STUDY AND RESEARCH IN MATHEMATICS
=601.= The first thing to be attended to in reading any algebraic
treatise is the gaining a perfect understanding of the different
processes there exhibited, and of their connection with one
another. This cannot be attained by the mere reading of the book,
however great the attention which may be given. It is impossible
in a mathematical work to fill up every process in the manner in
which it must be filled up in the mind of the student before he
can be said to have completely mastered it. Many results must be
given of which the details are suppressed, such are the
additions, multiplications, extractions of square roots, etc.,
with which the investigations abound. These must not be taken on
trust by the student, but must be worked out by his own pen,
which must never be out of his own hand while engaged in any
mathematical process.--DE MORGAN, A.
_Study and Difficulties of Mathematics
(Chicago, 1902), chap. 12._
=602.= The student should not lose any opportunity of exercising
himself in numerical calculation and particularly in the use of
logarithmic tables. His power of applying mathematics to
questions of practical utility is in direct proportion to the
facility which he possesses in computation.--DE MORGAN, A.
_Study and Difficulties of Mathematics
(Chicago, 1902), chap. 12._
=603.= The examples which a beginner should choose for practice
should be simple and should not contain very large numbers. The
powers of the mind cannot be directed to two things at once; if
the complexity of the numbers used requires all the student’s
attention, he cannot observe the principle of the rule which he
is following.--DE MORGAN, A.
_Study and Difficulties of Mathematics
(Chicago, 1902), chap. 3._
=604.= Euclid and Archimedes are allowed to be knowing, and to
have demonstrated what they say: and yet whosoever shall read
over their writings without perceiving the connection of their
proofs, and seeing what they show, though he may understand all
their words, yet he is not the more knowing. He may believe,
indeed, but does not know what they say, and so is not advanced
one jot in mathematical knowledge by all his reading of those
approved mathematicians.--LOCKE, JOHN.
_Conduct of the Understanding, sect.
24._
=605.= The student should read his author with the most sustained
attention, in order to discover the meaning of every sentence. If
the book is well written, it will endure and repay his close
attention: the text ought to be fairly intelligible, even without
illustrative examples. Often, far too often, a reader hurries
over the text without any sincere and vigorous effort to
understand it; and rushes to some example to clear up what ought
not to have been obscure, if it had been adequately considered.
The habit of scrupulously investigating the text seems to me
important on several grounds. The close scrutiny of language is a
very valuable exercise both for studious and practical life. In
the higher departments of mathematics the habit is indispensable:
in the long investigations which occur there it would be
impossible to interpose illustrative examples at every stage, the
student must therefore encounter and master, sentence by
sentence, an extensive and complicated argument.--TODHUNTER, ISAAC.
_Private Study of Mathematics; Conflict
of Studies and other Essays (London,
1873), p. 67._
=606.= It must happen that in some cases the author is not
understood, or is very imperfectly understood; and the question is
what is to be done. After giving a reasonable amount of attention
to the passage, let the student pass on, reserving the obscurity
for future efforts.... The natural tendency of solitary students,
I believe, is not to hurry away prematurely from a hard passage,
but to hang far too long over it; the just pride that does not
like to acknowledge defeat, and the strong will that cannot endure
to be thwarted, both urge to a continuance of effort even when
success seems hopeless. It is only by experience we gain the
conviction that when the mind is thoroughly fatigued it has
neither the power to continue with advantage its course in an
assigned direction, nor elasticity to strike out a new path; but
that, on the other hand, after being withdrawn for a time from the
pursuit, it may return and gain the desired end.--TODHUNTER, ISAAC.
_Private Study of Mathematics; Conflict
of Studies and other Essays (London,
1873), p. 68._
=607.= Every mathematical book that is worth reading must be read
“backwards and forwards,” if I may use the expression. I would
modify Lagrange’s advice a little and say, “Go on, but often
return to strengthen your faith.” When you come on a hard or
dreary passage, pass it over; and come back to it after you have
seen its importance or found the need for it further on.
--CHRYSTAL, GEORGE.
_Algebra, Part 2 (Edinburgh, 1889),
Preface, p. 8._
=608.= The large collection of problems which our modern
Cambridge books supply will be found to be almost an exclusive
peculiarity of these books; such collections scarcely exist in
foreign treatises on mathematics, nor even in English treatises
of an earlier date. This fact shows, I think, that a knowledge of
mathematics may be gained without the perpetual working of
examples.... Do not trouble yourselves with the examples, make it
your main business, I might almost say your exclusive business,
to understand the text of your author.--TODHUNTER, ISAAC.
_Private Study of Mathematics; Conflict
of Studies and other Essays (London,
1873), p. 74._
=609.= In my opinion the English excel in the art of writing
text-books for mathematical teaching; as regards the clear
exposition of theories and the abundance of excellent examples,
carefully selected, very few books exist in other countries which
can compete with those of Salmon and many other distinguished
English authors that could be named.--CREMONA, L.
_Projective Geometry [Leudesdorf]
(Oxford, 1885), Preface._
=610.= The solution of fallacies, which give rise to absurdities,
should be to him who is not a first beginner in mathematics an
excellent means of testing for a proper intelligible insight into
mathematical truth, of sharpening the wit, and of confining the
judgment and reason within strictly orderly limits.--VIOLA, J.
_Mathematische Sophismen (Wien, 1864),
Vorwort._
=611.= Success in the solution of a problem generally depends in
a great measure on the selection of the most appropriate method
of approaching it; many properties of conic sections (for
instance) being demonstrable by a few steps of pure geometry
which would involve the most laborious operations with trilinear
co-ordinates, while other properties are almost self-evident
under the method of trilinear co-ordinates, which it would
perhaps be actually impossible to prove by the old geometry.
--WHITWORTH, W. A.
_Modern Analytic Geometry (Cambridge,
1866), p. 154._
=612.= The deep study of nature is the most fruitful source of
mathematical discoveries. By offering to research a definite end,
this study has the advantage of excluding vague questions and
useless calculations; besides it is a sure means of forming
analysis itself and of discovering the elements which it most
concerns us to know, and which natural science ought always to
conserve.--- FOURIER, J.
_Théorie Analytique de la Chaleur,
Discours Préliminaire._
=613.= It is certainly true that all physical phenomena are
subject to strictly mathematical conditions, and mathematical
processes are unassailable in themselves. The trouble arises from
the data employed. Most phenomena are so highly complex that one
can never be quite sure that he is dealing with all the factors
until the experiment proves it. So that experiment is rather the
criterion of mathematical conclusions and must lead the way.
--DOLBEAR, A. E.
_Matter, Ether, Motion (Boston, 1894),
p. 89._
=614.= Students should learn to study at an early stage the great
works of the great masters instead of making their minds sterile
through the everlasting exercises of college, which are of no use
whatever, except to produce a new Arcadia where indolence is
veiled under the form of useless activity.... Hard study on the
great models has ever brought out the strong; and of such must be
our new scientific generation if it is to be worthy of the era to
which it is born and of the struggles to which it is destined.
--BELTRAMI.
_Giornale di matematiche, Vol. 11, p.
153. [Young, J. W.]_
=615.= The history of mathematics may be instructive as well as
agreeable; it may not only remind us of what we have, but may
also teach us to increase our store. Says De Morgan, “The early
history of the mind of men with regards to mathematics leads us
to point out our own errors; and in this respect it is well to
pay attention to the history of mathematics.” It warns us against
hasty conclusions; it points out the importance of a good
notation upon the progress of the science; it discourages
excessive specialization on the part of the investigator, by
showing how apparently distinct branches have been found to
possess unexpected connecting links; it saves the student from
wasting time and energy upon problems which were, perhaps, solved
long since; it discourages him from attacking an unsolved problem
by the same method which has led other mathematicians to failure;
it teaches that fortifications can be taken by other ways than by
direct attack, that when repulsed from a direct assault it is
well to reconnoitre and occupy the surrounding ground and to
discover the secret paths by which the apparently unconquerable
position can be taken.--CAJORI, F.
_History of Mathematics (New York,
1897), pp. 1-2._
=616.= The history of mathematics is important also as a valuable
contribution to the history of civilization. Human progress is
closely identified with scientific thought. Mathematical and
physical researches are a reliable record of intellectual
progress.--CAJORI, F.
_History of Mathematics (New York,
1897), p. 4._
=617.= It would be rash to say that nothing remains for discovery
or improvement even in elementary mathematics, but it may be
safely asserted that the ground has been so long and so
thoroughly explored as to hold out little hope of profitable
return for a casual adventurer.--TODHUNTER, ISAAC.
_Private Study of Mathematics; Conflict
of Studies and other Essays (London,
1873), p. 73._
=618.= We do not live in a time when knowledge can be extended
along a pathway smooth and free from obstacles, as at the time of
the discovery of the infinitesimal calculus, and in a measure
also when in the development of projective geometry obstacles
were suddenly removed which, having hemmed progress for a long
time, permitted a stream of investigators to pour in upon virgin
soil. There is no longer any browsing along the beaten paths; and
into the primeval forest only those may venture who are equipped
with the sharpest tools.--BURKHARDT, H.
_Mathematisches und wissenschaftliches
Denken; Jahresbericht der Deutschen
Mathematiker Vereinigung, Bd. 11, p.
55._
=619.= Though we must not without further consideration condemn a
body of reasoning merely because it is easy, nevertheless we must
not allow ourselves to be lured on merely by easiness; and we
should take care that every problem which we choose for attack,
whether it be easy or difficult, shall have a useful purpose,
that it shall contribute in some measure to the up-building of
the great edifice.--SEGRE, CORRADI.
_Some Recent Tendencies in Geometric
Investigation; Rivista di Matematica
(1891), p. 63. Bulletin American
Mathematical Society, 1904, p. 465.
[Young, J. W.]._
=620.= No mathematician now-a-days sets any store on the
discovery of isolated theorems, except as affording hints of an
unsuspected new sphere of thought, like meteorites detached from
some undiscovered planetary orb of speculation.--SYLVESTER, J. J.
_Notes to the Exeter Association
Address; Collected Mathematical Papers
(Cambridge, 1908), Vol. 2, p. 715._
=621.= Isolated, so-called “pretty theorems” have even less value
in the eyes of a modern mathematician than the discovery of a new
“pretty flower” has to the scientific botanist, though the layman
finds in these the chief charm of the respective sciences.
--HANKEL, HERMANN.
_Die Entwickelung der Mathematik in den
letzten Jahrhunderten (Tübingen, 1884),
p. 15._
=622.= It is, so to speak, a scientific tact, which must guide
mathematicians in their investigations, and guard them from
spending their forces on scientifically worthless problems and
abstruse realms, a tact which is closely related to _esthetic
tact_ and which is the only thing in our science which cannot be
taught or acquired, and is yet the indispensable endowment of
every mathematician.--HANKEL, HERMANN.
_Die Entwickelung der Mathematik in den
letzten Jahrhunderten (Tübingen, 1884),
p. 21._
=623.= The mathematician requires tact and good taste at every
step of his work, and he has to learn to trust to his own
instinct to distinguish between what is really worthy of his
efforts and what is not; he must take care not to be the slave of
his symbols, but always to have before his mind the realities
which they merely serve to express. For these and other reasons
it seems to me of the highest importance that a mathematician
should be trained in no narrow school; a wide course of reading
in the first few years of his mathematical study cannot fail to
influence for good the character of the whole of his subsequent
work.--GLAISHER, J. W. L.
_Presidential Address British
Association for the Advancement of
Science, Section A, (1890); Nature, Vol.
42, p. 467._
=624.= As long as a branch of science offers an abundance of
problems, so long it is alive; a lack of problems foreshadows
extinction or the cessation of independent development.
--HILBERT, D.
_Mathematical Problems; Bulletin
American Mathematical Society, Vol. 8,
p. 438._
=625.= In mathematics as in other fields, to find one self lost
in wonder at some manifestation is frequently the half of a new
discovery.--DIRICHLET, P. G. L.
_Werke, Bd. 2 (Berlin, 1897), p. 233._
=626.= The student of mathematics often finds it hard to throw
off the uncomfortable feeling that his science, in the person of
his pencil, surpasses him in intelligence,--an impression which
the great Euler confessed he often could not get rid of. This
feeling finds a sort of justification when we reflect that the
majority of the ideas we deal with were conceived by others,
often centuries ago. In a great measure it is really the
intelligence of other people that confronts us in science.
--MACH, ERNST.
_Popular Scientific Lectures (Chicago,
1910), p. 196._
=627.= It is probably this fact [referring to the circumstance
that the problems of the parallel axiom, the squaring of the
circle, the solution of the equation of the fifth degree, have
finally found fully satisfactory and rigorous solutions] along
with other philosophical reasons that gives rise to the
conviction (which every mathematician shares, but which no one
has yet supported by proof) that every definite mathematical
problem must necessarily be susceptible of an exact settlement,
either in the form of an actual answer to the question asked, or
by the proof of the impossibility of its solution and therewith
the necessary failure of all attempts.... This conviction of the
solvability of every mathematical problem is a powerful incentive
to the worker. We hear within us the perpetual call: There is the
problem. Seek its solution. You can find it by pure reason, for
in mathematics there is no _ignorabimus_.--HILBERT, D.
_Mathematical Problems; Bulletin
American Mathematical Society, Vol. 8,
pp. 444-445._
=628.= He who seeks for methods without having a definite problem
in mind seeks for the most part in vain.--HILBERT, D.
_Mathematical Problems; Bulletin
American Mathematical Society, Vol. 8,
p. 444._
=629.= A mathematical problem should be difficult in order to
entice us, yet not completely inaccessible, lest it mock at our
efforts. It should be to us a guide post on the mazy paths to
hidden truths, and ultimately a reminder of our pleasure in the
successful solution.--HILBERT, D.
_Mathematical Problems; Bulletin
American Mathematical Society, Vol. 8,
p. 438._
=630.= The great mathematicians have acted on the principle
“_Divinez avant de demontrer_,” and it is certainly true that
almost all important discoveries are made in this fashion.
--KASNER, EDWARD.
_The Present Problems in Geometry;
Bulletin American Mathematical Society,
Vol. 11, p. 285._
=631.= “Divide _et impera_” is as true in algebra as in
statecraft; but no less true and even more fertile is the maxim
“auge _et impera_.” The more to do or to prove, the easier the
doing or the proof.--SYLVESTER, J. J.
_Proof of the Fundamental Theorem of
Invariants; Philosophic Magazine (1878),
p. 186; Collected Mathematical Papers,
Vol. 3, p. 126._
=632.= As in the domains of practical life so likewise in science
there has come about a division of labor. The individual can no
longer control the whole field of mathematics: it is only
possible for him to master separate parts of it in such a manner
as to enable him to extend the boundaries of knowledge by
creative research.--LAMPE, E.
_Die reine Mathematik in den Jahren
1884-1899, p. 10._
=633.= With the extension of mathematical knowledge will it not
finally become impossible for the single investigator to embrace
all departments of this knowledge? In answer let me point out how
thoroughly it is ingrained in mathematical science that every
real advance goes hand in hand with the invention of sharper
tools and simpler methods which at the same time assist in
understanding earlier theories and to cast aside some more
complicated developments. It is therefore possible for the
individual investigator, when he makes these sharper tools and
simpler methods his own, to find his way more easily in the
various branches of mathematics than is possible in any other
science.--HILBERT, D.
_Mathematical Problems; Bulletin
American Mathematical Society, Vol. 8,
p. 479._
=634.= It would seem at first sight as if the rapid expansion of
the region of mathematics must be a source of danger to its
future progress. Not only does the area widen but the subjects
of study increase rapidly in number, and the work of the
mathematician tends to become more and more specialized. It is,
of course, merely a brilliant exaggeration to say that no
mathematician is able to understand the work of any other
mathematician, but it is certainly true that it is daily becoming
more and more difficult for a mathematician to keep himself
acquainted, even in a general way, with the progress of any of
the branches of mathematics except those which form the field of
his own labours. I believe, however, that the increasing extent
of the territory of mathematics will always be counteracted by
increased facilities in the means of communication. Additional
knowledge opens to us new principles and methods which may
conduct us with the greatest ease to results which previously
were most difficult of access; and improvements in notation may
exercise the most powerful effects both in the simplification and
accessibility of a subject. It rests with the worker in
mathematics not only to explore new truths, but to devise the
language by which they may be discovered and expressed; and the
genius of a great mathematician displays itself no less in the
notation he invents for deciphering his subject than in the
results attained.... I have great faith in the power of
well-chosen notation to simplify complicated theories and to
bring remote ones near and I think it is safe to predict that the
increased knowledge of principles and the resulting improvements
in the symbolic language of mathematics will always enable us to
grapple satisfactorily with the difficulties arising from the
mere extent of the subject.--GLAISHER, J. W. L.
_Presidential Address British
Association for the Advancement of
Science, Section A., (1890), Nature,
Vol. 42, p. 466._
=635.= Quite distinct from the theoretical question of the manner
in which mathematics will rescue itself from the perils to which
it is exposed by its own prolific nature is the practical problem
of finding means of rendering available for the student the
results which have been already accumulated, and making it
possible for the learner to obtain some idea of the present state
of the various departments of mathematics.... The great mass of
mathematical literature will be always contained in Journals and
Transactions, but there is no reason why it should not be
rendered far more useful and accessible than at present by means
of treatises or higher text-books. The whole science suffers from
want of avenues of approach, and many beautiful branches of
mathematics are regarded as difficult and technical merely
because they are not easily accessible.... I feel very strongly
that any introduction to a new subject written by a competent
person confers a real benefit on the whole science. The number of
excellent text-books of an elementary kind that are published in
this country makes it all the more to be regretted that we have
so few that are intended for the advanced student. As an example
of the higher kind of text-book, the want of which is so badly
felt in many subjects, I may mention the second part of Prof.
Chrystal’s “Algebra” published last year, which in a small
compass gives a great mass of valuable and fundamental knowledge
that has hitherto been beyond the reach of an ordinary student,
though in reality lying so close at hand. I may add that in any
treatise or higher text-book it is always desirable that
references to the original memoirs should be given, and, if
possible, short historic notices also. I am sure that no subject
loses more than mathematics by any attempt to dissociate it from
its history.--GLAISHER, J. W. L.
_Presidential Address British
Association for the Advancement of
Science, Section A (1890); Nature, Vol.
42, p. 466._
=636.= The more a science advances, the more will it be possible
to understand immediately results which formerly could be
demonstrated only by means of lengthy intermediate considerations:
a mathematical subject cannot be considered as finally completed
until this end has been attained.--GORDAN, PAUL.
_Formensystem binärer Formen (Leipzig,
1875), p. 2._
=637.= An old French geometer used to say that a mathematical
theory was never to be considered complete till you had made it
so clear that you could explain it to the first man you met in
the street.--SMITH, H. J. S.
_Nature, Vol. 8 (1873), p. 452._
=638.= In order to comprehend and fully control arithmetical
concepts and methods of proof, a high degree of abstraction is
necessary, and this condition has at times been charged against
arithmetic as a fault. I am of the opinion that all other fields
of knowledge require at least an equally high degree of
abstraction as mathematics,--provided, that in these fields the
foundations are also everywhere examined with the rigour and
completeness which is actually necessary.--HILBERT, D.
_Die Theorie der algebraischen
Zahlkorper, Vorwort; Jahresbericht der
Deutschen Mathematiker Vereinigung, Bd.
4._
=639.= The anxious precision of modern mathematics is necessary
for accuracy, ... it is necessary for research. It makes for
clearness of thought and for fertility in trying new combinations
of ideas. When the initial statements are vague and slipshod, at
every subsequent stage of thought, common sense has to step in to
limit applications and to explain meanings. Now in creative
thought common sense is a bad master. Its sole criterion for
judgment is that the new ideas shall look like the old ones, in
other words it can only act by suppressing originality.
--WHITEHEAD, A. N.
_Introduction to Mathematics (New York,
1911), p. 157._
=640.= Mathematicians attach great importance to the elegance of
their methods and their results. This is not pure dilettantism.
What is it indeed that gives us the feeling of elegance in a
solution, in a demonstration? It is the harmony of the diverse
parts, their symmetry, their happy balance; in a word it is all
that introduces order, all that gives unity, that permits us to
see clearly and to comprehend at once both the _ensemble_ and the
details. But this is exactly what yields great results, in fact
the more we see this aggregate clearly and at a single glance,
the better we perceive its analogies with other neighboring
objects, consequently the more chances we have of divining the
possible generalizations. Elegance may produce the feeling of the
unforeseen by the unexpected meeting of objects we are not
accustomed to bring together; there again it is fruitful, since
it thus unveils for us kinships before unrecognized. It is
fruitful even when it results only from the contrast between the
simplicity of the means and the complexity of the problem set; it
makes us then think of the reason for this contrast and very
often makes us see that chance is not the reason; that it is to
be found in some unexpected law. In a word, the feeling of
mathematical elegance is only the satisfaction due to any
adaptation of the solution to the needs of our mind, and it is
because of this very adaptation that this solution can be for us
an instrument. Consequently this esthetic satisfaction is bound
up with the economy of thought.--POINCARÉ, H.
_The Future of Mathematics; Monist, Vol.
20, p. 80. [Halsted]._
=641.= The importance of a result is largely relative, is judged
differently by different men, and changes with the times and
circumstances. It has often happened that great importance has
been attached to a problem merely on account of the difficulties
which it presented; and indeed if for its solution it has been
necessary to invent new methods, noteworthy artifices, etc., the
science has gained more perhaps through these than through the
final result. In general we may call important all investigations
relating to things which in themselves are important; all those
which have a large degree of generality, or which unite under a
single point of view subjects apparently distinct, simplifying
and elucidating them; all those which lead to results that
promise to be the source of numerous consequences; etc.
--SEGRE, CORRADI.
_Some Recent Tendencies in Geometric
Investigations. Rivista di Matematica,
Vol. 1, p. 44. Bulletin American
Mathematical Society, 1904, p. 444.
[Young, J. W.]._
=642.= Geometric writings are not rare in which one would seek in
vain for an idea at all novel, for a result which sooner or later
might be of service, for anything in fact which might be
destined to survive in the science; and one finds instead
treatises on trivial problems or investigations on special forms
which have absolutely no use, no importance, which have their
origin not in the science itself but in the caprice of the
author; or one finds applications of known methods which have
already been made thousands of times; or generalizations from
known results which are so easily made that the knowledge of the
latter suffices to give at once the former. Now such work is not
merely useless; it is actually harmful because it produces a real
incumbrance in the science and an embarrassment for the more
serious investigators; and because often it crowds out certain
lines of thought which might well have deserved to be studied.
--SEGRE, CORRADI.
_On some Recent Tendencies in Geometric
Investigations; Rivista di Matematica,
1891, p. 43. Bulletin American
Mathematical Society, 1904, p. 443
[Young, J. W.]._
=643.= A student who wishes now-a-days to study geometry by
dividing it sharply from analysis, without taking account of the
progress which the latter has made and is making, that student no
matter how great his genius, will never be a whole geometer. He
will not possess those powerful instruments of research which
modern analysis puts into the hands of modern geometry. He will
remain ignorant of many geometrical results which are to be
found, perhaps implicitly, in the writings of the analyst. And
not only will he be unable to use them in his own researches, but
he will probably toil to discover them himself, and, as happens
very often, he will publish them as new, when really he has only
rediscovered them.--SEGRE, CORRADI.
_On some recent Tendencies in
Geometrical Investigations; Rivista di
Matematica, 1891, p. 43. Bulletin
American Mathematical Society, 1904, p.
443 [Young, J. W.]._
=644.= Research may start from definite problems whose importance
it recognizes and whose solution is sought more or less directly
by all forces. But equally legitimate is the other method of
research which only selects the field of its activity and,
contrary to the first method, freely reconnoitres in the search
for problems which are capable of solution. Different individuals
will hold different views as to the relative value of these two
methods. If the first method leads to greater penetration it is
also easily exposed to the danger of unproductivity. To the second
method we owe the acquisition of large and new fields, in which
the details of many things remain to be determined and explored by
the first method.--CLEBSCH, A.
_Zum Gedächtniss an Julius Plücker;
Göttinger Abhandlungen, 16, 1871,
Mathematische Classe, p. 6._
=645.= During a conversation with the writer in the last weeks of
his life, _Sylvester_ remarked as curious that notwithstanding he
had always considered the bent of his mind to be rather
analytical than geometrical, he found in nearly every case that
the solution of an analytical problem turned upon some quite
simple geometrical notion, and that he was never satisfied until
he could present the argument in geometrical language.
--MACMAHON, P. A.
_Proceedings London Royal Society, Vol.
63, p. 17._
=646.= The origin of a science is usually to be sought for not in
any systematic treatise, but in the investigation and solution of
some particular problem. This is especially the case in the
ordinary history of the great improvements in any department of
mathematical science. Some problem, mathematical or physical, is
proposed, which is found to be insoluble by known methods. This
condition of insolubility may arise from one of two causes:
Either there exists no machinery powerful enough to effect the
required reduction, or the workmen are not sufficiently expert to
employ their tools in the performance of an entirely new piece of
work. The problem proposed is, however, finally solved, and in
its solution some new principle, or new application of old
principles, is necessarily introduced. If a principle is brought
to light it is soon found that in its application it is not
necessarily limited to the particular question which occasioned
its discovery, and it is then stated in an abstract form and
applied to problems of gradually increasing generality.
Other principles, similar in their nature, are added, and the
original principle itself receives such modifications and
extensions as are from time to time deemed necessary. The same
is true of new applications of old principles; the application is
first thought to be merely confined to a particular problem, but
it is soon recognized that this problem is but one, and generally
a very simple one, out of a large class, to which the same
process of investigation and solution are applicable. The result
in both of these cases is the same. A time comes when these
several problems, solutions, and principles are grouped together
and found to produce an entirely new and consistent method; a
nomenclature and uniform system of notation is adopted, and the
principles of the new method become entitled to rank as a
distinct science.--CRAIG, THOMAS.
_A Treatise on Projection, Preface. U.
S. Coast and Geodetic Survey, Treasury
Department Document, No. 61._
=647.= The aim of research is the discovery of the equations
which subsist between the elements of phenomena.--MACH, ERNST.
_Popular Scientific Lectures (Chicago,
1910), p. 205._
=648.= Let him [the author] be permitted also in all humility to
add ... that in consequence of the large arrears of algebraical
and arithmetical speculations waiting in his mind their turn to
be called into outward existence, he is driven to the alternative
of leaving the fruits of his meditations to perish (as has been
the fate of too many foregone theories, the still-born progeny of
his brain, now forever resolved back again into the primordial
matter of thought), or venturing to produce from time to time
such imperfect sketches as the present, calculated to evoke the
mental co-operation of his readers, in whom the algebraical
instinct has been to some extent developed, rather than to
satisfy the strict demands of rigorously systematic exposition.
--SYLVESTER, J. J.
_Philosophic Magazine (1863), p. 460._
=649.= In other branches of science, where quick publication
seems to be so much desired, there may possibly be some excuse
for giving to the world slovenly or ill-digested work, but there
is no such excuse in mathematics. The form ought to be as
perfect as the substance, and the demonstrations as rigorous as
those of Euclid. The mathematician has to deal with the most
exact facts of Nature, and he should spare no effort to render
his interpretation worthy of his subject, and to give to his work
its highest degree of perfection. “_Pauca sed matura_” was
Gauss’s motto.--GLAISHER, J. W. L.
_Presidential Address British
Association for the Advancement of
Science, Section A, (1890); Nature, Vol.
42, p. 467._
=650.= It is the man not the method that solves the problem.
--MASCHKE, H.
_Present Problems of Algebra and
Analysis; Congress of Arts and Sciences
(New York and Boston, 1905), Vol. 1, p.
530._
=651.= Today it is no longer questioned that the principles of
the analysts are the more far-reaching. Indeed, the synthesists
lack two things in order to engage in a general theory of
algebraic configurations: these are on the one hand a definition
of imaginary elements, on the other an interpretation of general
algebraic concepts. Both of these have subsequently been
developed in synthetic form, but to do this the essential
principle of synthetic geometry had to be set aside. This
principle which manifests itself so brilliantly in the theory of
linear forms and the forms of the second degree, is the
possibility of immediate proof by means of visualized
constructions.--KLEIN, FELIX.
_Riemannsche Flächen (Leipzig, 1906),
Bd. 1, p. 234._
=652.= Abstruse mathematical researches ... are ... often abused for
having no obvious physical application. The fact is that the most
useful parts of science have been investigated for the sake of
truth, and not for their usefulness. A new branch of mathematics,
which has sprung up in the last twenty years, was denounced by the
Astronomer Royal before the University of Cambridge as doomed to
be forgotten, on account of its uselessness. Now it turns out that
the reason why we cannot go further in our investigations of
molecular action is that we do not know enough of this branch of
mathematics.--CLIFFORD, W. K.
_Conditions of Mental Development;
Lectures and Essays (London, 1901), Vol.
1, p. 115._
=653.= In geometry, as in most sciences, it is very rare that an
isolated proposition is of immediate utility. But the theories
most powerful in practice are formed of propositions which
curiosity alone brought to light, and which long remained useless
without its being able to divine in what way they should one day
cease to be so. In this sense it may be said, that in real
science, no theory, no research, is in effect useless.--VOLTAIRE.
_A Philosophical Dictionary, Article
“Geometry”; (Boston, 1881), Vol. 1, p.
374._
=654.= Scientific subjects do not progress necessarily on the
lines of direct usefulness. Very many applications of the
theories of pure mathematics have come many years, sometimes
centuries, after the actual discoveries themselves. The weapons
were at hand, but the men were not able to use them.
--FORSYTH, A. R.
_Perry’s Teaching of Mathematics
(London, 1902), p. 35._
=655.= It is no paradox to say that in our most theoretical moods
we may be nearest to our most practical applications.
--WHITEHEAD, A. N.
_Introduction to Mathematics (New York),
p. 100._
=656.= Although with the majority of those who study and practice
in these capacities [engineers, builders, surveyors, geographers,
navigators, hydrographers, astronomers], second-hand acquirements,
trite formulas, and appropriate tables are sufficient for ordinary
purposes, yet these trite formulas and familiar rules were
originally or gradually deduced from the profound investigations
of the most gifted minds, from the dawn of science to the present
day.... The further developments of the science, with its possible
applications to larger purposes of human utility and grander
theoretical generalizations, is an achievement reserved for a few
of the choicest spirits, touched from time to time by Heaven to
these highest issues. The intellectual world is filled with latent
and undiscovered truth as the material world is filled with latent
electricity.--EVERETT, EDWARD.
_Orations and Speeches, Vol. 3 (Boston,
1870), p. 513._
=657.= If we view mathematical speculations with reference to
their use, it appears that they should be divided into two
classes. To the first belong those which furnish some marked
advantage either to common life or to some art, and the value of
such is usually determined by the magnitude of this advantage.
The other class embraces those speculations which, though
offering no direct advantage, are nevertheless valuable in that
they extend the boundaries of analysis and increase our resources
and skill. Now since many investigations, from which great
advantage may be expected, must be abandoned solely because of
the imperfection of analysis, no small value should be assigned
to those speculations which promise to enlarge the field of
analysis.--EULER.
_Novi Comm. Petr., Vol. 4, Preface._
=658.= The discovery of the conic sections, attributed to Plato,
first threw open the higher species of form to the contemplation
of geometers. But for this discovery, which was probably regarded
in Plato’s time and long after him, as the unprofitable amusement
of a speculative brain, the whole course of practical philosophy
of the present day, of the science of astronomy, of the theory
of projectiles, of the art of navigation, might have run in
a different channel; and the greatest discovery that has ever
been made in the history of the world, the law of universal
gravitation, with its innumerable direct and indirect consequences
and applications to every department of human research and
industry, might never to this hour have been elicited.
--SYLVESTER, J. J.
_A Probationary Lecture on Geometry;
Collected Mathematical Papers, Vol. 2
(Cambridge, 1908), p. 7._
=659.= No more impressive warning can be given to those who would
confine knowledge and research to what is apparently useful, than
the reflection that conic sections were studied for eighteen
hundred years merely as an abstract science, without regard to
any utility other than to satisfy the craving for knowledge on
the part of mathematicians, and that then at the end of this long
period of abstract study, they were found to be the necessary
key with which to attain the knowledge of the most important laws
of nature.--WHITEHEAD, A. N.
_Introduction to Mathematics (New York,
York, 1911), pp. 136-137._
=660.= The Greeks in the first vigour of their pursuit of
mathematical truth, at the time of Plato and soon after, had by
no means confined themselves to those propositions which had a
visible bearing on the phenomena of nature; but had followed out
many beautiful trains of research concerning various kinds of
figures, for the sake of their beauty alone; as for instance in
their doctrine of Conic Sections, of which curves they had
discovered all the principal properties. But it is curious to
remark, that these investigations, thus pursued at first as mere
matters of curiosity and intellectual gratification, were
destined, two thousand years later, to play a very important part
in establishing that system of celestial motions which succeeded
the Platonic scheme of cycles and epicycles. If the properties of
conic sections had not been demonstrated by the Greeks and thus
rendered familiar to the mathematicians of succeeding ages,
Kepler would probably not have been able to discover those laws
respecting the orbits and motions of planets which were the
occasion of the greatest revolution that ever happened in the
history of science.--WHEWELL, W.
_History of Scientific Ideas, Bk. 2,
chap. 14, sect. 3._
=661.= The greatest mathematicians, as Archimedes, Newton, and
Gauss, always united theory and applications in equal measure.
--KLEIN, FELIX.
_Elementarmathematik vom höheren
Standpunkte aus (Leipzig, 1909), Bd. 2,
p. 392._
=662.= We may see how unexpectedly recondite parts of pure
mathematics may bear upon physical science, by calling to mind
the circumstance that Fresnel obtained one of the most curious
confirmations of the theory (the laws of Circular Polarization by
reflection) through an interpretation of an algebraical
expression, which, according to the original conventional meaning
of the symbols, involved an impossible quantity.--WHEWELL, W.
_History of Scientific Ideas, Bk. 2,
chap. 14, sect. 8._
=663.= A great department of thought must have its own inner life,
however transcendent may be the importance of its relations to the
outside. No department of science, least of all one requiring so
high a degree of mental concentration as Mathematics, can be
developed entirely, or even mainly, with a view to applications
outside its own range. The increased complexity and specialisation
of all branches of knowledge makes it true in the present, however
it may have been in former times, that important advances in such
a department as Mathematics can be expected only from men who are
interested in the subject for its own sake, and who, whilst
keeping an open mind for suggestions from outside, allow their
thought to range freely in those lines of advance which are
indicated by the present state of their subject, untrammelled by
any preoccupation as to applications to other departments of
science. Even with a view to applications, if Mathematics is to be
adequately equipped for the purpose of coping with the intricate
problems which will be presented to it in the future by Physics,
Chemistry and other branches of physical science, many of these
problems probably of a character which we cannot at present
forecast, it is essential that Mathematics should be allowed to
develop freely on its own lines.--HOBSON, E. W.
_Presidential Address British
Association for the Advancement of
Science, Section A, (1910); Nature, Vol.
84, p. 286._
=664.= To emphasize this opinion that mathematicians would be
unwise to accept practical issues as the sole guide or the chief
guide in the current of their investigations, ... let me take one
more instance, by choosing a subject in which the purely
mathematical interest is deemed supreme, the theory of functions
of a complex variable. That at least is a theory in pure
mathematics, initiated in that region, and developed in that
region; it is built up in scores of papers, and its plan
certainly has not been, and is not now, dominated or guided by
considerations of applicability to natural phenomena. Yet what
has turned out to be its relation to practical issues? The
investigations of Lagrange and others upon the construction of
maps appear as a portion of the general property of conformal
representation; which is merely the general geometrical method
of regarding functional relations in that theory. Again, the
interesting and important investigations upon discontinuous
two-dimensional fluid motion in hydrodynamics, made in the last
twenty years, can all be, and now are all, I believe, deduced
from similar considerations by interpreting functional relations
between complex variables. In the dynamics of a rotating heavy
body, the only substantial extension of our knowledge since the
time of Lagrange has accrued from associating the general
properties of functions with the discussion of the equations of
motion. Further, under the title of conjugate functions, the
theory has been applied to various questions in electrostatics,
particularly in connection with condensors and electrometers.
And, lastly, in the domain of physical astronomy, some of the
most conspicuous advances made in the last few years have been
achieved by introducing into the discussion the ideas, the
principles, the methods, and the results of the theory of
functions ... the refined and extremely difficult work of
Poincaré and others in physical astronomy has been possible only
by the use of the most elaborate developments of some purely
mathematical subjects, developments which were made without a
thought of such applications.--FORSYTH, A. R.
_Presidential Address British
Association for the Advancement of
Science, Section A, (1897); Nature, Vol.
56, p. 377._