ARITHMETIC
=1601.= There is no problem in all mathematics that cannot be
solved by direct counting. But with the present implements of
mathematics many operations can be performed in a few minutes
which without mathematical methods would take a lifetime.
--MACH, ERNST.
_Popular Scientific Lectures [McCormack]
(Chicago, 1898), p. 197._
=1602.= There is no inquiry which is not finally reducible to a
question of Numbers; for there is none which may not be conceived
of as consisting in the determination of quantities by each
other, according to certain relations.--COMTE, A.
_Positive Philosophy [Martineau], Bk. 1,
chap. 1._
=1603.= Pythagoras says that number is the origin of all things,
and certainly the law of number is the key that unlocks the
secrets of the universe. But the law of number possesses an
immanent order, which is at first sight mystifying, but on a more
intimate acquaintance we easily understand it to be intrinsically
necessary; and this law of number explains the wondrous
consistency of the laws of nature.--CARUS, PAUL.
_Reflections on Magic Squares; Monist,
Vol. 16 (1906), p. 139._
=1604.= An ancient writer said that arithmetic and geometry are
the _wings of mathematics_; I believe one can say without
speaking metaphorically that these two sciences are the
foundation and essence of all the sciences which deal with
quantity. Not only are they the foundation, they are also, as it
were, the capstones; for, whenever a result has been arrived at,
in order to use that result, it is necessary to translate it into
numbers or into lines; to translate it into numbers requires the
aid of arithmetic, to translate it into lines necessitates the
use of geometry.--LAGRANGE.
_Leçons Elémentaires sur les
Mathématiques, Leçon seconde._
=1605.= It is number which regulates everything and it is measure
which establishes universal order.... A quiet peace, an
inviolable order, an inflexible security amidst all change and
turmoil characterize the world which mathematics discloses and
whose depths it unlocks.--DILLMANN, E.
_Die Mathematik die Fackelträgerin einer
neuen Zeit (Stuttgart, 1889), p. 12._
=1606.=
Number, the inducer of philosophies,
The synthesis of letters, ....
--AESCHYLUS.
_Quoted in, Thomson, J. A., Introduction
to Science, chap. 1 (London)._
=1607.= Amongst all the ideas we have, as there is none suggested
to the mind by more ways, so there is none more simple, than that
of _unity_, or one: it has no shadow of variety or composition in
it; every object our senses are employed about; every idea in our
understanding; every thought of our minds, brings this idea along
with it. And therefore it is the most intimate to our thoughts,
as well as it is, in its agreement to all other things, _the most
universal idea we have_.--LOCKE, JOHN.
_An Essay concerning Human
Understanding, Bk. 2, chap. 16, sect.
1._
=1608.= The _simple modes_ of _number_ are of all other the most
distinct; every the least variation, which is an unit, making
each combination as clearly different from that which approacheth
nearest to it, as the most remote; two being as distinct from
one, as two hundred; and the idea of two as distinct from the
idea of three, as the magnitude of the whole earth is from that
of a mite.--LOCKE, JOHN.
_An Essay concerning Human
Understanding, Bk. 2, chap. 16, sect.
3._
=1609.= The number of a class is the class of all classes similar
to the given class.--RUSSELL, BERTRAND.
_Principles of Mathematics (Cambridge,
1903), p. 115._
=1610.= Number is that property of a group of distinct things
which remains unchanged during any change to which the group may
be subjected which does not destroy the distinctness of the
individual things.--FINE, H. B.
_Number-system of Algebra (Boston and
New York, 1890), p. 3._
=1611.= The science of arithmetic may be called the science of
exact limitation of matter and things in space, force, and time.
--PARKER, F. W.
_Talks on Pedagogics (New York, 1894),
p. 64._
=1612.=
Arithmetic is the science of the Evaluation
of Functions,
Algebra is the science of the Transformation
of Functions.
--HOWISON, G. H.
_Journal of Speculative Philosophy, Vol.
5, p. 175._
=1613.= That _arithmetic_ rests on pure intuition of _time_ is not
so obvious as that geometry is based on pure intuition of space,
but it may be readily proved as follows. All counting consists in
the repeated positing of unity; only in order to know how often it
has been posited, we mark it each time with a different word:
these are the numerals. Now repetition is possible only through
succession: but succession rests on the immediate intuition of
_time_, it is intelligible only by means of this latter concept:
hence counting is possible only by means of time.--This dependence
of counting on _time_ is evidenced by the fact that in all
languages multiplication is expressed by “times” [mal], that
is, by a concept of time; sexies, ἑξακις, six fois, six times.
--SCHOPENHAUER, A.
_Die Welt als Vorstellung und Wille;
Werke (Frauenstaedt) (Leipzig, 1877),
Bd. 3, p. 39._
=1614.= The miraculous powers of modern calculation are due to
three inventions: the Arabic Notation, Decimal Fractions and
Logarithms.--CAJORI, F.
_History of Mathematics (New York,
1897), p. 161._
=1615.= The grandest achievement of the Hindoos and the one
which, of all mathematical investigations, has contributed most
to the general progress of intelligence, is the invention of the
principle of position in writing numbers.--CAJORI, F.
_History of Mathematics (New York,
1897), p. 87._
=1616.= The invention of logarithms and the calculation of the
earlier tables form a very striking episode in the history of
exact science, and, with the exception of the _Principia_ of
Newton, there is no mathematical work published in the country
which has produced such important consequences, or to which so
much interest attaches as to Napier’s Descriptio.
--GLAISHER, J. W. L.
_Encyclopedia Britannica, 9th Edition;
Article “Logarithms.”_
=1617.= All minds are equally capable of attaining the science of
numbers: yet we find a prodigious difference in the powers of
different men, in that respect, after they have grown up, because
their minds have been more or less exercised in it.
--JOHNSON, SAMUEL.
_Boswell’s Life of Johnson, Harper’s
Edition (1871), Vol. 2, p. 33._
=1618.= The method of arithmetical teaching is perhaps the best
understood of any of the methods concerned with elementary
studies.--BAIN, ALEXANDER.
_Education as a Science (New York,
1898), p. 288._
=1619.= What a benefite that onely thyng is, to haue the witte
whetted and sharpened, I neade not trauell to declare, sith all
men confesse it to be as greate as maie be. Excepte any witlesse
persone thinke he maie bee to wise. But he that most feareth
that, is leaste in daunger of it. Wherefore to conclude, I see
moare menne to acknowledge the benefite of nomber, than I can
espie willying to studie, to attaine the benefites of it. Many
praise it, but fewe dooe greatly practise it: onlesse it bee for
the vulgare practice, concernying Merchaundes trade. Wherein the
desire and hope of gain, maketh many willying to sustaine some
trauell. For aide of whom, I did sette forth the first parte of
_Arithmetike_. But if thei knewe how faree this seconde parte,
doeeth excell the firste parte, thei would not accoumpte any
tyme loste, that were emploied in it. Yea thei would not thinke
any tyme well bestowed till thei had gotten soche habilitie by
it, that it might be their aide in al other studies.
--RECORDE, ROBERT.
_Whetstone of Witte (London, 1557)._
=1620.= You see then, my friend, I observed, that our real need
of this branch of science [arithmetic] is probably because it
seems to compel the soul to use our intelligence in the search
after pure truth.
Aye, remarked he, it does this to a remarkable extent.
Have you ever noticed that those who have a turn for arithmetic
are, with scarcely an exception, naturally quick in all sciences;
and that men of slow intellect, if they be trained and exercised
in this study ... become invariably quicker than they were
before?
Exactly so, he replied.
And, moreover, I think you will not easily find that many things
give the learner and student more trouble than this.
Of course not.
On all these accounts, then, we must not omit this branch of
science, but those with the best of talents should be instructed
therein.--PLATO.
_Republic [Davis], Bk. 7, chap. 8._
=1621.= Arithmetic has a very great and elevating effect,
compelling the soul to reason about abstract number, and if
visible or tangible objects are obtruding upon the argument,
refusing to be satisfied.--PLATO.
_Republic [Jowett], Bk. 7, p. 525._
=1622.= Good arithmetic contributes powerfully to purposive
effort, to concentration, to tenacity of purpose, to generalship,
to faith in right, and to the joy of achievement, which are the
elements that make up efficient citizenship.... Good arithmetic
exalts thinking, furnishes intellectual pleasure, adds appreciably
to love of right, and subordinates pure memory.--MYERS, GEORGE.
_Monograph on Arithmetic in Public
Education (Chicago), p. 21._
=1623.= On the one side we may say that the purpose of number
work is to put a child in possession of the machinery of
calculation; on the other side it is to give him a better mastery
of the world through a clear (mathematical) insight into the
varied physical objects and activities. The whole world, from one
point of view, can be definitely interpreted and appreciated by
mathematical measurements and estimates. Arithmetic in the common
school should give a child this point of view, the ability to see
and estimate things with a mathematical eye.--MCMURRAY, C. A.
_Special Method in Arithmetic_ (_New
York, 1906_), _p. 18._
=1624.= We are so accustomed to hear arithmetic spoken of as
one of the three fundamental ingredients in all schemes of
instruction, that it seems like inquiring too curiously to ask
why this should be. Reading, Writing, and Arithmetic--these
three are assumed to be of co-ordinate rank. Are they indeed
co-ordinate, and if so on what grounds?
In this modern “trivium” the art of reading is put first. Well,
there is no doubt as to its right to the foremost place. For
reading is the instrument of all our acquisition. It is
indispensable. There is not an hour in our lives in which it does
not make a great difference to us whether we can read or not.
And the art of Writing, too; that is the instrument of all
communication, and it becomes, in one form or other, useful to us
every day. But Counting--doing sums,--how often in life does this
accomplishment come into exercise? Beyond the simplest additions,
and the power to check the items of a bill, the arithmetical
knowledge required of any well-informed person in private life is
very limited. For all practical purposes, whatever I may have
learned at school of fractions, or proportion, or decimals, is,
unless I happen to be in business, far less available to me in
life than a knowledge, say, of history of my own country, or the
elementary truths of physics. The truth is, that regarded as
practical _arts_, reading, writing, and arithmetic have no right
to be classed together as co-ordinate elements of education; for
the last of these is considerably less useful to the average man
or woman not only than the other two, but than many others that
might be named. But reading, writing, and such mathematical or
logical exercise as may be gained in connection with the
manifestation of numbers, _have_ a right to constitute the
primary elements of instruction. And I believe that arithmetic,
if it deserves the high place that it conventionally holds in our
educational system, deserves it mainly on the ground that it is
to be treated as a logical exercise. It is the only branch of
mathematics which has found its way into primary and early
education; other departments of pure science being reserved for
what is called higher or university instruction. But all the
arguments in favor of teaching algebra and trigonometry to
advanced students, apply equally to the teaching of the
principles or theory of arithmetic to schoolboys. It is
calculated to do for them exactly the same kind of service, to
educate one side of their minds, to bring into play one set of
faculties which cannot be so severely or properly exercised in
any other department of learning. In short, relatively to the
needs of a beginner, Arithmetic, as a science, is just as
valuable--it is certainly quite as intelligible--as the higher
mathematics to a university student.--FITCH, J. G.
_Lectures on Teaching (New York, 1906),
pp. 267-268._
=1625.= What mathematics, therefore are expected to do for the
advanced student at the university, Arithmetic, if taught
demonstratively, is capable of doing for the children even of the
humblest school. It furnishes training in reasoning, and
particularly in deductive reasoning. It is a discipline in
closeness and continuity of thought. It reveals the nature of
fallacies, and refuses to avail itself of unverified assumptions.
It is the one department of school-study in which the sceptical
and inquisitive spirit has the most legitimate scope; in which
authority goes for nothing. In other departments of instruction
you have a right to ask for the scholar’s confidence, and to
expect many things to be received on your testimony with
the understanding that they will be explained and verified
afterwards. But here you are justified in saying to your pupil
“Believe nothing which you cannot understand. Take nothing for
granted.” In short, the proper office of arithmetic is to serve
as elementary training in logic. All through your work as
teachers you will bear in mind the fundamental difference between
knowing and thinking; and will feel how much more important
relatively to the health of the intellectual life the habit of
thinking is than the power of knowing, or even facility of
achieving visible results. But here this principle has special
significance. It is by Arithmetic more than by any other subject
in the school course that the art of thinking--consecutively,
closely, logically--can be effectually taught.--FITCH, J. G.
_Lectures on Teaching (New York, 1906),
pp. 292-293._
=1626.= Arithmetic and geometry, those wings on which the
astronomer soars as high as heaven.--BOYLE, ROBERT.
_Usefulness of Mathematics to Natural
Philosophy; Works (London, 1772), Vol.
3, p. 429._
=1627.= Arithmetical symbols are written diagrams and geometrical
figures are graphic formulas.--HILBERT, D.
_Mathematical Problems; Bulletin
American Mathematical Society, Vol. 8
(1902), p. 443._
=1628.= Arithmetic and geometry are much more certain than the
other sciences, because the objects of them are in themselves so
simple and so clear that they need not suppose anything which
experience can call in question, and both proceed by a chain of
consequences which reason deduces one from another. They are also
the easiest and clearest of all the sciences, and their object is
such as we desire; for, except for want of attention, it is
hardly supposable that a man should go astray in them. We must
not be surprised, however, that many minds apply themselves by
preference to other studies, or to philosophy. Indeed everyone
allows himself more freely the right to make his guess if the
matter be dark than if it be clear, and it is much easier to have
on any question some vague ideas than to arrive at the truth
itself on the simplest of all.--DESCARTES.
_Rules for the Direction of the Mind;
Torrey’s Philosophy of Descartes (New
York, 1892), p. 63._
=1629.=
Why are _wise_ few, _fools_ numerous in the
excesse?
’Cause, wanting _number_, they are
_numberlesse_.
--LOVELACE.
_Noah Bridges: Vulgar Arithmetike
(London, 1659), p. 127._
=1630.= The clearness and distinctness of each mode of number
from all others, even those that approach nearest, makes me apt
to think that demonstrations in numbers, if they are not more
evident and exact than in extension, yet they are more general in
their use, and more determinate in their application. Because the
ideas of numbers are more precise and distinguishable than in
extension; where every equality and excess are not so easy to be
observed or measured; because our thoughts cannot in space arrive
at any determined smallness beyond which it cannot go, as an
unit; and therefore the quantity or proportion of any the least
excess cannot be discovered.--LOCKE, JOHN.
_An Essay concerning Human
Understanding, Bk. 2, chap. 16, sect.
4._
=1631.= Battalions of figures are like battalions of men, not
always as strong as is supposed.--SAGE, M.
_Mrs. Piper and the Society for
Psychical Research [Robertson] (New
York, 1909), p. 151._
=1632.= Number was born in superstition and reared in mystery,...
numbers were once made the foundation of religion and philosophy, and
the tricks of figures have had a marvellous effect on a credulous
people.--PARKER, F. W.
_Talks on Pedagogics (New York, 1894),
P. 64._
=1633.= A rule to trick th’ arithmetic.--KIPLING, R.
_To the True Romance._
=1634.= God made integers, all else is the work of man.
--KRONECKER, L.
_Jahresberichte der Deutschen
Mathematiker Vereinigung, Bd. 2, p. 19._
=1635.= Plato said “ἀεὶ ὁ θεὸς γεωμέτρε.” Jacobi changed this to
“ἀεὶ ὁ θεὸς ἀριθμητίζει.” Then came Kronecker and created the
memorable expression “Die ganzen Zahlen hat Gott gemacht, alles
andere ist Menschenwerk.”--KLEIN, F.
_Jahresbericht der Deutschen
Mathematiker Vereinigung, Bd. 6, p.
136._
=1636.= Integral numbers are the fountainhead of all mathematics.
--MINKOWSKI, H.
_Diophantische Approximationen (Leipzig,
1907), Vorrede._
=1637.= The “Disquisitiones Arithmeticae” that great book with
seven seals.--MERZ, J. T.
_A History of European Thought in the
Nineteenth Century (Edinburgh and
London, 1908), p. 721._
=1638.= It may fairly be said that the germs of the modern
algebra of linear substitutions and concomitants are to be found
in the fifth section of the _Disquisitiones Arithmeticae_; and
inversely, every advance in the algebraic theory of forms is an
acquisition to the arithmetical theory.--MATHEWS, G. B.
_Theory of Numbers (Cambridge, 1892),
Part 1, sect. 48._
=1639.= Strictly speaking, the theory of numbers has nothing to
do with negative, or fractional, or irrational quantities, _as
such._ No theorem which cannot be expressed without reference to
these notions is purely arithmetical: and no proof of an
arithmetical theorem, can be considered finally satisfactory if
it intrinsically depends upon extraneous analytical theories.
--MATHEWS, G. B.
_Theory of Numbers (Cambridge, 1892),
Part 1, sect. 1._
=1640.= Many of the greatest masters of the mathematical sciences
were first attracted to mathematical inquiry by problems relating
to numbers, and no one can glance at the periodicals of the
present day which contain questions for solution without noticing
how singular a charm such problems still continue to exert. The
interest in numbers seems implanted in the human mind, and it is
a pity that it should not have freer scope in this country. The
methods of the theory of numbers are peculiar to itself, and are
not readily acquired by a student whose mind has for years been
familiarized with the very different treatment which is
appropriate to the theory of continuous magnitude; it is
therefore extremely desirable that some portion of the theory
should be included in the ordinary course of mathematical
instruction at our University. From the moment that Gauss, in his
wonderful treatise of 1801, laid down the true lines of the
theory, it entered upon a new day, and no one is likely to be
able to do useful work in any part of the subject who is
unacquainted with the principles and conceptions with which he
endowed it.--GLAISHER, J. W. L.
_Presidential Address British
Association for the Advancement of
Science (1890); Nature, Vol. 42, p.
467._
=1641.= Let us look for a moment at the general significance of
the fact that calculating machines actually exist, which relieve
mathematicians of the purely mechanical part of numerical
computations, and which accomplish the work more quickly and with
a greater degree of accuracy; for the machine is not subject to
the slips of the human calculator. The existence of such a
machine proves that computation is not concerned with the
significance of numbers, but that it is concerned essentially
only with the formal laws of operation; for it is only these that
the machine can obey--having been thus constructed--an intuitive
perception of the significance of numbers being out of the
question.--KLEIN, F.
_Elementarmathematik vom höheren
Standpunkte aus. (Leipzig, 1908), Bd. 1,
p. 53._
=1642.= Mathematics is the queen of the sciences and arithmetic
the queen of mathematics. She often condescends to render service
to astronomy and other natural sciences, but in all relations she
is entitled to the first rank.--GAUSS.
_Sartorius von Waltershausen: Gauss zum
Gedächtniss. (Leipzig, 1866), p. 79._
=1643.=
Zu Archimedes kam ein wissbegieriger Jüngling,
Weihe mich, sprach er zu ihm, ein in die
göttliche Kunst,
Die so herrliche Dienste der Sternenkunde
geleistet,
Hinter dem Uranos noch einen Planeten entdeckt.
Göttlich nennst Du die Kunst, sie ist’s,
versetzte der Weise,
Aber sie war es, bevor noch sie den Kosmos
erforscht,
Ehe sie herrliche Dienste der Sternenkunde
geleistet,
Hinter dem Uranos noch einen Planeten entdeckt.
Was Du im Kosmos erblickst, ist nur der
Göttlichen Abglanz,
In der Olympier Schaar thronet die ewige
Zahl.
--JACOBI, C. G. J.
_Journal für Mathematik, Bd. 101 (1887),
p. 338._
To Archimedes came a youth intent upon
knowledge,
Quoth he, “Initiate me into the science divine
Which to astronomy, lo! such excellent service
has rendered,
And beyond Uranus’ orb a hidden planet
revealed.”
“Call’st thou the science divine? So it is,”
the wise man responded,
“But so it was long before its light on the
Cosmos it shed,
Ere in astronomy’s realm such excellent service
it rendered,
And beyond Uranus’ orb a hidden planet
revealed.
Only reflection divine is that which Cosmos
discloses,
Number herself sits enthroned among Olympia’s
hosts.”
=1644.= The higher arithmetic presents us with an inexhaustible
store of interesting truths,--of truths too, which are not
isolated, but stand in a close internal connexion, and between
which, as our knowledge increases, we are continually discovering
new and sometimes wholly unexpected ties. A great part of its
theories derives an additional charm from the peculiarity that
important propositions, with the impress of simplicity upon them,
are often easily discoverable by induction, and yet are of so
profound a character that we cannot find their demonstration
till after many vain attempts; and even then, when we do succeed,
it is often by some tedious and artificial process, while the
simpler methods may long remain concealed.--GAUSS, C. F.
_Preface to Eisenstein’s Mathematische
Abhandlungen (Berlin, 1847), [H. J. S.
Smith]._
=1645.= The Theory of Numbers has acquired a great and increasing
claim to the attention of mathematicians. It is equally
remarkable for the number and importance of its results, for the
precision and rigorousness of its demonstrations, for the variety
of its methods, for the intimate relations between truths
apparently isolated which it sometimes discloses, and for the
numerous applications of which it is susceptible in other parts
of analysis.--SMITH, H. J. S.
_Report on the Theory of Numbers,
British Association, 1859; Collected
Mathematical Papers, Vol. 1, p. 38._
=1646.= The invention of the symbol ≡ by Gauss affords a striking
example of the advantage which may be derived from an appropriate
notation, and marks an epoch in the development of the science of
arithmetic.--MATHEWS, G. B.
_Theory of Numbers (Cambridge, 1892),
Part 1, sect. 29._
=1647.= As Gauss first pointed out, the problem of cyclotomy, or
division of the circle into a number of equal parts, depends in a
very remarkable way upon arithmetical considerations. We have
here the earliest and simplest example of those relations of the
theory of numbers to transcendental analysis, and even to pure
geometry, which so often unexpectedly present themselves, and
which, at first sight, are so mysterious.--MATHEWS, G. B.
_Theory of Numbers (Cambridge, 1892),
Part 1, sect. 167._
=1648.= I have sometimes thought that the profound mystery which
envelops our conceptions relative to prime numbers depends upon
the limitations of our faculties in regard to time, which like
space may be in its essence poly-dimensional, and that this and
such sort of truths would become self-evident to a being
whose mode of perception is according to _superficially_ as
distinguished from our own limitation to _linearly_ extended
time.--SYLVESTER, J. J.
_Collected Mathematical Papers, Vol. 4,
p. 600, footnote._
CHAPTER XVII
ALGEBRA
=1701.= The science of algebra, independently of any of its uses,
has all the advantages which belong to mathematics in general as
an object of study, and which it is not necessary to enumerate.
Viewed either as a science of quantity, or as a language of
symbols, it may be made of the greatest service to those who are
sufficiently acquainted with arithmetic, and who have sufficient
power of comprehension to enter fairly upon its difficulties.
--DE MORGAN, A.
_Elements of Algebra (London, 1837),
Preface._
=1702.= Algebra is generous, she often gives more than is asked
of her.--D’ALEMBERT.
_Quoted in Bulletin American Mathematical
Society, Vol. 2 (1905), p. 285._
=1703.= The operations of symbolic arithmetick seem to me to
afford men one of the clearest exercises of reason that I ever
yet met with, nothing being there to be performed without strict
and watchful ratiocination, and the whole method and progress of
that appearing at once upon the paper, when the operation is
finished, and affording the analyst a lasting, and, as it were,
visible ratiocination.--BOYLE, ROBERT.
_Works (London, 1772), Vol. 3, p. 426._
=1704.= The human mind has never invented a labor-saving machine
equal to algebra.--
_The Nation, Vol. 33, p. 237._
=1705.= They that are ignorant of Algebra cannot imagine the
wonders in this kind are to be done by it: and what further
improvements and helps advantageous to other parts of knowledge
the sagacious mind of man may yet find out, it is not easy to
determine. This at least I believe, that the _ideas of quantity_
are not those alone that are capable of demonstration and
knowledge; and that other, and perhaps more useful, parts of
contemplation, would afford us certainty, if vices, passions, and
domineering interest did not oppose and menace such endeavours.
--LOCKE, JOHN.
_An Essay concerning Human Understanding, Bk.
4, chap. 3, sect. 18._
=1706.= Algebra is but written geometry and geometry is but
figured algebra.--GERMAIN, SOPHIE.
_Mémoire sur la surfaces élastiques._
=1707.= So long as algebra and geometry proceeded separately
their progress was slow and their application limited, but when
these two sciences were united, they mutually strengthened each
other, and marched together at a rapid pace toward perfection.
--LAGRANGE.
_Leçons élémentaires sur les Mathématiques,
Leçon Cinquième._
=1708.= The laws of algebra, though suggested by arithmetic, do
not depend on it. They depend entirely on the conventions by
which it is stated that certain modes of grouping the symbols are
to be considered as identical. This assigns certain properties to
the marks which form the symbols of algebra. The laws regulating
the manipulation of algebraic symbols are identical with those of
arithmetic. It follows that no algebraic theorem can ever
contradict any result which could be arrived at by arithmetic;
for the reasoning in both cases merely applies the same general
laws to different classes of things. If an algebraic theorem can
be interpreted in arithmetic, the corresponding arithmetical
theorem is therefore true.--WHITEHEAD, A. N.
_Universal Algebra (Cambridge, 1898), p. 2._
=1709.= That a formal science like algebra, the creation of our
abstract thought, should thus, in a sense, dictate the laws of
its own being, is very remarkable. It has required the experience
of centuries for us to realize the full force of this appeal.
--MATHEWS, G. B.
_F. Spencer: Chapters on Aims and Practice of
Teaching (London, 1899), p. 184._
=1710.= The rules of algebra may be investigated by its own
principles, without any aid from geometry; and although in many
cases the two sciences may serve to illustrate each other, there
is not now the least necessity in the more elementary parts to
call in the aid of the latter in expounding the former.
--CHRYSTAL, GEORGE.
_Encyclopedia Britannica, 9th Edition;
Article “Algebra.”_
=1711.= Algebra, as an art, can be of no use to any one in the
business of life; certainly not as taught in the schools. I
appeal to every man who has been through the school routine
whether this be not the case. Taught as an art it is of little
use in the higher mathematics, as those are made to feel who
attempt to study the differential calculus without knowing more
of the principles than is contained in books of rules.
--DE MORGAN, A.
_Elements of Algebra (London, 1837),
Preface._
=1712.= We may always depend upon it that algebra, which cannot
be translated into good English and sound common sense, is bad
algebra.--CLIFFORD, W. K.
_Common Sense in the Exact Sciences (London,
1885), chap. 1, sect. 7._
=1713.= The best review of arithmetic consists in the study of
algebra.--CAJORI, F.
_Teaching and History of Mathematics in U. S.
(Washington, 1896), p. 110._
=1714.= [Algebra] has for its object the resolution of equations;
taking this expression in its full logical meaning, which
signifies the transformation of implicit functions into
equivalent explicit ones. In the same way arithmetic may be
defined as destined to the determination of the values of
functions.... We will briefly say that _Algebra is the Calculus
of Functions_, and _Arithmetic the Calculus of Values_.--COMTE, A.
_Philosophy of Mathematics [Gillespie] (New
York, 1851), p. 55._
=1715.= ... the subject matter of algebraic science is the
abstract notion of time; divested of, or not yet clothed with,
any actual knowledge which we may possess of the real Events of
History, or any conception which we may frame of Cause and Effect
in Nature; but involving, what indeed it _cannot_ be divested of,
the thought of _possible_ Succession, or of pure, _ideal_
Progression.--HAMILTON, W. R.
_Graves’ Life of Hamilton (New York,
1882-1889), Vol. 3, p. 633._
=1716.= ... instead of seeking to attain consistency and
uniformity of system, as some modern writers have attempted, by
banishing this thought of time from the _higher_ Algebra, I seek
to _attain_ the same object, by systematically introducing it
into the _lower_ or earlier parts of the science.--HAMILTON, W. R.
_Graves’ Life of Hamilton (New York,
1882-1889), Vol. 3, p. 634._
=1717.= The circumstances that algebra has its origin in
arithmetic, however widely it may in the end differ from that
science, led Sir Isaac Newton to designate it “Universal
Arithmetic,” a designation which, vague as it is, indicates its
character better than any other by which it has been attempted to
express its functions--better certainly, to ordinary minds, than
the designation which has been applied to it by Sir William Rowan
Hamilton, one of the greatest mathematicians the world has seen
since the days of Newton--“the Science of Pure Time;” or even
than the title by which De Morgan would paraphrase Hamilton’s
words--“the Calculus of Succession.”--CHRYSTAL, GEORGE.
_Encyclopedia Britannica, 9th Edition;
Article “Algebra.”_
=1718.= Time is said to have only _one dimension_, and space to
have _three dimensions_.... The mathematical _quaternion_
partakes of _both_ these elements; in technical language it may
be said to be “time plus space,” or “space plus time:” and in
this sense it has, or at least involves a reference to, _four
dimensions_....
And how the One of Time, of Space the Three,
Might in the Chain of Symbols girdled be.
--HAMILTON, W. R.
_Graves’ Life of Hamilton (New York,
1882-1889), Vol. 3, p. 635._
=1719.= It is confidently predicted, by those best qualified to
judge, that in the coming centuries Hamilton’s Quaternions will
stand out as the great discovery of our nineteenth century. Yet
how silently has the book taken its place upon the shelves of the
mathematician’s library! Perhaps not fifty men on this side of
the Atlantic have seen it, certainly not five have read it.
--HILL, THOMAS.
_North American Review, Vol. 85, p.
223._
=1720.= I think the time may come when double algebra will be the
beginner’s tool; and quaternions will be where double algebra is
now. The Lord only knows what will come above the quaternions.
--DE MORGAN, A.
_Graves’ Life of Hamilton (New York,
1882-1889), Vol. 3, p. 493._
=1721.= Quaternions came from Hamilton after his really good work
had been done; and though beautifully ingenious, have been an
unmixed evil to those who have touched them in any way, including
Clerk Maxwell.--THOMSON, WILLIAM.
_Thompson, S. P.: Life of Lord Kelvin
(London, 1910), p. 1138._
=1722.= The whole affair [quaternions] has in respect to
mathematics a value not inferior to that of “Volapuk” in respect
to language.--THOMSON, WILLIAM.
_Thompson, S. P.: Life of Lord Kelvin
(London, 1910), p. 1138._
=1723.= A quaternion of maladies! Do send me some formula by help
of which I may so doctor them that they may all become imaginary
or positively equal to nothing.--SEDGWICK.
_Graves’ Life of Hamilton (New York,
1882-1889), Vol. 3, p. 2._
=1724.= If nothing more could be said of Quaternions than that
they enable us to exhibit in a singularly compact and elegant
form, whose meaning is obvious at a glance on account of the
utter inartificiality of the method, results which in the
ordinary Cartesian co-ordinates are of the utmost complexity, a
very powerful argument for their use would be furnished. But it
would be unjust to Quaternions to be content with such a
statement; for we are fully entitled to say that in _all_ cases,
even in those to which the Cartesian methods seem specially
adapted, they give as simple an expression as any other method;
while in the great majority of cases they give a vastly simpler
one. In the common methods a judicious choice of co-ordinates is
often of immense importance in simplifying an investigation; in
Quaternions there is usually _no choice_, for (except when they
degrade to mere scalars) they are in general utterly independent
of any particular directions in space, and select of themselves
the most natural reference lines for each particular problem.
--TAIT, P. G.
_Presidential Address British
Association for the Advancement of
Science (1871); Nature, Vol. 4, p. 270._
=1725.= Comparing a Quaternion investigation, no matter in what
department, with the equivalent Cartesian one, even when the
latter has availed itself to the utmost of the improvements
suggested by Higher Algebra, one can hardly help making the remark
that they contrast even more strongly than the decimal notation
with the binary scale, or with the old Greek arithmetic--or than
the well-ordered subdivisions of the metrical system with the
preposterous no-systems of Great Britain, a mere fragment of which
(in the form of Table of Weights and Measures) form, perhaps the
most effective, if not the most ingenious, of the many instruments
of torture employed in our elementary teaching.--TAIT, P. G.
_Presidential Address British
Association for the Advancement of
Science (1871); Nature, Vol. 4, p. 271._
=1726.= It is true that, in the eyes of the pure mathematician,
Quaternions have one grand and fatal defect. They cannot be
applied to space of _n_ dimensions, they are contented to deal
with those poor three dimensions in which mere mortals are doomed
to dwell, but which cannot bound the limitless aspirations of a
Cayley or a Sylvester. From the physical point of view this,
instead of a defect, is to be regarded as the greatest possible
recommendation. It shows, in fact, Quaternions to be the special
instrument so constructed for application to the _Actual_ as to
have thrown overboard everything which is not absolutely
necessary, without the slightest consideration whether or no it
was thereby being rendered useless for application to the
_Inconceivable_.--TAIT, P. G.
_Presidential Address British
Association for the Advancement of
Science (1871); Nature, Vol. 4, p. 271._
=1727.= There is an old epigram which assigns the empire of the
sea to the English, of the land to the French, and of the clouds
to the Germans. Surely it was from the clouds that the Germans
fetched + and −; the ideas which these symbols have generated are
much too important to the welfare of humanity to have come from
the sea or from the land.--WHITEHEAD, A. N.
_An Introduction to Mathematics (New
York, 1911), p. 86._
=1728.= Now as to what pertains to these Surd numbers (which, as
it were by way of reproach and calumny, having no merit of their
own are also styled Irrational, Irregular, and Inexplicable) they
are by many denied to be numbers properly speaking, and are wont
to be banished from arithmetic to another Science, (which yet is
no science) viz. algebra.--BARROW, ISAAC.
_Mathematical Lectures (London, 1734),
p. 44._
=1729.= If it is true as Whewell says, that the essence of the
triumphs of science and its progress consists in that it enables
us to consider evident and necessary, views which our ancestors
held to be unintelligible and were unable to comprehend, then the
extension of the number concept to include the irrational, and we
will at once add, the imaginary, is the greatest forward step
which pure mathematics has ever taken.--HANKEL, HERMANN.
_Theorie der Complexen Zahlen (Leipzig,
1867), p. 60._
=1730.= That this subject [of imaginary magnitudes] has hitherto
been considered from the wrong point of view and surrounded by a
mysterious obscurity, is to be attributed largely to an ill-adapted
notation. If for instance, +1,−1, √−1 had been called direct,
inverse, and lateral units, instead of positive, negative, and
imaginary (or even impossible) such an obscurity would have been
out of question.--GAUSS, C. F.
_Theoria residiorum biquadraticorum,
Commentatio secunda; Werke, Bd. 2
(Goettingen, 1863), p. 177._
=1731.= ... the imaginary, this bosom-child of complex mysticism.
--DÜHRING, EUGEN.
_Kritische Geschichte der allgemeinen
Principien der Mechanik (Leipzig, 1877),
p. 517._
=1732.= Judged by the only standards which are admissible in a
pure doctrine of numbers _i_ is imaginary in the same sense as
the negative, the fraction, and the irrational, but in no other
sense; all are alike mere symbols devised for the sake of
representing the results of operations even when these results
are not numbers (positive integers).--FINE, H. B.
_The Number-System of Algebra (Boston,
1890), p. 36._
=1733.= This symbol [√-1] is restricted to a precise signification
as the representative of perpendicularity in quaternions, and this
wonderful algebra of space is intimately dependent upon the
special use of the symbol for its symmetry, elegance, and power.
The immortal author of quaternions has shown that there are other
significations which may attach to the symbol in other cases. But
the strongest use of the symbol is to be found in its magical
power of doubling the actual universe, and placing by its side an
ideal universe, its exact counterpart, with which it can be
compared and contrasted, and, by means of curiously connecting
fibres, form with it an organic whole, from which modern analysis
has developed her surpassing geometry.--PEIRCE, BENJAMIN.
_On the Uses and Transformations of
Linear Algebras; American Journal of
Mathematics, Vol. 4 (1881), p. 216._
=1734.= The conception of the inconceivable [imaginary], this
measurement of what not only does not, but cannot exist, is one
of the finest achievements of the human intellect. No one can
deny that such imaginings are indeed imaginary. But they lead to
results grander than any which flow from the imagination of the
poet. The imaginary calculus is one of the masterkeys to physical
science. These realms of the inconceivable afford in many places
our only mode of passage to the domains of positive knowledge.
Light itself lay in darkness until this imaginary calculus threw
light upon light. And in all modern researches into electricity,
magnetism, and heat, and other subtile physical inquiries, these
are the most powerful instruments.--HILL, THOMAS.
_North American Review, Vol. 85, p.
235._
=1735.= All the fruitful uses of imaginaries, in Geometry, are those
which begin and end with real quantities, and use imaginaries only
for the intermediate steps. Now in all such cases, we have a real
spatial interpretation at the beginning and end of our argument,
where alone the spatial interpretation is important; in the
intermediate links, we are dealing in purely algebraic manner with
purely algebraic quantities, and may perform any operations which
are algebraically permissible. If the quantities with which we end
are capable of spatial interpretation, then, and only then, our
results may be regarded as geometrical. To use geometrical
language, in any other case, is only a convenient help to the
imagination. To speak, for example, of projective properties which
refer to the circular points, is a mere _memoria technica_ for
purely algebraical properties; the circular points are not to be
found in space, but only in the auxiliary quantities by which
geometrical equations are transformed. That no contradictions
arise from the geometrical interpretation of imaginaries is not
wonderful; for they are interpreted solely by the rules of
Algebra, which we may admit as valid in their interpretation to
imaginaries. The perception of space being wholly absent, Algebra
rules supreme, and no inconsistency can arise.--RUSSELL, BERTRAND.
_Foundations of Geometry (Cambridge,
1897), p. 45._
=1736.= Indeed, if one understands by algebra the application of
arithmetic operations to composite magnitudes of all kinds, whether
they be rational or irrational number or space magnitudes, then
the learned Brahmins of Hindostan are the true inventors of
algebra.--HANKEL, HERMANN.
_Geschichte der Mathematik im Altertum
und Mittelalter (Leipzig, 1874), p.
195._
=1737.= It is remarkable to what extent Indian mathematics enters
into the science of our time. Both the form and the spirit of the
arithmetic and algebra of modern times are essentially Indian and
not Grecian.--CAJORI, F.
_History of Mathematics (New York,
1897), p. 100._
=1738.= There are many questions in this science [algebra] which
learned men have to this time in vain attempted to solve; and
they have stated some of these questions in their writings, to
prove that this science contains difficulties, to silence those
who pretend they find nothing in it above their ability, to warn
mathematicians against undertaking to answer every question that
may be proposed, and to excite men of genius to attempt their
solution. Of these I have selected seven.
1. To divide 10 into two parts, such, that when each part is
added to its square-root and the sums multiplied together, the
product is equal to the supposed number.
2. What square is that, which being increased or diminished by
10, the sum and remainder are both square numbers?
3. A person said he owed to Zaid 10 all but the square-root of
what he owed to Amir, and that he owed Amir 5 all but the
square-root of what he owed Zaid.
4. To divide a cube number into two cube numbers.
5. To divide 10 into two parts such, that if each is divided by
the other, and the two quotients are added together, the sum is
equal to one of the parts.
6. There are three square numbers in continued geometric
proportion, such, that the sum of the three is a square number.
7. There is a square, such, that when it is increased and
diminished by its root and 2, the sum and the difference are
squares.--KHULASAT-AL-HISAB.
_Algebra; quoted in Hutton: A
Philosophical and Mathematical
Dictionary (London, 1815), Vol. 1, p.
70._
=1739.= The solution of such questions as these [referring to the
solution of cubic equations] depends on correct judgment, aided
by the assistance of God.--BIJA GANITA.
_Quoted in Hutton: A Philosophical and
Mathematical Dictionary (London, 1815),
Vol. 1, p. 65._
=1740.= For what is the theory of determinants? It is an algebra
upon algebra; a calculus which enables us to combine and foretell
the results of algebraical operations, in the same way as algebra
itself enables us to dispense with the performance of the special
operations of arithmetic. All analysis must ultimately clothe
itself under this form.--SYLVESTER, J. J.
_Philosophical Magazine, Vol. 1, (1851),
p. 300; Collected Mathematical Papers,
Vol. 1, p. 247._
=1741.=
Fuchs. Fast möcht’ ich nun _moderne Algebra_ studieren.
Meph. Ich wünschte nicht euch irre zu führen.
Was diese Wissenschaft betrifft,
Es ist so schwer, die leere Form zu meiden,
Und wenn ihr es nicht recht begrifft,
Vermögt die Indices ihr kaum zu unterscheiden.
Am Besten ist’s, wenn ihr nur _Einem_ traut
Und auf des Meister’s Formeln baut.
Im Ganzen--haltet euch an die _Symbole_.
Dann geht ihr zu der Forschung Wohle
Ins sichre Reich der Formeln ein.
Fuchs. Ein Resultat muss beim Symbole sein?
Meph. Schon gut! Nur muss man sich nicht alzu ängstlich
quälen.
Denn eben, wo die Resultate fehlen,
Stellt ein Symbol zur rechten Zeit sich ein.
Symbolisch lässt sich alles schreiben,
Müsst nur im Allgemeinen bleiben.
Wenn man der Gleichung Lösung nicht erkannte,
Schreibt man sie als Determinante.
Schreib’ was du willst, nur rechne _nie_ was aus.
Symbole lassen trefflich sich traktieren,
Mit einem Strich ist alles auszuführen,
Und mit Symbolen kommt man immer aus.
--LASSWITZ, KURD.
_Der Faust-Tragödie (-n)ter Teil;
Zeitschrift für mathematischen und
naturwissenschaftlichen Unterricht, Bd.
14, p. 317._
Fuchs. To study _modern algebra_ I’m most persuaded.
Meph. ’Twas not my wish to lead thee astray.
But as concerns this science, truly
’Tis difficult to avoid the empty form,
And should’st thou lack clear comprehension,
Scarcely the indices thou’ll know apart.
’Tis safest far to trust but _one_
And built upon your master’s formulas.
On the whole--cling closely to your _symbols_.
Then, for the weal of research you may gain
An entrance to the formula’s sure domain.
Fuchs. The symbol, it must lead to some result?
Meph. Granted. But never worry about results,
For, mind you, just where the results are wanting
A symbol at the nick of time appears.
To symbolic treatment all things yield,
Provided we stay in the general field.
Should a solution prove elusive,
Write the equation in determinant form.
Write what you please, but _never_ calculate.
Symbols are patient and long suffering,
A single stroke completes the whole affair.
Symbols for every purpose do suffice.
=1742.= As all roads are said to lead to Rome, so I find, in my
own case at least, that all algebraic inquiries sooner or later
end at the Capitol of Modern Algebra over whose shining portal
is inscribed “Theory of Invariants.”--SYLVESTER, J. J.
_On Newton’s Rule for the Discovery of
Imaginary Roots; Collected Mathematical
Papers, Vol. 2, p. 380._
=1743.= If we consider the beauty of the theorem [Sylvester’s
Theorem on Newton’s Rule for the Discovery of Imaginary Roots]
which has now been expounded, the interest which belongs to the
rule associated with the great name of Newton, and the long lapse
of years during which the reason and extent of that rule remained
undiscovered by mathematicians, among whom Maclaurin, Waring and
Euler are explicitly included, we must regard Professor
Sylvester’s investigations made to the Theory of Equations in
modern times, justly to be ranked with those of Fourier, Sturm
and Cauchy.--TODHUNTER, I.
_Theory of Equations (London, 1904), p.
250._
=1744.= Considering the remarkable elegance, generality, and
simplicity of the method [Homer’s Method of finding the numerical
values of the roots of an equation], it is not a little
surprising that it has not taken a more prominent place in
current mathematical textbooks.... As a matter of fact, its
spirit is purely arithmetical; and its beauty, which can only be
appreciated after one has used it in particular cases, is of that
indescribably simple kind, which distinguishes the use of
position in the decimal notation and the arrangement of the
simple rules of arithmetic. It is, in short, one of those things
whose invention was the creation of a commonplace.
--CHRYSTAL, GEORGE.
_Algebra (London and Edinburgh, 1893),
Vol. 1, chap. 15, sect. 25._
=1745.= _To a missing member of a family group of terms in an
algebraical formula._
Lone and discarded one! divorced by fate,
Far from thy wished-for fellows--whither art
flown?
Where lingerest thou in thy bereaved estate,
Like some lost star, or buried meteor stone?
Thou mindst me much of that presumptuous one
Who loth, aught less than greatest, to be
great,
From Heaven’s immensity fell headlong down
To live forlorn, self-centred, desolate:
Or who, like Heraclid, hard exile bore,
Now buoyed by hope, now stretched on rack of
fear,
Till throned Astæa, wafting to his ear
Words of dim portent through the Atlantic roar,
Bade him “the sanctuary of the Muse revere
And strew with flame the dust of Isis’
shore.”
--SYLVESTER, J. J.
_Inaugural Lecture, Oxford, 1885;
Nature, Vol. 33, p. 228._
=1746.= In every subject of inquiry there are certain entities,
the mutual relations of which, under various conditions, it is
desirable to ascertain. A certain combination of these entities
are submitted to certain processes or are made the subjects of
certain operations. The theory of invariants in its widest
scientific meaning determines these combinations, elucidates
their properties, and expresses results when possible in terms of
them. Many of the general principles of political science and
economics can be represented by means of invariantive relations
connecting the factors which enter as entities into the special
problems. The great principle of chemical science which asserts
that when elementary or compound bodies combine with one another
the total weight of the materials is unchanged, is another case
in point. Again, in physics, a given mass of gas under the
operation of varying pressure and temperature has the well-known
invariant, pressure multiplied by volume and divided by absolute
temperature.... In mathematics the entities under examination may
be arithmetical, algebraical, or geometrical; the processes to
which they are subjected may be any of those which are met with
in mathematical work.... It is the _principle_ which is so
valuable. It is the _idea_ of invariance that pervades today all
branches of mathematics.--MACMAHON, P. A.
_Presidential Address British
Association for the Advancement of
Science (1901); Nature, Vol. 64, p.
481._
=1747.= [The theory of invariants] has invaded the domain of
geometry, and has almost re-created the analytical theory; but it
has done more than this for the investigations of Cayley have
required a full reconsideration of the very foundations of
geometry. It has exercised a profound influence upon the theory
of algebraic equations; it has made its way into the theory of
differential equations; and the generalisation of its ideas is
opening out new regions of most advanced and profound functional
analysis. And so far from its course being completed, its
questions fully answered, or its interest extinct, there is no
reason to suppose that a term can be assigned to its growth and
its influence.--FORSYTH, A. R.
_Presidential Address British
Association for the Advancement of
Science (1897); Nature, Vol. 56, p.
378._
=1748.= ... the doctrine of Invariants, a theory filling the heavens
like a light-bearing ether, penetrating all the branches of
geometry and analysis, revealing everywhere abiding configurations
in the midst of change, everywhere disclosing the eternal reign of
the law of form.--KEYSER, C. J.
_Lectures on Science, Philosophy and Art
(New York, 1908), p. 28._
=1749.= It is in the mathematical doctrine of Invariance, the
realm wherein are sought and found configurations and types of
being that, amidst the swirl and stress of countless hosts of
transformations remain immutable, and the spirit dwells in
contemplation of the serene and eternal reign of the subtile laws
of Form, it is there that Theology may find, if she will, the
clearest conceptions, the noblest symbols, the most inspiring
intimations, the most illuminating illustrations, and the surest
guarantees of the object of her teaching and her quest, an
Eternal Being, unchanging in the midst of the universal flux.
--KEYSER, C. J.
_Lectures on Science, Philosophy and Art
(New York, 1908), p. 42._
=1750.= I think that young chemists desirous of raising their
science to its proper rank would act wisely in making themselves
master betimes of the theory of algebraic forms. What mechanics
is to physics, that I think is algebraic morphology, founded at
option on the theory of partitions or ideal elements, or both, is
destined to be to the chemistry of the future ... invariants and
isomerism are sister theories.--SYLVESTER, J. J.
_American Journal of Mathematics, Vol. 1
(1878), p. 126._
=1751.= The great notion of Group, ... though it had barely merged
into consciousness a hundred years ago, has meanwhile become a
concept of fundamental importance and prodigious fertility, not
only affording the basis of an imposing doctrine--the Theory of
Groups--but therewith serving also as a bond of union, a kind of
connective tissue, or rather as an immense cerebro-spinal system,
uniting together a large number of widely dissimilar doctrines as
organs of a single body.--KEYSER, C. J.
_Lectures on Science, Philosophy and Art
(New York, 1908), p. 12._
=1752.= In recent times the view becomes more and more prevalent
that many branches of mathematics are nothing but the theory of
invariants of special groups.--LIE, SOPHUS.
_Continuierliche Gruppen--Scheffers
(Leipzig, 1893), p. 665._
=1753.= Universal Algebra has been looked on with some suspicion
by many mathematicians, as being without intrinsic mathematical
interest and as being comparatively useless as an engine of
investigation.... But it may be shown that Universal Algebra has
the same claim to be a serious subject of mathematical study as
any other branch of mathematics.--WHITEHEAD, A. N.
_Universal Algebra (Cambridge, 1898),
Preface, p. vi._
=1754.= [Function] theory was, in effect, founded by Cauchy; but,
outside his own investigations, it at first made slow and
hesitating progress. At the present day, its fundamental ideas
may be said almost to govern most departments of the analysis of
continuous quantity. On many of them, it has shed a completely
new light; it has educed relations between them before unknown.
It may be doubted whether any subject is at the present day so
richly endowed with variety of method and fertility of resource;
its activity is prodigious, and no less remarkable than its
activity is its freshness.--FORSYTH, A. R.
_Presidential Address British
Association for the Advancement of
Science (1897); Nature, Vol. 56, p.
378._
=1755.= Let me mention one other contribution which this theory
[Theory of functions of a complex variable] has made to knowledge
lying somewhat outside our track. During the rigorous revision to
which the foundations of the theory have been subjected in its
re-establishment by Weierstrass, new ideas as regards number and
continuity have been introduced. With him and with others
influenced by him, there has thence sprung a new theory of higher
arithmetic; and with its growth, much has concurrently been
effected in the elucidation of the general notions of number and
quantity.... It thus appears to be the fact that, as with Plato,
or Descartes, or Leibnitz, or Kant, the activity of pure
mathematics is again lending some assistance to the better
comprehension of those notions of time, space, number, quantity,
which underlie a philosophical conception of the universe.
--FORSYTH, A. R.
_Presidential Address British
Association for the Advancement of
Science (1897); Nature, Vol. 56, p.
378._
CHAPTER XVIII
GEOMETRY
=1801.= The science of figures is most glorious and beautiful.
But how inaptly it has received the name geometry!--FRISCHLINUS, N.
_Dialog 1._
=1802.= Plato said that God geometrizes continually.--PLUTARCH.
_Convivialium disputationum, liber 8,
2._
=1803.= μηδεὶς ἐγεωμέτρητος εἰσίτω μοῦ τὴν στέγην. [Let no one
ignorant of geometry enter my door.]--PLATO.
_Tzetzes, Chiliad, 8, 972._
=1804.= All the authorities agree that he [Plato] made a study of
geometry or some exact science an indispensable preliminary to
that of philosophy. The inscription over the entrance to his
school ran “Let none ignorant of geometry enter my door,” and on
one occasion an applicant who knew no geometry is said to have
been refused admission as a student.--BALL, W. W. R.
_History of Mathematics (London, 1901),
p. 45._
=1805.= Form and size constitute the foundation of all search for
truth.--PARKER, F. W.
_Talks on Pedagogics (New York, 1894),
p. 72._
=1806.= At present the science [of geometry] is in flat
contradiction to the language which geometricians use, as will
hardly be denied by those who have any acquaintance with the
study: for they speak of finding the side of a square, and
applying and adding, and so on, as if they were engaged in some
business, and as if all their propositions had a practical end in
view: whereas in reality the science is pursued wholly for the
sake of knowledge.
Certainly, he said.
Then must not a further admission be made?
What admission?
The admission that this knowledge at which geometry aims is of
the eternal, and not of the perishing and transient.
That may be easily allowed. Geometry, no doubt, is the knowledge
of what eternally exists.
Then, my noble friend, geometry will draw the soul towards truth,
and create the mind of philosophy, and raise up that which is now
unhappily allowed to fall down.--PLATO.
_Republic [Jowett-Davies], Bk. 7, p.
527._
=1807.= Among them [the Greeks] geometry was held in highest
honor: nothing was more glorious than mathematics. But we have
limited the usefulness of this art to measuring and calculating.
--CICERO.
_Tusculanae Disputationes, 1, 2, 5._
=1808.=
Geometria,
Through which a man hath the sleight
Of length, and brede, of depth, of height.
--GOWER, JOHN.
_Confessio Amantis, Bk. 7._
=1809.= Geometrical truths are in a way asymptotes to physical truths,
that is to say, the latter approach the former indefinitely near
without ever reaching them exactly.--D’ALEMBERT.
_Quoted in Rebière: Mathématiques et
Mathématiciens (Paris, 1898), p. 10._
=1810.= Geometry exhibits the most perfect example of logical
stratagem.--BUCKLE, H. T.
_History of Civilization in England (New
York, 1891), Vol. 2, p. 342._
=1811.= It is the glory of geometry that from so few principles,
fetched from without, it is able to accomplish so much.--NEWTON.
_Philosophiae Naturalis Principia
Mathematica, Praefat._
=1812.= Geometry is the application of strict logic to those
properties of space and figure which are self-evident, and which
therefore cannot be disputed. But the rigor of this science is
carried one step further; for no property, however evident it may
be, is allowed to pass without demonstration, if that can be
given. The question is therefore to demonstrate all geometrical
truths with the smallest possible number of assumptions.
--DE MORGAN, A.
_On the Study and Difficulties of
Mathematics (Chicago, 1902), p. 231._
=1813.= Geometry is a true natural science:--only more simple,
and therefore more perfect than any other. We must not suppose
that, because it admits the application of mathematical analysis,
it is therefore a purely logical science, independent of
observation. Every body studied by geometers presents some
primitive phenomena which, not being discoverable by reasoning,
must be due to observation alone.--COMTE, A.
_Positive Philosophy [Martineau], Bk. 1,
chap. 3._
=1814.= Geometry in every proposition speaks a language which
experience never dares to utter; and indeed of which she but half
comprehends the meaning. Experience sees that the assertions are
true, but she sees not how profound and absolute is their truth.
She unhesitatingly assents to the laws which geometry delivers,
but she does not pretend to see the origin of their obligation.
She is always ready to acknowledge the sway of pure scientific
principles as a matter of fact, but she does not dream of
offering her opinion on their authority as a matter of right;
still less can she justly claim to herself the source of that
authority.--WHEWELL, WILLIAM.
_The Philosophy of the Inductive
Sciences, Part 1, Bk. 1, chap. 6, sect.
1 (London, 1858)._
=1815.= Geometry is the science created to give understanding
and mastery of the external relations of things; to make easy
the explanation and description of such relations and the
transmission of this mastery.--HALSTED, G. B.
_Proceedings of the American Association
for the Advancement of Science (1904),
p. 359._
=1816.= A mathematical point is the most indivisible and unique
thing which art can present.--DONNE, JOHN.
_Letters, 21._
=1817.= It is certain that from its completeness, uniformity and
faultlessness, from its arrangement and progressive character,
and from the universal adoption of the completest and best line
of argument, Euclid’s “Elements” stand pre-eminently at the head
of all human productions. In no science, in no department of
knowledge, has anything appeared like this work: for upward of
2000 years it has commanded the admiration of mankind, and that
period has suggested little toward its improvement.--KELLAND, P.
_Lectures on the Principles of
Demonstrative Mathematics (London,
1843), p. 17._
=1818.= In comparing the performance in Euclid with that in
Arithmetic and Algebra there could be no doubt that Euclid had
made the deepest and most beneficial impression: in fact it might
be asserted that this constituted by far the most valuable part
of the whole training to which such persons [students, the
majority of which were not distinguished for mathematical taste
and power] were subjected.--TODHUNTER, I.
_Essay on Elementary Geometry; Conflict
of Studies and other Essays (London,
1873), p. 167._
=1819.= In England the geometry studied is that of Euclid, and I
hope it never will be any other; for this reason, that so much
has been written on Euclid, and all the difficulties of geometry
have so uniformly been considered with reference to the form in
which they appear in Euclid, that the study of that author is a
better key to a great quantity of useful reading than any other.
--DE MORGAN, A.
_Elements of Algebra (London, 1837),
Introduction._
=1820.= This book [Euclid] has been for nearly twenty-two
centuries the encouragement and guide of that scientific thought
which is one thing with the progress of man from a worse to a
better state. The encouragement; for it contained a body of
knowledge that was really known and could be relied on, and that
moreover was growing in extent and application. For even at the
time this book was written--shortly after the foundation of the
Alexandrian Museum--Mathematics was no longer the merely ideal
science of the Platonic school, but had started on her career of
conquest over the whole world of Phenomena. The guide; for the
aim of every scientific student of every subject was to bring his
knowledge of that subject into a form as perfect as that which
geometry had attained. Far up on the great mountain of Truth,
which all the sciences hope to scale, the foremost of that sacred
sisterhood was seen, beckoning for the rest to follow her. And
hence she was called, in the dialect of the Phythagoreans, “the
purifier of the reasonable soul.”--CLIFFORD, W. K.
_Lectures and Essays (London, 1901),
Vol. 1, p. 354._
=1821.= [Euclid] at once the inspiration and aspiration of
scientific thought.--CLIFFORD, W. K.
_Lectures and Essays (London, 1901), Vol
1, p. 355._
=1822.= The “elements” of the Great Alexandrian remain for all
time the first, and one may venture to assert, the _only_ perfect
model of logical exactness of principles, and of rigorous
development of theorems. If one would see how a science can be
constructed and developed to its minutest details from a very
small number of intuitively perceived axioms, postulates, and
plain definitions, by means of rigorous, one would almost say
chaste, syllogism, which nowhere makes use of surreptitious or
foreign aids, if one would see how a science may thus be
constructed one must turn to the elements of Euclid.--HANKEL, H.
_Die Entwickelung der Mathematik in den
letzten Jahrhunderten (Tübingen, 1884),
p. 7._
=1823.= If we consider him [Euclid] as meaning to be what
his commentators have taken him to be, a model of the most
unscrupulous formal rigour, we can deny that he has altogether
succeeded, though we admit that he made the nearest approach.
--DE MORGAN, A.
_Smith’s Dictionary of Greek and Roman
Biography and Mythology (London, 1902);
Article “Eucleides.”_
=1824.= The Elements of Euclid is as small a part of mathematics
as the Iliad is of literature; or as the sculpture of Phidias is
of the world’s total art.--KEYSER, C. J.
_Lectures on Science, Philosophy and Art
(New York, 1908), p. 8._
=1825.= I should rejoice to see ... Euclid honourably shelved
or buried “deeper than did ever plummet sound” out of the
schoolboys’ reach; morphology introduced into the elements of
algebra; projection, correlation, and motion accepted as aids to
geometry; the mind of the student quickened and elevated and his
faith awakened by early initiation into the ruling ideas of
polarity, continuity, infinity, and familiarization with the
doctrines of the imaginary and inconceivable.--SYLVESTER, J. J.
_A Plea for the Mathematician; Nature,
Vol. 1, p. 261._
=1826.= The early study of Euclid made me a hater of geometry,
... and yet, in spite of this repugnance, which had become a
second nature in me, whenever I went far enough into any
mathematical question, I found I touched, at last, a geometrical
bottom.--SYLVESTER, J. J.
_A Plea for the Mathematician; Nature,
Vol. 1, p. 262._
=1827.= Newton had so remarkable a talent for mathematics that
Euclid’s Geometry seemed to him “a trifling book,” and he wondered
that any man should have taken the trouble to demonstrate
propositions, the truth of which was so obvious to him at the
first glance. But, on attempting to read the more abstruse
geometry of Descartes, without having mastered the elements of the
science, he was baffled, and was glad to come back again to his
Euclid.--PARTON, JAMES.
_Sir Isaac Newton._
=1828.= As to the need of improvement there can be no question
whilst the reign of Euclid continues. My own idea of a useful
course is to begin with arithmetic, and then not Euclid but
algebra. Next, not Euclid, but practical geometry, solid as well
as plane; not demonstration, but to make acquaintance. Then not
Euclid, but elementary vectors, conjoined with algebra, and
applied to geometry. Addition first; then the scalar product.
Elementary calculus should go on simultaneously, and come into
the vector algebraic geometry after a bit. Euclid might be an
extra course for learned men, like Homer. But Euclid for children
is barbarous.--HEAVISIDE, OLIVER.
_Electro-Magnetic Theory (London, 1893),
Vol. 1, p. 148._
=1829.= Geometry is nothing if it be not rigorous, and the whole
educational value of the study is lost, if strictness of
demonstration be trifled with. The methods of Euclid are, by
almost universal consent, unexceptionable in point of rigour.
--SMITH, H. J. S.
_Nature, Vol. 8, p. 450._
=1830.= To seek for proof of geometrical propositions by an
appeal to observation proves nothing in reality, except that the
person who has recourse to such grounds has no due apprehension
of the nature of geometrical demonstration. We have heard of
persons who convince themselves by measurement that the
geometrical rule respecting the squares on the sides of a
right-angles triangle was true: but these were persons whose
minds had been engrossed by practical habits, and in whom
speculative development of the idea of space had been stifled by
other employments.--WHEWELL, WILLIAM.
_The Philosophy of the Inductive
Sciences, (London, 1858), Part 1, Bk. 2,
chap. 1, sect. 4._
=1831.= No one has ever given so easy and natural a chain of
geometrical consequences [as Euclid]. There is a never-erring
truth in the results.--DE MORGAN, A.
_Smith’s Dictionary of Greek and Roman
Biography and Mythology (London, 1902);
Article “Eucleides.”_
=1832.= Beyond question, Egyptian geometry, such as it was, was
eagerly studied by the early Greek philosophers, and was the germ
from which in their hands grew that magnificent science to which
every Englishman is indebted for his first lessons in right
seeing and thinking.--GOW, JAMES.
_A Short History of Greek Mathematics
(Cambridge, 1884), p. 131._
=1833.=
A figure and a step onward:
Not a figure and a florin.
--MOTTO OF THE PYTHAGOREAN BROTHERHOOD.
_W. B. Frankland: Story of Euclid
(London, 1902), p. 33._
=1834.= The doctrine of proportion, as laid down in the fifth
book of Euclid, is, probably, still unsurpassed as a masterpiece
of exact reasoning; although the cumbrousness of the forms of
expression which were adopted in the old geometry has led to the
total exclusion of this part of the elements from the ordinary
course of geometrical education. A zealous defender of Euclid
might add with truth that the gap thus created in the elementary
teaching of mathematics has never been adequately supplied.
--SMITH, H. J. S.
_Presidential Address British
Association for the Advancement of
Science (1873); Nature, Vol. 8, p. 451._
=1835.= The Definition in the Elements, according to Clavius, is
this: Magnitudes are said to be in the same Reason [ratio], a
first to a second, and a third to a fourth, when the Equimultiples
of the first and third according to any Multiplication whatsoever
are both together either short of, equal to, or exceed the
Equimultiples of the second and fourth, if those be taken,
which answer one another.... Such is Euclid’s Definition of
Proportions; that _scare_-Crow at which the over modest or
slothful Dispositions of Men are generally affrighted: they are
modest, who distrust their own Ability, as soon as a Difficulty
appears, but they are slothful that will not give some Attention
for the learning of Sciences; as if while we are involved in
Obscurity we could clear ourselves without Labour. Both of which
Sorts of Persons are to be admonished, that the former be not
discouraged, nor the latter refuse a little Care and Diligence
when a Thing requires some Study.--BARROW, ISAAC.
_Mathematical Lectures (London, 1734),
p. 388._
=1836.= Of all branches of human knowledge, there is none which,
like it [geometry] has sprung a completely armed Minerva from the
head of Jupiter; none before whose death-dealing Aegis doubt
and inconsistency have so little dared to raise their eyes.
It escapes the tedious and troublesome task of collecting
experimental facts, which is the province of the natural sciences
in the strict sense of the word: the sole form of its scientific
method is deduction. Conclusion is deduced from conclusion, and
yet no one of common sense doubts but that these geometrical
principles must find their practical application in the real
world about us. Land surveying, as well as architecture, the
construction of machinery no less than mathematical physics, are
continually calculating relations of space of the most varied
kinds by geometrical principles; they expect that the success of
their constructions and experiments shall agree with their
calculations; and no case is known in which this expectation has
been falsified, provided the calculations were made correctly and
with sufficient data.--HELMHOLTZ, H.
_The Origin and Significance of
Geometrical Axioms; Popular Scientific
Lectures [Atkinson], Second Series (New
York, 1881), p. 27._
=1837.= The amazing triumphs of this branch of mathematics
[geometry] show how powerful a weapon that form of deduction is
which proceeds by an artificial reparation of facts, in
themselves inseparable.--BUCKLE, H. T.
_History of Civilization in England (New
York, 1891), Vol. 2, p. 343._
=1838.= Every theorem in geometry is a law of external nature,
and might have been ascertained by generalizing from observation
and experiment, which in this case resolve themselves into
comparisons and measurements. But it was found practicable, and
being practicable was desirable, to deduce these truths by
ratiocination from a small number of general laws of nature, the
certainty and universality of which was obvious to the most
careless observer, and which compose the first principles and
ultimate premises of the science.--MILL, J. S.
_System of Logic, Bk. 3, chap. 24, sect.
7._
=1839.= All such reasonings [natural philosophy, chemistry,
agriculture, political economy, etc.] are, in comparison with
mathematics, very complex; requiring so much _more_ than that
does, beyond the process of merely deducing the conclusion
logically from the premises: so that it is no wonder that the
longest mathematical demonstration should be much more easily
constructed and understood, than a much shorter train of just
reasoning concerning real facts. The former has been aptly
compared to a long and steep, but even and regular, flight of
steps, which tries the breath, and the strength, and the
perseverance only; while the latter resembles a short, but rugged
and uneven, ascent up a precipice, which requires a quick eye,
agile limbs, and a firm step; and in which we have to tread now
on this side, now on that--ever considering as we proceed,
whether this or that projection will afford room for our foot, or
whether some loose stone may not slide from under us. There are
probably as many steps of pure reasoning in one of the longer of
Euclid’s demonstrations, as in the whole of an argumentative
treatise on some other subject, occupying perhaps a considerable
volume.--WHATELY, R.
_Elements of Logic, Bk. 4, chap. 2,
sect. 5._
=1840.=
[Geometry] that held acquaintance with the stars,
And wedded soul to soul in purest bond
Of reason, undisturbed by space or time.
--WORDSWORTH.
_The Prelude, Bk. 5._
=1841.= The statement that a given individual has received a
sound geometrical training implies that he has segregated from
the whole of his sense impressions a certain set of these
impressions, that he has eliminated from their consideration all
irrelevant impressions (in other words, acquired a subjective
command of these impressions), that he has developed on the basis
of these impressions an ordered and continuous system of logical
deduction, and finally that he is capable of expressing the
nature of these impressions and his deductions therefrom in terms
simple and free from ambiguity. Now the slightest consideration
will convince any one not already conversant with the idea, that
the same sequence of mental processes underlies the whole career
of any individual in any walk of life if only he is not concerned
entirely with manual labor; consequently a full training in the
performance of such sequences must be regarded as forming an
essential part of any education worthy of the name. Moreover the
full appreciation of such processes has a higher value than is
contained in the mental training involved, great though this be,
for it induces an appreciation of intellectual unity and beauty
which plays for the mind that part which the appreciation of
schemes of shape and color plays for the artistic faculties; or,
again, that part which the appreciation of a body of religious
doctrine plays for the ethical aspirations. Now geometry is not
the sole possible basis for inculcating this appreciation. Logic
is an alternative for adults, provided that the individual is
possessed of sufficient wide, though rough, experience on which
to base his reasoning. Geometry is, however, highly desirable in
that the objective bases are so simple and precise that they can
be grasped at an early age, that the amount of training for the
imagination is very large, that the deductive processes are not
beyond the scope of ordinary boys, and finally that it affords a
better basis for exercise in the art of simple and exact
expression than any other possible subject of a school course.
--CARSON, G. W. L.
_The Functions of Geometry as a Subject
of Education (Tonbridge, 1910), p. 3._
=1842.= It seems to me that the thing that is wanting in the
education of women is not the acquaintance with any facts, but
accurate and scientific habits of thought, and the courage to
think that true which appears unlikely. And for supplying this
want there is a special advantage in geometry, namely that it
does not require study of a physically laborious kind, but rather
that rapid intuition which women certainly possess; so that it is
fit to become a scientific pursuit for them.--CLIFFORD, W. K.
_Quoted by Pollock in Clifford’s
Lectures and Essays (London, 1901), Vol.
1, Introduction, p. 43._
=1843.=
On the lecture slate
The circle rounded under female hands
With flawless demonstration.
--TENNYSON.
_The Princess, II, l. 493._
=1844.= It is plain that that part of geometry which bears upon
strategy does concern us. For in pitching camps, or in occupying
positions, or in closing or extending the lines of an army, and
in all the other manœuvres of an army whether in battle or on
the march, it will make a great difference to a general, whether
he is a geometrician or not.--PLATO.
_Republic, Bk. 7, p. 526._
=1845.= Then nothing should be more effectually enacted, than
that the inhabitants of your fair city should learn geometry.
Moreover the science has indirect effects, which are not small.
Of what kind are they? he said.
There are the military advantages of which you spoke, I said; and
in all departments of study, as experience proves, any one who
has studied geometry is infinitely quicker of apprehension.--PLATO.
_Republic [Jowett], Bk. 7, p. 527._
=1846.= It is doubtful if we have any other subject that does so
much to bring to the front the danger of carelessness, of
slovenly reasoning, of inaccuracy, and of forgetfulness as this
science of geometry, which has been so polished and perfected as
the centuries have gone on.--SMITH, D. E.
_The Teaching of Geometry (Boston,
1911), p. 12._
=1847.= The culture of the geometric imagination, tending to
produce precision in remembrance and invention of visible forms
will, therefore, tend directly to increase the appreciation of
works of belles-letters.--HILL, THOMAS.
_The Uses of Mathesis; Bibliotheca
Sacra, Vol. 32, p. 504._
=1848.=
Yet may we not entirely overlook
The pleasures gathered from the rudiments
Of geometric science. Though advanced
In these inquiries, with regret I speak,
No farther than the threshold, there I found
Both elevation and composed delight:
With Indian awe and wonder, ignorance pleased
With its own struggles, did I meditate
On the relations those abstractions bear
To Nature’s laws.
* * * * *
More frequently from the same source I drew
A pleasure quiet and profound, a sense
Of permanent and universal sway,
And paramount belief; there, recognized
A type, for finite natures, of the one
Supreme Existence, the surpassing life
Which to the boundaries of space and time,
Of melancholy space and doleful time,
Superior and incapable of change,
Nor touched by welterings of passion--is,
And hath the name of God. Transcendent peace
And silence did wait upon these thoughts
That were a frequent comfort to my youth.
* * * * *
Mighty is the charm
Of those abstractions to a mind beset
With images and haunted by himself,
And specially delightful unto me
Was that clear synthesis built up aloft
So gracefully; even then when it appeared
Not more than a mere plaything, or a toy
To sense embodied: not the thing it is
In verity, an independent world,
Created out of pure intelligence.
--WORDSWORTH.
_The Prelude, Bk. 6._
=1849.=
’Tis told by one whom stormy waters threw,
With fellow-sufferers by the shipwreck spared,
Upon a desert coast, that having brought
To land a single volume, saved by chance,
A treatise of Geometry, he wont,
Although of food and clothing destitute,
And beyond common wretchedness depressed,
To part from company, and take this book
(Then first a self taught pupil in its truths)
To spots remote, and draw his diagrams
With a long staff upon the sand, and thus
Did oft beguile his sorrow, and almost
Forget his feeling:
--WORDSWORTH.
_The Prelude, Bk. 6._
=1850.= We study art because we receive pleasure from the great
works of the masters, and probably we appreciate them the more
because we have dabbled a little in pigments or in clay. We do
not expect to be composers, or poets, or sculptors, but we wish
to appreciate music and letters and the fine arts, and to derive
pleasure from them and be uplifted by them....
So it is with geometry. We study it because we derive pleasure
from contact with a great and ancient body of learning that has
occupied the attention of master minds during the thousands of
years in which it has been perfected, and we are uplifted by it.
To deny that our pupils derive this pleasure from the study is to
confess ourselves poor teachers, for most pupils do have positive
enjoyment in the pursuit of geometry, in spite of the tradition
that leads them to proclaim a general dislike for all study. This
enjoyment is partly that of the game,--the playing of a game that
can always be won, but that cannot be won too easily. It is
partly that of the aesthetic, the pleasure of symmetry of form,
the delight of fitting things together. But probably it lies
chiefly in the mental uplift that geometry brings, the contact
with absolute truth, and the approach that one makes to the
Infinite. We are not quite sure of any one thing in biology; our
knowledge of geology is relatively very slight, and the economic
laws of society are uncertain to every one except some individual
who attempts to set them forth; but before the world was
fashioned the square on the hypotenuse was equal to the sum of
the squares on the other two sides of a right triangle, and it
will be so after this world is dead; and the inhabitant of Mars,
if he exists, probably knows its truth as we know it. The uplift
of this contact with absolute truth, with truth eternal, gives
pleasure to humanity to a greater or less degree, depending upon
the mental equipment of the particular individual; but it
probably gives an appreciable amount of pleasure to every student
of geometry who has a teacher worthy of the name.--SMITH, D. E.
_The Teaching of Geometry (Boston,
1911), p. 16._
=1851.= No other person can judge better of either [the merits of
a writer and the merits of his works] than himself; for none have
had access to a closer or more deliberate examination of them. It
is for this reason, that in proportion that the value of a work
is intrinsic, and independent of opinion, the less eagerness will
the author feel to conciliate the suffrages of the public. Hence
that inward satisfaction, so pure and so complete, which the
study of geometry yields. The progress which an individual makes
in this science, the degree of eminence which he attains in it,
all this may be measured with the same rigorous accuracy as the
methods about which his thoughts are employed. It is only when we
entertain some doubts about the justness of our own standard,
that we become anxious to relieve ourselves from our uncertainty,
by comparing it with the standard of another. Now, in all matters
which fall under the cognizance of taste, this standard is
necessarily somewhat variable; depending on a sort of gross
estimate, always a little arbitrary, either in whole or in part;
and liable to continual alteration in its dimensions, from
negligence, temper, or caprice. In consequence of these
circumstances I have no doubt, that if men lived separate from
each other, and could in such a situation occupy themselves about
anything but self-preservation, they would prefer the study of
the exact sciences to the cultivation of the agreeable arts. It
is chiefly on account of others, that a man aims at excellence in
the latter, it is on his own account that he devotes himself to
the former. In a desert island, accordingly, I should think that
a poet could scarcely be vain; whereas a geometrician might still
enjoy the pride of discovery.--D’ALEMBERT.
_Essai sur les Gens Lettres; Melages
(Amsterdam 1764), t. 1, p. 334._
=1852.= If it were required to determine inclined planes of
varying inclinations of such lengths that a free rolling body
would descend on them in equal times, any one who understands the
mechanical laws involved would admit that this would necessitate
sundry preparations. But in the circle the proper arrangement
takes place of its own accord for an infinite variety of
positions yet with the greatest accuracy in each individual case.
For all chords which meet the vertical diameter whether at its
highest or lowest point, and whatever their inclinations, have
this in common: that the free descent along them takes place in
equal times. I remember, one bright pupil, who, after I had
stated and demonstrated this theorem to him, and he had caught
the full import of it, was moved as by a miracle. And, indeed,
there is just cause for astonishment and wonder when one beholds
such a strange union of manifold things in accordance with such
fruitful rules in so plain and simple an object as the circle.
Moreover, there is no miracle in nature, which because of its
pervading beauty or order, gives greater cause for astonishment,
unless it be, for the reason that its causes are not so clearly
comprehended, marvel being a daughter of ignorance.--KANT.
_Der einzig mögliche Beweisgrund zu
einer Demonstration des Daseins Gottes;
Werke (Hartenstein), Bd. 2, p. 137._
=1853.= These examples [taken from the geometry of the circle]
indicate what a countless number of other such harmonic relations
obtain in the properties of space, many of which are manifested
in the relations of the various classes of curves in higher
geometry, all of which, besides exercising the understanding
through intellectual insight, affect the emotion in a similar or
even greater degree than the occasional beauties of nature.--KANT.
_Der einzig mögliche Beweisgrund zu
einer Demonstration des Daseins Gottes;
Werke (Hartenstein), Bd. 2, p. 138._
=1854.= But neither thirty years, nor thirty centuries, affect
the clearness, or the charm, of Geometrical truths. Such a
theorem as “the square of the hypotenuse of a right-angled
triangle is equal to the sum of the squares of the sides” is as
dazzlingly beautiful now as it was in the day when Pythagoras
first discovered it, and celebrated its advent, it is said, by
sacrificing a hecatomb of oxen--a method of doing honor to
Science that has always seemed to me _slightly_ exaggerated and
uncalled-for. One can imagine oneself, even in these degenerate
days, marking the epoch of some brilliant scientific discovery by
inviting a convivial friend or two, to join one in a beefsteak
and a bottle of wine. But a _hecatomb_ of oxen! It would produce
a quite inconvenient supply of beef.--DODGSON, C. L.
_A New Theory of Parallels (London,
1895), Introduction, p. 16._
=1855.= After Pythagoras discovered his fundamental theorem he
sacrificed a hecatomb of oxen. Since that time all dunces[10]
[Ochsen] tremble whenever a new truth is discovered.--BOERNE.
_Quoted in Moszkowski: Die unsterbliche
Kiste (Berlin, 1908), p. 18._
[10] In the German vernacular a dunce or blockhead
is called an ox.
=1856.=
_Vom Pythagorieschen Lehrsatz._
Die Wahrheit, sie besteht in Ewigkeit,
Wenn erst die blöde Welt ihr Licht erkannt:
Der Lehrsatz, nach Pythagoras benannt,
Gilt heute, wie er galt in seiner Zeit.
Ein Opfer hat Pythagoras geweiht
Den Göttern, die den Lichtstrahl ihm gesandt;
Es thaten kund, geschlachtet und verbrannt,
Ein hundert Ochsen seine Dankbarkeit.
Die Ochsen seit den Tage, wenn sie wittern,
Dass eine neue Wahrheit sich enthülle,
Erheben ein unmenschliches Gebrülle;
Pythagoras erfüllt sie mit Entsetzen;
Und machtlos, sich dem Licht zu wiedersetzen,
Verschiessen sie die Augen und erzittern.
--CHAMISSO, ADELBERT VON.
_Gedichte, 1835 (Haushenbusch), (Berlin,
1889), p. 302._
Truth lasts throughout eternity,
When once the stupid world its light discerns:
The theorem, coupled with Pythagoras’ name,
Holds true today, as’t did in olden times.
A splendid sacrifice Pythagoras brought
The gods, who blessed him with this ray divine;
A great burnt offering of a hundred kine,
Proclaimed afar the sage’s gratitude.
Now since that day, all cattle [blockheads] when they
scent
New truth about to see the light of day,
In frightful bellowings manifest their dismay;
Pythagoras fills them all with terror;
And powerless to shut out light by error,
In sheer despair they shut their eyes and tremble.
=1857.= To the question “Which is the signally most beautiful of
geometrical truths?” Frankland replies: “One star excels another
in brightness, but the very sun will be, by common consent, a
property of the circle [Euclid, Book 3, Proposition 31] selected
for particular mention by Dante, that greatest of all exponents
of the beautiful.”--FRANKLAND, W. B.
_The Story of Euclid (London, 1902), p.
70._
=1858.=
As one
Who vers’d in geometric lore, would fain
Measure the circle; and, though pondering long
And deeply, that beginning, which he needs,
Finds not; e’en such was I, intent to scan
The novel wonder, and trace out the form,
How to the circle fitted, and therein
How plac’d: but the flight was not for my wing;
--DANTE.
_Paradise [Carey] Canto 33, lines
122-129._
=1859.= If geometry were as much opposed to our passions and
present interests as is ethics, we should contest it and violate
it but little less, notwithstanding all the demonstrations of
Euclid and of Archimedes, which you would call dreams and believe
full of paralogisms; and Joseph Scaliger, Hobbes, and others, who
have written against Euclid and Archimedes, would not find
themselves in such a small company as at present.--LEIBNITZ.
_New Essays concerning Human
Understanding [Langley], Bk. 1, chap. 2,
sect. 12._
=1860.= I have no fault to find with those who teach geometry.
That science is the only one which has not produced sects; it is
founded on analysis and on synthesis and on the calculus; it does
not occupy itself with probable truth; moreover it has the same
method in every country.--FREDERICK THE GREAT.
_Oeuvres (Decker), t. 7, p. 100._
=1861.= There are, undoubtedly, the most ample reasons for
stating both the principles and theorems [of geometry] in their
general form, .... But, that an unpractised learner, even in
making use of one theorem to demonstrate another, reasons rather
from particular to particular than from the general proposition,
is manifest from the difficulty he finds in applying a theorem to
a case in which the configuration of the diagram is extremely
unlike that of the diagram by which the original theorem was
demonstrated. A difficulty which, except in cases of unusual
mental powers, long practice can alone remove, and removes
chiefly by rendering us familiar with all the configurations
consistent with the general conditions of the theorem.--MILL, J. S.
_System of Logic, Bk. 2, chap. 3, sect.
3._
=1862.= The reason why I impute any defect to geometry, is,
because its original and fundamental principles are deriv’d
merely from appearances; and it may perhaps be imagin’d, that
this defect must always attend it, and keep it from ever reaching
a greater exactness in the comparison of objects or ideas, than
what our eye or imagination alone is able to attain. I own that
this defect so far attends it, as to keep it from ever aspiring
to a full certainty. But since these fundamental principles
depend on the easiest and least deceitful appearances, they
bestow on their consequences a degree of exactness, of which
these consequences are singly incapable.--HUME, D.
_A Treatise of Human Nature, Part 3,
sect. 1._
=1863.= I have already observed, that geometry, or the art, by
which we fix the proportions of figures, tho’ it much excels both
in universality and exactness, the loose judgments of the senses
and imagination; yet never attains a perfect precision and
exactness. Its first principles are still drawn from the general
appearance of the objects; and that appearance can never afford
us any security, when we examine the prodigious minuteness of
which nature is susceptible....
There remain, therefore, algebra and arithmetic as the only
sciences, in which we can carry on a chain of reasoning to any
degree of intricacy, and yet preserve a perfect exactness and
certainty.--HUME, D.
_A Treatise of Human Nature, Part 3,
sect. 1._
=1864.= All geometrical reasoning is, in the last resort,
circular: if we start by assuming points, they can only be
defined by the lines or planes which relate them; and if we start
by assuming lines or planes, they can only be defined by the
points through which they pass.--RUSSELL, BERTRAND.
_Foundations of Geometry (Cambridge,
1897), p. 120._
=1865.= The description of right lines and circles, upon which
Geometry is founded, belongs to Mechanics. Geometry does not
teach us to draw these lines, but requires them to be drawn....
it requires that the learner should first be taught to describe
these accurately, before he enters upon Geometry; then it shows
how by these operations problems may be solved. To describe right
lines and circles are problems, but not geometrical problems. The
solution of these problems is required from Mechanics; by
Geometry the use of them, when solved, is shown.... Therefore
Geometry is founded in mechanical practice, and is nothing but
that part of universal Mechanics which accurately proposes and
demonstrates the art of measuring. But since the manual arts are
chiefly conversant in the moving of bodies, it comes to pass
that Geometry is commonly referred to their magnitudes, and
Mechanics to their motion.--NEWTON.
_Philosophiae Naturalis Principia
Mathematica, Praefat._
=1866.= We must, then, admit ... that there is an independent
science of geometry just as there is an independent science of
physics, and that either of these may be treated by mathematical
methods. Thus geometry becomes the simplest of the natural
sciences, and its axioms are of the nature of physical laws, to
be tested by experience and to be regarded as true only within
the limits of error of observation--BÔCHER, MAXIME.
_Bulletin American Mathematical Society,
Vol. 2 (1904), p. 124._
=1867.= Geometry is not an experimental science; experience forms
merely the occasion for our reflecting upon the geometrical ideas
which pre-exist in us. But the occasion is necessary, if it did
not exist we should not reflect, and if our experiences were
different, doubtless our reflections would also be different.
Space is not a form of sensibility; it is an instrument which
serves us not to represent things to ourselves, but to reason
upon things.--POINCARÉ, H.
_On the Foundations of Geometry; Monist,
Vol. 9 (1898-1899), p. 41._
=1868.= It has been said that geometry is an instrument. The
comparison may be admitted, provided it is granted at the same
time that this instrument, like Proteus in the fable, ought
constantly to change its form.--ARAGO.
_Oeuvres, t. 2 (1854), p. 694._
=1869.= It is essential that the treatment [of geometry] should
be rid of everything superfluous, for the superfluous is an
obstacle to the acquisition of knowledge; it should select
everything that embraces the subject and brings it to a focus,
for this is of the highest service to science; it must have great
regard both to clearness and to conciseness, for their opposites
trouble our understanding; it must aim to generalize its
theorems, for the division of knowledge into small elements
renders it difficult of comprehension.--PROCLUS.
_Quoted in D. E. Smith: The Teaching of
Geometry (Boston, 1911), p. 71._
=1870.= Many are acquainted with mathematics, but mathesis few
know. For it is one thing to know a number of propositions and to
make some obvious deductions from them, by accident rather than
by any sure method of procedure, another thing to know clearly
the nature and character of the science itself, to penetrate into
its inmost recesses, and to be instructed by its universal
principles, by which facility in working out countless problems
and their proofs is secured. For as the majority of artists, by
copying the same model again and again, gain certain technical
skill in painting, but no other knowledge of the art of painting
than what their eyes suggest, so many, having read the books of
Euclid and other geometricians, are wont to devise, in imitation
of them and to prove some propositions, but the most profound
method of solving more difficult demonstrations and problems they
are utterly ignorant of.--LAFAILLE, J. C.
_Theoremata de Centro Gravitatis
(Anvers, 1632), Praefat._
=1871.= The elements of plane geometry should precede algebra for
every reason known to sound educational theory. It is more
fundamental, more concrete, and it deals with things and their
relations rather than with symbols.--BUTLER, N. M.
_The Meaning of Education etc. (New
York, 1905), p. 171._
=1872.= The reason why geometry is not so difficult as algebra,
is to be found in the less general nature of the symbols
employed. In algebra a general proposition respecting numbers is
to be proved. Letters are taken which may represent any of the
numbers in question, and the course of the demonstration, far
from making use of a particular case, does not even allow that
any reasoning, however general in its nature, is conclusive,
unless the symbols are as general as the arguments.... In
geometry on the contrary, at least in the elementary parts, any
proposition may be safely demonstrated on reasonings on any one
particular example.... It also affords some facility that the
results of elementary geometry are in many cases sufficiently
evident of themselves to the eye; for instance, that two sides of
a triangle are greater than the third, whereas in algebra many
rudimentary propositions derive no evidence from the senses; for
example, that a³−b³ is always divisible without a remainder by
a−b.--DE MORGAN, A.
_On the Study and Difficulties of
Mathematics (Chicago, 1902), chap. 13._
=1873.= The principal characteristics of the ancient geometry
are:--
(1) A wonderful clearness and definiteness of its concepts and an
almost perfect logical rigour of its conclusions.
(2) A complete want of general principles and methods.... In the
demonstration of a theorem, there were, for the ancient
geometers, as many different cases requiring separate proof as
there were different positions of the lines. The greatest
geometers considered it necessary to treat all possible cases
independently of each other, and to prove each with equal
fulness. To devise methods by which all the various cases could
all be disposed of with one stroke, was beyond the power of the
ancients.--CAJORI, F.
_History of Mathematics (New York,
1897), p. 62._
=1874.= It has been observed that the ancient geometers made use
of a kind of analysis, which they employed in the solution of
problems, although they begrudged to posterity the knowledge of
it.--DESCARTES.
_Rules for the Direction of the Mind;
The Philosophy of Descartes [Torrey]
(New York, 1892), p. 68._
=1875.= The ancients studied geometry with reference to the
_bodies_ under notice, or specially: the moderns study it with
reference to the _phenomena_ to be considered, or generally. The
ancients extracted all they could out of one line or surface,
before passing to another; and each inquiry gave little or no
assistance in the next. The moderns, since Descartes, employ
themselves on questions which relate to any figure whatever. They
abstract, to treat by itself, every question relating to the
same geometrical phenomenon, in whatever bodies it may be
considered. Geometers can thus rise to the study of new
geometrical conceptions, which, applied to the curves
investigated by the ancients, have brought out new properties
never suspected by them.--COMTE.
_Positive Philosophy [Martineau], Bk. 1,
chap. 3._
=1876.= It is astonishing that this subject [projective geometry]
should be so generally ignored, for mathematics offers nothing
more attractive. It possesses the concreteness of the ancient
geometry without the tedious particularity, and the power of the
analytical geometry without the reckoning, and by the beauty of
its ideas and methods illustrates the esthetic generality which
is the charm of higher mathematics, but which the elementary
mathematics generally lacks.
_Report of the Committee of Ten on
Secondary School Studies (Chicago,
1894), p. 116._
=1877.= There exist a small number of very simple fundamental
relations which contain the scheme, according to which the
remaining mass of theorems [in projective geometry] permit of
orderly and easy development.
By a proper appropriation of a few fundamental relations one
becomes master of the whole subject; order takes the place of
chaos, one beholds how all parts fit naturally into each other,
and arrange themselves serially in the most beautiful order, and
how related parts combine into well-defined groups. In this
manner one arrives, as it were, at the elements, which nature
herself employs in order to endow figures with numberless
properties with the utmost economy and simplicity.--STEINER, J.
_Werke, Bd. 1 (1881), p. 233._
=1878.= Euclid once said to his king Ptolemy, who, as is easily
understood, found the painstaking study of the “Elements”
repellant, “There exists no royal road to mathematics.” But we
may add: Modern geometry is a royal road. It has disclosed “the
organism, by means of which the most heterogeneous phenomena in
the world of space are united one with another” (Steiner), and
has, as we may say without exaggeration, almost attained to the
scientific ideal.--HANKEL, H.
_Die Entwickelung der Mathematik in den
letzten Jahrhunderten (Tübingen, 1869)._
=1879.= The two mathematically fundamental things in projective
geometry are anharmonic ratio, and the quadrilateral construction.
Everything else follows mathematically from these two.
--RUSSELL, BERTRAND.
_Foundations of Geometry (Cambridge,
1897), p. 122._
=1880.= ... Projective Geometry: a boundless domain of countless
fields where reals and imaginaries, finites and infinites, enter
on equal terms, where the spirit delights in the artistic balance
and symmetric interplay of a kind of conceptual and logical
counterpoint,--an enchanted realm where thought is double and
flows throughout in parallel streams.--KEYSER, C. J.
_Lectures on Science, Philosophy and
Arts (New York, 1908), p. 2._
=1881.= The ancients, in the early days of the science,
made great use of the graphic method, even in the form of
construction; as when Aristarchus of Samos estimated the distance
of the sun and moon from the earth on a triangle constructed as
nearly as possible in resemblance to the right-angled triangle
formed by the three bodies at the instant when the moon is in
quadrature, and when therefore an observation of the angle at the
earth would define the triangle. Archimedes himself, though he
was the first to introduce calculated determinations into
geometry, frequently used the same means. The introduction of
trigonometry lessened the practice; but did not abolish it. The
Greeks and Arabians employed it still for a great number of
investigations for which we now consider the use of the Calculus
indispensable.--COMTE, A.
_Positive Philosophy [Martineau], Bk. 1,
chap. 3._
=1882.= A mathematical problem may usually be attacked by what is
termed in military parlance the method of “systematic approach;”
that is to say, its solution may be gradually felt for, even
though the successive steps leading to that solution cannot be
clearly foreseen. But a Descriptive Geometry problem must be seen
through and through before it can be attempted. The entire scope
of its conditions, as well as each step toward its solution, must
be grasped by the imagination. It must be “taken by assault.”
--CLARKE, G. S.
_Quoted in W. S. Hall: Descriptive
Geometry (New York, 1902), chap. 1._
=1883.= The grand use [of Descriptive Geometry] is in its
application to the industrial arts;--its few abstract problems,
capable of invariable solution, relating essentially to the
contacts and intersections of surfaces; so that all the
geometrical questions which may arise in any of the various arts
of construction,--as stone-cutting, carpentry, perspective,
dialing, fortification, etc.,--can always be treated as simple
individual cases of a single theory, the solution being certainly
obtainable through the particular circumstances of each case.
This creation must be very important in the eyes of philosophers
who think that all human achievement, thus far, is only a first
step toward a philosophical renovation of the labours of mankind;
towards that precision and logical character which can alone
ensure the future progression of all arts.... Of Descriptive
Geometry, it may further be said that it usefully exercises the
student’s faculty of Imagination,--of conceiving of complicated
geometrical combinations in space; and that, while it belongs to
the geometry of the ancients by the character of its solutions,
it approaches to the geometry of the moderns by the nature of the
questions which compose it.--COMTE, A.
_Positive Philosophy [Martineau], Bk. 1,
chap. 3._
=1884.= There is perhaps nothing which so occupies, as it were,
the middle position of mathematics, as trigonometry.
--HERBART, J. F.
_Idee eines ABC der Anschauung; Werke
(Kehrbach) (Langensalza, 1890), Bd. 1,
p. 174._
=1885.= Trigonometry contains the science of continually
undulating magnitude: meaning magnitude which becomes alternately
greater and less, without any termination to succession of
increase and decrease.... All trigonometric functions are not
undulating: but it may be stated that in common algebra nothing
but infinite series undulate: in trigonometry nothing but
infinite series do not undulate.--DE MORGAN, A.
_Trigonometry and Double Algebra
(London, 1849), Bk. 1, chap. 1._
=1886.= Sin²φ is odious to me, even though Laplace made use of it;
should it be feared that sinφ² might become ambiguous, which would
perhaps never occur, or at most very rarely when speaking of sin
(φ²), well then, let us write (sinφ)², but not sin²φ, which by
analogy should signify sin(sinφ).--GAUSS.
_Gauss-Schumacher Briefwechsel, Bd. 3,
p. 292; Bd. 4, p. 63._
=1887.= Perhaps to the student there is no part of elementary
mathematics so repulsive as is spherical trigonometry.--TAIT, P. G.
_Encyclopedia Britannica, 9th Edition;
Article “Quaternions.”_
=1888.= “Napier’s Rule of circular parts” is perhaps the happiest
example of artificial memory that is known.--CAJORI, F.
_History of Mathematics (New York,
1897), p. 165._
=1889.= The analytical equations, unknown to the ancients, which
Descartes first introduced into the study of curves and surfaces,
are not restricted to the properties of figures, and to those
properties which are the object of rational mechanics; they apply
to all phenomena in general. There cannot be a language more
universal and more simple, more free from errors and obscurities,
that is to say, better adapted to express the invariable
relations of nature.--FOURIER.
_Théorie Analytique de la Chaleur,
Discours Préliminaire._
=1890.= It is impossible not to feel stirred at the thought of
the emotions of men at certain historic moments of adventure and
discovery--Columbus when he first saw the Western shore, Pizarro
when he stared at the Pacific Ocean, Franklin when the electric
spark came from the string of his kite, Galileo when he first
turned his telescope to the heavens. Such moments are also
granted to students in the abstract regions of thought, and high
among them must be placed the morning when Descartes lay in bed
and invented the method of co-ordinate geometry.--WHITEHEAD, A. N.
_An Introduction to Mathematics (New
York, 1911), p. 122._
=1891.= It is often said that an equation contains only what has
been put into it. It is easy to reply that the new form under
which things are found often constitutes by itself an important
discovery. But there is something more: analysis, by the simple
play of its symbols, may suggest generalizations far beyond the
original limits.--PICARD, E.
_Bulletin American Mathematical Society,
Vol. 2 (1905), p. 409._
=1892.= It is not the Simplicity of the Equation, but the
Easiness of the Description, which is to determine the Choice of
our Lines for the Constructions of Problems. For the Equation
that expresses a Parabola is more simple than that that expresses
the Circle, and yet the Circle, by its more simple Construction,
is admitted before it.--NEWTON.
_The Linear Constructions of Equations;
Universal Arithmetic (London, 1769),
Vol. 2, p. 468._
=1893.= The pursuit of mathematics unfolds its formative power
completely only with the transition from the elementary subjects
to analytical geometry. Unquestionably the simplest geometry and
algebra already accustom the mind to sharp quantitative thinking,
as also to assume as true only axioms and what has been proven.
But the representation of functions by curves or surfaces reveals
a new world of concepts and teaches the use of one of the most
fruitful methods, which the human mind ever employed to increase
its own effectiveness. What the discovery of this method by Vieta
and Descartes brought to humanity, that it brings today to
every one who is in any measure endowed for such things: a
life-epoch-making beam of light [Lichtblick]. This method has its
roots in the farthest depths of human cognition and so has an
entirely different significance, than the most ingenious artifice
which serves a special purpose.--BOIS-REYMOND, EMIL DU.
_Reden, Bd. 1 (Leipzig, 1885), p. 287._
=1894.=
_Song of the Screw._
A moving form or rigid mass,
Under whate’er conditions
Along successive screws must pass
Between each two positions.
It turns around and slides along--
This is the burden of my song.
The pitch of screw, if multiplied
By angle of rotation,
Will give the distance it must glide
In motion of translation.
Infinite pitch means pure translation,
And zero pitch means pure rotation.