Newtonian method (Methodus
fluxionem) whenever they can be referred to number or subjected
to measurement (real or imaginary). By the aid of Newton’s
calculus the mode of action of natural changes from moment to
moment can be portrayed as faithfully as these words represent
the thoughts at present in my mind. From this, the law which
controls the whole process can be determined with unmistakable
certainty by pure calculation.--MELLOR, J. W.

                         _Higher Mathematics for Students of
                         Chemistry and Physics (London, 1902),
                         Prologue._


=1913.= The calculus is the greatest aid we have to the
appreciation of physical truth in the broadest sense of the word.

                                                  --OSGOOD, W. F.

                         _Bulletin American Mathematical Society,
                         Vol. 13 (1907), p. 467._


=1914.= [Infinitesimal] analysis is the most powerful weapon of
thought yet devised by the wit of man.--SMITH, W. B.

                         _Infinitesimal Analysis (New York,
                         1898), Preface, p. vii._


=1915.= The method of Fluxions is the general key by help whereof
the modern mathematicians unlock the secrets of Geometry, and
consequently of Nature. And, as it is that which hath enabled
them so remarkably to outgo the ancients in discovering theorems
and solving problems, the exercise and application thereof is
become the main if not sole employment of all those who in this
age pass for profound geometers.--BERKELEY, GEORGE.

                         _The Analyst, sect. 3._


=1916.= I have at last become fully satisfied that the language
and idea of infinitesimals should be used in the most elementary
instruction--under all safeguards of course.--DE MORGAN, A.

                         _Graves’ Life of W. R. Hamilton (New
                         York, 1882-1889), Vol. 3, p. 479._


=1917.= Pupils should be taught how to differentiate and how to
integrate simple algebraic expressions before we attempt to
teach them geometry and these other complicated things. The
dreadful fear of the symbols is entirely broken down in those
cases where at the beginning the teaching of the calculus is
adopted. Then after the pupil has mastered those symbols you may
begin geometry or anything you please. I would also abolish out
of the school that thing called geometrical conics. There is a
great deal of superstition about conic sections. The student
should be taught the symbols of the calculus and the simplest use
of these symbols at the earliest age, instead of these being left
over until he has gone to the College or University.

                                                --THOMPSON, S. P.

                         _Perry’s Teaching of Mathematics
                         (London, 1902), p. 49._


=1918.= Every one versed in the matter will agree that even the
elements of a scientific study of nature can be understood only by
those who have a knowledge of at least the elements of the
differential and integral calculus, as well as of analytical
geometry--i.e. the so-called lower part of the higher mathematics....
We should raise the question, whether sufficient time could not be
reserved in the curricula of at least the science high schools
[Realanstalten] to make room for these subjects....

The first consideration would be to entirely relieve from the
mathematical requirements of the university [Hochschule]
certain classes of students who can get along without extended
mathematical knowledge, or to make the necessary mathematical
knowledge accessible to them in a manner which, for various
reasons, has not yet been adopted by the university. Among such
students I would count architects, also the chemists and in
general the students of the so-called descriptive natural
sciences. I am moreover of the opinion--and this has been for
long a favorite idea of mine--, that it would be very useful to
medical students to acquire such mathematical knowledge as
is indicated by the above described modest limits; for it
seems impossible to understand far-reaching physiological
investigations, if one is terrified as soon as a differential or
integration symbol appears.--KLEIN, F.

                         _Jahresbericht der Deutschen
                         Mathematiker Vereinigung, Bd. 2 (1902),
                         p. 131._


=1919.= Common integration is only the _memory of
differentiation_ ... the different artifices by which integration
is effected, are changes, not from the known to the unknown, but
from forms in which memory will not serve us to those in which it
will.--DE MORGAN, A.

                         _Transactions Cambridge Philosophical
                         Society, Vol. 8 (1844), p. 188._


=1920.= Given for one instant an intelligence which could
comprehend all the forces by which nature is animated and the
respective positions of the beings which compose it, if moreover
this intelligence were vast enough to submit these data to
analysis, it would embrace in the same formula both the movements
of the largest bodies in the universe and those of the lightest
atom: to it nothing would be uncertain, and the future as the
past would be present to its eyes. The human mind offers a feeble
outline of that intelligence, in the perfection which it has
given to astronomy. Its discoveries in mechanics and in geometry,
joined to that of universal gravity, have enabled it to
comprehend in the same analytical expressions the past and future
states of the world system.--LAPLACE.

                         _Théorie Analytique des Probabilités,
                         Introduction; Oeuvres, t. 7 (Paris,
                         1886), p. 6._


=1921.= There is perhaps the same relation between the action of
natural selection during one generation and the accumulated
result of a hundred thousand generations, that there exists
between differential and integral. How seldom are we able to
follow completely this latter relation although we subject it to
calculation. Do we on that account doubt the correctness of our
integrations?--BOIS-REYMOND, EMIL DU.

                         _Reden, Bd. 1 (Leipzig, 1885), p. 228._


=1922.= It seems to be expected of every pilgrim up the slopes of
the mathematical Parnassus, that he will at some point or other
of his journey sit down and invent a definite integral or two
towards the increase of the common stock.--SYLVESTER, J. J.

                         _Notes to the Meditation on Poncelet’s
                         Theorem; Mathematical Papers, Vol. 2, p.
                         214._


=1923.= The experimental verification of a theory concerning
any natural phenomenon generally rests on the result of an
integration.--MELLOR, J. W.

                         _Higher Mathematics for Students of
                         Chemistry and Physics (New York, 1902),
                         p. 150._


=1924.= Among all the mathematical disciplines the theory of
differential equations is the most important.... It furnishes the
explanation of all those elementary manifestations of nature
which involve time....--LIE, SOPHUS.

                         _Leipziger Berichte, 47 (1895);
                         Math.-phys. Classe, p. 262._


=1925.= If the mathematical expression of our ideas leads to
equations which cannot be integrated, the working hypothesis will
either have to be verified some other way, or else relegated to
the great repository of unverified speculations.--MELLOR, J. W.

                         _Higher Mathematics for Students of
                         Chemistry and Physics (New York, 1902),
                         p. 157._


=1926.= It is well known that the central problem of the whole of
modern mathematics is the study of the transcendental functions
defined by differential equations.--KLEIN, F.

                         _Lectures on Mathematics (New York,
                         1911), p. 8._


=1927.= Every one knows what a curve is, until he has studied
enough mathematics to become confused through the countless
number of possible exceptions.... A curve is the totality of
points, whose co-ordinates are functions of a parameter which may
be differentiated as often as may be required.--KLEIN, F.

                         _Elementar Mathematik vom höheren
                         Standpunkte aus. (Leipzig. 1909) Vol. 2,
                         p. 354._


=1928.= Fourier’s theorem is not only one of the most beautiful
results of modern analysis, but it may be said to furnish an
indispensable instrument in the treatment of nearly every
recondite question in modern physics. To mention only sonorous
vibrations, the propagation of electric signals along telegraph
wires, and the conduction of heat by the earth’s crust, as
subjects in their generality intractable without it, is to give
but a feeble idea of its importance.--THOMSON AND TAIT.

                         _Elements of Natural Philosophy, chap.
                         1._


=1929.= The principal advantage arising from the use of
hyperbolic functions is that they bring to light some curious
analogies between the integrals of certain irrational functions.

                                                  --BYERLY, W. E.

                         _Integral Calculus (Boston, 1890), p.
                         30._


=1930.= Hyperbolic functions are extremely useful in every branch
of pure physics and in the applications of physics whether to
observational and experimental sciences or to technology. Thus
whenever an entity (such as light, velocity, electricity, or
radio-activity) is subject to gradual absorption or extinction,
the decay is represented by some form of hyperbolic functions.
Mercator’s projection is likewise computed by hyperbolic
functions. Whenever mechanical strains are regarded great enough
to be measured they are most simply expressed in terms of
hyperbolic functions. Hence geological deformations invariably
lead to such expressions....--WALCOTT, C. D.

                         _Smithsonian Mathematical Tables,
                         Hyperbolic Functions (Washington, 1909),
                         Advertisement._


=1931.= Geometry may sometimes appear to take the lead over
analysis, but in fact precedes it only as a servant goes before
his master to clear the path and light him on the way. The
interval between the two is as wide as between empiricism and
science, as between the understanding and the reason, or as
between the finite and the infinite.--SYLVESTER, J. J.

                         _Philosophic Magazine, Vol. 31 (1866),
                         p. 521._


=1932.= Nature herself exhibits to us measurable and observable
quantities in definite mathematical dependence; the conception of
a function is suggested by all the processes of nature where we
observe natural phenomena varying according to distance or to
time. Nearly all the “known” functions have presented themselves
in the attempt to solve geometrical, mechanical, or physical
problems.--MERZ, J. T.

                         _A History of European Thought in the
                         Nineteenth Century (Edinburgh and
                         London, 1903), p. 696._


=1933.= That flower of modern mathematical thought--the notion of
a function.--MCCORMACK, THOMAS J.

                         _On the Nature of Scientific Law and
                         Scientific Explanation, Monist, Vol. 10
                         (1899-1900), p. 555._


=1934.=

        Fuchs. Ich bin von alledem so consterniert,
               Als würde mir ein Kreis im Kopfe quadriert.

        Meph.  Nachher vor alien andern Sachen
               Müsst ihe euch an die Funktionen-Theorie machen.
               Da seht, dass ihr tiefsinnig fasst,
               Was sich zu integrieren nicht passt.
               An Theoremen wird’s euch nicht fehlen,
               Müsst nur die Verschwindungspunkte zählen,
               Umkehren, abbilden, auf der Eb’ne ’rumfahren
               Und mit den Theta-Produkten nicht sparen.
                                               --LASSWITZ, KURD.

                         _Der Faust-Tragödie (-n)ter Tiel;
                         Zeitschrift für den math.-natur.
                         Unterricht, Bd. 14 (1883), p. 316._

        Fuchs. Your words fill me with an awful dread,
               Seems like a circle were squared in my head.

        Meph.  Next in order you certainly ought
               On function-theory bestow your thought,
               And penetrate with contemplation
               What resists your attempts at integration.
               You’ll find no dearth of theorems there--
               To vanishing-points give proper care--
               Enumerate, reciprocate,
               Nor forget to delineate,
               Traverse the plane from end to end,
               And theta-functions freely spend.


=1935.= The student should avoid _founding results_ upon
divergent series, as the question of their legitimacy is disputed
upon grounds to which no answer commanding anything like general
assent has yet been given. But they may be used as means of
discovery, provided that their results be verified by other means
before they are considered as established.--DE MORGAN, A.

                         _Trigonometry and Double Algebra
                         (London, 1849), p. 55._


=1936.= There is nothing now which ever gives me any thought or
care in algebra except divergent series, which I cannot follow
the French in rejecting.--DE MORGAN, A.

                         _Graves’ Life of W. R. Hamilton (New
                         York, 1882-1889), Vol. 3, p. 249._


=1937.= It is a strange vicissitude of our science that these
[divergent] series which early in the century were supposed to be
banished once and for all from rigorous mathematics should at its
close be knocking at the door for readmission.--PIERPONT, J.

                         _Congress of Arts and Sciences (Boston
                         and New York, 1905), Vol. 1, p. 476._


=1938.= Zeno was concerned with three problems.... These are the
problem of the infinitesimal, the infinite, and continuity....
From him to our own day, the finest intellects of each generation
in turn attacked these problems, but achieved broadly speaking
nothing.... Weierstrass, Dedekind, and Cantor, ... have
completely solved them. Their solutions ... are so clear as to
leave no longer the slightest doubt of difficulty. This
achievement is probably the greatest of which the age can
boast.... The problem of the infinitesimal was solved by
Weierstrass, the solution of the other two was begun by Dedekind
and definitely accomplished by Cantor.--RUSSELL, BERTRAND.

                         _International Monthly, Vol. 4 (1901),
                         p. 89._


=1939.= It was not till Leibnitz and Newton, by the discovery of
the differential calculus, had dispelled the ancient darkness
which enveloped the conception of the infinite, and had clearly
established the conception of the continuous and continuous
change, that a full and productive application of the newly-found
mechanical conceptions made any progress.--HELMHOLTZ, H.

                         _Aim and Progress of Physical Science;
                         Popular Lectures [Flight] (New York,
                         1900), p. 372._


=1940.= The idea of an infinitesimal involves no
contradiction.... As a mathematician, I prefer the method of
infinitesimals to that of limits, as far easier and less infested
with snares.--PIERCE, C. F.

                         _The Law of Mind; Monist, Vol. 2
                         (1891-1892), pp. 543, 545._


=1941.= The chief objection against all _abstract_ reasonings is
derived from the ideas of space and time; ideas, which, in common
life and to a careless view, are very clear and intelligible, but
when they pass through the scrutiny of the profound sciences (and
they are the chief object of these sciences) afford principles,
which seem full of obscurity and contradiction. No priestly
_dogmas_, invented on purpose to tame and subdue the rebellious
reason of mankind, ever shocked common sense more than the
doctrine of the infinite divisibility of extension, with
its consequences; as they are pompously displayed by all
geometricians and metaphysicians, with a kind of triumph and
exultation. A real quantity, infinitely less than any finite
quantity, containing quantities infinitely less than itself, and
so on _in infinitum_; this is an edifice so bold and prodigious,
that it is too weighty for any pretended demonstration to
support, because it shocks the clearest and most natural
principles of human reason. But what renders the matter more
extraordinary, is, that these seemingly absurd opinions are
supported by a chain of reasoning, the clearest and most natural;
nor is it possible for us to allow the premises without admitting
the consequences. Nothing can be more convincing and satisfactory
than all the conclusions concerning the properties of circles and
triangles; and yet, when these are once received, how can we
deny, that the angle of contact between a circle and its tangent
is infinitely less than any rectilineal angle, that as you may
increase the diameter of the circle _in infinitum_, this angle of
contact becomes still less, even _in infinitum_, and that the
angle of contact between other curves and their tangents may be
infinitely less than those between any circle and its tangent,
and so on, _in infinitum_? The demonstration of these principles
seems as unexceptionable as that which proves the three angles
of a triangle to be equal to two right ones, though the
latter opinion be natural and easy, and the former big with
contradiction and absurdity. Reason here seems to be thrown into
a kind of amazement and suspense, which, without the suggestion
of any sceptic, gives her a diffidence of herself, and of the
ground on which she treads. She sees a full light, which
illuminates certain places; but that light borders upon the most
profound darkness. And between these she is so dazzled and
confounded, that she scarcely can pronounce with certainty and
assurance concerning any one object.--HUME, DAVID.

                         _An Inquiry concerning Human
                         Understanding, Sect. 12, part 2._


=1942.= He who can digest a second or third fluxion, a second or
third difference, need not, methinks, be squeamish about any
point in Divinity.--BERKELEY, G.

                         _The Analyst, sect. 7._


=1943.= And what are these fluxions? The velocities of evanescent
increments. And what are these same evanescent increments? They
are neither finite quantities, nor quantities infinitely small,
nor yet nothing. May we not call them ghosts of departed
quantities?--BERKELEY, G.

                         _The Analyst, sect. 35._


=1944.= It is said that the minutest errors are not to be
neglected in mathematics; that the fluxions are celerities, not
proportional to the finite increments, though ever so small; but
only to the moments or nascent increments, whereof the proportion
alone, and not the magnitude, is considered. And of the aforesaid
fluxions there be other fluxions, which fluxions of fluxions are
called second fluxions. And the fluxions of these second fluxions
are called third fluxions: and so on, fourth, fifth, sixth, etc.,
_ad infinitum_. Now, as our Sense is strained and puzzled with
the perception of objects extremely minute, even so the
Imagination, which faculty derives from sense, is very much
strained and puzzled to frame clear ideas of the least particle
of time, or the least increment generated therein: and much more
to comprehend the moments, or those increments of the flowing
quantities in _status nascenti_, in their first origin or
beginning to exist, before they become finite particles. And it
seems still more difficult to conceive the abstracted velocities
of such nascent imperfect entities. But the velocities of the
velocities, the second, third, fourth, and fifth velocities,
etc., exceed, if I mistake not, all human understanding. The
further the mind analyseth and pursueth these fugitive ideas the
more it is lost and bewildered; the objects, at first fleeting
and minute, soon vanishing out of sight. Certainly, in any sense,
a second or third fluxion seems an obscure Mystery. The incipient
celerity of an incipient celerity, the nascent augment of a
nascent augment, i.e. of a thing which hath no magnitude; take it
in what light you please, the clear conception of it will, if I
mistake not, be found impossible; whether it be so or no I appeal
to the trial of every thinking reader. And if a second fluxion be
inconceivable, what are we to think of third, fourth, fifth
fluxions, and so on without end.--BERKELEY, G.

                         _The Analyst, sect, 4._


=1945.= The _infinite_ divisibility of _finite_ extension, though
it is not expressly laid down either as an axiom or theorem in
the elements of that science, yet it is throughout the same
everywhere supposed and thought to have so inseparable and
essential a connection with the principles and demonstrations in
Geometry, that mathematicians never admit it into doubt, or make
the least question of it. And, as this notion is the source
whence do spring all those amusing geometrical paradoxes which
have such a direct repugnancy to the plain common sense of
mankind, and are admitted with so much reluctance into a mind not
yet debauched by learning; so it is the principal occasion of all
that nice and extreme subtility which renders the study of
Mathematics so difficult and tedious.--BERKELEY, G.

                         _On the Principles of Human Knowledge,
                         Sect. 123._


=1946.= To avoid misconception, it should be borne in mind that
infinitesimals are not regarded as being actual quantities in the
ordinary acceptation of the words, or as capable of exact
representation. They are introduced for the purpose of abridgment
and simplification of our reasonings, and are an ultimate phase
of magnitude when it is conceived by the mind as capable of
diminution below any assigned quantity, however small....
Moreover such quantities are neglected, not, as Leibnitz stated,
because they are infinitely small in comparison with those that
are retained, which would produce an infinitely small error, but
because they must be neglected to obtain a rigorous result; since
such result must be definite and determinate, and consequently
independent of these _variable indefinitely small quantities_. It
may be added that the precise principles of the infinitesimal
calculus, like those of any other science, cannot be thoroughly
apprehended except by those who have already studied the science,
and made some progress in the application of its principles.

                                                 --WILLIAMSON, B.

                         _Encyclopedia Britannica, 9th Edition;
                         Article “Infinitesimal Calculus,” Sect.
                         12, 14._


=1947.= We admit, in geometry, not only infinite magnitudes, that
is to say, magnitudes greater than any assignable magnitude, but
infinite magnitudes infinitely greater, the one than the other.
This astonishes our dimension of brains, which is only about six
inches long, five broad, and six in depth, in the largest heads.

                                                      --VOLTAIRE.

                         _A Philosophical Dictionary; Article
                         “Infinity.” (Boston, 1881)._


=1948.= Infinity is the land of mathematical hocus pocus. There
Zero the magician is king. When Zero divides any number he
changes it without regard to its magnitude into the infinitely
small [great?], and inversely, when divided by any number he
begets the infinitely great [small?]. In this domain the
circumference of the circle becomes a straight line, and then the
circle can be squared. Here all ranks are abolished, for Zero
reduces everything to the same level one way or another. Happy is
the kingdom where Zero rules!--CARUS, PAUL.

                         _Logical and Mathematical Thought;
                         Monist, Vol. 20 (1909-1910), p. 69._


=1949.=

        Great fleas have little fleas upon their backs
            to bite ’em,
        And little fleas have lesser fleas, and so _ad
            infinitum._
        And the great fleas themselves, in turn, have
            greater fleas to go on;
        While these again have greater still, and
            greater still, and so on.
                                             --DE MORGAN, A.

                         _Budget of Paradoxes (London, 1872), p.
                         377._


=1950.= We have adroitly defined the infinite in arithmetic by a
loveknot, in this manner ∞; but we possess not therefore the
clearer notion of it.--VOLTAIRE.

                         _A Philosophical Dictionary; Article
                         “Infinity.” (Boston, 1881)._


=1951.= I protest against the use of infinite magnitude as
something completed, which in mathematics is never permissible.
Infinity is merely a _facon de parler_, the real meaning being a
limit which certain ratios approach indefinitely near, while
others are permitted to increase without restriction.--GAUSS.

                         _Brief an Schumacher (1831); Werke, Bd.
                         8 p. 216._


=1952.= In spite of the essential difference between the
conceptions of the _potential_ and the _actual_ infinite, the
former signifying a _variable_ finite magnitude increasing beyond
all finite limits, while the latter is a _fixed_, _constant_
quantity lying beyond all finite magnitudes, it happens only too
often that the one is mistaken for the other.... Owing to a
justifiable aversion to such _illegitimate_ actual infinities and
the influence of the modern epicuric-materialistic tendency, a
certain _horror infiniti_ has grown up in extended scientific
circles, which finds its classic expression and support in the
letter of Gauss [see 1951], yet it seems to me that the
consequent uncritical rejection of the legitimate actual infinite
is no lesser violation of the nature of things, which must be
taken as they are.--CANTOR, G.

                         _Zum Problem des actualen Unendlichen;
                         Natur und Offenbarung, Bd. 32 (1886), p.
                         226._


=1953.= The Infinite is often confounded with the Indefinite, but
the two conceptions are diametrically opposed. Instead of being a
quantity with unassigned yet assignable limits, the Infinite is
not a quantity at all, since it neither admits of augmentation
nor diminution, having no assignable limits; it is the operation
of continuously _withdrawing_ any limits that may have been
assigned: the endless addition of new quantities to the old: the
flux of continuity. The Infinite is no more a quantity than Zero
is a quantity. If Zero is the sign of a vanished quantity, the
Infinite is a sign of that continuity of Existence which has been
ideally divided into discrete parts in the affixing of limits.

                                                   --LEWES, G. H.

                         _Problems of Life and Mind (Boston,
                         1875), Vol. 2, p. 384._


=1954.= A great deal of misunderstanding is avoided if it be
remembered that the terms _infinity_, _infinite_, _zero_,
_infinitesimal_ must be interpreted in connexion with their
context, and admit a variety of meanings according to the way in
which they are defined.--MATHEWS, G. B.

                         _Theory of Numbers (Cambridge, 1892),
                         Part 1, sect. 104._


=1955.= This further is observable in number, that it is that
which the mind makes use of in measuring all things that by us
are measurable, which principally are _expansion_ and _duration_;
and our idea of infinity, even when applied to those, seems to be
nothing but the infinity of number. For what else are our ideas
of Eternity and Immensity, but the repeated additions of certain
ideas of imagined parts of duration and expansion, with the
infinity of number; in which we can come to no end of addition?

                                                   --LOCKE, JOHN.

                         _An Essay concerning Human
                         Understanding, Bk. 2, chap. 16, sect.
                         8._


=1956.= But of all other ideas, it is number, which I think
furnishes us with the clearest and most distinct idea of infinity
we are capable of.--LOCKE, JOHN.

                         _An Essay concerning Human
                         Understanding, Bk. 2, chap. 17, sect.
                         9._


=1957.=

        Willst du ins Unendliche schreiten?
        Geh nur im Endlichen nach allen Seiten!
        Willst du dich am Ganzen erquicken,
        So musst du das Ganze im Kleinsten erblicken.
                                                   --GOETHE.

                         _Gott, Gemüt und Welt (1815)._

        [Would’st thou the infinite essay?
         The finite but traverse in every way.
         Would’st in the whole delight thy heart?
         Learn to discern the whole in its minutest part.]


=1958.=

        Ich häufe ungeheure Zahlen,
        Gebürge Millionen auf,
        Ich setze Zeit auf Zeit und Welt auf Welt zu Hauf,
        Und wenn ich von der grausen Höh’
        Mit Schwindeln wieder nach dir seh,’
        Ist alle Macht der Zahl, vermehrt zu tausendmalen,
        Noch nicht ein Theil von dir.
        _Ich zieh’ sie ab, und du liegst ganz vor mir._
                                        --HALLER, ALBR. VON.

                         _Quoted in Hegel: Wissenschaft der
                         Logik, Buch 1, Abschnitt 2, Kap. 2, C,
                         b._

        [Numbers upon numbers pile,
         Mountains millions high,
         Time on time and world on world amass,
         Then, if from the dreadful hight, alas!
         Dizzy-brained, I turn thee to behold,
         All the power of number, increased thousandfold,
         Not yet may match thy part.
         _Subtract what I will, wholly whole thou art._]


=1959.= A collection of terms is infinite when it contains as
parts other collections which have just as many terms in it as it
has. If you can take away some of the terms of a collection,
without diminishing the number of terms, then there is an
infinite number of terms in the collection.--RUSSELL, BERTRAND.

                         _International Monthly, Vol. 4 (1901),
                         p. 93._


=1960.= An assemblage (ensemble, collection, group, manifold) of
elements (things, no matter what) is infinite or finite according
as it has or has not a part to which the whole is just _equivalent_
in the sense that between the elements composing that part and
those composing the whole there subsists a unique and reciprocal
(one-to-one) correspondence.--KEYSER, C. J.

                         _The Axioms of Infinity; Hibbert
                         Journal, Vol. 2 (1903-1904), p. 539._


=1961.= Whereas in former times the Infinite betrayed its
presence not indeed to the faculties of Logic but only to the
spiritual Imagination and Sensibility, mathematics has shown ...
that the structure of Transfinite Being is open to exploration
by the organon of Thought.--KEYSER, C. J.

                         _Lectures on Science, Philosophy and Art
                         (New York, 1908), p. 42._


=1962.= The mathematical theory of probability is a science which
aims at reducing to calculation, where possible, the amount of
credence due to propositions or statements, or to the occurrence
of events, future or past, more especially as contingent or
dependent upon other propositions or events the probability of
which is known.--CROFTON, M. W.

                         _Encyclopedia Britannica, 9th Edition;
                         Article, “Probability.”_


=1963.= The theory of probabilities is at bottom nothing but
common sense reduced to calculus; it enables us to appreciate
with exactness that which accurate minds feel with a sort of
instinct for which ofttimes they are unable to account. If we
consider the analytical methods to which this theory has given
birth, the truth of the principles on which it is based, the fine
and delicate logic which their employment in the solution of
problems requires, the public utilities whose establishment rests
upon it, the extension which it has received and which it may
still receive through its application to the most important
problems of natural philosophy and the moral sciences; if again
we observe that, even in matters which cannot be submitted to the
calculus, it gives us the surest suggestions for the guidance of
our judgments, and that it teaches us to avoid the illusions
which often mislead us, then we shall see that there is no
science more worthy of our contemplations nor a more useful one
for admission to our system of public education.--LAPLACE.

                         _Théorie Analytique des Probabilitiés,
                         Introduction; Oeuvres, t. 7 (Paris,
                         1886), p. 153._


=1964.= It is a truth very certain that, when it is not in our
power to determine what is true, we ought to follow what is most
probable.--DESCARTES.

                         _Discourse on Method, Part 3._


=1965.= As _demonstration_ is the showing the agreement or
disagreement of two ideas, by the intervention of one or more
proofs, which have a constant, immutable, and visible connexion
one with another; so _probability_ is nothing but the appearance
of such an agreement or disagreement, by the intervention of
proofs, whose connexion is not constant and immutable, or at
least is not perceived to be so, and it is enough to induce the
mind to judge the proposition to be true or false, rather than
contrary.--LOCKE, JOHN.

                         _An Essay concerning Human
                         Understanding, Bk. 4, chap. 15, sect.
                         1._


=1966.= The difference between necessary and contingent truths is
indeed the same as that between commensurable and incommensurable
numbers. For the reduction of commensurable numbers to a common
measure is analogous to the demonstration of necessary truths, or
their reduction to such as are identical. But as, in the case of
surd ratios, the reduction involves an infinite process, and yet
approaches a common measure, so that a definite but unending
series is obtained, so also contingent truths require an infinite
analysis, which God alone can accomplish.--LEIBNITZ.

                         _Philosophische Schriften [Gerhardt] Bd.
                         7 (Berlin, 1890), p. 200._


=1967.= The theory in question [theory of probability] affords an
excellent illustration of the application of the theory of
permutation and combinations which is the fundamental part of the
algebra of discrete quantity; it forms in the elementary parts an
excellent logical exercise in the accurate use of terms and in
the nice discrimination of shades of meaning; and, above all, it
enters into the regulation of some of the most important
practical concerns of modern life.--CHRYSTAL, GEORGE.

                         _Algebra, Vol. 2 (Edinburgh, 1889),
                         chap. 36, sect. 1._


=1968.= There is possibly no branch of mathematics at once so
interesting, so bewildering, and of so great practical importance
as the theory of probabilities. Its history reveals both the
wonders that can be accomplished and the bounds that cannot be
transcended by mathematical science. It is the link between rigid
deduction and the vast field of inductive science. A complete
theory of probabilities would be the complete theory of the
formation of belief. It is certainly a pity then, that, to quote
M. Bertrand, “one cannot well understand the calculus of
probabilities without having read Laplace’s work,” and that “one
cannot read Laplace’s work without having prepared oneself for it
by the most profound mathematical studies.”--DAVIS, E. W.

                         _Bulletin American Mathematical Society,
                         Vol. 1 (1894-1895), p. 16._


=1969.= The most important questions of life are, for the most
part, really only problems of probability. Strictly speaking one
may even say that nearly all our knowledge is problematical; and
in the small number of things which we are able to know with
certainty, even in the mathematical sciences themselves,
induction and analogy, the principal means for discovering truth,
are based on probabilities, so that the entire system of human
knowledge is connected with this theory.--LAPLACE.

                         _Théorie Analytique des Probabilitiés,
                         Introduction; Oeuvres, t. 7 (Paris,
                         1886), p. 5._


=1970.= There is no more remarkable feature in the mathematical
theory of probability than the manner in which it has been found
to harmonize with, and justify, the conclusions to which mankind
have been led, not by reasoning, but by instinct and experience,
both of the individual and of the race. At the same time it has
corrected, extended, and invested them with a definiteness and
precision of which these crude, though sound, appreciations of
common sense were till then devoid.--CROFTON, M. W.

                         _Encyclopedia Britannica, 9th Edition;
                         Article “Probability.”_


=1971.= It is remarkable that a science [probabilities] which
began with the consideration of games of chance, should have
become the most important object of human knowledge.--LAPLACE.

                         _Théorie Analytique des Probabilitiés,
                         Introduction; Oeuvres, t. 7 (Paris,
                         1886), p. 152._


=1972.= Not much has been added to the subject [of probability]
since the close of Laplace’s career. The history of science
records more than one parallel to this abatement of activity.
When such a genius has departed, the field of his labours seems
exhausted for the time, and little left to be gleaned by his
successors. It is to be regretted that so little remains to us of
the inner workings of such gifted minds, and of the clue by which
each of their discoveries was reached. The didactic and synthetic
form in which these are presented to the world retains but faint
traces of the skilful inductions, the keen and delicate
perception of fitness and analogy, and the power of imagination
... which have doubtless guided such a master as Laplace or
Newton in shaping out such great designs--only the minor details
of which have remained over, to be supplied by the less cunning
hand of commentator and disciple.--CROFTON, M. W.

                         _Encyclopedia Britannica, 9th Edition;
                         Article “Probability.”_


=1973.= The theory of errors may be defined as that branch of
mathematics which is concerned, first, with the expression of the
resultant effect of one or more sources of error to which
computed and observed quantities are subject; and, secondly, with
the determination of the relation between the magnitude of an
error and the probability of its occurrence.--WOODWARD, R. S.

                         _Probability and Theory of Errors (New
                         York, 1906), p. 30._


=1974.= Of all the applications of the doctrine of probability
none is of greater utility than the theory of errors. In
astronomy, geodesy, physics, and chemistry, as in every science
which attains precision in measuring, weighing, and computing, a
knowledge of the theory of errors is indispensable. By the aid of
this theory the exact sciences have made great progress during
the nineteenth century, not only in the actual determinations of
the constants of nature, but also in the fixation of clear ideas
as to the possibilities of future conquests in the same
direction. Nothing, for example, is more satisfactory and
instructive in the history of science than the success with which
the unique method of least squares has been applied to the
problems presented by the earth and the other members of the
solar system. So great, in fact, are the practical value and
theoretical importance of least squares, that it is frequently
mistaken for the whole theory of errors, and is sometimes
regarded as embodying the major part of the doctrine of
probability itself.--WOODWARD, R. S.

                         _Probability and Theory of Errors (New
                         York, 1906), pp. 9-10._


=1975.= Direct and inverse ratios have been applied by an
ingenious author to measure human affections, and the moral worth
of actions. An eminent Mathematician attempted to ascertain by
calculation, the ratio in which the evidence of facts must
decrease in the course of time, and fixed the period when the
evidence of the facts on which Christianity is founded shall
become evanescent, and when in consequence no faith shall be
found on the earth.--REID, THOMAS.

                         _Essays on the Powers of the Human Mind
                         (Edinburgh, 1812), Vol. 2, p. 408._



  CHAPTER XX

  THE FUNDAMENTAL CONCEPTS, TIME AND SPACE


=2001.= Kant’s Doctrine of Time.

I. Time is not an empirical concept deduced from any experience,
for neither co-existence nor succession would enter into our
perception, if the representation of time were not given _a
priori_. Only when this representation _a priori_ is given,
can we imagine that certain things happen at the same time
(simultaneously) or at different times (successively).

II. Time is a necessary representation on which all intuitions
depend. We cannot take away time from phenomena in general,
though we can well take away phenomena out of time. In time alone
is reality of phenomena possible. All phenomena may vanish, but
time itself (as the general condition of their possibility)
cannot be done away with.

III. On this _a priori_ necessity depends also the possibility of
apodictic principles of the relations of time, or of axioms of
time in general. Time has one dimension only; different times are
not simultaneous, but successive, while different spaces are
never successive, but simultaneous. Such principles cannot be
derived from experience, because experience could not impart to
them absolute universality nor apodictic certainty....

IV. Time is not a discursive, or what is called a general
concept, but a pure form of sensuous intuition. Different times
are parts only of one and the same time....

V. To say that time is infinite means no more than that every
definite quantity of time is possible only by limitations of one
time which forms the foundation of all times. The original
representation of time must therefore be given as unlimited. But
when the parts themselves and every quantity of an object can be
represented as determined by limitation only, the whole
representation cannot be given by concepts (for in that case the
partial representation comes first), but must be founded on
immediate intuition.--KANT, I.

                         _Critique of Pure Reason [Max Müller]
                         (New York, 1900), pp. 24-25._


=2002.= Kant’s Doctrine of Space.

I. Space is not an empirical concept which has been derived from
external experience. For in order that certain sensations should
be referred to something outside myself, i.e. to something in a
different part of space from that where I am; again, in order
that I may be able to represent them as side by side, that
is, not only as different, but as in different places, the
representation of space must already be there....

II. Space is a necessary representation _a priori_, forming the
very foundation of all external intuitions. It is impossible to
imagine that there should be no space, though one might very well
imagine that there should be space without objects to fill it.
Space is therefore regarded as a condition of the possibility of
phenomena, not as a determination produced by them; it is a
representation _a priori_ which necessarily precedes all external
phenomena.

III. On this necessity of an _a priori_ representation of space
rests the apodictic certainty of all geometrical principles, and
the possibility of their construction _a priori_. For if the
intuition of space were a concept gained _a posteriori_, borrowed
from general external experience, the first principles of
mathematical definition would be nothing but perceptions. They
would be exposed to all the accidents of perception, and there
being but one straight line between two points would not be a
necessity, but only something taught in each case by experience.
Whatever is derived from experience possesses a relative
generality only, based on induction. We should therefore not be
able to say more than that, so far as hitherto observed, no space
has yet been found having more than three dimensions.

IV. Space is not a discursive or so-called general concept of
the relations of things in general, but a pure intuition. For,
first of all, we can imagine one space only, and if we speak of
many spaces, we mean parts only of one and the same space. Nor
can these parts be considered as antecedent to the one and
all-embracing space and, as it were, its component parts out
of which an aggregate is formed, but they can be thought of
as existing within it only. Space is essentially one; its
multiplicity, and therefore the general concept of spaces in
general, arises entirely from limitations. Hence it follows that,
with respect to space, an intuition _a priori_, which is not
empirical, must form the foundation of all conceptions of
space....

V. Space is represented as an infinite given quantity. Now it is
quite true that every concept is to be thought as a representation,
which is contained in an infinite number of different possible
representations (as their common characteristic), and therefore
comprehends them: but no concept, as such, can be thought as if
it contained in itself an infinite number of representations.
Nevertheless, space is so thought (for all parts of infinite space
exist simultaneously). Consequently, the original representation
of space is an _intuition a priori_, and not a concept.--KANT, I.

                         _Critique of Pure Reason [Max Müller]
                         (New York, 1900), pp. 18-20 and
                         Supplement 8._


=2003.=

             _Schopenhauer’s Predicabilia a priori._[11]

              OF TIME                            OF SPACE

   1. There is but _one time_, all    1. There is but _one space_,
      different times are parts of       all different spaces are
      it.                                parts of it.

   2. Different times are not         2. Different spaces are not
      simultaneous but successive.       successive but
                                         simultaneous.

   3. Everything in time may be       3. Everything in space may be
      thought of as non-existent,        thought of as non-existent,
      but not time.                      but not space.

   4. Time has three divisions:       4. Space has three dimensions:
      past, present and future,          height, breadth, and
      which form two directions          length.
      with a point of indifference.

   5. Time is infinitely              5. Space is infinitely
      divisible.                         divisible.

   6. Time is homogeneous and a       6. Space is homogeneous and a
      continuum: i.e. no part is         continuum: i.e. no part
      different from another, nor        is different from another,
      separated by something             nor separated by something
      which is not time.                 which is not space.

   7. Time has no beginning nor       7. Space has no limits
      end, but all beginning and         [Gränzen], but all limits
      end is in time.                    are in space.

   8. Time makes counting             8. Space makes measurement
      possible.                          possible.

   9. Rhythm exists only in time.     9. Symmetry exists only in
                                         space.

  10. The laws of time are _a        10. The laws of space are _a
      priori_ conceptions.               priori_ conceptions.

  11. Time is perceptible _a         11. Space is immediately
      priori_, but only by a             perceptible _a priori_.
      means of a line-image.

  12. Time has no permanence but     12. Space never passes but is
      passes the moment it is            permanent throughout
      present.                           all time.

  13. Time never rests.              13. Space never moves.

  14. Everything in time has         14. Everything in space has
      duration.                          position.

  15. Time has no duration, but      15. Space has no motion, but
      all duration is in time;           all motion is in space;
      time is the persistence of         space is the change in
      what is permanent in               position of that which
      contrast with its restless         moves in contrast to its
      course.                            imperturbable rest.

  16. Motion is only possible in     16. Motion is only possible in
      time.                              space.

  17. Velocity, the space being      17. Velocity, the time being
      the same, is in the inverse        the same, is in the direct
      ratio of the time.                 ratio of the space.

  18. Time is not directly           18. Space is measurable directly
      measurable by means of             through itself and
      itself but only by means of        indirectly through motion
      motion which takes place in        which takes place in both
      both space and time....            time and space....

  19. Time is omnipresent: each      19. Space is eternal: each
      part of it is everywhere.          part of it exists always.

  20. In time alone all things       20. In space alone all things
      are successive.                    are simultaneous.

  21. Time makes possible the        21. Space makes possible the
      change of accidents.               endurance of substance.

  22. Each part of time contains     22. No part of space contains
      all substance.                     the same substance as
                                         another.

  23. Time is the _principium        23. Space is the _principium
      individuationis_.                  individuationis_.

  24. The now is without             24. The point is without
      duration.                          extension.

  25. Time of itself is empty and    25. Space is of itself empty
      indeterminate.                     and indeterminate.

  26. Each moment is conditioned     26. The relation of each
      by the one which precedes          boundary in space to every
      it, and only so far as this        other is determined by its
      one has ceased to exist.           relation to any one.
      (Principle of sufficient           (Principle of sufficient
      reason of being in time.)          reason of being in space.)

  27. Time makes Arithmetic          27. Space makes Geometry
      possible.                          possible.

  28. The simple element of          28. The element of Geometry
      Arithmetic is unity.               is the point.
                                               --SCHOPENHAUER, A.

                         _Die Welt als Vorstellung und Wille;
                         Werke (Frauenstädt) (Leipzig, 1877), Bd.
                         2, p. 55._

    [11]  Schopenhauer’s table contains a third column
          headed “of matter” which has here been omitted.


=2004.= The clear possession of the Idea of Space is the first
requisite for all geometrical reasoning; and this clearness of
idea may be tested by examining whether the axioms offer
themselves to the mind as evident.--WHEWELL, WILLIAM.

                         _The Philosophy of the Inductive
                         Sciences, Part 1, Bk. 2, chap. 4, sect.
                         4 (London, 1858)._


=2005.= Geometrical axioms are neither synthetic _a priori_
conclusions nor experimental facts. They are conventions:
our choice, amongst all possible conventions, is guided by
experimental facts; but it remains free, and is only limited by
the necessity of avoiding all contradiction.... In other words,
axioms of geometry are only definitions in disguise.

That being so what ought one to think of this question: Is the
Euclidean Geometry true?

The question is nonsense. One might as well ask whether the
metric system is true and the old measures false; whether
Cartesian co-ordinates are true and polar co-ordinates false.

                                                   --POINCARÉ, H.

                         _Non-Euclidean Geometry; Nature, Vol 45
                         (1891-1892), p. 407._


=2006.= I do in no wise share this view [that the axioms are
arbitrary propositions which we assume wholly at will, and that
in like manner the fundamental conceptions are in the end only
arbitrary symbols with which we operate] but consider it the
death of all science: in my judgment the axioms of geometry are
not arbitrary, but reasonable propositions which generally have
the origin in space intuition and whose separate content and
sequence is controlled by reasons of expediency.--KLEIN, F.

                         _Elementarmathematik vom höheren
                         Standpunkte aus (Leipzig, 1909), Bd. 2,
                         p. 384._


=2007.= Euclid’s Postulate 5 [The Parallel Axiom].

That, if a straight line falling on two straight lines make the
interior angles on the same side less than two right angles, the
two straight lines, if produced indefinitely, meet on that side
on which are the angles less than the two right angles.--EUCLID.

                         _The Thirteen Books of Euclid’s Elements
                         [T. L. Heath] Vol. 1 (Cambridge, 1908),
                         p. 202._


=2008.= It must be admitted that Euclid’s [Parallel] Axiom is
unsatisfactory as the basis of a theory of parallel straight
lines. It cannot be regarded as either simple or self-evident,
and it therefore falls short of the essential characteristics of
an axiom....--HALL, H. S. and STEVENS, F. H.

                         _Euclid’s Elements (London, 1892), p.
                         55._


=2009.= We may still well declare the parallel axiom the simplest
assumption which permits us to represent spatial relations, and
so it will be true generally, that concepts and axioms are not
immediate facts of intuition, but rather the idealizations of
these facts chosen for reasons of expediency.--KLEIN, F.

                         _Elementarmathematik vom, höheren
                         Standpunkte aus (Leipzig, 1909), Bd. 2,
                         p. 382._


=2010.= The characteristic features of our space are not
necessities of thought, and the truth of Euclid’s axioms, in so
far as they specially differentiate our space from other
conceivable spaces, must be established by experience and by
experience only.--BALL, R. S.

                         _Encyclopedia Britannica, 9th Edition;
                         Article “Measurement.”_


=2011.= Mathematical and physiological researches have shown that
the space of experience is simply an _actual_ case of many
conceivable cases, about whose peculiar properties experience
alone can instruct us.--MACH, ERNST.

                         _Popular Scientific Lectures (Chicago,
                         1910), p. 205._


=2012.= The familiar definition: An axiom is a self-evident
truth, means if it means anything, that the proposition which we
call an axiom has been approved by us in the light of our
experience and intuition. In this sense mathematics has no
axioms, for mathematics is a formal subject over which formal and
not material implication reigns.--WILSON, E. B.

                         _Bulletin American Mathematical Society,
                         Vol. 2 (1904-1905), p. 81._


=2013.= The proof of self-evident propositions may seem, to the
uninitiated, a somewhat frivolous occupation. To this we might
reply that it is often by no means self-evident that one obvious
proposition follows from another obvious proposition; so that we
are really discovering new truths when we prove what is evident
by a method which is not evident. But a more interesting retort
is, that since people have tried to prove obvious propositions,
they have found that many of them are false. Self-evidence is
often a mere will-o’-the-wisp, which is sure to lead us astray if
we take it as our guide.--RUSSELL, BERTRAND.

                         _Recent Work on the Principles of
                         Mathematics; International Monthly, Vol.
                         4 (1901), p. 86._


=2014.= The problem [of Euclid’s Parallel Axiom] is now at a par
with the squaring of the circle and the trisection of an angle by
means of ruler and compass. So far as the mathematical public is
concerned, the famous problem of the parallel is settled for all
time.--YOUNG, JOHN WESLEY.

                         _Fundamental Concepts of Algebra and
                         Geometry (New York, 1911), p. 32._


=2015.= If the Euclidean assumptions are true, the constitution
of those parts of space which are at an infinite distance from
us, “geometry upon the plane at infinity,” is just as well known
as the geometry of any portion of this room. In this infinite and
thoroughly well-known space the Universe is situated during at
least some portion of an infinite and thoroughly well-known time.
So that here we have real knowledge of something at least that
concerns the Cosmos; something that is true throughout the
Immensities and the Eternities. That something Lobatchewsky and
his successors have taken away. The geometer of to-day knows
nothing about the nature of the actually existing space at an
infinite distance; he knows nothing about the properties of this
present space in a past or future eternity. He knows, indeed,
that the laws assumed by Euclid are true with an accuracy that no
direct experiment can approach, not only in this place where we
are, but in places at a distance from us that no astronomer has
conceived; but he knows this as of Here and Now; beyond this
range is a There and Then of which he knows nothing at present,
but may ultimately come to know more.--CLIFFORD, W. K.

                         _Lectures and Essays (New York, 1901),
                         Vol. 1, pp. 358-359._


=2016.= The truth is that other systems of geometry are possible,
yet after all, these other systems are not spaces but other
methods of space measurements. There is one space only, though we
may conceive of many different manifolds, which are contrivances
or ideal constructions invented for the purpose of determining
space.--CARUS, PAUL.

                         _Science, Vol. 18 (1903), p. 106._


=2017.= As I have formerly stated that from the philosophic side
Non-Euclidean Geometry has as yet not frequently met with full
understanding, so I must now emphasize that it is universally
recognized in the science of mathematics; indeed, for many
purposes, as for instance in the modern theory of functions, it
is used as an extremely convenient means for the visual
representation of highly complicated arithmetical relations.

                                                      --KLEIN, F.

                         _Elementarmathematik vom höheren
                         Standpunkte aus (Leipzig, 1909), Bd. 2,
                         p. 377._


=2018.= Everything in physical science, from the law of
gravitation to the building of bridges, from the spectroscope to
the art of navigation, would be profoundly modified by any
considerable inaccuracy in the hypothesis that our actual space
is Euclidean. The observed truth of physical science, therefore,
constitutes overwhelming empirical evidence that this hypothesis
is very approximately correct, even if not rigidly true.

                                             --RUSSELL, BERTRAND.

                         _Foundations of Geometry (Cambridge,
                         1897), p. 6._


=2019.= The most suggestive and notable achievement of the last
century is the discovery of Non-Euclidean geometry.--HILBERT, D.

                         _Quoted by G. D. Fitch in Manning’s “The
                         Fourth Dimension Simply Explained,” (New
                         York, 1910), p. 58._


=2020.= Non-Euclidean geometry--primate among the emancipators of
the human intellect....--KEYSER, C. J.

                         _The Foundations of Mathematics; Science
                         History of the Universe, Vol. 8 (New
                         York, 1909), p. 192._


=2021.= Every high school teacher [Gymnasial-lehrer] must of
necessity know something about non-euclidean geometry, because it
is one of the few branches of mathematics which, by means of
certain catch-phrases, has become known in wider circles, and
concerning which any teacher is consequently liable to be asked
at any time. In physics there are many such matters--almost every
new discovery is of this kind--which, through certain catch-words
have become topics of common conversation, and about which
therefore every teacher must of course be informed. Think of a
teacher of physics who knows nothing of Roentgen rays or of
radium; no better impression would be made by a mathematician who
is unable to give information concerning non-euclidean geometry.

                                                      --KLEIN, F.

                         _Elementarmathematik vom höheren
                         Standpunkte_ aus _(Leipzig, 1909), Bd.
                         2, p. 378._


=2022.= What Vesalius was to Galen, what Copernicus was to
Ptolemy, that was Lobatchewsky to Euclid. There is, indeed, a
somewhat instructive parallel between the last two cases.
Copernicus and Lobatchewsky were both of Slavic origin. Each of
them has brought about a revolution in scientific ideas so great
that it can only be compared with that wrought by the other. And
the reason of the transcendent importance of these two changes is
that they are changes in the conception of the Cosmos.... And in
virtue of these two revolutions the idea of the Universe, the
Macrocosm, the All, as subject of human knowledge, and therefore
of human interest, has fallen to pieces.--CLIFFORD, W. K.

                         _Lectures and Essays (New York, 1901),
                         Vol. 1, pp. 356, 358._


=2023.= I am exceedingly sorry that I have failed to avail myself
of our former greater proximity to learn more of your work on
the foundations of geometry; it surely would have saved me
much useless effort and given me more peace, than one of my
disposition can enjoy so long as so much is left to consider in a
matter of this kind. I have myself made much progress in this
matter (though my other heterogeneous occupations have left me
but little time for this purpose); though the course which I have
pursued does not lead as much to the desired end, which you
assure me you have reached, as to the questioning of the truth of
geometry. It is true that I have found much which many would
accept as proof, but which in my estimation proves _nothing_, for
instance, if it could be shown that a rectilinear triangle is
possible, whose area is greater than that of any given surface,
then I could rigorously establish the whole of geometry. Now most
people, no doubt, would grant this as an axiom, but not I; it is
conceivable that, however distant apart the vertices of the
triangle might be chosen, its area might yet always be below a
certain limit. I have found several other such theorems, but none
of them satisfies me.--GAUSS.

                         _Letter to Bolyai (1799); Werke, Bd. 8
                         (Göttingen, 1900), p. 159._


=2024.= On the supposition that Euclidean geometry is not valid,
it is easy to show that similar figures do not exist; in that
case the angles of an equilateral triangle vary with the side in
which I see no absurdity at all. The angle is a function of the
side and the sides are functions of the angle, a function which,
of course, at the same time involves a constant length. It seems
somewhat of a paradox to say that a constant length could be
given a priori as it were, but in this again I see nothing
inconsistent. Indeed, it would be desirable that Euclidean
geometry were not valid, for then we should possess a general a
priori standard of measure.--GAUSS.

                         _Letter to Gerling (1816); Werke, Bd. 8
                         (Göttingen, 1900), p. 169._


=2025.= I am convinced more and more that the necessary truth of
our geometry cannot be demonstrated, at least not _by_ the
_human_ intellect _to_ the human understanding. Perhaps in
another world we may gain other insights into the nature of space
which at present are unattainable to us. Until then we must
consider geometry as of equal rank not with arithmetic, which is
purely a priori, but with mechanics.--GAUSS.

                         _Letter to Olbers (1817); Werke, Bd. 8
                         (Göttingen, 1900), p. 177._


=2026.= There is no doubt that it can be rigorously established
that the sum of the angles of a rectilinear triangle cannot
exceed 180°. But it is otherwise with the statement that the sum
of the angles cannot be less than 180°; this is the real Gordian
knot, the rocks which cause the wreck of all.... I have been
occupied with the problem over thirty years and I doubt if anyone
has given it more serious attention, though I have never
published anything concerning it. The assumption that the angle
sum is less than 180° leads to a peculiar geometry, entirely
different from the Euclidean, but throughout consistent with
itself. I have developed this geometry to my own satisfaction so
that I can solve every problem that arises in it with the
exception of the determination of a certain constant which cannot
be determined a priori. The larger one assumes this constant the
more nearly one approaches the Euclidean geometry, an infinitely
large value makes the two coincide. The theorems of this geometry
seem in part paradoxical, and to the unpracticed absurd; but on a
closer and calm reflection it is found that in themselves they
contain nothing impossible.... All my efforts to discover some
contradiction, some inconsistency in this Non-Euclidean geometry
have been fruitless, the one thing in it that seems contrary to
reason is that space would have to contain a _definitely
determinate_ (though to us unknown) linear magnitude. However, it
seems to me that notwithstanding the meaningless word-wisdom of
the metaphysicians we know really too little, or nothing,
concerning the true nature of space to confound what appears
unnatural with the _absolutely impossible._ Should Non-Euclidean
geometry be true, and this constant bear some relation to
magnitudes which come within the domain of terrestrial or
celestial measurement, it could be determined a posteriori.

                                                         --GAUSS.

                         _Letter to Taurinus (1824); Werke, Bd. 8
                         (Göttingen, 1900), p. 187._


=2027.= There is also another subject, which with me is nearly
forty years old, to which I have again given some thought during
leisure hours, I mean the foundations of geometry.... Here, too,
I have consolidated many things, and my conviction has, if
possible become more firm that geometry cannot be completely
established on a priori grounds. In the mean time I shall
probably not for a long time yet put my _very extended_
investigations concerning this matter in shape for publication,
possibly not while I live, for I fear the cry of the Bœotians
which would arise should I express my whole view on this
matter.--It is curious too, that besides the known gap in
Euclid’s geometry, to fill which all efforts till now have been
in vain, and which will never be filled, there exists another
defect, which to my knowledge no one thus far has criticised and
which (though possible) it is by no means easy to remove. This is
the definition of a plane as a surface which wholly contains the
line joining any two points. This definition contains more than
is necessary to the determination of the surface, and tacitly
involves a theorem which demands proof.--GAUSS.

                         _Letter to Bessel (1829); Werke, Bd. 8
                         (Göttingen, 1900), p. 200._


=2028.= I will add that I have recently received from Hungary a
little paper on Non-Euclidean geometry, in which I rediscover all
_my own ideas_ and _results_ worked out with great elegance, ....
The writer is a very young Austrian officer, the son of one of my
early friends, with whom I often discussed the subject in 1798,
although my ideas were at that time far removed from the
development and maturity which they have received through the
original reflections of this young man. I consider the young
geometer v. Bolyai a genius of the first rank.--GAUSS.

                         _Letter to Gerling (1832); Werke, Bd. 8
                         (Göttingen, 1900), p. 221._


=2029.= Think of the image of the world in a convex mirror.... A
well-made convex mirror of moderate aperture represents the
objects in front of it as apparently solid and in fixed positions
behind its surface. But the images of the distant horizon and of
the sun in the sky lie behind the mirror at a limited distance,
equal to its focal length. Between these and the surface of the
mirror are found the images of all the other objects before it,
but the images are diminished and flattened in proportion to the
distance of their objects from the mirror.... Yet every straight
line or plane in the outer world is represented by a straight [?]
line or plane in the image. The image of a man measuring with a
rule a straight line from the mirror, would contract more and
more the farther he went, but with his shrunken rule the man in
the image would count out exactly the same number of centimeters
as the real man. And, in general, all geometrical measurements of
lines and angles made with regularly varying images of real
instruments would yield exactly the same results as in the outer
world, all lines of sight in the mirror would be represented by
straight lines of sight in the mirror. In short, I do not see how
men in the mirror are to discover that their bodies are not rigid
solids and their experiences good examples of the correctness of
Euclidean axioms. But if they could look out upon our world as
we look into theirs without overstepping the boundary, they must
declare it to be a picture in a spherical mirror, and would speak
of us just as we speak of them; and if two inhabitants of the
different worlds could communicate with one another, neither, as
far as I can see, would be able to convince the other that he had
the true, the other the distorted, relation. Indeed I cannot see
that such a question would have any meaning at all, so long as
mechanical considerations are not mixed up with it.--HELMHOLTZ, H.

                         _On the Origin and Significance of
                         Geometrical Axioms; Popular Scientific
                         Lectures, second series (New York,
                         1881), pp. 57-59._


=2030.= That space conceived of as a locus of points has but
three dimensions needs no argument from the mathematical point of
view; but just as little can we from this point of view prevent
the assertion that space has really four or an infinite number of
dimensions though we perceive only three. The theory of
multiply-extended manifolds, which enters more and more into the
foreground of mathematical research, is from its very nature
perfectly independent of such an assertion. But the form of
expression, which this theory employs, has indeed grown out of
this conception. Instead of referring to the individuals of a
manifold, we speak of the points of a higher space, etc. In
itself this form of expression has many advantages, in that it
facilitates comprehension by calling up geometrical intuition.
But it has this disadvantage, that in extended circles,
investigations concerning manifolds of any number of dimensions
are considered singular alongside the above-mentioned conception
of space. This view is without the least foundation. The
investigations in question would indeed find immediate geometric
applications if the conception were valid but its value and
purpose, being independent of this conception, rests upon its
essential mathematical content.--KLEIN, F.

                         _Mathematische Annalen, Bd. 43 (1893),
                         p. 95._


=2031.= We are led naturally to extend the language of geometry
to the case of any number of variables, still using the word
_point_ to designate any system of values of n variables (the
coördinates of the point), the word _space_ (of n dimensions) to
designate the totality of all these points or systems of values,
_curves_ or _surface_ to designate the spread composed of
points whose coördinates are given functions (with the proper
restrictions) of one or two parameters (the _straight line_ or
_plane_, when they are linear fractional functions with the same
denominator), etc. Such an extension has come to be a necessity
in a large number of investigations, in order as well to give
them the greatest generality as to preserve in them the intuitive
character of geometry. But it has been noted that in such use of
geometric language we are no longer constructing truly a
geometry, for the forms that we have been considering are
essentially analytic, and that, for example, the general
projective geometry constructed in this way is in substance
nothing more than the algebra of linear transformations.

                                                --SEGRE, CORRADI.

                         _Rivista di Matematica, Vol. I (1891),
                         p. 59. [J. W. Young.]_


=2032.= Those who can, in common algebra, find a square root of
−1, will be at no loss to find a fourth dimension in space in
which ABC may become ABCD: or, if they cannot find it, they have
but to imagine it, and call it an _impossible_ dimension, subject
to all the laws of the three we find possible. And just as √−1 in
common algebra, gives all its _significant_ combinations _true_,
so would it be with any number of dimensions of space which the
speculator might choose to call into _impossible_ existence.

                                                   --DE MORGAN, A.

                         _Trigonometry and Double Algebra
                         (London, 1849), Part 2, chap. 3._


=2033.= The doctrine of non-Euclidean spaces and of hyperspaces
in general possesses the highest intellectual interest, and it
requires a far-sighted man to foretell that it can never have any
practical importance.--SMITH, W. B.

                         _Introductory Modern Geometry (New York,
                         1893), p. 274._


=2034.= According to his frequently expressed view, Gauss considered
the three dimensions of space as specific peculiarities of the
human soul; people, which are unable to comprehend this, he
designated in his humorous mood by the name Bœotians. We could
imagine ourselves, he said, as beings which are conscious of but
two dimensions; higher beings might look at us in a like manner,
and continuing jokingly, he said that he had laid aside certain
problems which, when in a higher state of being, he hoped to
investigate geometrically.--SARTORIUS, W. V. WALTERSHAUSEN.

                         _Gauss zum Gedächtniss (Leipzig, 1856),
                         p. 81._


=2035.= _There is many a rational logos_, and the mathematician
has high delight in the contemplation of _in_consistent _systems_
of _consistent relationships_. There are, for example, a
Euclidean geometry and more than one species of non-Euclidean. As
theories of a given space, these are not compatible. If our
universe be, as Plato thought, and nature-science takes for
granted, a space-conditioned, geometrised affair, one of these
geometries may be, none of them may be, not all of them can be,
valid in it. But in the vaster world of thought, all of them are
valid, there they co-exist, and interlace among themselves and
others, as differing component strains of a higher, strictly
supernatural, hypercosmic, harmony.--KEYSER, C. J.

                         _The Universe and Beyond; Hibbert
                         Journal, Vol. 3 (1904-1905), p. 313._


=2036.= The introduction into geometrical work of conceptions
such as the infinite, the imaginary, and the relations of
hyperspace, none of which can be directly imagined, has a
psychological significance well worthy of examination. It gives a
deep insight into the resources and working of the human mind. We
arrive at the borderland of mathematics and psychology.

                                                    --MERZ, J. T.

                         _History of European Thought in the
                         Nineteenth Century (Edinburgh and
                         London, 1903), p. 716._


=2037.= Among the splendid generalizations effected by modern
mathematics, there is none more brilliant or more inspiring or
more fruitful, and none more commensurate with the limitless
immensity of being itself, than that which produced the great
concept designated ... hyperspace or multidimensional space.

                                                  --KEYSER, C. J.

                         _Mathematical Emancipations; Monist,
                         Vol. 16 (1906), p. 65._


=2038.= The great generalization [of hyperspace] has made it
possible to enrich, quicken and beautify analysis with the terse,
sensuous, artistic, stimulating language of geometry. On the
other hand, the hyperspaces are in themselves immeasurably
interesting and inexhaustibly rich fields of research. Not only
does the geometrician find light in them for the illumination of
otherwise dark and undiscovered properties of ordinary spaces of
intuition, but he also discovers there wondrous structures quite
unknown to ordinary space.... It is by creation of hyperspaces
that the rational spirit secures release from limitation. In them
it lives ever joyously, sustained by an unfailing sense of
infinite freedom.--KEYSER, C. J.

                         _Mathematical Emancipations; Monist,
                         Vol. 16 (1906), p. 83._


=2039.= Mathematicians who busy themselves a great deal with the
formal theory of four-dimensional space, seem to acquire a capacity
for imagining this form as easily as the three-dimensional form
with which we are all familiar.--OSTWALD, W.

                         _Natural Philosophy [Seltzer], (New
                         York, 1910), p. 77._


=2040.=

        Fuchs. Was soll ich nun aber denn studieren?

        Meph.  Ihr könnt es mit _analytischer Geometrie_ probieren.
               Da wird der Raum euch wohl dressiert,
               In Coordinaten eingeschnürt,
               Dass ihr nicht etwa auf gut Glück
               Von der Figur gewinnt ein Stück.
               Dann lehret man euch manchen Tag,
               Dass, was ihr sonst auf einen Schlag
               Construiertet im Raume frei,
               Eine Gleichung dazu nötig sei.
               Zwar war dem Menschen zu seiner Erbauung
               Die dreidimensionale Raumanschauung,
               Dass er sieht, was um ihn passiert,
               Und die Figuren sich construiert--
               Der Analytiker tritt herein
               Und beweist, das könnte auch anders sein.
               Gleichungen, die auf dem Papiere stehn,
               Die müsst’ man auch können im Raume sehn;
               Und könnte man’s nicht construieren,
               Da müsste man’s anders definieren.
               Denn was man formt nach Zahlengesetzen
               Müsst’ uns auch geometrisch erletzen.
               Drum in den unendlich fernen beiden
               Imaginären Punkten müssen sich schneiden
               Alle Kreise fein säuberlich,
               Auch Parallelen, die treffen sich,
               Und im Raume kann man daneben
               Allerlei Krümmungsmasse erleben.
               Die Formeln sind alle wahr und schön,
               Warum sollen sie nicht zu deuten gehn?
               Da preisen’s die Schüler aller Orten,
               Dass das Gerade ist krumm geworden.
               _Nicht-Euklidisch_ nennt’s die Geometrie,
               Spotted ihrer selbst, und weiss nicht wie.

        Fuchs. Kann euch nicht eben ganz verstehn.

        Meph.  Das soll den Philosophen auch so gehn.
               Doch wenn ihr lernt alles reducieren
               Und gehörig transformieren,
               Bis die Formeln den Sinn verlieren,
               Dann versteht ihr mathematish zu spekulieren.
                                               --LASSWITZ, KURD.

                         _Der Faust-Tragödie (-n)ter Teil;
                         Zeitschrift für den math-naturw.
                         Unterricht, Bd. 14 (1888), p. 316._

        [Fuchs. To what study then should I myself apply?

         Meph.  Begin with _analytical geometry_.
                There all space is properly trained,
                By coördinates well restrained,
                That no one by some lucky assay
                Carry some part of the figure away.
                Next thou’ll be taught to realize,
                Constructions won’t help thee to geometrize,
                And the result of a free construction
                Requires an equation for proper deduction.
                Three-dimensional space relation
                Exists for human edification,
                That he may see what about him transpires,
                And construct such figures as he requires.
                Enters the analyst. Forthwith you see
                That all this might otherwise be.
                Equations, written with pencil or pen,
                Must be visible in space, and when
                Difficulties in construction arise,
                We need only define it otherwise.
                For, what is formed after laws arithmetic
                Must also yield some delight geometric.
                Therefore we must not object
                That all circles intersect
                In the circular points at infinity.
                And all parallels, they declare,
                If produced must meet somewhere.
                So in space, it can’t be denied,
                Any old curvature may abide.
                The formulas are all fine and true,
                Then why should they not have a meaning too?
                Pupils everywhere praise their fate
                That that now is crooked which once was straight.
                Non-Euclidean, in fine derision,
                Is what it’s called by the geometrician.

         Fuchs. I do not fully follow thee.

         Meph.  No better does philosophy.
                To master mathematical speculation,
                Carefully learn to reduce your equation
                By an adequate transformation
                Till the formulas are devoid of interpretation.]



  CHAPTER XXI

  PARADOXES AND CURIOSITIES


=2101.= The pseudomath is a person who handles mathematics as a
monkey handles the razor. The creature tried to shave himself as
he had seen his master do; but, not having any notion of the
angle at which the razor was to be held, he cut his own throat.
He never tried it a second time, poor animal! but the pseudomath
keeps on in his work, proclaims himself clean shaved, and all the
rest of the world hairy.

The graphomath is a person who, having no mathematics, attempts
to describe a mathematician. Novelists perform in this way: even
Walter Scott now and then burns his fingers. His dreaming
calculator, Davy Ramsay, swears “by the bones of the immortal
Napier.” Scott thought that the philomaths worshipped relics: so
they do in one sense.--DE MORGAN, A.

                         _Budget of Paradoxes (London, 1872), p.
                         473._


=2102.= Proof requires a person who can give and a person who can
receive....

        A blind man said, As to the Sun,
        I’ll take my Bible oath there’s none;
        For if there had been one to show
        They would have shown it long ago.
        How came he such a goose to be?
        Did he not know he couldn’t see?
                             Not he.
                                             --DE MORGAN, A.

                         _Budget of Paradoxes (London, 1872), p.
                         262._


=2103.= Mathematical research, with all its wealth of hidden
treasure, is all too apt to yield nothing to our research: for it
is haunted by certain _ignes fatui_--delusive phantoms, that
float before us, and seem so fair, and are _all but_ in our
grasp, so nearly that it never seems to need more than _one_ step
further, and the prize shall be ours! Alas for him who has been
turned aside from real research by one of these spectres--who
has found a music in its mocking laughter--and who wastes his
life and energy in the desperate chase!--DODGSON, C. L.

                         _A new Theory of Parallels (London,
                         1895), Introduction._


=2104.= As lightning clears the air of impalpable vapours, so an
incisive paradox frees the human intelligence from the lethargic
influence of latent and unsuspected assumptions. Paradox is the
slayer of Prejudice.--SYLVESTER, J. J.

                         _On a Lady’s Fan etc. Collected
                         Mathematical Papers, Vol. 3, p. 36._


=2105.= When a paradoxer parades capital letters and diagrams
which are as good as Newton’s to all who know nothing about it,
some persons wonder why science does not rise and triturate the
whole thing. This is why: all who are fit to read the refutation
are satisfied already, and can, if they please, detect the
paradoxer for themselves. Those who are not fit to do this would
not know the difference between the true answer and the new
capitals and diagrams on which the delighted paradoxer would
declare that he had crumbled the philosophers, and not they him.

                                                  --DE MORGAN, A.

                         _A Budget of Paradoxes (London, 1872),
                         p. 484._


=2106.= Demonstrative reason never raises the cry of _Church in
Danger!_ and it cannot have any Dictionary of heresies except a
Budget of Paradoxes. Mistaken claimants are left to Time and his
extinguisher, with the approbation of all non-claimants: there is
no need of a succession of exposures. Time gets through the job
in his own workmanlike manner.--DE MORGAN, A.

                         _A Budget of Paradoxes (London, 1872),
                         p. 485._


=2107.= D’Israeli speaks of the “six follies of science,”--the
quadrature, the duplication, the perpetual motion, the philosopher’s
stone, magic, and astrology. He might as well have added the
trisection, to make the mystic number seven; but had he done so,
he would still have been very lenient; only seven follies in all
science, from mathematics to chemistry! Science might have said
to such a judge--as convicts used to say who got seven years,
expecting it for life, “Thank you, my Lord, and may you sit there
until they are over,”--may the Curiosities of Literature outlive
the Follies of Science!--DE MORGAN, A.

                         _A Budget of Paradoxes (London, 1872),
                         p. 71._


=2108.= Montucla says, speaking of France, that he finds three
notions prevalent among cyclometers: 1. That there is a large
reward offered for success; 2. That the longitude problem depends
on that success; 3. That the solution is the great end and object
of geometry. The same three notions are equally prevalent among
the same class in England. No reward has ever been offered by the
government of either country. The longitude problem in no way
depends upon perfect solution; existing approximations are
sufficient to a point of accuracy far beyond what can be wanted.
And geometry, content with what exists, has long passed on to
other matters. Sometimes a cyclometer persuades a skipper who has
made land in the wrong place that the astronomers are at fault,
for using a wrong measure of the circle; and the skipper thinks
it a very comfortable solution! And this is the utmost that the
problem has to do with longitude.--DE MORGAN, A.

                         _A Budget of Paradoxes (London, 1872),
                         p. 96._


=2109.= Gregory St. Vincent is the greatest of circle-squarers,
and his investigations led him into many truths: he found the
property of the arc of the hyperbola which led to Napier’s
logarithms being called hyperbolic. Montucla says of him, with
sly truth, that no one ever squared the circle with so much
genius, or, excepting his principal object, with so much success.

                                                  --DE MORGAN, A.

                         _A Budget of Paradoxes (London, 1872),
                         p. 70._


=2110.= When I reached geometry, and became acquainted with the
proposition the proof of which has been sought for centuries, I
felt irresistibly impelled to try my powers at its discovery. You
will consider me foolish if I confess that I am still earnestly
of the opinion to have succeeded in my attempt.--BOLZANO, BERNARD.

                         _Selbstbiographie (Wien, 1875), p. 19._


=2111.= The Theory of Parallels.

It is known that to complete the theory it is only necessary to
demonstrate the following proposition, which Euclid assumed as an
axiom:

Prop. If the sum of the interior angles ECF and DBC which two
straight lines EC and DB make with a third line CP is less than
two right angles, the lines, if sufficiently produced, will
intersect.

[Illustration: A geometrical drawing of parallel lines and
intersecting lines to accompany proof.]

Proof. Construct PCA equal to the supplement PBD of CBD, and ECF,
FCG, etc. each equal to ACE, so that ACF = 2.ACE, ACG = 3.ACE,
etc. Then however small the angle ACE may be, there exists some
number n such that n.ACE = ACH will be equal to or greater than
ACP.

Again, take BI, IL, etc. each equal to CB, and draw IK, LM, etc.
parallel to BD, then the figures ACBD, DBIK, KILM, etc. are
congruent, and ACIK = 2.ABCD, ACLM = 3.ACBD, etc.

Take ACNO = n.ACBD, n having the same value as in the expression
ACH = n.ACE, then ACNO is certainly less than ACP, since ACNO
must be increased by ONP to be equal to ACP. It follows that ACNO
is also less than ACH, and by taking the nth part of each of
these, that ACBD is less than ACE.

But if ACE is greater than ACBD, CE and BD must intersect, for
otherwise ACE would be a part of ACBD.

                         _Journal für Mathematik, Bd. 2 (1834),
                         p. 198._


=2112.= Are you sure that it is impossible to trisect the angle
by _Euclid_? I have not to lament a single hour thrown away on
the attempt, but fancy that it is rather a tact, a feeling, than
a proof, which makes us think that the thing cannot be done. But
would _Gauss’s_ inscription of the regular polygon of seventeen
sides have seemed, a century ago, much less an impossible thing,
by line and circle?--HAMILTON, W. R.

                         _Letter to De Morgan (1852)._


=2113.= One of the most curious of these cases [geometrical
paradoxers] was that of a student, I am not sure but a graduate,
of the University of Virginia, who claimed that geometers were in
error in assuming that a line had no thickness. He published
a school geometry based on his views, which received the
endorsement of a well-known New York school official and, on the
basis of this, was actually endorsed, or came very near being
endorsed, as a text-book in the public schools of New York.

                                                --NEWCOMB, SIMON.

                         _The Reminiscences of an Astronomer
                         (Boston and New York, 1903), p. 388._


=2114.= What distinguishes the straight line and circle more
than anything else, and properly separates them for the purpose
of elementary geometry? Their self-similarity. Every inch of
a straight line coincides with every other inch, and off a
circle with every other off the same circle. Where, then, did
Euclid fail? In not introducing the third curve, which has the
same property--the _screw_. The right line, the circle, the
screw--the representations of translation, rotation, and the two
combined--ought to have been the instruments of geometry. With
a screw we should never have heard of the impossibility of
trisecting an angle, squaring the circle, etc.--DE MORGAN, A.

                         _Quoted in Graves’ Life of Sir W. R.
                         Hamilton, Vol. 3 (New York, 1889), p.
                         342._


=2115.=

        Mad Mathesis alone was unconfined,
        Too mad for mere material chains to bind,
        Now to pure space lifts her ecstatic stare,
        Now, running round the circle, finds it square.
                                          --POPE, ALEXANDER.

                         _The Dunciad, Bk. 4, lines 31-34._


=2116.=

        Or is’t a tart idea, to procure
        An edge, and keep the practic soul in ure,
        Like that dear Chymic dust, or puzzling quadrature?
                                          --QUARLES, PHILIP.

                         _Quoted by De Morgan: Budget of
                         Paradoxes (London, 1872), p. 436._


=2117.=

        Quale è’l geometra che tutto s’ affige
        Per misurar lo cerchio, e non ritruova,
        Pensando qual principio ond’ egli indige.
                                                    --DANTE.

                         _Paradise, canto 33, lines 122-125._

        [As doth the expert geometer appear
         Who seeks to square the circle, and whose skill
         Finds not the law with which his course to steer.[12]]

                         _Quoted in Frankland’s Story of Euclid
                         (London, 1902), p. 101._

    [12]  For another rendition of these same lines see
          1858.


=2118.=

        In _Mathematicks_ he was greater
        Than _Tycho Brahe_, or _Erra Pater_:
        For he, by _Geometrick_ scale,
        Could take the size of _Pots of Ale_;
        Resolve by Signs and Tangents streight,
        If _Bread_ or _Butter_ wanted weight;
        And wisely tell what hour o’ th’ day
        The Clock doth strike, by _Algebra_.
                                           --BUTLER, SAMUEL.

                         _Hudibras, Part 1, canto 1, lines_
                         119-126.


=2119.= I have often been surprised that Mathematics, the
quintessence of truth, should have found admirers so few and so
languid. Frequent considerations and minute scrutiny have at
length unravelled the cause; viz. that though Reason is feasted,
Imagination is starved; whilst Reason is luxuriating in its
proper Paradise, Imagination is wearily travelling on a dreary
desert.--COLERIDGE, SAMUEL.

                         _A Mathematical Problem._


=2120.= At last we entered the palace, and proceeded into the
chamber of presence where I saw the king seated on his throne,
attended on each side by persons of prime quality. Before the
throne, was a large table filled with globes and spheres, and
mathematical instruments of all kinds. His majesty took not the
least notice of us, although our entrance was not without
sufficient noise, by the concourse of all persons belonging to
the court. But he was then deep in a problem, and we attended an
hour, before he could solve it. There stood by him, on each side,
a young page with flaps in their hands, and when they saw he was
at leisure, one of them gently struck his mouth, and the other
his right ear; at which he started like one awaked on the sudden,
and looking toward me and the company I was in, recollected the
occasion of our coming, whereof he had been informed before. He
spake some words, whereupon immediately a young man with a flap
came to my side, and flapt me gently on the right ear, but I made
signs, as well as I could, that I had no occasion for such an
instrument; which, as I afterwards found, gave his majesty, and
the whole court, a very mean opinion of my understanding. The
king, as far as I could conjecture, asked me several questions,
and I addressed myself to him in all the languages I had. When it
was found, that I could neither understand nor be understood, I
was conducted by his order to an apartment in his palace, (this
prince being distinguished above all his predecessors, for his
hospitality to strangers) where two servants were appointed to
attend me. My dinner was brought, and four persons of quality,
did me the honour to dine with me. We had two courses of three
dishes each. In the first course, there was a shoulder of mutton
cut into an equilateral triangle, a piece of beef into a
rhomboides, and a pudding into a cycloid. The second course, was,
two ducks trussed up in the form of fiddles; sausages and
puddings, resembling flutes and haut-boys, and a breast of veal
in the shape of a harp. The servants cut our bread into cones,
cylinders, parallelograms, and several other mathematical
figures.--SWIFT, JONATHAN.

                         _Gulliver’s Travels; A Voyage to Laputa;
                         Chap. 2._


=2121.= Those to whom the king had entrusted me, observing how
ill I was clad, ordered a taylor to come next morning, and take
measure for a suit of cloaths. This operator did his office
after a different manner, from those of his trade in Europe. He
first took my altitude by a quadrant, and then, with rule and
compasses, described the dimensions and outlines of my whole
body, all which he entered upon paper; and in six days, brought
my cloaths very ill made, and quite out of shape, by happening to
mistake a figure in the calculation. But my comfort was, that I
observed such accidents very frequent, and little regarded.

                                               --SWIFT, JONATHAN.

                         _Gulliver’s Travels; A Voyage to Laputa,
                         Chap. 2._


=2122.= The knowledge I had in mathematics, gave me great
assistance in acquiring their phraseology, which depended much
upon that science, and music; and in the latter I was not
unskilled. Their ideas are perpetually conversant in lines and
figures. If they would, for example, praise the beauty of a
woman, or any other animal, they describe it by rhombs, circles,
parallelograms, ellipses, and other geometrical terms, or by
words of art drawn from music, needless here to repeat. I
observed in the king’s kitchen all sorts of mathematical and
musical instruments, after the figures of which, they cut up the
joints that were served to his majesty’s table.--SWIFT, JONATHAN.

                         _Gulliver’s Travels; A Voyage to Laputa,
                         Chap. 2._


=2123.= I was at the mathematical school, where the master taught
his pupils, after a method, scarce imaginable to us in Europe.
The propositions, and demonstrations, were fairly written on a
thin wafer, with ink composed of a cephalic tincture. This, the
student was to swallow upon a fasting stomach, and for three days
following, eat nothing but bread and water. As the wafer
digested, the tincture mounted to his brain, bearing the
proposition along with it. But the success has not hitherto been
answerable, partly by some error in the _quantum_ or composition,
and partly by the perverseness of lads; to whom this bolus is so
nauseous, that they generally steal aside, and discharge it
upwards, before it can operate; neither have they been yet
persuaded to use so long an abstinence as the prescription
requires.--SWIFT, JONATHAN.

                         _Gulliver’s Travels; A Voyage to Laputa,
                         Chap. 5._


=2124.= It is worth observing that some of those who disparage
some branch of study in which they are deficient, will often
affect more contempt for it than they really feel. And not
unfrequently they will take pains to have it thought that they
are themselves well versed in it, or that they easily might be,
if they thought it worth while;--in short, that it is not from
hanging too high that the grapes are called sour.

Thus, Swift, in the person of Gulliver, represents himself, while
deriding the extravagant passion for Mathematics among the
Laputians, as being a good mathematician. Yet he betrays his
utter ignorance, by speaking “of a pudding in the _form of a
cycloid_:” evidently taking the cycloid for a _figure_, instead
of a _line_. This may help to explain the difficulty he is said
to have had in obtaining his Degree.--WHATELY, R.

                         _Annotations to Bacon’s Essays, Essay
                         L._


=2125.= It is natural to think that an abstract science cannot be
of much importance in the affairs of human life, because it has
omitted from its consideration everything of real interest. It
will be remembered that Swift, in his description of Gulliver’s
voyage to Laputa, is of two minds on this point. He describes the
mathematicians of that country as silly and useless dreamers,
whose attention has to be awakened by flappers. Also, the
mathematical tailor measures his height by a quadrant, and
deduces his other dimensions by a rule and compasses, producing a
suit of very ill-fitting clothes. On the other hand, the
mathematicians of Laputa, by their marvellous invention of the
magnetic island floating in the air, ruled the country and
maintained their ascendency over their subjects. Swift, indeed,
lived at a time peculiarly unsuited for gibes at contemporary
mathematicians. Newton’s _Principia_ had just been written, one
of the great forces which have transformed the modern world.
Swift might just as well have laughed at an earthquake.

                                               --WHITEHEAD, A. N.

                         _An Introduction to Mathematics (New
                         York, 1911), p. 10._


=2126.= [Illustration: A geometrical drawing including square and
four triangles to demonstrate a graphical proof of the theorem of
Pythagoras as described in the poem.]

        Here I am as you may see
        a² + b² - ab
        When two Triangles on me stand
        Square of hypothen^e is plann’d
        But if I stand on them instead,
        The squares of both the sides are read.
                                               --AIRY, G. B.

                         _Quoted in Graves’ Life of Sir W. R.
                         Hamilton, Vol. 3 (New York, 1889), p.
                         502._


=2127.= π = 3.141 592 653 589 793 238 462 643 383 279 ...

         3  1    4  1    5       9
        Now I, even I, would celebrate
         2    6     5     3    5
        In rhymes inapt, the great
           8         9         7        9
        Immortal Syracusan, rivaled nevermore,
         3   2  3      8      4
        Who in his wondrous lore,
           6    2    6
        Passed on before,
          4   3   3      8     3   2    7        9
        Left men his guidance how to circles mensurate.
                                                --ORR, A. C.

                         _Literary Digest, Vol. 32 (1906), p.
                         84._


=2128.= I take from a biographical dictionary the first five
names of poets, with their ages at death. They are

                    Aagard,  died at 48.
                    Abeille,  “   “  76.
                    Abulola,  “   “  84.
                    Abunowas, “   “  48.
                    Accords,  “   “  45.

These five ages have the following characters in common:--

1. The difference of the two digits composing the number divided
by _three_, leaves a remainder of _one_.

2. The first digit raised to the power indicated by the second,
and then divided by _three_, leaves a remainder of _one_.

3. The sum of the prime factors of each age, including _one_ as a
prime factor, is divisible by _three_.--PEIRCE, C. S.

                         _A Theory of Probable Inference; Studies
                         in Logic (Boston, 1883), p. 163._


=2129.= In view of the fact that the offered prize [for the
solution of the problem of Fermat’s Greater Theorem] is about
$25,000 and that lack of marginal space in his copy of Diophantus
was the reason given by Fermat for not communicating his proof,
one might be tempted to wish that one could send credit for a
dime back through the ages to Fermat and thus secure this coveted
prize, if it actually existed. This might, however, result more
seriously than one would at first suppose; for if Fermat had
bought on credit a dime’s worth of paper even during the year of
his death, 1665, and if this bill had been drawing compound
interest at the rate of six per cent, since that time, the bill
would now amount to more than seven times as much as the prize.

                                                  --MILLER, G. A.

                         _Some Thoughts on Modern Mathematical
                         Research; Science, Vol. 35 (1912), p.
                         881._


=2130.= _If the Indians hadn’t spent the $24._ In 1626 Peter
Minuit, first governor of New Netherland, purchased Manhattan
Island from the Indians for about $24. The rate of interest on
money is higher in new countries, and gradually decreases as
wealth accumulates. Within the present generation the legal rate
in the state has fallen from 7% to 6%. Assume for simplicity a
uniform rate of 7% from 1626 to the present, and suppose that the
Indians had put their $24 at interest at that rate (banking
facilities in New York being always taken for granted!) and had
added the interest to the principal yearly. What would be the
amount now, after 280 years? 24 × (1.07)^{280} = more than
4,042,000,000.

The latest tax assessment available at the time of writing gives
the realty for the borough of Manhattan as $3,820,754.181. This
is estimated to be 78% of the actual value, making the actual
value a little more than $4,898,400,000.

The amount of the Indians’ money would therefore be more than the
present assessed valuation but less than the actual valuation.

                                                   --WHITE, W. F.

                         _A Scrap-book of Elementary Mathematics
                         (Chicago, 1908), pp. 47-48._


=2131.= See Mystery to Mathematics fly!--POPE, ALEXANDER.

                         _The Dunciad, Bk. 4, line 647._


=2132.= The Pythagoreans and Platonists were carried further by
this love of simplicity. Pythagoras, by his skill in mathematics,
discovered that there can be no more than five regular solid
figures, terminated by plane surfaces which are all similar and
equal; to wit, the tetrahedron, the cube, the octahedron, the
dodecahedron, and the eicosihedron. As nature works in the most
simple and regular way, he thought that all elementary bodies
must have one or other of those regular figures; and that the
discovery of the properties and relations of the regular solids
must be a key to open the mysteries of nature.

This notion of the Pythagoreans and Platonists has undoubtedly
great beauty and simplicity. Accordingly it prevailed, at least
to the time of Euclid. He was a Platonic philosopher, and is said
to have wrote all the books of his Elements, in order to discover
the properties and relations of the five regular solids. The
ancient tradition of the intention of Euclid in writing his
elements, is countenanced by the work itself. For the last book
of the elements treats of the regular solids, and all the
preceding are subservient to the last.--REID, THOMAS.

                         _Essays on the Powers of the Human Mind
                         (Edinburgh, 1812), Vol. 2, p. 400._


=2133.= In the Timæus [of Plato] it is asserted that the
particles of the various elements have the forms of these [the
regular] solids. Fire has the Pyramid; Earth has the Cube; Water
the Octahedron; Air the Icosahedron; and the Dodecahedron is the
plan of the Universe itself. It was natural that when Plato had
learnt that other mathematical properties had a bearing upon
the constitution of the Universe, he should suppose that
the singular property of space, which the existence of this
limited and varied class of solids implied, should have some
corresponding property in the Universe, which exists in space.

                                                    --WHEWELL, W.

                         _History of the Inductive Sciences, 3rd
                         Edition, Additions to Bk. 2._


=2134.= The orbit of the earth is a circle: round the sphere to
which this circle belongs, describe a dodecahedron; the sphere
including this will give the orbit of Mars. Round Mars describe a
tetrahedron; the circle including this will be the orbit of
Jupiter. Describe a cube round Jupiter’s orbit; the circle
including this will be the orbit of Saturn. Now inscribe in the
earth’s orbit an icosahedron; the circle inscribed in it will be
the orbit of Venus. Inscribe an octahedron in the orbit of Venus;
the circle inscribed in it will be Mercury’s orbit. This is the
reason of the number of the planets.--KEPLER.

                         _Mysterium Cosmographicum [Whewell]._


=2135.= It will not be thought surprising that Plato expected
that Astronomy, when further advanced, would be able to render an
account of many things for which she has not accounted even to
this day. Thus, in the passage in the seventh Book of the
_Republic_, he says that the philosopher requires a reason for
the proportion of the day to the month, and the month to the
year, deeper and more substantial than mere observation can give.
Yet Astronomy has not yet shown us any reason why the proportion
of the times of the earth’s rotation on its axis, the moon’s
revolution round the earth, and the earth’s revolution round the
sun, might not have been made by the Creator quite different from
what they are. But in asking Mathematical Astronomy for reasons
which she cannot give, Plato was only doing what a great
astronomical discoverer, Kepler, did at a later period. One of
the questions which Kepler especially wished to have answered
was, why there are five planets, and why at such particular
distances from the sun? And it is still more curious that he
thought he had found the reason of these things, in the relation
of those five regular solids which Plato was desirous of
introducing into the philosophy of the universe.... Kepler
regards the law which thus determines the number and magnitude of
the planetary orbits by means of the five regular solids as a
discovery no less remarkable and certain than the Three Laws
which give his name its imperishable place in the history of
astronomy.--WHEWELL, W.

                         _History of the Inductive Sciences, 3rd
                         Edition, Additions to Bk. 3._


=2136.= Pythagorean philosophers ... maintained that of two
combatants, he would conquer, the sum of the numbers expressed by
the characters of whose names exceeded the sum of those expressed
by the other. It was upon this principle that they explained the
relative prowess and fate of the heroes in Homer, Πατροκλος, Ἑκτορ
and Αχιλλευς, the sum of the numbers in whose names are 861, 1225,
and 1276 respectively.--PEACOCK, GEORGE.

                         _Encyclopedia of Pure Mathematics
                         (London, 1847); Article “Arithmetic,”
                         sect. 38._


=2137.= Round numbers are always false.--JOHNSON, SAMUEL.

                         _Johnsoniana; Apothegms, Sentiment,
                         etc._


=2138.= Numero deus impare gaudet [God in number odd rejoices.]

                                                        --VIRGIL.

                         _Eclogue, 8, 77._


=2139.= Why is it that we entertain the belief that for every
purpose odd numbers are the most effectual?--PLINY.

                         _Natural History, Bk. 28, chap. 5._


=2140.=

        “Then here goes another,” says he, “to make sure,
         Fore there’s luck in odd numbers,” says Rory O’Moore.
                                                  --LOVER, S.

                         _Rory O’Moore._


=2141.= This is the third time; I hope, good luck lies in odd
numbers.... They say, there is divinity in odd numbers, either in
nativity, chance, or death.--SHAKESPEARE.

                         _The Merry Wives of Windsor, Act 5,
                         scene 1._


=2142.= To add to golden numbers, golden numbers.--DECKER, THOMAS.

                         _Patient Grissell, Act 1, scene 1._


=2143.=

        I’ve read that things inanimate have moved,
        And, as with living souls, have been inform’d,
        By magic numbers and persuasive sound.
                                        --CONGREVE, RICHARD.

                         _The Morning Bride, Act 1, scene 1._


=2144.= ... the Yancos on the Amazon, whose name for three is

                     Poettarrarorincoaroac,

of a length sufficiently formidable to justify the remark of La
Condamine: Heureusement pour ceux qui ont à faire avec eux, leur
Arithmetique ne va pas plus loin.--PEACOCK, GEORGE.

                         _Encyclopedia of Pure Mathematics
                         (London, 1847); Article “Arithmetic,”
                         sect. 32._


=2145.= There are three principal sins, avarice, luxury, and
pride; three sorts of satisfaction for sin, fasting, almsgiving,
and prayer; three persons offended by sin, God, the sinner
himself, and his neighbour; three witnesses in heaven, _Pater_,
_verbum_, and _spiritus sanctus_; three degrees of penitence,
contrition, confession, and satisfaction, which Dante has
represented as the three steps of the ladder that lead to
purgatory, the first marble, the second black and rugged stone,
and the third red porphyry. There are three sacred orders
in the church militant, _sub-diaconati_, _diaconiti_, and
_presbyterati_; there are three parts, not without mystery, of
the most sacred body made by the priest in the mass; and three
times he says _Agnus Dei_, and three times, _Sanctus_; and if we
well consider all the devout acts of Christian worship, they are
found in a ternary combination; if we wish rightly to partake of
the holy communion, we must three times express our contrition,
_Domine non sum dignus_; but who can say more of the ternary
number in a shorter compass, than what the prophet says, _tu
signaculum sanctae trinitatis_. There are three Furies in the
infernal regions; three Fates, Atropos, Lachesis, and Clotho.
There are three theological virtues: _Fides_, _spes_, and
_charitas_. _Tria sunt pericula mundi: Equum currere; navigare,
et sub tyranno vivere._ There are three enemies of the soul: the
Devil, the world, and the flesh. There are three things which are
of no esteem: the strength of a porter, the advice of a poor man,
and the beauty of a beautiful woman. There are three vows of the
Minorite Friars: poverty, obedience, and chastity. There are
three terms in a continued proportion. There are three ways in
which we may commit sin: _corde_, _ore_, _ope_. Three principal
things in Paradise: glory, riches, and justice. There are three
things which are especially displeasing to God: an avaricious
rich man, a proud poor man, and a luxurious old man. And all
things, in short, are founded in three; that is, in number, in
weight, and in measure.

 --PACIOLI, _Author of the first printed treatise on arithmetic._

                         _Quoted in Encyclopedia of Pure
                         Mathematics (London, 1847); Article
                         “Arithmetic,” sect. 90._


=2146.= Ah! why, ye Gods, should two and two make four?

                                               --POPE, ALEXANDER.

                         _The Dunciad, Bk. 2, line 285._


=2147.=

        By him who stampt _The Four_ upon the mind,--
        _The Four_, the fount of nature’s endless stream.
                                 --_Ascribed to_ PYTHAGORAS.

                         _Quoted in Whewell’s History of the
                         Inductive Sciences, Bk. 4, chap. 3._


=2148.=

        Along the skiey arch the goddess trode,
        And sought Harmonia’s august abode;
        The universal plan, the mystic Four,
        Defines the figure of the palace floor.
        Solid and square the ancient fabric stands,
        Raised by the labors of unnumbered hands.
                                                   --NONNUS.

                         _Dionysiac, 41, 275-280. [Whewell]._


=2149.= The number seventy-seven figures the abolition of all
sins by baptism.... The number ten signifies justice and
beatitude, resulting from the creature, which makes seven with
the Trinity, which is three: therefore it is that God’s
commandments are ten in number. The number eleven denotes sin,
because it _transgresses_ ten.... This number seventy-seven is
the product of eleven, figuring sin, multiplied by seven, and not
by ten, for seven is the number of the creature. Three represents
the soul, which is in some sort an image of Divinity; and four
represents the body, on account of its four qualities....

                                                 --ST. AUGUSTINE.

                         _Sermon 41, art. 23._


=2150.= Heliodorus says that the Nile is nothing else than the
year, founding his opinion on the fact that the numbers expressed
by the letters Νειλος, Nile, are in Greek arithmetic, Ν = 50; Ε =
5; Ι = 10; Λ = 30; Ο = 70; Σ = 200; and these figures make up
together 365, the number of days in the year.

                         _Littell’s Living Age, Vol. 117, p.
                         380._


=2151.= In treating 666, Bungus [Petri Bungi Bergomatis Numerorum
mysteria, Bergamo, 1591] a good Catholic, could not compliment
the Pope with it, but he fixes it on Martin Luther with a little
forcing. If from A to I represent 1-9, from K to S 10-90, and
from T to Z 100-500, we see--

               M A  R  T  I  N    L  U   T  E  R A
              30 1 80 100 9 40   20 200 100 5 80 1

which gives 666. Again in Hebrew, _Lulter_ [Hebraized form of
Luther] does the same:--

                        ר    ת    ל  י   ל
                       200  400  30  6  30
                                             --DE MORGAN, A.

                         _Budget of Paradoxes (London, 1872), p. 37._


=2152.= Stifel, the most acute and original of the early
mathematicians of Germany, ... relates ... that whilst a monk at
Esslingen in 1520, and when infected by the writings of Luther,
he was reading in the library of his convent the 13th Chapter of
_Revelations_, it struck his mind that the _Beast_ must signify
the Pope, Leo X.; He then proceeded in pious hope to make the
calculation of the sum of the numeral letters in _Leo decimus_,
which he found to be M, D, C, L, V, I; the sum which these formed
was too great by M, and too little by X; but he bethought him
again, that he has seen the name written Leo X., and that there
were ten letters in _Leo decimus_, from either of which he could
obtain the deficient number, and by interpreting the M to mean
_mysterium_, he found the number required, a discovery which gave
him such unspeakable comfort, that he believed that his
interpretation must have been an immediate inspiration of God.

                                               --PEACOCK, GEORGE.

                         _Encyclopedia of Pure Mathematics
                         (London, 1847); Article “Arithmetic,”
                         sect. 89._


=2153.= Perhaps the best anagram ever made is that by Dr. Burney
on Horatio Nelson, so happily transformed into the Latin sentence
so truthful of the great admiral, _Honor est a Nilo_. Reading
this, one is almost persuaded that the hit contained in it has a
meaning provided by providence or fate.

This is also amusingly illustrated in the case of the Frenchman
André Pujom, who, using j as i, found in his name the anagram,
Pendu à Riom. Riom being the seat of justice for the province of
Auvergne, the poor fellow, impelled by a sort of infatuation,
actually committed a capital offence in that province, and was
hanged at Riom, that the anagram might be fulfilled.

                         _New American Cyclopedia, Vol. 1;
                         Article “Anagram.”_


=2154.= The most remarkable pseudonym [of transposed names
adopted by authors] is the name of “Voltaire,” which the
celebrated philosopher assumed instead of his family name,
“François Marie Arouet,” and which is now generally allowed to be
an anagram of “Arouet, l. j.,” that is, Arouet the younger.

                         _Encyclopedia Britannica, 11th Edition;
                         Article “Anagram.”_


=2155.= Perhaps the most beautiful anagram that has ever been
composed is by Jablonsky, a former rector of the school at Lissa.
The occasion was the following: When while a young man king
Stanislaus of Poland returned from a journey, the whole house of
Lescinsky assembled to welcome the family heir. On this occasion
Jablonsky arranged for a school program, the closing number of
which consisted of a ballet by thirteen pupils impersonating
youthful heroes. Each of them carried a shield on which appeared
in gold one of the letters of the words _Domus Lescinia_. At the
end of the first dance the children were so arranged that the
letters on their shields spelled the words _Domus Lescinia_. At
the end of the second dance they read: _ades incolumis_ (sound
thou art here). After the third: _omnis es lucida_ (wholly
brilliant art thou); after the fourth: _lucida sis omen_ (bright
be the omen). Then: _mane sidus loci_ (remain our country’s
star); and again: _sis columna Dei_ (be a column of God); and
finally: _I! scande solium_ (Proceed, ascend the throne). This
last was the more beautiful since it proved a true prophecy.

Even more artificial are the anagrams which transform one verse
into another. Thus an Italian scholar beheld in a dream the line
from Horace: _Grata superveniet, quae non sperabitur, hora_. This
a friend changed to the anagram: _Est ventura Rhosina parataque
nubere pigro._ This induced the scholar, though an old man, to
marry an unknown lady by the name of Rosina.--HEIS, EDUARD.

                         _Algebraische Aufgaben (Köln, 1898), p.
                         331._


=2156.= The following verses read the same whether read forward
or backward:--

        Aspice! nam raro mittit timor arma, nec ipsa
          Si se mente reget, non tegeret Nemesis;[13]

also,

                Sator Arepo tenet opera rotas.
                                             --HEIS, EDUARD.

                         _Algebraische Aufgaben (Köln, 1898), p.
                         328._

    [13]  The beginning of a poem which Johannes a Lasco
          wrote on the count Karl von Südermanland.


=2157.= There is a certain spiral of a peculiar form on which a
point may have been approaching for centuries the center, and
have nearly reached it, before we discover that its rate of
approach is accelerated. The first thought of the observer, on
seeing the acceleration, would be to say that it would reach the
center sooner than he had before supposed. But as the point comes
near the center it suddenly, although still moving under the same
simple law as from the beginning, makes a very short turn upon
its path and flies off rapidly almost in a straight line, out to
an infinite distance. This illustrates that apparent breach of
continuity which we sometimes find in a natural law; that
apparently sudden change of character which we sometimes see in
man.--HILL, THOMAS.

                         _Uses of Mathesis; Bibliotheca Sacra,
                         Vol. 32, p. 521._


=2158.= One of the most remarkable of Babbage’s illustrations of
miracles has never had the consideration in the popular mind
which it deserves; the illustration drawn from the existence of
isolated points fulfilling the equation of a curve.... There are
definitions of curves which describe not only the positions
of every point in a certain curve, but also of one or more
perfectly isolated points; and if we should attempt to get by
induction the definition, from the observation of the points on
the curve, we might fail altogether to include these isolated
points; which, nevertheless, although standing alone, as
miracles to the observer of the course of the points in the
curve, are nevertheless rigorously included in the law of the
curve.--HILL, THOMAS.

                         _Uses of Mathesis; Bibliotheca Sacra,
                         Vol. 32, p. 516._


=2159.= Pure mathematics is the magician’s real wand.--NOVALIS.

                         _Schriften, Zweiter Teil (Berlin, 1901),
                         p. 223._


=2160.= Miracles, considered as antinatural facts, are
amathematical, but there are no miracles in this sense, and those
so called may be comprehended by means of mathematics, for to
mathematics nothing is miraculous.--NOVALIS.

                         _Schriften, Zweiter Teil (Berlin, 1911),
                         p. 222._



  INDEX

  =Black-faced numbers refer to authors=

  Abbreviations:--m. = mathematics, math. = mathematical,
                 math’n. = mathematician.


  Abbott, =1001=.

  Abstract method, Development of, 729.

  Abstract nature of m., Reason for, 638.

  Abstract reasoning, Objection to, 1941.

  Abstractness, math., Compared with logical, 1304.

  Adams, Henry, M. and history, =1599=.
    Math’ns practice freedom, =208=, =805=.

  Adams, John, Method in m., =226=.

  Aeneid, Euler’s knowledge of, 859.

  Aeschylus. On number, 1606.

  Aim in teaching m., 501-508, 517, 844.

  Airy, Pythagorean theorem, =2126=.

  Akenside, =1532=.

  Alexander, 901, 902.

  Algebra, Chapter XVII.
    Definitions of, 110, 1714, 1715.
    Problems in, 320, 530, 1738.
    Of use to grown men, 425.
    And geometry, 525-527, 1610, 1707.
    Advantages of, 1701, 1703, 1705.
    Laws of, 1708-1710.
    As an art, 1711.
    Review of, 1713.
    Designations of, 1717.
    Origin of, 1736.
    Burlesque on modern, 1741.
    Hume on, 1863.

  Algebraic notation, value of, 1213, 1214.

  Algebraic treatises, How to read, 601.

  Amusements in m., 904, 905.

  Anagrams, On De Morgan, 947.
    On Domus Lescinia, 2155.
    On Flamsteed, 968.
    On Macaulay, 996.
    On Nelson, 2153.
    On Newton, 1028.
    On Voltaire, 2154.

  Analysis, Invigorates the faculty of resolution, 416.
    Relation of geometry to, 1931.

  Analytical geometry, 1889, 1890, 1893.
    Method of, 310.
    Importance of, 949.
    Burlesque on, 2040.

  Ancient geometry,
    Characteristics of, 712, 714.
    Compared with modern, 1711-1716.
    Method of, 1425, 1873-1875.

  Ancients, M. among the, 321.

  Anecdotes, Chapters, IX, X.

  Anger, M. destroys predisposition to, 458.

  Angling like m., 739.

  Anglo-Danes, Aptitude for m., 836.

  Anglo-Saxons, Aptitude for m., 837.
    Newton as representative of, 1014.

  Anonymous, Song of the screw, 1894.

  Appolonius, 712, 714.

  Approximate m., Why not sufficient, 1518.

  Aptitude for m., 509, 510, 520, 836-838, 976, 1617.

  Arabic notation, 1614.

  Arago, M. the enemy of scientific romances, =267=.
    Euler, “analysis incarnate,” =961=.
    Euler as a computer, =962=.
    On Kepler’s discovery, =982=.
    Newton’s efforts superhuman, =1006=.
    On probabilities, =1591=.
    Geometry as an instrument, 1868.

  Arbuthnot, M. frees from prejudice, credulity and
      superstition, =449=.
    M. the friend of religion, =458=.
    M. compared to music, =1112=.
    On math, reasoning, =1503=.

  Archimedes, His machines, 903, 904.
    Estimate of math, appliances, 904-906, 908.
    Wordsworth on, 906.
    Schiller on, 907.
    And engineering, 908.
    Death of, 909.
    His tomb, 910.
    Compared with Newton, 911.
    Character of his work, 912, 913.
    Applied m., 1312.

  Architecture and m., 276.

  Archytas, 904.
    And Plato, 1427.

  Aristippus the Cyrenaic, 845.

  Aristotle, 914.
    On relation of m. to esthetics, =318=.

  Arithmetic, Chapter XVI.
    Definitions of, 106, 110, 1611, 1612, 1714.
    Emerson on advantage of study of, 408.
    Problems in, 528.
    A master-key, 1571.
    Based on concept of time, 1613.
    Method of teaching, 1618.
    Purpose of teaching, 454, 1624.
    As logic, 1624, 1625.
    The queen of m., 1642.
    Higher, 1755.
    Hume on, 1863.

  Arithmetical theorems, 1639.

  Art, M. as a fine, Chapter XI

  Arts, M. and the, 1568-1570, 1573.

  Astronomy and m., 1554, 1559, 1562-1567.

  “Auge et impera.,” 631.

  Authority in science, 1528.

  Axioms, 518, 2015.
    In geometry, 1812, 2004, 2006.
    Def. in disguise, 2005.
    Euclid’s, 2007-2010, 2014.
    Nature of, 2012.
    Proofs of, 2013.
    And the idea of space, 2004.


  Babbage, 923.

  Bacon, Lord,
    Classification of m., =106=.
    M. makes men subtile, =248=.
    View of m., 316, 915, 916.
    M. held in high esteem by ancients, =321=.
    On the generalizing power of m., =327=.
    On the value of math, studies, =410=.
    M. develops concentration of mind, =411=.
    M. cures distraction of mind, =412=.
    M. essential to study of nature, =436=.
    His view of m., =915=, =916=.
    His knowledge of m., 917, 918.
    M. and logic, =1310=.
    Growth of m., =1511=.

  Bacon, Roger, Neglect of m. works injury to all science,
      =310=.
    On the value of m., =1547=.

  Bain, Importance of m. in education, =442=.
    On the charm of the study of m., =453=.
    M. and science teaching, =522=.
    Teaching of arithmetic, =1618=.

  Ball, R. S., =2010=.

  Ball, W. W. R., On Babbage, =923=.
    On Demoivre’s death, =944=.
    De Morgan and the actuary, =945=.
    Gauss as astronomer, =971=.
    Laplace’s “It is easy to see.” =986=.
    Lagrange, Laplace and Gauss contrasted, =993=.
    Newton’s interest in chemistry and theology, =1015=.
    On Newton’s method of work, =1026=.
    On Newton’s discovery of the calculus, =1027=.
    Gauss’s estimate of Newton, =1029=.
    M. and philosophy, =1417=.
    Advance in physics, =1530=.
    Plato on geometry, 1804.
    Notation of the calculus, =1904=.

  Barnett, M. the type of perfect reasoning, =307=.

  Barrow, On the method of m., =213=, =227=.
    Eulogy of m., =330=.
    M. as a discipline of the mind, =402=.
    M. and eloquence, =830=.
    Philosophy and m., =1430=.
    Uses of m., =1572=.
    On surd numbers, =1728=.
    Euclid’s definition of proportion, =1835=.

  Beattie, =1431=.

  Beauty of m., 453, 824, 1208.
    Consists in simplicity, 242, 315.
    Sylvester on, 1101.
    Russell on, 1104.
    Young on, 1110.
    Kummer on, 1111.
    White on, 1119.
    And truth, 1114.
    Boltzmann on, 1116.

  Beltrami, On reading of the masters, =614=.

  Berkeley, On geometry as logic, =428=.
    On math. symbols, =1214=.
    On fluxions, =1915=, =1942-1944=.
    On infinite divisibility, =1945=.

  Bernoulli, Daniel, 919.

  Bernoulli, James,
    Legend for his tomb, 920, 922.
    Computation of sum of tenth powers of numbers, 921.
    Discussion of logarithmic spiral, 922.

  Berthelot, M. inspires respect for truth, =438=.

  Bija Ganita, Solution of problems, =1739=.

  Billingsley, M. beautifies the mind, =319=.

  Binary arithmetic, 991.

  Biology and m., 1579-1581.

  Biot, Laplace’s “It is easy to see,” 986.

  Bôcher, M. likened to painting, =1103=.
    Interrelation of m. and logic, =1313=.
    Geometry as a natural science, =1866=.

  Boerne, On Pythagoras, =1855=.

  Bois-Reymond, On the analytic method, =1893=.
    Natural selection and the calculus, =1921=.

  Boltzmann, On beauty in m., =1116=.

  Bolyai, Janos,
    Duel with officers, 924.
    Universal language, 925.
    Science absolute of space, 926.

  Bolyai, Wolfgang, 927.
    On Gauss, =972=.

  Bolzano, 928.
    Cured by Euclid, =929=.
    Parallel axiom, =2110=.

  Book-keeping, Importance of the art of, 1571.

  Boole, M. E. =719=.

  Boole’s Laws of Thought, 1318.

  Borda-Demoulins, Philosophy and m., =1405=.

  Boswell, =981=.

  Bowditch, On Laplace’s “Thus it plainly appears,” =985=.

  Boyle, Usefulness of m. to physics, =437=.
    M. and science, =1513=, =1533=.
    Ignorance of m., =1577=.
    M. and physiology, =1582=.
    Wings of m., =1626=.
    Advantages of algebra, =1703=.

  Brahmagupta, Estimate of m., =320=.

  Brewster, On Euler’s knowledge of the Aeneid, =959=.
    On Euler as a computer, =963=.
    On Newton’s fame, =1002=.

  Brougham, =1202=.

  Buckle, On geometry, =1810=, =1837=.

  Burke, On the value of m., =447=.

  Burkhardt, On discovery in m., =618=.
    On universal symbolism, =1221=.

  Butler, N. M., M. demonstrates the supremacy of the human
      reason, =309=.
    M. the most astounding intellectual creation, =707=.
    Geometry before algebra, =1871=.

  Butler, Samuel, =2118=.

  Byerly, On hyperbolic functions, =1929=.


  Cajori, On the value of the history of m., =615=.
    On Bolyai, =927=.
    Cayley’s view of Euclid, =936=.
    On the extent of Euler’s work, =960=.
    On Euler’s math. power, =964=.
    On the Darmstaetter prize, =967=.
    On Sylvester’s first class at Johns Hopkins, =1031=.
    On music and m. among the Pythagoreans, =1130=.
    On the greatest achievement of the Hindoos, =1615=.
    On modern calculation, =1614=.
    On review in arithmetic, =1713=.
    On Indian m., =1737=.
    On the characteristics of ancient geometry, =1873=.
    On Napier’s rule, =1888=.

  Calculating machines, =1641=.

  Calculation, Importance of, 602.
    Not the sole object of m., 268.

  Calculus, Chapter XIX. Foundation of 253.
    As a method, 309.
    May be taught at an early age, 519, 1917, 1918.

  Cambridge m., 836, 1210.

  Cantor, On freedom in m., =205=, =207=.
    On the character of Gauss’s writing, =975=.
    Zeno’s problem, 1938.
    On the infinite, =1952=.

  Carlisle life tables, 946.

  Carnot, On limiting ratios, =1908=.
    On the infinitesimal method, =1907=.

  Carson, Value of geometrical training, =1841=.

  Cartesian method, 1889, 1890.

  Carus, Estimate of m., =326=.
    M. reveals supernatural God, =460=.
    Number and nature, =1603=.
    Zero and infinity, =1948=.
    Non-euclidean geometry, =2016=.

  Cathedral, “Petrified mathematics,” 1110.

  Causation in m., 251, 254.

  Cayley, Advantage of modern geometry over ancient, =711=.
    On the imaginary, =722=.
    Sylvester on, 930.
    Noether on, 931.
    His style, 932.
    Forsyth on, 932-934.
    His method, 933.
    Compared with Euler, 934.
    Hermite on, 935.
    His view of Euclid, 936.
    His estimate of quaternions, 937.
    M. and philosophy, =1420=.

  Certainty of m., 222, 1440-1442, 1628, 1863.

  Chamisso, Pythagorean theorem, =1856=.

  Chancellor, M. develops observation, imagination and
      reason, =433=.

  Chapman, Different aspects of m., =265=.

  Characteristics of m., 225, 229, 247, 263.

  Characteristics of modern m., 720, 724-729.

  Charm in m., 1115, 1640, 1848.

  Chasles, Advantage of modern geometry over ancient, =712=.

  Checks in m., 230.

  Chemistry and m., 1520, 1560, 1561, 1750.

  Chess, M. like, 840.

  Chrystal, Definition of m., =113=.
    Definition of quantity, =115=.
    On problem solving, =531=.
    On modern text-books, =533=.
    How to read m., =607=.
    His algebra, 635.
    On Bernoulli’s numbers, =921=.
    On math. versus logical abstractness, =1304=.
    Rules of algebra, =1710=.
    On universal arithmetic, =1717=.
    On Horner’s method, =1744=.
    On probabilities, =1967=.

  Cicero, Decadence of geometry among Romans, =1807=.

  Circle, Properties of, 1852, 1857.

  Circle-squarers, 2108, 2109.

  Clarke, Descriptive geometry, =1882=.

  Classic problems, Hilbert on, 627.

  Clebsch, On math. research, =644=.

  Clifford, On direct usefulness of math. results, =652=.
    Correspondence the central idea of modern m., =726=.
    His vision, 938.
    His method, 939.
    His knowledge of languages, 940.
    His physical strength, 941.
    On Helmholtz, =979=.
    On m. and mineralogy, =1558=.
    On algebra and good English, =1712=.
    Euclid the encouragement and guide of scientific thought, =1820=.
    Euclid the inspiration and aspiration of scientific thought, =1821=.
    On geometry for girls, =1842=.
    On Euclid’s axioms, =2015=.
    On non-Euclidean geometry, =2022=

  Colburn, 967.

  Coleridge, On problems in m., =534=.
    Proposition, gentle maid, =1419=.
    M. the quintessence of truth, =2019=.

  Colton, On the effect of math. training, =417=.

  Commensurable numbers, 1966.

  Commerce and m., 1571.

  Committee of Ten, On figures in geometry, =524=.
    On projective geometry, =1876=.

  Common sense, M. the etherealization of, 312.

  Computation, Not m., 515.
    And m., 810.
    Not concerned with significance of numbers, 1641.

  Comte, On the object of m., =103=.
    On the business of concrete m., =104=.
    M. the indispensable basis of all education, =334=.
    Mill on, 942.
    Hamilton on, 943.
    M. and logic, =1308=, =1314=, =1325=.
    On Kant’s view of m., =1437=.
    Estimate of m., =1504=.
    M. essential to scientific education, =1505=.
    M. and natural philosophy, =1506=.
    M. and physics, =1535=, =1551=.
    M. and science, =1536=.
    M. and biology, =1578=, =1580=, =1581=.
    M. and social science, =1587=.
    Every inquiry reducible to a question of number, =1602=.
    Definition of algebra and arithmetic, =1714=.
    Geometry a natural science, =1813=.
    Ancient and modern methods, =1875=.
    On the graphic method, =1881=.
    On descriptive geometry, =1883=.
    Mill’s estimate of, 1903.

  Congreve, =2143=.

  Congruence, Symbol of, 1646.

  Conic sections, 658, 660, 1541, 1542.

  Conjecture, M. free from, 234.

  Contingent truths, 1966.

  Controversies in m., 215, 243, 1859.

  Correlation in m., 525-527, 1707, 1710.

  Correspondence, Concept of, 725, 726.

  Coulomb, 1516.

  Counting, Every problem can be solved by, 1601.

  Cournot, On the object of m., =268=.
    On algebraic notation, =1213=.
    Advantage of math, notation, =1220=.

  Craig, On the origin of a new science, =646=.

  Credulity, M. frees mind from, 450.

  Cremona, On English text-books, =609=.

  Crofton,
    On value of probabilities, =1590=.
    On probabilities, =1952=, =1970=,=1972=.

  Cromwell, On m. and public service, =328=.

  Curiosities, Chapter XXI.

  Curtius, M. and philosophy, =1409=.

  Curve, Definition of, 1927.

  Cyclometers, Notions of, 2108.

  Cyclotomy depends on number theory, 1647.


  D’Alembert, On rigor in m., =536=.
    Geometry as logic, =1311=.
    Algebra is generous, =1702=.
    Geometrical versus physical truths, =1809=.
    Standards in m., =1851=.

  Dante, =1858=, =2117=.

  Darmstaetter prize, 2129.

  Davis, On Sylvester’s method, =1035=.
    M. and science, =1510=.
    On probability, =1968=.

  Decimal fractions, 1217, 1614.

  Decker, =2142=.

  Dedekind, Zeno’s Problem, 1938.

  Deduction, Why necessary, 219.
    M. based on, 224.
    And Intuition, 1413.

  Dee, On the nature of m., =261=.

  Definitions of m., Chapter I.
    Also 2005.

  Democritus, 321.

  Demoivre, His death, 944.

  Demonstrations, Locke on, 236.
    Outside of m., 1312.
    In m., 1423.

  De Morgan, Imagination in m., =258=.
    M. as an exercise in reasoning, =430=.
    On difficulties in m., =521=.
    On correlation in m., =525=.
    On extempore lectures, =540=.
    On reading algebraic works, =601=.
    On numerical calculations, =602=.
    On practice problems, =603=.
    On the value of the history of m., =615=, =616=.
    On math’ns., =812=.
    On Bacon’s knowledge of m., =918=.
    And the actuary, 945.
    On life tables, =946=.
    Anagrams’ on his name, =947=.
    On translations of Euclid, =953=.
    Euclid’s elements compared with Newton’s Principia, =954=.
    Euler and Diderot, =966=.
    Lagrange and the parallel axiom, =984=.
    Anagram on Macaulay’s name, =996=.
    Anagrams on Newton’s name, =1028=.
    On math, notation, =1216=.
    Antagonism of m. and logic, =1315=.
    On German metaphysics, =1416=.
    On m. and science, =1537=.
    On m. and physics, =1538=.
    On the advantages of algebra, =1701=.
    On algebra as an art, =1711=.
    On double algebra and quaternions, =1720=.
    On assumptions in geometry, =1812=.
    On Euclid in schools, =1819=.
    Euclid not faultless, =1823=.
    On Euclid’s rigor, =1831=.
    Geometry before algebra, =1872=.
    On trigonometry, =1885=.
    On the calculus in elementary instruction, =1916.=
    On integration, =1919=.
    On divergent series, =1935=, =1936=.
    Ad infinitum, =1949=.
    On the fourth dimension, =2032=.
    Pseudomath and graphomath, =2101=.
    On proof, =2102=.
    On paradoxers, =2105=.
    Budget of paradoxes, =2106=.
    On D’Israeli’s six follies of science, =2107=.
    On notions of cyclometers, =2108=.
    On St. Vincent, =2109=.
    Where Euclid failed, =2114=.
    On the number of the beast, =2151=.

  Descartes, On the use of the term m., =102=.
    On intuition and deduction, =219=, =1413=.
    Math’ns alone arrive at proofs, =817=.
    The most completely math. type of mind, 948.
    Hankel on, 949.
    Mill on, 950.
    Hankel on, 1404.
    On m. and philosophy, =1425=, =1434=.
    Estimate of m., =1426=.
    Unpopularity of, =1501=.
    On the certainty of m., =1628=.
    On the method of the ancients, =1874=.
    On probable truth, =1964=.
    Descriptive geometry, 1882, 1883.

  Dessoir, M. and medicine, =1585=.

  Determinants, 1740, 1741.

  Diderot and Euler, 966.

  Differential calculus, Chapter XIX.
    And scientific physics, 1549.

  Differential equations, 1549-1552, 1924, 1926.

  Difficulties in m., 240, 521, 605-607, 634, 734, 735.

  Dillmann, M. a royal science, =204=.
    On m. as a high school subject, =401=.
    Ancient and modern geometry compared, =715=.
    On ignorance of, =807=.
    On m. as a language, =1204=.
    Number regulates all things, =1505=.

  Dirichlet, On math, discovery, =625=.
    As a student of Gauss, 977.

  Discovery in m., 617-622, 625.

  _D_-ism versus _dot_-age, 923.

  Disquisitiones Arithmeticae, 975, 977, 1637, 1638.

  D’Israeli, 2007.

  Divergent series, 1935-1937.

  “Divide et impera,” 631.

  Divine character of m., 325, 329.

  “Divinez avant de demontrer,” 630.

  Division of labor in m., 631, 632.

  Dodgson, On the charm of, =302=.
    Pythagorean theorem, =1854=.
    Ignes fatui in m., =2103=.

  Dolbear, On experiment in math. research, =613=.

  Domus Lescinia, Anagram on, 2155.

  Donne, =1816=.

  _Dot_-age versus _d_-ism, 923.

  Durfee, On Sylvester’s forgetfulness, =1038=.

  Dutton, On the ethical value of m., =446=.


  “Eadem mutata resurgo.” 920, 922.

  Echols, On the ethical value of m., =455=.

  Economics and m., 1593, 1594.

  Edinburgh Review, M. and astronomy, =1565=, =1566=.

  Education, Place of m. in, 334, 408.
    Study of arithmetic better than rhetoric, 408.
    M. as an instrument in, 413, 414.
    M. in primary, 431.
    M. as a common school subject, 432.
    Bain on m. in, 442.
    Calculus in elementary, 1916, 1917.

  Electricity, M. and the theory of, 1554.

  Elegance in m., 640, 728.

  Ellis, On precocity in m., =835=.
    On aptitude of Anglo-Danes for m., =836=.
    On Newton’s genius, =1014=.

  Emerson, On Newton and Laplace, =1003=.
    On poetry and m., =1124=.

  Endowment of math’ns, 818.

  Enthusiasm, 801.

  Equality, Grassmann’s definition of, =105=.

  Equations, 104, 526, 1891, 1892.

  Errors, Theory of, 1973, 1974.

  Esthetic element in m., 453-455, 640, 1102, 1105, 1852,
      1853.

  Esthetic tact, 622.

  Esthetic value of m., 1848, 1850.

  Esthetics, Relation of m. to, 318, 319, 439.

  Estimates of m., Chapter III.
    See also 1317, 1324, 1325, 1427, 1504, 1508.

  Ethical value of m., 402, 438, 446, 449, 455-457.

  Euclid, Bolzano cured by, 929.
    And Ptolemy, 951, 1878.
    And the student, 952.

  Euclid’s Elements,
    Translations of, 953.
    Compared with the Principia, 954.
    Greatness of, 955.
    Greatest of human productions, 1817.
    Performance in, 1818.
    In English schools, 1819.
    Encouragement and guide, 1820.
    Inspiration and aspiration, 1821.
    The only perfect model, 1822.
    Not altogether faultless, 1823.
    Only a small part of m., 1824.
    Not fitted for boys, 1825.
    Early study of, 1826.
    Newton and, 1827.
    Its place, 1828.
    Unexceptional in rigor, 1829.
    Origin of, 1831.
    Doctrine of proportion, 1834.
    Definition of proportion, 1835.
    Steps in demonstration, 1839.
    Parallel axiom, 2007.

  Euclidean geometry, 711, 713, 715.

  Eudoxus, 904.

  Euler, the myriad-minded, 255.
    Pencil outruns intelligence, 626.
    On theoretical investigations, 657.
    Merit of his work, 956.
    The creator of modern math. thought, 957.
    His general knowledge, 958.
    His knowledge of the Aeneid, 959.
    Extent of his work, 960.
    “Analysis incarnate,” 961.
    As a computer, 962, 963.
    His math. power, 964.
    His _Tentamen novae theorae musicae_, 965.
    And Diderot, 966.
    Error in Fermat’s law of prime numbers, 967.

  Eureka, 911, 917.

  Euripedes, 1568.

  Everett, Estimate of m., =325=.
    Value of math. training, =443=.
    Theoretical investigations, =656=.
    Arithmetic a master-key, =1571=.
    On m. and law, =1598=.

  Exactness, See precision.

  Examinations, 407.

  Examples, 422.

  Experiment in m., 612, 613, 1530, 1531.

  Extent of m., 737, 738.


  Fairbairn, 528.

  Fallacies, 610.

  Faraday, M. and physics, 1554.

  Fermat, 255, 967, 1902.

  Fermat’s theorem, 2129.

  Figures, Committee of Ten on, 524.
    Democritus view of, 321.
    Battalions of, 1631.

  Fine, Definition of number, =1610=.
    On the imaginary, =1732=.

  Fine Art, M. as a, Chapter XI.

  Fisher, M. and economics, =1594=.

  Fiske, Imagination in m., =256=.
    Advantage of m. as logic, =1324=.

  Fitch, Definition of m., =125=.
    M. in education, =429=.
    Purpose of teaching arithmetic, =1624=, =1625=.

  Fizi, Origin of the Liliwati, =995=.

  Flamsteed, Anagram on, =968=.

  Fluxions, 1911, 1915, 1942-1944.

  Fontenelle, Bernoulli’s tomb, =920=.

  Formulas, Compared to focus of a lens, 1515.

  Forsyth, On direct usefulness of math. results, =654=.
    On theoretical investigations, =664=.
    Progress of m. =704=.
    On Cayley, =932-934=.
    On m. and physics, =1539=.
    On m. and applications, =1540=.
    On invariants, =1747=.
    On function theory, =1754=, =1755=.

  Foster, On m. and physics, =1516=, =1522=.
    On experiment in m., =1531=.

  Foundations of m., 717.

  Four, The number, 2147, 2148.

  Fourier, Math, analysis co-extensive with nature, =218=.
    On math. research, =612=.
    Hamilton on, 969.
    On m. and physics, =1552=, =1553=.
    On the advantage of the Cartesian method, =1889=.

  Fourier’s theorem, 1928.

  Fourth dimension, 2032, 2039.

  Frankland, A., M. and chemistry, =1560=.

  Frankland, W. B., Motto of Pythagorean brotherhood, =1833=.
    The most beautiful truth in geometry, =1857=.

  Franklin, B., Estimate of m., =322=.
    On the value of the study of m., =323=.
    On the excellence of m., =324=.
    On m. as a logical exercise, =1303=.

  Franklin, F., On Sylvester’s weakness, =1033=.

  Frederick the Great, On geometry, =1860=.

  Freedom in m., 205-208, 805.

  French m., 1210.

  Fresnel, 662.

  Frischlinus, =1801=.

  Froebel, M. a mediator between man and nature, =262=.

  Function theory, 709, 1732, 1754, 1755.

  Functional exponent, 1210.

  Functionality, The central idea of modern m., 254.
    Correlated to life, 272.

  Functions, 1932, 1933.
    Concept not used by Sylvester, 1034.

  Fundamental concepts, Chapter XX.

  Fuss, On Euler’s _Tentamen novae theorae musicae_, =965=.


  Galileo, On authority in science, =1528=.

  Galton, 838.

  Gauss, His motto, 649.
    Mere math’ns, =820=.
    And Newton compared, 827.
    His power, 964.
    His favorite pursuits, 970.
    The first of theoretical astronomers, 971.
    The greatest of arithmeticians, 971.
    The math. giant, 972.
    Greatness of, 973.
    Lectures to three students, =974=.
    His style and method, 983.
    His estimate of Newton, 1029.
    On the advantage of new calculi, =1215=.
    M. and experiment, 1531.
    His _Disquisitiones Arithmeticae_, 1639, 1640.
    M. the queen of the sciences, =1642=.
    On number theory, =1644=.
    On imaginaries, =1730=.
    On the notation sin²φ, =1886=.
    On infinite magnitude, =1950=.
    On non-euclidean geometry, =2023-2028=.
    On the nature of space, 2034.

  Generalization in m., 245, 246, 252, 253, 327, 728.

  Genius, 819.

  Geometrical investigations, 642, 643.

  Geometrical training, Value of, 1841, 1842, 1844-1846.

  Geometry, Chapter XVIII.
    Bacon’s definition of, 106.
    Sylvester’s definition of, 110.
    Value to mankind, 332, 449.
    And patriotism, 332.
    An excellent logic, 428.
    Plato’s view of, 429.
    The fountain of all thought, 451.
    And algebra, 525-527.
    Lack of concreteness, 710.
    Advantage of modern over ancient, 711, 712.
    And music, 965.
    And arithmetic, 1604.
    Is figured algebra, 1706.
    Name inapt, 1801.
    And experience, 1814.
    Halsted’s definition of, 1815.
    And observation, 1830.
    Controversy in, 1859.
    A mechanical science, 1865.
    A natural science, 1866.
    Not an experimental science, 1867.
    Should come before algebra, 1767, 1871, 1872.
    And analysis, 1931.

  Germain, Algebra is written geometry, =1706=.

  Gilman, Enlist a great math’n, =808=.

  Glaisher, On the importance of broad training, =623=.
    On the importance of a well-chosen notation, =634=.
    On the expansion of the field of m., =634=.
    On the need of text-books on higher m., =635=.
    On the perfection of math. productions, =649=.
    On the invention of logarithms, =1616=.
    On the theory of numbers, =1640=.

  Goethe, On the exactness of m., =228=.
    M. an organ of the higher sense, =273=.
    Estimate of m., =311=.
    M. opens the fountain of all thought, =451=.
    Math’ns must perceive beauty of truth, =803=.
    Math’ns bear semblance of divinity, =804=.
    Math’ns like Frenchmen, =813=.
    His aptitude for m., =976=.
    M. like dialectics, =1307=.
    On the infinite, 1957.

  Golden age of m., 701, 702.
    Of art and m. coincident, 1134.

  Gordan, When a math. subject is complete, =636=.

  Gow, Origin of Euclid, =1832=.

  Gower, =1808=.

  Grammar and m. compared, 441.

  Grandeur of m., 325.

  Grassmann, Definition of m., =105=.
    Definition of magnitude, =105=.
    Definition of equality, =105=.
    On rigor in m., =538=.
    On the value of m., =1512=.

  Greek view of science, 1429.

  Graphic method, 1881.

  Graphomath, 2101.

  Group, Notion of, 1751.

  Growth of m., 209, 211, 703.


  Hall, G. S., M. the ideal and norm of all careful
      thinking, =304=.

  Hall and Stevens, On the parallel axiom, =2008=.

  Haller, On the infinite, =1958=.

  Halley, On Cartesian geometry, 716.

  Halsted, On Bolyai, =924-926=.
    On Sylvester, =1030=, =1039=.
    And Sylvester, =1031=, =1032=.
    On m. as logic, =1305=.
    Definition of geometry, =1815=.

  Hamilton, Sir William, His ignorance of m., 978.

  Hamilton, W. R., Importance of his quaternions, 333.
    Estimate of Comte’s ability, =943=.
    To the memory of Fourier, =969=.
    Discovery in light, 1558.
    On algebra as the science of time, =1715=, =1716=.
    On quaternions, =1718=.
    On trisection of an angle, =2112=.

  Hankel, Definition of m., =114=.
    On freedom in m., =206=.
    On the permanency of math. knowledge, =216=.
    On aim in m., =508=.
    On isolated theorems, =621=.
    On tact in m., =622=.
    On geometry, 714.
    Ancient and modern m. compared, =718=, =720=.
    Variability the central idea in modern m., =720=.
    Characteristics of modern m., =728=.
    On Descartes, =949=.
    On Euler’s work, =956=.
    On philosophy and m., =1404=.
    On the origin of m., =1412=.
    On irrationals and imaginaries, =1729=.
    On the origin of algebra, =1736=.
    Euclid the only perfect model, =1822=.
    Modern geometry a royal road, =1878=.

  Harmony, 326, 1208.

  Harris, M. gives command over nature, =434=.

  Hathaway, On Sylvester, =1036=.

  Heat, M. and the theory of, 1552, 1553.

  Heath, Character of Archimedes’ work, =913=.

  Heaviside, The place of Euclid, =1828=.

  Hebrew and Latin races, Aptitude for m., 838.

  Hegel, =1417=.

  Heiss, Famous anagrams, =2055=.
    Reversible verses, =2056=.

  Helmholtz, M. the purest form of logical activity, =231=.
    M. requires perseverance and great caution, =240=.
    M. should take more important place in education, =441=.
    Clifford on, =979=.
    M. the purest logic, =1302=.
    M. and applications, =1445=.
    On geometry, =1836=.
    On the importance of the calculus, =1939=.
    A non-euclidean world, =2029=.

  Herbart, Definition of m., =117=.
    M. the predominant science, =209=.
    On the method of m., =212=, =1576=.
    M. the priestess of definiteness and clearness, =217=.
    On the importance of checks, =230=.
    On imagination in m, =257=.
    M. and invention, =406=.
    M. the chief subject for common schools, =432=.
    On aptitude for m., =509=.
    On the teaching of m., =516=.
    M. the greatest blessing, =1401=.
    M. and philosophy, =1408=.
    If philosophers understood m., =1415=.
    M. indispensable to science, =1502=.
    M. and psychology, =1583=, =1684=.
    On trigonometry, =1884=.

  Hermite, On Cayley, =935=.

  Herschel, M. and astronomy, =1564=.
    On probabilities, =1592=.

  Hiero, 903, 904.

  Higher m., Mellor’s definition of, =108=.

  Hilbert, On the nature of m., =266=.
    On rigor in m., =537=.
    On the importance of problems, =624=, =628=.
    On the solvability of problems, =627=.
    Problems should be difficult, =629=.
    On the abstract character of m., =638=.
    On arithmetical symbols, =1627=.
    On non-euclidean geometry, =2019=.

  Hill, Aaron, On Newton, =1009=.

  Hill, Thomas, On the spirit of mathesis, =274=.
    M. expresses thoughts of God, =275=.
    Value of m., =332=.
    Estimate of Newton’s work, =333=.
    Math’ns difficult to judge, =841=.
    Math’ns indifferent to ordinary interests of life, =842=.
    A geometer must be tried by his peers, =843=.
    On Bernoulli’s spiral, =922=.
    On mathesis and poetry, =1125=.
    On poesy and m., =1126=.
    On m. as a language, =1209=.
    Math, language untranslatable, =1210=.
    On quaternions, =1719=.
    On the imaginary, =1734=.
    On geometry and literature, =1847=.
    M. and miracles, =2157=, =2158=.

  Hindoos, Grandest achievement of, 1615.

  History and m., 1599.

  History of m., 615, 616, 625, 635.

  Hobson, Definition of m., =118=.
    On the nature of m., =252=.
    Functionality the central idea of m., =264=.
    On theoretical investigations, =663=.
    On the growth of m., =703=.
    A great math’n a great artist, =1109=.
    On m. and science, =1508=.
    Hoffman, Science and poetry not antagonistic, =1122=.

  Holzmüller, On the teaching of m., =518=.

  Hooker, =1432=.

  Hopkinson, M. a mill, =239=.

  Horner’s method, 1744.

  Howison, Definition of m., =134=, =135=.
    Definition of arithmetic, =1612=.

  Hudson, On the teaching of m., =612=.

  Hughes, On science for its own sake, =1546=.

  Humboldt, M. and astronomy, =1567=.

  Hume, On the advantage of math, science, =1438=.
    On geometry, =1862=.
    On certainty in m., =1863=.
    Objection to abstract reasoning, =1941=.

  Humor in m., 539.

  Hutton, On Bernoulli, =919=.
    On Euler’s knowledge, =958=.
    On the method of fluxions, =1911=.

  Huxley, Negative qualities of m., =250=.

  Hyper-space, 2030, 2031, 2033, 2036-2038.

  Hyperbolic functions, 1929, 1930.


  Ignes fatui in m., 2103.

  Ignorabimus, None in m., 627.

  Ignorance of m., 310, 331, 807, 1537, 1577.

  Imaginaries, 722, 1729-1735.

  Imagination in m., 246, 251, 253, 256-258, 433, 1883.

  Improvement of elementary m., 617.

  Incommensurable numbers, contingent truths like, 1966.

  Indian m., 1736, 1737.

  Induction in m., 220-223, 244.
    And analogy, 724.

  Infinite collection, Definition of, 1959, 1960.

  Infinite divisibility, 1945.

  Infinitesimal analysis, 1914.

  Infinitesimals, 1905-1907, 1940, 1946, 1954.

  Infinitum, Ad, 1949.

  Infinity and infinite magnitude, 723, 928, 1947, 1948,
      1950-1958.

  Integers, Kronecker on, 1634, 1635.

  Integral numbers, Minkowsky on, 1636.

  Integrals, Invention of, 1922.

  Integration, 1919-1921, 1923, 1925.

  International Commission on m., =501=, =502=, =938=.

  Intuition and deduction, 1413.

  Invariance, Correlated to life, 272.
    MacMahon on, 1746.
    Keyser on, 1749.

  Invariants, Changeless in the midst of change, 276.
    Importance of concept of, 727.
    Sylvester on, 1742.
    Forsyth on, 1747.
    Keyser on, 1748.
    Lie on, 1752.

  Invention in m., 251, 260.

  Inverse process, 1207.

  Investigations, See research.

  Irrationals, 1729.

  Isolated theorems in m., 620, 621.

  “It is easy to see,” 985, 986, 1045.


  Jacobi, His talent for philology, 980.
    Aphorism, =1635=.
    Die “Ewige Zahl,” =1643=.

  Jefferson, On m. and law, =1597=.

  Johnson, His recourse to m., 981.
    Aptitude for numbers, =1617=.
    On round numbers, =2137=.

  Journals and transactions, 635.

  Jowett, M. as an instrument in education, =413=.

  Judgment, M. requires, 823.

  Jupiter’s eclipses, 1544.

  Justitia, The goddess, 824.

  Juvenal, Nemo mathematicus etc., =831=.


  Kant, On the a priori nature of m., =130=.
    M. follows the safe way of science, =201=.
    On the origin of scientific m., 201.
    On m. in primary education, =431=.
    M. the embarrassment of metaphysics, =1402=.
    His view of m., =1436=, =1437=.
    On the difference between m. and philosophy, =1436=.
    On m. and science, =1508=.
    Esthetic elements in m., =1852=, =1853=.
    Doctrine of time, =2001=.
    Doctrine of space, =2003=.

  Karpinsky, M. and efficiency, =1673=.

  Kasner, “Divinez avant de demontrer,” =630=.
    On modern geometry, =710=.

  Kelland, On Euclid’s elements, =1817=.

  Kelvin, Lord, See William Thomson.

  Kepler, His method, 982.
    Planetary orbits and the regular solids, =2134=.

  Keyser, Definition of m., =132=.
    Three characteristics of m., =225=.
    On the method of m., =244=.
    On ratiocination, =246=.
    M. not detached from life, =273=.
    On the spirit of mathesis, =276=.
    Computation not m., =515=.
    Math, output of present day, =702=.
    Modern theory of functions, =709=.
    M. and journalism, =731=.
    Difficulty of m., =735=.
    M. appeals to whole mind, =815=.
    Endowment of math’ns, =818=.
    Math’ns in public service, =823=.
    The aim of the math’n, =844=.
    On Bolzano, =929=.
    On Lie, =992=.
    On symbolic logic, =1321=.
    On the emancipation of logic, =1322=.
    On the Principia Mathematica, =1326=.
    On invariants, =1728=.
    On invariance, =1729=.
    On the notion of group, =1751=.
    On the elements of Euclid, =1824=.
    On protective geometry, =1880=.
    Definition of infinite assemblage, =1960=.
    On the infinite, =1961=.
    On non-euclidean geometry, =2035=.
    On hyper-space, =2037=, =2038=.

  Khulasat-al-Hisab, Problems, =1738=.

  Kipling, =1633=.

  Kirchhoff, Artistic nature of his works, 1116.

  Klein, Definition of m., =123=.
    M. a versatile science, =264=.
    Aim in teaching, =507=, =517=.
    Analysts versus synthesists, =651=.
    On theory and practice, =661=.
    Math, aptitudes of various races, =838=.
    Lie’s final aim, =993=.
    Lie’s genius, =994=.
    On m. and science, =1520=.
    Famous aphorisms, =1635=.
    Calculating machines, =1641=.
    Calculus for high schools, =1918=.
    On differential equations, =1926=.
    Definition of a curve, =1927=.
    On axioms of geometry, =2006=.
    On the parallel axiom, =2009=.
    On non-euclidean geometry, =2017=, =2021=.
    On hyper-space, =2030=.

  Kronecker, On the greatness of Gauss, =973=.
    God made integers etc., =1634=.

  Kummer, On Dirichlet, =977=.
    On beauty in m., =1111=.


  LaFaille, Mathesis few know, =1870=.

  Lagrange, On correlation of algebra and geometry, =527=.
    His style and method, 983.
    And the parallel axiom, 984.
    On Newton, =1011=.
    Wings of m., =1604=.
    Union of algebra and geometry, =1707=.
    On the infinitesimal method, =1906=.

  Lalande, M. in French army, =314=.

  Langley, M. in Prussia, 513.

  Lampe, On division of labor in m., =632=.
    On Weierstrass, =1049=.
    Weierstrass and Sylvester, =1050=.
    Qualities common to math’ns and artists, =1113=.
    Charm of m., =1115=.
     Golden age of art and m. coincident, =1134=.

  Language, Chapter XII. See also 311, 419, 443, 1523, 1804,
      1889.

  Laplace, On instruction in m., =220=.
    His style and method, =983=.
    “Thus it plainly appears,” 985, 986.
    Emerson on, 1003.
    On Leibnitz, =991=.
    On the language of analysis, =1222=.
    On m. and nature, =1525=.
    On the origin of the calculus, =1902=.
    On the exactitude of the differential calculus, =1910=.
    The universe in a single formula, =1920=.
    On probability, =1963=, =1969=, =1971=.

  Laputa, Math’ns of, 2120-2122,
    Math. school of, 2123.

  Lasswitz, On modern algebra, =1741=.
    On function theory, =1934=.
    On non-euclidean geometry, =2040=.

  Latin squares, 252.

  Latta, On Leibnitz’s logical calculus, =1317=.

  Law and m., 1597, 1598.

  Laws of thought, 719, 1318.

  Leadership, M. as training for, 317.

  Lecture, Preparation of, 540.

  Lefevre, M. hateful to weak minds, =733=.
    Logic and m., =1309=.

  Leibnitz, On difficulties in m., =241=.
    His greatness, 987.
    His influence, 988.
    The nature of his work, 989.
    His math. tendencies, 990.
    His binary arithmetic, 991.
    On Newton, =1010=.
    On demonstrations outside of m., =1312=.
    Ars characteristica, =1316=.
    His logical calculus, 1317.
    Union of philosophical and m. productivity, 1404.
    M. and philosophy, =1435=.
    On the certainty of math. knowledge, =1442=.
    On controversy in geometry, =1859=.
    His differential calculus, 1902.
    His notation of the calculus, 1904.
    On necessary and contingent truth, =1966=.

  Leverrier, Discovery of Neptune, 1559.

  Lewes, On the infinite, =1953=.

  Lie, On central conceptions in modern m., =727=.
    Endowment of math’ns, =818=.
    The comparative anatomist, 992.
    Aim of his work, 993.
    His genius, 994.
    On groups, =1752=.
    On the origin of the calculus, =1901=.
    On differential equations, =1924=.

  Liliwati, Origin of, 995.

  Limitations of math. science, 1437.

  Limits, Method of, 1905, 1908, 1909, 1940.

  Lindeman, On m. and science, =1523=.

  Liouville, 822.

  Lobatchewsky, =2022=.

  Locke, On the method of m., =214=, =235=.
    On proofs and demonstrations, =236=.
    On the unpopularity of m., =271=.
    On m. as a logical exercise, =423=, =424=.
    M. cures presumption, =425=.
    Math, reasoning of universal application, =426=.
    On reading of classic authors, =604=.
    On Aristotle, =914=.
    On m. and philosophy, =1433=.
    On m. and moral science, =1439=, =1440=.
    On the certainty of math. knowledge, =1440=, =1441=.
    On unity, =1607=.
    On number, =1608=.
    On demonstrations in numbers, =1630=.
    On the advantages of algebra, =1705=.
    On infinity, =1955=, =1957=.
    On probability, =1965=.

  Logarithmic spiral, 922.

  Logarithmic tables, 602.

  Logarithms, 1526, 1614, 1616.

  Logic and m., Chapter XIII.
    See also 423-430, 442.

  Logical calculus, 1316, 1317.

  Longevity of math’ns, 839.

  Lovelace, Why are wise few etc., =1629=.

  Lover, =2140=.


  Macaulay, Plato and Bacon, =316=.
    On Archimedes, =905=.
    Bacon’s view of m., =915=, =916=.
    Anagram on his name, =996=.
    Plato and Archytas, =1427=.
    On the power of m., =1527=.

  Macfarlane, On Tait, Maxwell, Thomson, =1042=.
    On Tait and Hamilton’s quaternions, =1044=.

  Mach, On thought-economy in m., =203=.
    M. seems possessed of intelligence, =626=.
    On aim of research, =647=.
    On m. and counting, =1601=.
    On the space of experience, =2011=.

  MacMahon, Latin squares, 252.
    On Sylvester’s bend of mind, =645=.
    On Sylvester’s style, =1040=.
    On the idea of invariance, =1746=.

  Magnitude, Grassmann’s definition, 105.

  Magnus, On the aim in teaching m., =505=.

  Manhattan Island, Cost of, 2130.

  Marcellus, Estimate of Archimedes, =909=.

  Maschke, Man above method, =650=.

  Masters, On the reading of the, 614.

  Mathematic, Sylvester on use of term, 101.
    Bacon’s use of term, 106.

  Mathematical faculty, Frequency of, 832.

  Mathematical mill, The, 239, 1891.

  Mathematical productions, 648, 649.

  Mathematical theory, When complete, 636, 637.

  Mathematical training, 443, 444.
    Maxims of math’ns, 630, 631, 649.
    Not a computer, 1211.
    Intellectual habits of math’ns, 1428.
    The place of the, 1529.
    Characteristics of the mind of a, 1534.

  Mathematician, The, Chapter VIII.

  Mathematics, Definitions of, Chapter I.
    Objects of, Chapter I.
    Nature of, Chapter II.
    Estimates of, Chapter III.
    Value of, Chapter IV.
    Teaching of, Chapter V.
    Study of, Chapter VI.
    Research in, Chapter VI.
    Modern, Chapter VII.
    As a fine art, Chapter XI.
    As a language, Chapter XII.
      Also 445, 1814.
    And logic, Chapter XIII.
    And philosophy, Chapter XIV.
    And science, Chapter XV.
    And applications, Chapter XV.
    Knowledge most in, 214.
    Suppl. brevity of life, 218.
    The range of, 269.
    Compared to French language, 311.
    The care of great men, 322.
    And professional education, 429.
    And science teaching, 522.
    The queen of the sciences, 975.
    Advantage over philosophy, 1436, 1438.
    As an instrument, 1506.
    For its own sake, 1540, 1541, 1545, 1546.
    The wings of, 1604.

  Mathesis, 274, 276, 1870, 2015.

  Mathews, On Disqu. Arith. =1638=.
    On number theory, =1639=.
    The symbol ≡, =1646=.
    On Cyclotomy, =1647=.
    Laws of algebra, =1709=.
    On infinite, zero, infinitesimal, =1954=.

  Maxims of great math’ns, 630, 631, 649.

  Maxwell, 1043, 1116.

  McCormack, On the unpopularity of m., =270=.
    On function, =1933=.

  Méchanique céleste, 985, 986.

  Medicine, M. and the study of, 1585, 1918.

  Mellor, Definition of higher m., =108=.
    Conclusions involved in premises, =238=.
    On m. and science, =1561=.
    On the calculus, =1912=.
    On integration, =1923=, =1925=.

  Memory in m., 253.

  Menæchmus, 901.

  Mere math’ns, 820, 821.

  Merz, On the transforming power of m., =303=.
    On the dominant ideas in m., =725=.
    On extreme views in m., =827=.
    On Leibnitz’s work, =989=.
    On the math. tendency of Leibnitz, =990=.
    On m. as a lens, =1515=.
    M. extends knowledge, =1524=.
    Disquisitiones Arithmeticae, =1637=.
    On functions, =1932=.
    On hyper-space, =2036=.

  Metaphysics, M. the only true, 305.

  Meteorology and m., 1557.

  Method of m. 212-215, 226, 227, 230, 235, 244, 806, 1576.

  Metric system, 1725.

  Military training, M. in, 314, 418, 1574.

  Mill, On induction in m., =221=, =222=.
    On generalization in m., =245=.
    On math. studies, =409=.
    On m. in a scientific education, =444=.
    Math’ns hard to convince, =811=.
    Math’ns require genius, =819=.
    On Comte, =942=.
    On Descartes, =942=, =948=.
    On Sir William Hamilton’s ignorance of m., =978=.
    On Leibnitz, =987=.
    On m. and philosophy, =1421=.
    On m. as training for philosophers, =1422=.
    M. indispensable to science, =1519=.
    M. and social science, =1595=.
    On the nature of geometry, =1838=.
    On geometrical method, =1861=.
    On the calculus, =1903=.

  Miller, On the Darmstaetter prize, =2129=.

  Milner, Geometry and poetry, =1118=.

  Minchin, On English text-books, =539=.

  Mineralogy and m., 1558.

  Minkowski, On integral numbers, =1636=.

  Miracles and m., 2157, 2158, 2160.

  Mixed m., Bacon’s definition of, 106.
    Whewell’s definition of, 107.

  Modern algebra, 1031, 1032, 1638, 1741.

  Modern geometry, 1710-1713, 715, 716, 1878.

  Modern m., Chapter VII.

  Moebius, Math’ns constitute a favorite class, =809=.
    M. a fine art, =1107=.

  Moral science and m., 1438-1440.

  Moral value of m., See ethical value.

  Mottoes, Of math’ns, 630, 631, 649.
    Of Pythagoreans, 1833.

  Murray, Definition of m., =116=.

  Music and m., 101, 276, 965, 1107, 1112, 1116, 1127, 1128,
      1130-1133, 1135, 1136.

  Myers, On m. as a school subject, =403=.
    On pleasure in m., =454=.
    On the ethical value of m., =457=.
    On the value of arithmetic, =1622=.

  Mysticism and numbers, 2136-2141, 2143.


  Napier’s rule, 1888.

  Napoleon, M. and the welfare of the state, =313=.
    His interest in m., 314, 1001.

  Natural science and m., Chapter XV.
    Also 244, 444, 445, 501.

  Natural selection, 1921.

  Nature of m., Chapter II.
    See also 815, 1215, 1308, 1426, 1525, 1628.

  Nature, Study of, 433-436, 514, 516, 612.

  Navigation and m., 1543, 1544.

  Nelson, Anagram on, 2153.

  Neptune, Discovery of, 1554, 1559.

  Newcomb, On geometrical paradoxers, =2113=.

  Newton,
    Importance of his work, 333.
    On correlation in m., =526=.
    On problems in algebra, =530=.
    And Gauss compared, 827.
    His fame, 1002.
    Emerson on, 1003.
    Whewell on, 1004, 1005.
    Arago on, 1006.
    Pope on, 1007.
    Southey on, 1008.
    Hill on, 1009.
    Leibnitz on, 1010.
    Lagrange on, 1011.
    No monument to, 1012.
    Wilson on, 1012, 1013.
    His genius, 1014.
    His interest in chemistry and theology, 1015.
    And alchemy, 1016, 1017.
    His first experiment, 1018.
    As a lecturer, 1019.
    As an accountant, 1020.
    His memorandum-book, 1021.
    His absent-mindedness, 1022.
    Estimate of himself, =1023-1025=.
    His method of work, 1026.
    Discovery of the calculus, 1027.
    Anagrams on, 1028.
    Gauss’s estimate of, 1029.
    On geometry, =1811=.
    Compared with Euclid, 1827.
    Geometry a mechanical science, =1865=.
    Test of simplicity, =1892=.
    Method of fluxions, 1902.

  Newton’s rule, 1743.

  Nile, Origin of name, 2150.

  Noether, On Cayley, =931=.
    On Sylvester, =1034=, =1041=.

  Non-euclidean geometry, 1322, 2016-2029, 2033, 2035, 2040.

  Nonnus, On the mystic four, =2148=.

  Northrup, On Lord Kelvin, =1048=.

  Notation, Importance of, 634, 1222, 1646.
    Value of algebraic, 1213, 1214.
    Criterion of good, 1216.
    On Arabic, 1217, 1614.
    Advantage of math., 1220.
    See also symbolism.

  Notions, Cardinal of m., 110.
    Indefinable, 1219.

  Novalis, Definition of pure m., =112=.
    M. the life supreme, =329=.
    Without enthusiasm no m., =801=.
    Method is the essence of m., =806=.
    Math’ns not good computers, =810=.
    Music and algebra, =1128=.
    Philosophy and m., =1406=.
    M. and science, =1507=, =1526=.
    M. and historic science, =1599=.
    M. and magic, =2159=.
    M. and miracles, =2160=.

  Number, Every inquiry reducible to a question of, 1602.
    And nature, 1603.
    Regulates all things, 1605.
    Aeschylus on, 1606.
    Definition of, 1609, 1610.
    And superstition, 1632.
    Distinctness of, 1707.
    Of the beast, 2151, 2152.

  Number-theory,
    The queen of m., 975.
    Nature of, 1639.
    Gauss on, 1644.
    Smith on, 1645.
    Notation in, 1646.
    Aid to geometry, 1647.
    Mystery in, 1648.

  Number-work, Purpose of, 1623.

  Numbers, Pythagoras’ view of, 321.
    Mighty are, 1568.
    Aptitude for, 1617.
    Demonstrations in, 1630.
    Prime, 1648.
    Necessary truths like, 1966.
    Round, 2137.
    Odd, 2138-2141.
    Golden, 2142.
    Magic, 2143.


  Obscurity in m. and philosophy, 1407.

  Observation in m., 251-253, 255, 433, 1830.

  Obviousness in m., 985, 986, 1045.

  Olney, On the nature of m., =253=.

  Oratory and m., 829, 830.

  Order and arrangement, 725.

  Origin of m., 1412.

  Orr, Memory verse for π, =2127=.

  Osgood, On the calculus, =1913=.

  Ostwald, On four-dimensional space, 2039.


  π. In actuarial formula, 945.
    Memory verse for, 2127.

  Pacioli, On the number three, =2145=.

  Painting and m., 1103, 1107.

  Papperitz, On the object of pure m., 111.

  Paradoxes, Chapter XXI.

  Parallel axiom, Proof of, 984, 2110, 2111.
    See also non-euclidean geometry.

  Parker, Definition of arithmetic, =1611=.
    Number born in superstition, =1632=.
    On geometry, =1805=.

  Parton, On Newton, =1917-1919=, =1021=, =1022=, =1827=.

  Pascal, Logic and m., =1306=.

  Peacock, On the mysticism of Greek philosophers, =2136=.
    The Yankos word for three, =2144=.
    The number of the beast, =2152=.

  Pearson, M. and natural selection, =834=.

  Peirce, Benjamin, Definition of m., =120=.
    M. as an arbiter, =210=.
    Logic dependent on m., =1301=.
    On the symbol √-1, =1733=.

  Peirce, C. S. Definition of m., =133=.
    On accidental relations, =2128=.

  Perry, On the teaching of m., =510=, =511=, =619=, =837=.

  Persons and anecdotes, Chapters IX and X.

  Philosophy and m., Chapter XIV.
    Also 332, 401, 414, 444, 445, 452.

  Physics and m., 129, 437, 1516, 1530, 1535, 1538, 1539,
      1548, 1549, 1550, 1555, 1556.

  Physiology and m., 1578, 1581, 1582.

  Picard, On the use of equations, =1891=.

  Pierce, On infinitesimals, =1940=.

  Pierpont, Golden age of m., =701=.
    On the progress of m., =708=.
    Characteristics of modern m., =717=.
    On variability, =721=.
    On divergent series, =1937=.

  Plato, His view of m., 316, 429.
    M. a study suitable for freemen, =317=.
    His conic sections, 332.
    And Archimedes, 904.
    Union of math. and philosophical productivity, 1404.
    Diagonal of square, =1411=.
    And Archytas, 1427.
    M. and the arts, =1567=.
    On the value of m., =1574=.
    On arithmetic, =1620=, =1621=.
    God geometrizes, 1635, 1636. 1702.
    On geometry, 429, 1803, 1804, =1806=, =1844=, =1845=.

  Pleasure, Element of in m., 1622, 1629, 1848, 1850, 1851.

  Pliny, =2039=.

  Plus and minus signs, 1727.

  Plutarch, On Archimedes, =903=, =904=, =908-910=, =912=.
    God geometrizes, =1802=.

  Poe, =417=.

  Poetry and m.,
    Weierstrass on, 802.
    Pringsheim on, 1108.
    Wordsworth on, 1117.
    Milner on, 1118.
    Workman on, 1120.
    Pollock on, 1121.
    Hoffman on, 1122.
    Thoreau on, 1123.
    Emerson on, 1124.
    Hill on, 1125, 1126.
    Shakespeare on, 1127.

  Poincaré, On elegance in m., =640=.
    M. has a triple end, =1102=.
    M. as a language, =1208=.
    Geometry not an experimental science, =1867=.
    On geometrical axioms, =2005=.

  Point, 1816.

  Political science, M. and, 1201, 1324.

  Politics, Math’ns and, 814.

  Pollock, On Clifford, =938-941=, =1121=.

  Pope, =907=, =2015=, =2031=, =2046=.

  Precision in m., 228, 639, 728.

  Precocity in m., 835.

  Predicabilia a priori, 2003.

  Press, M. ignored by daily, 731, 732.

  Price, Characteristics of m., =247=.
    On m. and physics, =1550=.

  Prime numbers, Sylvester on, 1648.

  Principia Mathematica, 1326.

  Pringsheim, M. the science of the self-evident, =232=.
    M. should be studied for its own sake, =439=.
    On the indirect value of m., =448=.
    On rigor in m., =535=.
    On m. and journalism, =732=.
    On math’ns in public service, =824=.
    Math’n somewhat of a poet, =1108=.
    On music and m., =1132=.
    On the language of m., =1211=.
    On m. and physics, =1548=.

  Probabilities, 442, 823, 1589, 1590-1592, 1962-1972, 1975.

  Problem solving, 531, 532.

  Problems, In m., 523, 534.
    In arithmetic, 528.
    In algebra, 530.
    Should be simple, 603.
    In Cambridge texts, 608.
    On solution of, 611.
    On importance of, 624, 628.
    What constitutes good, 629.
    Aid to research, 644.
    Of modern m., 1926.

  Proclus, Ptolemy and Euclid, =951=.
    On characteristics of geometry, =1869=.

  Progress in m., 209, 211, 212, 216, 218, 702-705, 708.

  Projective geometry, 1876, 1877, 1879, 1880.

  Proportion, Euclid’s doctrine of, 1834.
    Euclid’s definition of, 1835.

  Proposition, 1219, 1419.

  Prussia, M. in, 513.

  Pseudomath, Defined, 2101.

  Psychology and m., 1576, 1583, 1584.

  Ptolemy and Euclid, 951.

  Public service, M. and, 823, 824, 1303, 1574.

  Public speaking, M. and, 420, 829, 830.

  Publications, Math. of present day, 702, 703.

  Pure M., Bacon’s definition of, 106.
    Whewell’s definition of, 107.
    On the object of, 111, 129.
    Novalis’ conception of, 112.
    Hobson’s definition of, 118.
    Russell’s definition of, 127, 128.

  Pursuit of m., 842.

  Pythagoras,
    Number the nature of things, 321.
    Union of math, and philosophical productivity, 1404.
    The number four, =2147=.

  Pythagorean brotherhood, Motto of, 1833.

  Pythagorean theorem, 1854-1856, 2026.

  Pythagoreans, Music and M., 1130.


  Quadrature, See Squaring of the circle.

  Quantity, Chrystal’s definition of, =115=.

  Quarles, On quadrature, =2116=.

  Quaternions, 333, 841, 937, 1044, 1210, 1718-1726.

  Quetelet, Growth of m., =1514=.


  Railway-making, 1570.

  Reading of m., 601, 604-606.

  Reason, M. most solid fabric of human, 308.
    M. demonstrates supremacy of human, 309.

  Reasoning, M. a type of perfect, 307.
    M. as an exercise in, 423-427, 429, 430, 1503.

  Recorde, Value of arithmetic, 1619.

  Regiomontanus, 1543.

  Regular solids, 2132-2135.

  Reid, M. frees from sophistry, =215=.
    Conjecture has no place in m., =234=.
    M. the most solid fabric, =308=.
    On Euclid’s elements, =955=.
    M. manifests what is impossible =1414=.
    On m. and philosophy, =1423=.
    Probability and Christianity, =1975=.
    On Pythagoras and the regular solids, =2132=.

  Reidt, M, as an exercise in language, =419=.
    On the ethical value of m., =456=.
    On aim in math. instruction, =506=.

  Religion and m., 274-276, 459, 460, 1013.

  Research in m., Chapter VI.

  Reversible verses, 2156.

  Reye, Advantages of modern over ancient geometry, =714=.

  Rhetoric and m., 1599.

  Riemann, On m. and physics, 1549.

  Rigor in m., 535-538.

  Rosanes, On the unpopularity of m., =730=.

  Royal road, 201, 901, 951, 1774.

  Royal science, M. a, 204.

  Rudio, On Euler, =957=.
    M. and great artists, =1105=.
    On m. and navigation, =1543=.

  Rush, M. cures predisposition to anger, =458=.

  Russell, Definition of m., =127=, =128=.
    On nineteenth century m., =705=.
    Chief triumph of modern m., =706=.
    On the infinite, =723=.
    On beauty in m., =1104=.
    On the value of symbols, =1219=.
    On Boole’s Laws of Thought, =1318=.
    Principia Mathematica, 1326.
    On geometry and philosophy, =1410=.
    Definition of number, =1609=.
    Fruitful uses of imaginaries, =1735=.
    Geometrical reasoning circular, =1864=.
    On projective geometry, =1879=.
    Zeno’s problems, =1938=.
    Definition of infinite collection, =1959=.
    On proofs of axioms, =2013=.
    On non-euclidean geometry, =2018=.


  Safford, On aptitude for m., =520=.
    On m. and science, =1509=.

  Sage, Battalions of figures, =1631=.

  Sartorius, Gauss on the nature of space, =2034=.

  Scepticism, 452, 811.

  Schellbach, Estimate of m., =306=.
    On truth, =1114=.

  Schiller, Archimedes and the youth, =907=.

  Schopenhauer, Arithmetic rests on the concept of time,
      =1613=.
    Predicabilia a priori, =2003=.

  Schröder, M. as a branch of logic, =1323=.

  Schubert, Three characteristics of m., =229=.
    On controversies in m., =243=.
    Characteristics of m., =263=.
    M. an exclusive science, =734=.

  Science and m., Chapter XV.
    M. an indispensible tool of, 309.
    Neglect of m. works injury to, 310.
    Craig on origin of new, 646.
    Greek view of, 1429.
    Six follies of, 2107.
    See also 433, 436, 437, 461, 725.

  Scientific education, Math. training indispensable basis
      of, =444=.

  Screw, The song of the, 1894.
    As an instrument in geometry, 2114.

  Sedgwick, Quaternion of maladies, =1723=.

  Segre, On research in m., =619=.
    What kind of investigations are important, =641=.
    On the worthlessness of certain investigations, =642,
        643=.
    On hyper-space, =2031=.

  Seneca, Alexander and geometry, =902=.

  Seventy-seven, The number, 2149.

  Shakespeare, 1127, 1129, 2141.

  Shaw, J. B., M. like game of chess, =840=.

  Shaw, W. H., M. and professional life, =1596=.

  Sherman, M. and rhetoric, =1599=.

  Smith, Adam, 1324.

  Smith, D. E., On problem solving, =532=.
    Value of geometrical training, =1846=.
    Reason for studying geometry, =1850=.

  Smith, H. J. S., When a math. theory is completed, =637=.
    On the growth of m., =1521=.
    On m. and science, =1542=.
    On m. and physics, =1556=.
    On m. and meteorology, =1557=.
    On number theory, =1645=.
    Rigor in Euclid, =1829=.
    On Euclid’s doctrine of proportion, =1834=.

  Smith, W. B., Definition of m., =121=.
    On infinitesimal analysis, =1914=.
    On non-euclidean and hyperspaces, =2033=.

  Simon, On beauty and truth, =1114=.

  Simplicity in m., 315, 526.

  Sin²φ, On the notation of, 1886.

  Six hundred sixty-six, The number, 2151, 2152.

  Social science and m., 1201, 1586, 1587.

  Social service, M. as an aid to, 313, 314, 328.

  Social value of m., 456, 1588.

  Solitude and m., 1849, 1851.

  Sophistry, M. free from, 215.

  Sound, M. and the theory of, 1551.

  Southey, On Newton, =1008=.

  Space, Of experience, 2011.
    Kant’s doctrine of, 2003.
    Schopenhauer’s predicabilia, 2004.
    Whewell, On the idea of, 2004.
    Non-euclidean, 2015, 2016, 2018.
    Hyper-, 2030, 2031, 2033, 2036-2038.

  Spedding, On Bacon’s knowledge of m., =917=.

  Speer, On m. and nature-study, =514=.

  Spence, On Newton, =1016=, =1020=.

  Spencer, On m. in the arts, =1570=.

  Spherical trigonometry, 1887.

  Spira mirabilis, 922.

  Spottiswoode, On the kingdom of m., =269=.

  Squaring the circle, 1537, 1858, 1934, 1948, 2115-2117.

  St. Augustine, The number seventy seven, =2149=.

  St. Vincent, As a circle-squarer, 2109.

  Steiner, On projective geometry, =1877=.

  Stewart, M. and facts, =237=.
    On beauty in m., =242=.
    What we most admire in m., =315=.
    M. for its own sake, =440=.
    M. the noblest instance of force of the human mind,
        =452=.
    Math’ns and applause, =816=.
    Mere math’ns, =821=.
    Shortcomings of math’ns, =828=.
    On the influence of Leibnitz, =988=.
    Reason supreme, =1424=.
    M. and philosophy compared, =1428=.
    M. and natural philosophy, =1555=.

  Stifel, The number of the beast, =2152=.

  Stobæus, Alexander and Menæchmus, =901=.
    Euclid and the student, =952=.

  Study of m., Chapter VI.

  Substitution, Concept of, 727.

  Superstition, M. frees mind from, 450.
    Number was born in, 1632.

  Surd numbers, 1728.

  Surprises, M. rich in, 202.

  Swift, On m. and politics, =814=.
    The math’ns of Laputa, =2120-2122=.
    The math. school of Laputa, =2123=.
    His ignorance of m., 2124, 2125.

  Sylvester, On the use of the terms mathematic and
      mathematics, =101=.
    Order and arrangement the basic ideas of m., =109=,
        =110=.
    Definition of algebra, =110=.
    Definition of arithmetic, =110=.
    Definition of geometry, =110=.
    On the object of pure m., =129=.
    M. requires harmonious action of all the faculties,
        =202=.
    Answer to Huxley, =251=.
    On the nature of m., =251=.
    On observation in m., =255=.
    Invention in m., =260=.
    M. entitled to human regard, =301=.
    On the ethical value of m., =449=.
    On isolated theorems, =620=.
    “Auge _et impera._” =631=.
    His bent of mind, =645=.
    Apology for imperfections, =648=.
    On theoretical investigations, =658=.
    Characteristics of modern m., =724=.
    Invested m. with halo of glory, 740.
    M. and eloquence, =829=.
    On longevity of math’ns, =839=.
    On Cayley, =930=.
    His view of Euclid, 936.
    Jacobi’s talent for philology, =980=.
    His eloquence, 1030.
    Researches in quantics, =1032=.
    His weakness, 1033, 1036, 1037.
    One-sided character of his work, 1034.
    His method, 1035, 1036, 1041.
    His forgetfulness, 1037, 1038.
    Relations with students, 1039.
    His style, 1040, 1041.
    His characteristics, 1041.
    His enthusiasm, 1041.
    The math. Adam, =1042=.
    And Weierstrass, 1050.
    On divine beauty and order in m., =1101=.
    M. among the fine arts, =1106=.
    On music and m., =1131=.
    M. the quintessence of language, =1205=.
    M. the language of the universe, =1206=.
    On prime numbers, =1648=.
    On determinants, =1740=.
    On invariants, =1742=.
    Contribution to theory of equations, 1743.
    To a missing member etc., =1745=.
    Invariants and isomerism, =1750=.
    His dislike for Euclid, =1826=.
    On the invention of integrals, =1922=.
    On geometry and analysis, =1931=.
    On paradoxes, =2104=.

  Symbolic language, M. as a, 1207, 1212.
    Use of, 1573.

  Symbolic logic, 1316-1321.

  Symbolism, On the nature of math., 1210.
    Difficulty of math., 1218.
    Universal impossible, 1221.
    See also notation.

  Symbols, Burlesque on, 1741.

  Symbols, M. leads to mastery of, 421.
    Value of math., 1209, 1212, 1219.
    Essential to demonstration, 1316.
    Arithmetical, 1627.


  Tact in m., 622, 623.

  Tait, On the unpopularity of m., =740=.
    And Thomson, 1043.
    And Hamilton, 1044.
    On quaternions, =1724-1726=.
    On spherical trigonometry, =1887=.

  Talent, Math’ns men of, 825.

  Teaching of m., Chapter V.

  Tennyson, 1843.

  Teutonic race, Aptitude for m., 838.

  Text-books, Chrystal on, 533.
    Minchin on, 539.
    Cremona on English, 609.
    Glaisher on need of, 635.

  Thales, 201.

  Theoretical investigations, 652-664.

  Theory and practice, 661.

  Thompson, Sylvanus, Lord Kelvin’s definition of a math’n,
      =822=.
    Cayley’s estimate of quaternions, =937=.
    Thomson’s “It is obvious that,” =1045=.
    Anecdote of Lord Kelvin, =1046=, =1047=.
    On the calculus for beginners, =1917=.

  Thomson, Sir William,
    M. the only true metaphysics, =305=.
    M. not repulsive to common sense, =312=.
    What is a math’n? =822=.
    And Tait, 1043.
    “It is obvious that,” 1045.
    Anecdotes concerning, 1046, 1047, 1048.
    On m. and astronomy, =1562=.
    On quaternions, =1721, 1722=.

  Thomson and Tait, 1043.
    On Fourier’s theorem, =1928=.

  Thoreau, On poetry and m., =1123=.

  Thought-economy in m., 203, 1209, 1704.

  Three, The Yankos word for, 2144.
    Pacioli on the number, 2145.

  Time, Arithmetic rests on notion of 1613.
    As a concept in algebra, 1715, 1716, 1717.
    Kant’s doctrine of, 2001.
    Schopenhauer’s predicabilia, 2003.

  Todhunter, On m. as a university subject, =405=.
    On m. as a test of performance, =408=.
    On m. as an instrument in education, =414=.
    M. requires voluntary exertion, =415=.
    On exercises, =422=.
    On problems, =523=, =608=.
    How to read m., =605=, =606=.
    On discovery in elementary m., =617=.
    On Sylvester’s theorem, =1743=.
    On performance in Euclid, =1818=.

  Transformation, Concept of, 727.

  Trigonometry, 1881, 1884-1889.

  Trilinear co-ordinates, 611.

  Trisection of angle, 2112.

  Truth, and m., 306.
    Math’ns must perceive beauty of, 803.
      And beauty, 1114.

  Tzetzes, Plato on geom., =1803=.


  Unity, Locke on the idea of, 1607.

  Universal algebra, 1753.

  Universal arithmetic, 1717.

  Universal language, 925.

  Unpopularity of m., 270, 271, 730-736, 738, 740, 1501,
      1628.

  Usefulness, As a principle in research, 652-655, 659, 664.

  Uses of m., See value of m.


  Value of m., Chapter IV.
    See also 330, 333, 1414, 1422, 1505, 1506, 1512, 1523,
        1526, 1527, 1533, 1541, 1542, 1543, 1547-1576,
        1619-1626, 1841, 1844-1851.

  Variability, The central idea of modern m., 720, 721.

  Venn, On m. as a symbolic language, =1207=.
    M. the only gate, =1517=.

  Viola, On the use of fallacies, =610=.

  Virgil, =2138=.

  Voltaire, Archimedes more imaginative than Homer, =259=.
    M. the staff of the blind, =461=.
    On direct usefulness of results, =653=.
    On infinite magnitudes, =1947=.
    On the symbol, =1950=.
    Anagram on, 2154.


  Walcott, On hyperbolic functions, =1930=.

  Walker, On problems in arithmetic, =528=.
    On the teaching of geometry, =529=.

  Wallace, On the frequency of the math. faculty, =832=.
    On m. and natural selection, =833, 834=.
    Parallel growth of m. and music, =1135=.

  Walton, Angling like m., =739=.

  Weber, On m. and physics, =1549=.

  Webster, Estimate of m., =331=.

  Weierstrass, Math’ns are poets, =802=.
    Anecdote concerning, 1049.
    And Sylvester, 1050.
    Problem of infinitesimals, 1938.

  Weismann, On the origin of the math. faculty, =1136=.

  Wells, On m. as a world language, =1201=.

  Whately, On m. as an exercise, =427=.
    On m. and navigation, =1544=.
    On geometrical demonstrations, =1839=.
    On Swift’s ignorance of m., =2124=.

  Whetham, On symbolic logic, =1319=.

  Whewell, On mixed and pure math., =107=.
    M. not an inductive science, =223=.
    Nature of m., 224.
    Value of geometry, 445.
    On theoretical investigations, =660=, =662=.
    Math’ns men of talent, =825=.
    Fame of math’ns, =826=.
    On Newton’s greatness, =1004=.
    On Newton’s theory, =1005=.
    On Newton’s humility, =1025=.
    On symbols, 1212.
    On philosophy and m., =1429=.
    On m. and science, =1534=.
    Quotation from R. Bacon, =1547=.
    On m. and applications, =1541=.
    Geometry and experience, 1814.
    Geometry not an inductive science, 1830.
    On limits, 1909.
    On the idea of space, 2004.
    On Plato and the regular solids, =2133=,
        =2135=.

  White, H. S., On the growth of m., =211=.

  White, W. F., Definition of m., =131=, =1203=.
    M. as a prerequisite for public speaking, =420=.
    On beauty in m., =1119=.
    The place of the math’n, =1529=.
    On m. and social science, =1586=.
    The cost of Manhattan island, =2130=.

  Whitehead, On the ideal of m., =119=.
    Definition of m., =122=.
    On the scope of m., =126=.
    On the nature of m., =233=.
    Precision necessary in m., =639=.
    On practical applications, =655=.
    On theoretical investigations, =659=.
    Characteristics of ancient geometry, =713=.
    On the extent of m., =737=.
    Archimedes compared with Newton, =911=.
    On the Arabic notation, =1217=.
    Difficulty of math. notation, =1218=.
    On symbolic logic, =1320=.
    Principia Mathematica, 1326.
    On philosophy and m., =1403=.
    On obscurity in m. and philosophy, =1407=.
    On the laws of algebra, =1708=.
    On + and − signs, =1727=.
    On universal algebra, =1753=.
    On the Cartesian method, =1890=.
    On Swift’s ignorance of m., =2125=.

  Whitworth, On the solution of problems, =611=.

  Williamson, On the value of m., =1575=.
    Infinitesimals and limits, =1905=.
    On infinitesimals, =1946=.

  Wilson, E. B., On the social value of m., =1588=.
    On m. and economics, =1593=.
    On the nature of axioms, =2012=.

  Wilson, John, On Newton and Shakespeare, =1012=.
    Newton and Linnæus, =1013=.

  Woodward, On probabilities, =1589=.
    On the theory of errors, =1973=, =1974=.

  Wordsworth, W., On Archimedes, =906=.
    On poetry and geometric truth, =1117=.
    On geometric rules, =1418=.
    On geometry, =1840=, =1848=.
    M. and solitude, =1859=.

  Workman, On the poetic nature of m., =1120=.


  Young, C. A., On the discovery of Neptune, =1559=.

  Young, C. W., Definition of m., =124=.

  Young, J. W. A., On m. as type a of thought, =404=.
    M. as preparation for science study, =421=.
    M. essential to comprehension of nature, =435=.
    Development of abstract methods, =729=.
    Beauty in m., =1110=.
    On Euclid’s axiom, =2014=.


  Zeno, His problems, 1938.

  Zero, 1948, 1954.


       *       *       *       *       *



  Transcriber’s Notes

  Punctuation has been standardised.

  Characters in small caps have been replaced by all caps.

  Italic text has been denoted by _underscores_ and bold text
      by =equal signs=.

  Em-dash added before all attribution names for consistency.

  The two omitted illustrations have been identified by an
      [Illustration:] tag with a short description.

  Mis-alphabetized entries in the Index have been
      corrected

  Non-printable superscripts are represented by a
      caret followed by the character, i.e. x^n.

  Non-printable subscripts have been represented by an
      underscore followed by the subscript in braces
      i.e. _{a}.

  Book was written in a period when many words had not
      become standarized in their spelling. Numerous words
      have multiple spelling variations in the text. These
      have been left unchanged unless noted below:

         §230  - “elmenetary” corrected to “elementary”
                 (the most elementary use of)

         §437  - “Mathematiks” corrected to “Mathematicks”
                 (The Usefulness of Mathematicks)
                 as in the quoted text.

         §511  - Block number shown as 517

         §517  - “hoheren” corrected to “höheren”
                 (höheren Schulen) for consistency

         §540  - duplicate word “the” removed
                 (let the mind)

         §657  - “anaylsis” corrected to “analysis”
                 (field of analysis.)

         §729  - “Geomtry” corrected to “Geometry”
                 (Algebra and Geometry)

         §822  - end of quote not identified;
                 placement unclear.

         §823  - “heros” corrected to “heroes”
                 (many of the major heroes)

         §917  - “εὓυρηκα” corrected to “εὔυρηκα”
                 (speaks of the εὔυρηκα)

         §986  - added missing end quote

         §1132 - “Vereiningung” corrected to “Vereinigung”
                 ( Deutschen Mathematiker Vereinigung)

         §1325 - “Philosphy” corrected to “Philosophy”
                 (Positive Philosophy)

         §1421 - “1427” corrected to block “1421”
                 (=1421.=)

         §1503 - “Todhunder’s” corrected to “Todhunter’s”
                 (Todhunter’s History of)

         §1535 - “uses” corrected to “use”
                 (the use of analysis)

         §1803 - “τὴυ” corrected to “τὴν”
                 (μοῦ τὴν στέγην)

         §1874 - “anaylsis” corrected to “analysis”
                 (a kind of analysis)

         §1930 - “Hyberbolic” corrected to “Hyperbolic”
                 (Mathematical Tables, Hyperbolic Functions)

         §2009 - “Stanfpunkte” corrected to “Standpunkte”
                 (höheren Standpunkte aus)

         §2126 - Omitted block number added

         §2135 - “astromomy” corrected to “astronomy”
                 (history of astronomy)

         §2151 - “10” corrected to “9”
                 (A to I represent 1-9)

    Appolonius - also spelled “Apollonius” but not referenced
                 at §523 and §917

        Bôcher - “Bocher” corrected to “Bôcher”
                 as given in text

       Halsted - “Slyvester” corrected to “Sylvester”
                 (And Sylvester)

     Jefferson - “Om” corrected to “On”
                 (On m. and law)

       Peacock - “philosphers” corrected to “philosophers”
                 (Greek philosophers)





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