Memorabilia Mathematica; or, the Philomath's Quotation-Book
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Title: Memorabilia Mathematica; or, the Philomath's Quotation-Book
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*** START OF THE PROJECT GUTENBERG EBOOK MEMORABILIA MATHEMATICA; OR, THE PHILOMATH'S QUOTATION-BOOK ***
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
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