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There have been printed seven hundred and fifty sets
of which this is copy

INTERNATIONAL CONGRESS
OF ARTS AND SCIENCE

ARCHIMEDES

Hand-painted Photogravure from the Painting by Niccolo Barabino

The most important services of Archimedes were rendered to pure Geometry,
but his popular fame rests chiefly on his application of mathematical theory
to mechanics. He invented the water-screw and discovered the principle of the
lever. Conerning the latter the famous saying is attributed to him : " Give
me where 1 may stand and I will move the world." He first established the
truth that a body plunged in a fluid loses as much of its weight as is
equal to the weight of an equal volume of fluid. This is known as the
" Principle of Archimedes,'" and is one of the most important discoveries in the
science of Hydrostatics. It was by this law that he determined how much
alloy the goldsmith, whom King Hiero had commissioned to make a crown of
pure gold, had fraudulently mixed with the metal. The solution of the
problem suggested itself to Archimedes as he was entering the bath, and he
is reported to have been so overjoyed that he ran through the streets without
waiting to dress, exclaiming, "Eureka! Eureka!" (I have found it!). He was
killed at the age of seventy-five, during the capture of Syracuse by Marcellus
in 287 B.C. The original painting of Archimedes by Niccolo Barabino is in
the Orsini palace, Genoa.

INTERNATIONAL CONGRESS

OF

ARTS AND SCIENCE

EDITED BY

HOWARD J. ROGERS, A.M., LL.D.

DIRECTOR OF CONGRESSES

VOLUME II

AESTHETICS AND MATHEMATICS

COMPRISING

Lectures on the Relation of Aesthetics to Psychology

and Philosophy, Problems of Ethics, History of

Mathematics in the Nineteenth Century,

Algebra, Analysis, Geometry, and

Mathematical Physics.

UNIVERSITY ALLIANCE

LONDON

NEW YORK

COPYRIGHT 1906 BY HOUGHTON, MIFFLIN & Co.

ALL RIGHTS RESERVED
COPYRIGHT 1008 BY UNIVERSITY ALLIANCE

ILLUSTRATIONS

VOLUME II

FACING
PAGE

ARCHIMEDES Frontispiece

Photogravure from the painting by NICCOI.O BARABINO

MENTAL EDUCATION OF A GREEK YOUTH 389

Photogravure from the painting by OTTO KNILLE

PORTRAITS OF DR. CHARLES EMILE PICARD, DR. HEINRICH MASCHKE AND

DR. E. H. MOORE . 452

Photogravure from a photograph

GERMAN UNIVERSITY STUDENTS 404

Photogravure from the painting by CARL HEYDEN

TABLE OF CONTENTS

VOLUME II

ETHICS.

The Relations of Ethics 391

BY PROF. WILLIAM RITCHIE SOBLEY, LL.D.

Problems of Ethics 403

BY PROF. PAUL HENSEL, PH.D.

ESTHETICS.

The Relation of Esthetics to Psychology and Philosophy . . . 417
BY HENRY RUTGERS MARSHALL, L.H.D.

The Fundamental Questions of Contemporary Esthetics . . . 434
BY PROF. MAX DESSOIR, M.D., PH.D.

Special Bibliography prepared by Professor Dessoir for his Address . 447
General Bibliography for Department of Philosophy . . . 449

MATHEMATICS.

The Fundamental Conceptions and Methods of Mathematics . . 456
BY PROF. MAXIME BOCHER, PH.D.

The History of Mathematics in the Nineteenth Century . . . 474
BY PROF. JAMES P. PIERPONT

ALGEBRA AND ANALYSIS.

On the Development of Mathematical Analysis and its Relations to

Some Other Sciences ........ 497

BY PROF. CHARLES EMILE PICARD, LL.D.

On Present Problems of Algebra and Analysis .... 518

BY PROF. HEINRICH MASCHKE, PH.D.

GEOMETRY.

A Study of the Development of Geometric Methods .... 535
BY PROF. JEAN GASTON DARBOUX, DR. Sc., LL.D.

TABLE OF CONTENTS

The Present Problems of Geometry ....... 559

BY PROF. EDWARD KASNEB, PH.D.

APPLIED MATHEMATICS.

The Relations of Applied Mathematics ...... 591

BY PROF. LUDWIG BOLTZMANN, PH.D.

The Principles of Mathematical Physics ...... 604

BY PROF. HENRI POINCARE, DR. Sc.

General Bibliography of the Department of Mathematics . . 623

Special Bibliography accompanying Professor Boltzmann's Address . 625

Science and Hypothesis ......... 629

BY PROF. HENRI POINCARE, DR. Sc.

MENTAL EDUCATION OF A GREEK YOUTH

Photogravure from the Painting by Otto Knille

Greek youths were carefully trained by educators who gave equal attention
to the physical and the mental needs of their charges, severity of ordeal being
characteristic of both. The picture opposite is a reproduction of a section of
a frieze painted by Knille for the library of the Berlin University, in which
by a series of four pictures the artist very admirably depicted the prime
features that distinguished the process of Greek education.

i

SECTION E — ETHICS

SECTION E — ETHICS

(Hall 6, September 23, 10 a. TO.)

CHAIRMAN: PROFESSOR GEORGE H. PALMER, Harvard University.
SPEAKERS: PROFESSOR WILLIAM R. SORLEY, University of Cambridge.

PROFESSOR PAUL HENSEL, University of Erlangen.
SECRETARY: PROFESSOR F. C. SHARP, University of Wisconsin.

THE RELATIONS OF ETHICS

BY WILLIAM RITCHIE SORLEY

[William Ritchie Sorley, Knightbridge Professor of Moral Philosophy in the
University of Cambridge; Fellow of the British Academy, b. Selkirk, Scot-
land, 1855. M.A. Edinburgh; Litt.D. Cambridge; Hon. LL.D. Edinburgh.
Post-Graduate, Shaw Fellow, Edinburgh University, 1878; Fellow, Trinity
College, Cambridge, 1883; Lecturer, Local Lectures Syndicate and for the
Moral Science Board, Cambridge, 1882-86; Deputy for the Professor of
Philosophy, University College, London, 1886-87; Professor of Philosophy,
University College, Cardiff, 1888-94; Regius Professor Moral Philosophy,
Aberdeen, 1894-1900. Author of Ethics of Naturalism, 1885 (new ed. 1904);
Mining Royalties, 1889; Recent Tendencies in Ethics, 1904; Edition of
Adamson Development of Modern Philosophy, 1903.]

THERE are many departments of inquiry whose scope is so well
defined by the consensus of experts that one may proceed, almost
without preliminary, to mark off the boundaries of one science from
other departments, to investigate the relations in which it stands
to them, and to exhibit the place which each occupies in the whole
scheme of human knowledge. In other departments opinion differs
not only regarding special problems and results, but concerning the
whole nature of the science and its relation to connected subjects.
The study of ethics still belongs to this latter group. In it there is no
consensus of experts. Competent scholars hold diametrically opposed
views as to its scope. They differ not merely in the answers they
give to ethical questions, but in their views as to what the fundamen-
tal question of ethics is. And this opposition of opinion as to its
nature is connected with a difference of view regarding the relation
of ethics to the sciences. By many investigators it is set in line
with the sciences of biology, psychology, and sociology; and its
problems are formulated and discussed by the application of the same
historical method as those sciences employ. On the other hand, it is
maintained that ethics implies and requires a concept so different
from the concepts used by the historical and natural sciences as to
give its problem an altogether distinct character and to indicate

392 ETHICS

for it a far more significant position in the whole scheme of human
thought.

The question of the relation of ethics to the sciences implies a view
of the nature of ethics itself and, in particular, of the fundamental
concept used in ethical judgments. If the nature of this concept and
its relation to the concepts employed in other branches of inquiry
can be determined, the relations of ethics will become clear of them-
selves. The problem of this paper will receive its most adequate
solution — so far as the time at my disposal permits — by an in-
dependent inquiry into the nature of the ethical concept in relation
to the concepts used in other sciences.

The immediate judgments of experience fall into two broadly
contrasted classes, which may be described in brief as judgments
of fact and judgments of worth. The former are the foundations
on which the whole edifice of science (as the term is commonly used)
is built. Science has no other object than to understand the relations
of facts as exhibited in historical sequence, in causal interconnection,
or in the logical interdependence which may be discovered amongst
their various aspects. In its beginnings it may have arisen as an aid
to the attainment of practical purposes: it is still everywhere yoked
to the chariot of man's desires and aims. But it has for long
vindicated an independent position for itself. It may be turned to
what uses you will; but its essential spirit stands aloof from these
uses. .It has one interest only, — to know what happens and how.
Otherwise it is indifferent to all purposes alike. It studies with
equal mind the slow growth of a plant or the swift destruction
wrought by the torpedo, the reign of a Caligula or of a Victoria; it
takes no side, but observes and describes all "just as if the question
were of lines, planes, and solids." Mathematical method does not
limit its range, but it typifies its attitude of indifference to every
interest save one, — that of knowing the what and how of things.

We can conceive an intelligence of this nature, a pure intelligence,
or mere intelligence, to whose understanding all the relations of
things are evident, with the prophetic power of the Laplacian Demon
and the gift of tongues to make its knowledge clear, and yet unable to
distinguish between good and evil or to see beauty or ugliness in
nature. We can cor reive such an intelligence; but it is an unreality,
a mere abstraction from the scientific aspect of human intelligence.
Pure intelligence of this sort does not exist in man, and we have no
grounds for asserting its existence anywhere. In the experience
which forms the basis of mental life, judgments of reality are every-
where combined with and colored by judgments of worth. And the
latter are as insistent as the former, and make up as large a part of
our experience. If we go back to the original judgments of experi-
ence, we find that they are not only of the form "it is here or there,"

THE RELATIONS OF ETHICS 393

"it is of this nature or that," "it has such and such effects;" just
as a large part of our experience is of another order which may be
expressed in judgments of the form "it is good or evil," "it is fair or
foul."

Nor does the way in which scientific judgments are elaborated
give any rationale of the distinction between good and evil. If we
ask of science "What is good?" it can give no relevant answer to the
question. Strictly speaking, it does not understand the meaning of
the question at all. The ball has gone out of bounds; and science can-
not touch it until it has been thrown back into the field. It can say
what is, and what will happen, and it can describe the methods or
laws by which things come to pass; that is all; it has only one law
for the just and the unjust.

But science is very resourceful, and is able to deal with judgments
of worth from its own point of view. For these judgments also are
facts of individual experience: they are formed by human minds
under certain conditions, betray certain relations to the judgments
of fact with which they are associated, and are connected with an
environment of social institutions and physical conditions of life:
they have a history therefore. And in these respects they become
part of the material for science: and a description of them can be
given by psychological and historical methods.

The general nature and results of the application of these methods
to ethics are too well known to need further comment, too well estab-
lished to require defense. But these results may be exaggerated and
have been exaggerated. When all has been said and done that the
historical method can say and do, the question "What is good?"
is found to remain exactly where it was. We may have learned much
as to the way in which certain kinds of conduct in certain circum-
stances promote certain ends, and as to the gradual changes which
men's ideas about good and evil, virtue and vice, have passed through;
but we have not touched the fundamental question which ethics has
to face — the question of the nature of worth or goodness or duty.

And yet it is this question only which gives significance to the
problems on which historical evolution has been able to throw light.
Moral ideas and moral institutions have all along been effective
factors in human development, as well as the subject of development
themselves. And the secret of their power has lain in this that men
have believed in those ideas as expressing a moral imperative or a
moral end, and that they have looked upon moral institutions as
embodiments of something which has worth for man or a moral
claim upon his devotion. These ideas and institutions would have
had no power apart from this belief in their validity.

But was this belief true? Were the ideas or institutions valid?
This question the man of science, as sociologist or historian, does not

394 ETHICS

answer and has no means of answering. He can show their adapta-
tion or want of adaptation to certain ends, but he can say nothing
about the validity of these ends themselves. It is implied in their
efficiency that these ends were conceived as having moral value or
moral authority. But to what ends does this moral value or authority
truly belong? and what is its significance? — these are questions
which the positive sciences (such as psychology and sociology) can-
not touch and which must be answered by other methods than those
which they employ.

The moral concept is expressed in various ways and by a variety
of terms, — right, duty, merit, virtue, goodness, worth. And these
different terms indicate different aspects opened up by a single new
point of view. Thus " right " seems to imply correspondence with a
standard or rule, which standard or rule is some moral law or ideal
of goodness; and "merit" indicates performance of the right,
perhaps in victory over some conflicting desire; and "virtue" means
a trait of character in which performance of this sort has become
habitual. The term "worth" has conveniences which have led to
its having considerable vogue in ethical treatises since the time of
Herbart; it lends itself easily to psychological manipulation; but
it does not seem to refer to a concept fundamentally distinct from
goodness. But between "goodness" and "duty" there seems to be
this difference at any rate, that the latter term refers definitely to
something to be done by a voluntary agent, whereas, in calling some-
thing "good," we may have no thought of action at all, but only
see and name a quality.

There lies here therefore a difference which is not a mere difference
of expression.

On the one hand it may be held that good is a quality which be-
longs to certain things and has no special and immediate reference
to volition: that we say this or that is good as we say that some-
thing else is heavy or green or positively electrified. No relation to
human life at all may be implied in the one form of judgment any
more than in the other. That relation will only follow by way of
application to circumstances. Just as a piece of lead may serve as
a letter-weight because it is heavy, so certain actions may come to
be our duty because they lead to the realization of something which
is objectively good in quality.

According to the other view goodness has reference in its primary
meaning to free self-conscious agency. The good is that which
ought to be brought into existence: goodness is a quality of things,
but only in a derivative regard because these things are produced
by a good will. It is objective, too, inasmuch as it unites the individual
will with a law or ideal which has a claim upon the will; but it does
not in its primary meaning indicate something out of relation to the

THE RELATIONS OF ETHICS 395

will: if there were no will there would be no law; apart from con-
scious agency good and evil would disappear.

The question thus raised is one of real and fundamental import-
ance. " Ethics " by its very name may seem to have primary refer-
ence to conduct; and that is the view which most moralists have,
in one way or another, adopted. But the other view which gives to
the concept "good" an independence of all relation to volition is not
always definitely excluded, even by these moralists; by others it
has been definitely maintained: it seems implied in Plato's idealism,
at one stage of its development; and quite recently a doctrine of
the principles of ethics has been worked out which is based on its
explicit recognition.1

If we would attempt to decide between these two conflicting
views of the ethical concept, we must, in the first place, imitate the
procedure of science and examine the facts on which the concept
is based. To get to the meaning of such scientific concepts as "mass,"
"energy," or the like, we begin by a consideration of the facts which
the concepts are introduced to describe. These facts are in the last-
resort the objects of sense perception. No examination of these
sense percepts will, as we have seen, yield the content of the ethical
concept; good and evil are not given in sense perception — they are
themselves an estimate of, or way of regarding, the immediate
material of experience. Moral experience is thus in a manner reflex,
as so many of the English moralists have called it. Its attitude to
things is not merely receptive; and the concepts to which it gives
rise have not mere understanding in view. Objects are perceived as
they occur; and experience of them is the groundwork of science.
There is also, at the same time, an attitude of approbation or dis-
approbation; this attitude is the special characteristic of moral
experience; and from moral experience the ethical concept is formed.

This reflex experience, or reflex attitude to experience, is exhibited
in different ways. There is, to begin with, the appreciation of beauty
in its various kinds and degrees and the corresponding depreciation
of ugliness or deformity. These give rise to the concepts and judg-
ments of esthetics. They are closely related to moral approbation
and disapprobation, so closely that there has always been a tendency
amongst a school of moralists to strain the facts by identifying them.
A certain looseness in our use of terms favors this tendency. For
we do often use good of a work of art or even scene in nature when
we mean beautiful. But if we reflect on and compare our mental
attitudes in commending, say, a sunset and self-sacrifice, it seems
to me that there can be no doubt that the two attitudes are different.
Both objects may be admired; but both are not, in the same sense,
approved. It is hard to express this difference otherwise than by

1 Principia Ethica, by G. E. Moore (1903).

396 ETHICS

saying that the moral attitude is present in the one and absent in the
other. But the difference is brought out by the fact that our aesthet-
ical and moral attitudes towards the same experience may diverge
from one another. We may admire the beauty of that which we
condemn as immoral. De Quincey saw a fine art in certain cases
of murder; the finish and perfection of wickedness may often stir
a certain artistic admiration, especially if we lull the moral sense to
sleep. And, on the other hand, moral approval is often tempered
by a certain aesthetic depreciation of those noble character who do
good awkwardly, without the ease and grace of a gentleman. John
Knox and Mary Queen of Scots (if I may assume for the moment
an historical judgment which may need qualification) will each have
his or her admirers according as the moral or aesthetic attitude
preponderates — the harsh tones of the one appealing to the law
of truth and goodness, the other an embodiment of the beauty and
gaiety of life, "without a moral sense, however feeble."

Nor is aesthetic appreciation the only other reflex attitude which
has a place in our experience side by side with the moral. Judgments
about matters of fact and relations of ideas are discriminated as
true or false; an ideal of truth is formed; and conditions of its
realization are laid down. Here again we have a concept and class
of judgments analogous to our sesthetical and ethical concepts and
judgments, but not the same as them, and not likely to be confused
with them.

Beside these may be put a whole class of judgments of worth
which may be described as judgments of utility. We estimate and
approve or disapprove various facts of experience according to their
tendency to promote or interfere with certain ends or objects of
desire. That moral judgments are to be identified with a special
class of these judgments of utility is a thesis too well known to
require discussion here, and too important to admit of discussion in
a few words. But it may be pointed out that it is only in a very
special and restricted sense of the term "utility" that judgments
of utility have ever been identified with moral judgments. The
" jimmy " is useful to the burglar, as his instruments are useful to
the surgeon; and they are in both cases appreciated by the same
kind of reflective judgment. Judgments of utility are all of them,
properly speaking, judgments about means to ends; and the ends
may and do differ; while it is only by a forced interpretation that all
these ends are sometimes and somehow made to resolve themselves
into pleasure.

t is enough, however, for my present purpose to recognize the
prima facie distinction of moral judgments or judgments of goodness
from other judgments of worth, such as those of utility, of beauty,
and of truth (in the sense in which these last also are judgments of

THE RELATIONS OF ETHICS 397

worth). Had the question of the origin and history of the moral
judgment been before us, a great deal more might have been neces-
sary. For our present purpose what has been already said may be
sufficient: it was required in order to enable us to approach the
consideration of the question already raised concerning the applica-
tion and meaning of the moral concept.

The question is, Does our moral experience support the assignment
of the predicate "good" or "bad" to things regarded as quite inde-
pendent of volition or consciousness? At first sight it may seem
easy to answer the question in the affirmative. We do talk of sun-
shine and gentle rain and fertile land as good, and of tornadoes and
disease and death as bad. But I think that when we do so, in nine
cases out of ten, our "good" or "bad" is not a moral good or
bad; they are predicates of utility or sometimes aesthetic predicates,
not moral predicates; and we recognize this in recognizing their
relativity: the fertile land is called good because its fertility makes
it useful to man's primary needs; but the barren and rocky moun-
tain may be better in the eyes of the tourist, though the farmer
would call it bad land. There is an appreciation, a judgment of
worth in the most general sense, in such experiences; but they are
in most cases without the special feature of moral approbation or
disapprobation.

There remains, however, the tenth case in which the moral predi-
cate does seem to be applied to the unconscious. One may instance
J. S. Mill's passionate impeachment of the course of nature, in which
"habitual injustice" and "nearly all the things which men are
hanged or imprisoned for doing to one another" are spoken of as
"nature's every-day performances;"1 and a similar indictment
was brought by Professor Huxley, twenty years after the publica-
tion of Mill's essay, against the cosmic process for its encourage-
ment of selfishness and ferocity.2 These are only examples. Litera-
ture is full of similar reflections on the indiscriminate slaughter
wrought by the earthquake or the hurricane, and on the sight of the
wicked flourishing or of the righteous begging his bread; and these
reflections find an echo in the experience of most men.

But the nature of this experience calls for remark.

In the first place, if we look more closely at the arguments of Mill
or Huxley, we see that both are cases of criticism of a philosophical
theory. Mill was refuting a view which he held (and rightly held)
to have influence still on popular thought, though it might have
ceased to be a living ethical theory — the doctrine that the standard
of right and wrong was to be found in nature; it was in keeping
with his purpose, therefore, to speak of the operations of nature as

1 J. S. Mill, Three Essays on Religion, pp. 35, 38.

* T. H. Huxley, Evolution and Ethics (Romanes Lecture).

398 ETHICS

if they were properly the subject of moral praise or blame. In the
same way, when Huxley wrote, the old doctrine which Mill regarded
as philosophically extinct and only surviving as a popular error had
been revived by the impetus which the theory of evolution had
given to every branch of study; and Huxley was criticising the evo-
lutionist ethics of Spencer and others who looked for moral guidance
to the course of evolution. He, therefore, was led to speak of the
cosmic process as a possible subject of moral predicates, not neces-
sarily because he thought that application appropriate, but in order
to demonstrate the hollowness of the ethics of evolution by showing
that if the moral predicate could be applied at all, then the appro-
priate adjective would be not "good" but "bad."

Perhaps there is more than this in Huxley; and Mill's expressions
often betray a direct and genuine moral condemnation of the methods
of nature as methods of wickedness; and, still more clearly, this
immediate moral disapproval may be found in expressions of common
experience as yet uncolored by philosophy. But if we examine these
we find that, while there is no reference to philosophical theories
about nature, the things approved or condemned are yet looked upon
as implying consciousness. In the lower stages of development this
implication is simply animistic; at a later period it becomes theo-
logical. But throughout experience moral judgments upon nature are
not passed upon mere nature. Its forces are regarded as expressing a
purpose or mind; and it is this that is condemned or approved. The
primitive man and the child do not merely condemn the misdoings of
inanimate objects; they wreak their vengeance upon them or punish
them: and this is a consequence of their animistic interpretation of
natural forces. Gradually, in the mental growth of the child, this ani-
mistic interpretation of things gives place to an understanding of the
natural laws of their working; and at the same time and by the same
degrees, the child ceases to inflict punishment upon the chair that
has fallen on him or to condemn its misdemeanor. Here the moral
judgment is displaced by the causal judgment; and the reason of its
displacement is the disappearance of mind or purpose from amongst
the phenomena. When the child comes to understand that the
chair falls by "laws of nature" which are not the expression of will,
like the acts done by himself or his companions, he ceases to disap-
prove or to resent, though he does not cease to feel pain or to im-
prove the circumstances by setting the chair firmly on the floor.
The recognition of natural causation as all that there is in the case
leaves no room for the moral attitude. So true is this that the same
result is sometimes thought to be a consequence of the scientific
understanding even of what is called moral causation, "tout com-
prendre c'est tout pardonner" — as if knowledge of motive and cir-
cumstances were sufficient to dispense with praise or blame.

THE RELATIONS OF ETHICS 399

Moral judgments of a more mature kind on the constitution and
course of nature form the material for optimistic and pessimistic
views of the world — at least, when these views rise above the asser-
tion of a preponderance of pleasure or of pain in life. But, so far as
I can see, in such moral judgments nature is never looked upon as
consisting of dead mechanical sequences. It is because it is looked
upon as the expression of a living will or as in some way — perhaps
very vaguely conceived — animated by purpose or consciousness, that
we regard it as morally good or evil. Apart from some such theological
conception, it does not seem to me that the nature of things calls out
the attitude of moral approval or disapproval. Things are estimated
as useful for this or that end, they are seen and appreciated as
beautiful or the reverse, without any reference to them as due to an
inspiring or originating mind ; and in one or other of these references
the terms "good" or "bad" may be used. But when we use the
term good in its specifically moral signification, we do not apply
it to the inanimate, except in a derivate way, on account of the
relation in which these inanimate ' things stand to the moral ends
and character of conscious beings.

So far, therefore, as the evidence of moral experience goes, it
does not support the view that the "good" is a quality which be-
longs to things out of relation to self-conscious activity. And, in so
far, the peculiarity of the moral experience would seem to be better
brought out by the conception "ought" than by the conception
"good."

But here a difficulty arises at once. For how can we say that any-
thing ought to be done or to be except on the assumption that it is
antecedently good? Is not such antecedent and independent good-
ness necessary in order to justify the assertion that any one ought
to produce it?

The question undoubtedly points to a difficulty; and if that diffi-
culty can be solved it may help to bring out the true significance of
the moral concept. The judgment which assigns the duty of an indi-
vidual — according to which I or any one ought to adopt a certain
course of action — involves a special application of the moral con-
cept. It binds the individual to a certain objective rule or end. The
individual's desires as mere facts of experience may point in an
altogether different direction; the purpose or volition contemplated
and approved by the moral judgment has in view the union of indi-
vidual striving with an end which is objective and, as objective, uni-
versal. This union involves an adaptation of two things which may
fall asunder, and which in every case of evil volition do fall asunder.
And the adaptation may be regarded from either side: on the side
of the individual, application to his individuality is implied; the
duty of one man is not just the same as the duty of any other; he

400 ETHICS

has his own special place and calling. But he is connected with
a larger purpose which in his consciousness becomes both an ideal
and a law, while its application is not limited to his individuality or
his circumstances.

All this is implied in the moral judgment. It is not limited to one
individual consciousness or volition. But it does not follow that the
predicate "good," in the ethical meaning of the term, is or can be
applied out of relation to consciousness altogether. At the earliest
stages of moral development we find it applied unhesitatingly
wherever conscious activity is supposed to be present — to anything
that is regarded as the embodiment of spirit; and it is applied to the
universe as a whole when the universe is thought of as the product
of mind. " Good " is not even limited to an actual existent; it neither
implies nor denies actual existence. "Such and such, if it existed,
would be a good " is as legitimate though not so primitive an expres-
sion of the moral judgment as "this existent is good." But it does
imply a relation to existence. It does not even seem possible to
distinguish except verbally between "good" and "ought to be."
And this " ought " seems to imply a reference to a purpose through
which the idea is to be realized.

This conception "ought to be" is not the same as the concept
"ought to be done by me." The latter is an application of the more
general concept to a special individual in special circumstances;
and this is the common meaning of the concept duty. The former
is the more general concept of "goodness." It may be called object-
ive, because it does not refer to any individual state of mind; it is
universal because independent of the judgments and desires of the
individual ; and when the goodness is not due to its tendency towards
some further end, it may also be called absolute.

The point of the whole argument can thus be made clear if we
bear in mind the familiar distinction between "good in itself" and
"good for me now." That the latter has always a relation to con-
sciousness is obvious: it is something to be done or experienced by
me. But there must be some ground why anything is to be or ought
to be done or experienced by me at any time. Present individual
activity must rest upon or be connected with some wider or objective
basis. What is good for me points to and depends upon something
which is not merely relatively good, but good in itself or absolutely.
Yet it does not follow that this good in itself is necessarily absolute
in the sense of having significance apart altogether from conscious-
ness. Its absoluteness consists in independence of individual con-
sciousness or feeling, not in independence of consciousness altogether.
It is objective rather than absolute in the literal sense of the term.
The good in itself, like the relative good, is one aspect which can only
belong to a consciousness — to purpose. The moral judgment on

THE RELATIONS OF ETHICS 401

things — either on the universe as a whole, or on anything in the
universe which is not regarded as due to the will of man — is only
justified if we regard these things as in some way expressing con-
sciousness; either as directly due to it, or as aiding it, or as in con-
flict with it. From any other point of view, to speak of things as good
or evil (unless in some non-ethical sense of these terms) seems out
of place, and is unsupported by the mode of application which be-
longs to the immediate judgments of the moral consciousness. If
the moral concept has significance beyond the range of the feelings
and desires of men, it is because the objects to which it applies are
the expression of mind.

This is not put forward as a vindication of a spiritual idealism.
It is only a small contribution towards the meaning of "good." A
comprehensive idealism may not be the only view of reality with
which the conclusions reached so far will harmonize. But it is the
view with which they harmonize most simply. The conception of a
purpose to which all the events of the world are related is a form in
which the essential feature of idealism may be expressed; the view
of this purpose as good makes the idealism at the same time a moral
interpretation of reality, and allows of our classing each distinguish-
able event as good or evil according as it tends to the furtherance or
hindrance of that purpose.

This doctrine of the significance and application of the ethical
concept would enable us to reach a definite view of the nature of
ethics and of the way in which it is related to the sciences and to
metaphysics. The ethical concept is based upon the primary facts
of the moral consciousness, just as scientific concepts have as their
basis the facts of direct experience. The primary facts of the moral
consciousness are themselves of the nature of judgment — they are
approbations or disapprobations. But all facts of experience involve
judgments, though these judgments may be only of the form "it is
here" or "it is of this or that nature." Again, tne primary ethical
facts or judgments cannot be assumed to be of unquestionable val-
idity: we may approve what is not worthy of approval, or disap-
prove what ought to have been approved. Our moral judgments
claim validity; and their claim is of the nature of an assertion, not
that one simply feels in such and such a way, but that something
ought or ought not to be. They imply an objective standard. But
the objective standard, when more clearly understood, may modify
or even reverse them. Our primary ethical judgments — all our
ethical judgments, indeed — stand in need of revision and criti-
cism; and they receive this revision and criticism in the course of
the elaboration of the ethical concept and of its application to the
worlds of fact and possibility. In the same way it may be contended
that the direct judgments of experience upon which science is based

402 ETHICS

need criticism and correction; though their variation may be less in
amount than the variation of moral judgments. The color-blind
man identifies red with green, and his judgment on this point has to
be reversed; the hypersensitive subject often confuses images with
percepts; exact observation needs a highly trained capacity. The
correction and criticism which is needed come from objective stand-
ards; and these are the result of the comparison of many experiences
and the work of many minds.

It is no otherwise in the case of ethics. Criticism brings to light
inconsistencies in the primary judgments of approbation and disap-
probation as well as in the later developments of the moral judgment.
And these inconsistencies must be dealt with in a way similar to that
in which we deal with inconsistencies in the judgments of perception
and of science. The objective standard is not itself given once for
all; it has to be formed by accumulation and comparison of moral
experiences. Like the experiences on which science is based, these
have to be made as far as possible harmonious, and analysis has to
be employed to bring out the element of identity which often lurks
behind apparent contradiction. They have also to be made as com-
prehensive as possible, so that they may be capable of application to
all relevant facts, and that the scattered details of the moral con-
sciousness may be welded into an harmonious system. In these
general respects the criticism of ethical concepts proceeds upon the
same lines as the criticism of scientific concepts. The difference lies
in the concepts themselves, for ethics involves a point of view to
which science must always remain a stranger.

BIBLIOGRAPHICAL NOTE

The relations of ethics are discussed in almost every ethical treatise; special
reference may be made to the writers who have worked out the theory of worth
or value, especially von Ehrenfels, System der Wert-theorie (1897); Meinong, Psy-
chologisch-Ethische Untersuchungen zur Werth-theorie (1894), and an article in
Archiv fur Syst. Phil. 1895; Krueger, Begriff der Absolut WertvoUen (1808); also
to articles by Standinger and by Natorp in Archiv fur Syst. Phil. (1896); by
Wentscher, Archiv fur Syst. Phil. 1899; by Westermarck, Mind, 1900; and by
Belot, Revue de Metaphysique et de Morale, 1905. — W. R. S.

PROBLEMS OF ETHICS

BY PAUL HENSEL
(Translated from the German by Professor J. H. Woods, Harvard University)

[Paul Hensel, Professor of Systematic Philosophy, University of Erlangen,
since 1902. b. May 17, 1860, Great Barten, East Prussia. Ph.D. Freiburg,
Baden, 1885. Privat-docent, Strassburg, 1888-95; Special Professor, Strass-
burg, 1895-98. Author of The Ethical Basis and Ethical Transactions; Car-
lyle; The Principal Problems of Ethics.]

SINCE the appearance of the three chief works of Kant a certain
rhythm in the treatment of philosophical problems, first of all in
Germany, but also, in less degree, in other civilized countries, is un-
mistakable. After an intense occupation with theoretical problems
a flood of ethical discussion usually follows; and this then is usually
resolved into a renewed revision of aesthetical problems. If I am
not deceived, we are now at the period of transition from the second
to the third epoch; so much the more favorable is the time to re-
view the present condition of ethical problems. In the first place,
then, it seems rather remarkable that recent ethical discussion, so
intensely carried on, has resulted in a definite victory for neither one
school nor the other. One thing alone, however, may with some
accuracy be said, that the school of utilitarianism of the older inter-
pretation by Bentham, which earlier prevailed almost alone in
England with a fairly strong representation in France and Germany,
seems to be withdrawn from the field. Not as if there were no men
to-day who in other times would have sworn by Bentham 's flag,
rather we are here facing a fact that a theory which formerly ap-
peared in independence, now may be deemed a special case of a
more inclusive theory, which with the help of its wider horizon can
remove a whole series of difficulties, which apparently raised insolv-
able problems for the special theory. Utilitarianism, since it had
started with the examination of the individual, could not, even in the
master-hand of Bentham, transfer itself without remainder into the
greatest happiness of the greatest number; the interest paid on
the sacrifice offered to fellow men, again and again seemed dubitable
and probable; again and again the best calculation seemed to con-
sist in egoism pure and simple. The impossibility of an exact calcula-
tion of consequences in pleasure and in pain was likewise repeatedly
emphasized by opponents; the suggestion that we do not count the
shrewd calculator so good as the man who acts impulsively was also
not lacking: all these were difficulties, which, on the ground of the
older utilitarianism, could be evaded but not quite entirely put out
of the world.

404 ETHICS

It is then easily understood that the further combinations into
which evolution was able to advance ethical questions have resulted
in the cessation of utilitarianism as an independent system. Around
the huge system of thought of Herbert Spencer one of the great camps
of ethical workers is collected. It is not correct to count Herbert
Spencer as systematizer of Darwin's thoughts; his main thoughts
were finished, before a line of Darwin had appeared. But it is correct
that the wonderful inductions of Darwin were precisely that which
Spencer's system needed in order to begin its triumphal march
through the civilized world. Here the case is the reverse of that of
Copernicus and Giordano Bruno: the systematizer precedes the
man of special research. It is superfluous on American soil to give
a description of Spencer's thoughts; they have become parts of the
general consciousness. So it may suffice to emphasize a few character-
istic features, to which my remarks shall be attached, since, other-
wise, in view of the richness of the system, there might easily be
other sides of it in the mind of my hearers than those to which I
have here to attach importance.

The characteristic feature of the system of Spencer is its unity and
compactness. Just as every picture has a definite point from which
it should be seen, so also the system of Spencer is a view of the world
from a quite definite point of view, — that of evolution. Systems
of evolution had already occurred in philosophy, — I mention the
vast performance of Hegel only, — but that which gives Spencer's
system its characteristic significance is that here evolution is con-
ceived not as logical, but as biological; while in the case of Hegel
nature is the vestibule of the realm of purpose, and therein alone
has its significance, Spencer takes nature as his point of departure,
and the realm of human activity represents itself to him merely as
the finest conformation of natural events. Here the whole evolution
from the nebula in world-space to the most delicate relations between
man and man are comprehended in one grand conception. The same
amount of force which then existed in world-space exists still to-day,
only in infinitely more differentiated form. The new which is pro-
duced is nothing else than the transformed old, but transformed in
an essential relation, in the direction towards constantly increasing
complexity of relations in which single things and centres of force
stand to each other.

If it be asked what this principle is which is the ground for this
differentiation, a glance at the behavior of organisms informs us. In
them we can most clearly recognize effects which result, with the
necessity of laws of nature, from increasing differentiation. The
undifferentiated individual is powerless in the presence of every
change of his environment. Banished to its accidental place, the
plant must wait for what happens to it. Only within a narrow limit

PROBLEMS OF ETHICS 405

can it maintain its existence. Better equipped we find the animal,
especially when it has gathered into social groups, either for pro-
tection against carnivora or for the breeding of progeny in common.
The young steer has an infinitely better prospect to maintain itself,
to grow up, than the single egg in the spawn of the sturgeon.

So it is, before all else, the fact of social combination which attracts
to itself the attention of the revolutionary ethicist. His ethics is
social ethics. The analysis of the historical development of mankind
forms the standard, in which the social combinations have resulted,
and in which greater and world-inclusive formations have replaced
those earlier, smaller, and smallest, usually engaged in war with
each other. It is a long way from the time when hospes was equivalent
to hostis to international expositions, and the single stages of this
way reflect themselves in the moral behavior of the individuals.
The old question, which in so many ways agitated the English
ethics of the seventeenth and eighteenth centuries, the question,
whether man should be regarded as an originally egoistic being, or
whether equally original, benevolent instincts must be ascribed to
him, is transferred by evolutionists of to-day beyond the realm of
man to that of his animal ancestors and, in this case, in favor of the
originality of egoism. But long before man appeared as an inde-
pendent species the effects of the life of the horde must have shown
themselves in him, since those communities only in which the single
members were bound to each other by sympathy had any prospect
of survival. It is therefore possible to speak of animal ethics. The
interesting attempts which Darwin had made in this field were taken
by Spencer, as a whole, into his system. It must, however, be con-
ceded that we must observe the full development of this process,
first of all, in man, and the tendency then consists in a constant
decrease of egoistic, as compared with altruistic, actions. How it
was possible that the individual was ever willing to renounce the
amounts of pleasure, which he could obtain, in favor of others,
Spencer skillfully tried to explain by the introduction of the egoistic-
altruistic feelings. These give the impulse to actions which are useful
to the community, but which give to the doer honor and distinction,
and thus, from egoistic motives, make actions which promote the
welfare of the community commendable. But those actions which
damage the community are visited with punishment of all kinds.
The theory of sanctions in Bentham and Mill here passes over into
the more extensive system of evolution. For modern theory of
evolution, by the broader biological foundation of its system, suc-
ceeds in explaining why even, in the case of those who cannot over-
look the consequences of such actions as are injurious to their own
person, these consequences are still ignored. The fact of the con-
science, for the consistent Benthamite a negligible quantity, forms

406 ETHICS

the keystone of Spencei^s ethics, and affords the chance of making
the theory of heredity applicable in a new field of ethical speculation.

It is, as a matter of fact, impossible for the single individual to
calculate, by Bentham's receipt, all the consequences of pleasure or
of pain which result from the actions for his own welfare. The
inaividual need not, however, undertake this calculation at all. He
does not begin at the .beginning of making his experiences in this
world; he enjoys the heaped-up treasure of experiences which, before
him, long-forgotten generations of ancestors had made; and the
sum of these experiences he calls his conscience. This voice of the
conscience restrains the individual from anti-social actions, which,
in accordance with experience, must lead to an injury to his own
person; in accordance, of course, with the experience not of single
ancestors but of the whole line. Here, again, a selective process in the
struggle for existence is being completed. Men with no conscience at
all or with an only imperfectly developed conscience have to contend
with disadvantages similar to those in whom the corporal adjustment
to the modern conditions of civilization have proved defective; they
are exterminated by seclusion in prison or by execution, as the others
by diseases which their bodies cannot resist. The criminal of to-day
might perhaps have been, in primitive times, a respected member of
his horde, perhaps, even a great chief. To-day he can be regarded
only as an atavistic survivor, who fits into our conditions as little
as a living ichthyosaurus into this lecture-hall. Again, it is to be
hoped, it is even definitely to be predicted, that many who to-day
are quite irreproachable in moral respects, in later times will no
longer succeed in satisfying the requirements in the form of their
grandson or great-grandson. For the progress is a biological necessity;
and he who cannot attach himself to its ways is submerged.

It is small disparagement for this vast construction of the connec-
tion between the moral life of the individual and the total evolution
of the associations of men, of organisms, of the whole, that, now espe-
cially in English ethics, a bitter strife has broken forth, which we may
regard as the one-sided elaboration of the individualistic parts of
Spencer's ethics on the one side, of the social on the other side.
While the orthodox disciples of Spencer insist that such progress
only can be kept in aim which must assure to the individual, to the
fit the most unrestricted possible amount of free movement, while
the whole rigor of the process of selection must fall upon the unad-
justed and the unfit, the socialist tendencies of our time tend to
advocate a reversal of this harsh result and to advocate both the
united struggle of human society, by suppressing over-energetic
individuals, and the preservation of the economically weak. Though
it would be interesting to trace this division to its final grounds, I
must limit myself to note the fact that the socialist movement

PROBLEMS OF ETHICS 407

seems here also to be in advance, — at least, so far as European
movements of thought are concerned; and that they are in the
condition to compensate for their departure from the teachings of
the master by an appeal to the main thoughts of his system, con-
cerns me just here. Doubtless socialistic thought is on the whole
in advance when compared with liberal and individualistic thought.
And, under these circumstances, the inference for every disciple of
Darwin's theory of evolution is simple; that here again is a case
of survival of the fittest; that socialistic ideals represent a higher
form of adjustment; that just by the fact of their victory the ne-
cessity and justification of this victory is placed beyond doubt. It
helped little that the venerable thinker himself in the last years of
his rich and active life descended into the arena of the contest and
warned his beloved England against the dangers of this socialistic
tendency. It was inconsistent that he tried to brand these thoughts
as a retrograde movement, as a step backward, since his own system
with its powerful optimism affords no possibility for victorious
retrograde movements. Even imperfection and evil has for Spencer
only the significance of an imperfect progress; and the thought
that imperfection could even win the victory over the perfect, that
must be warned against it, could only be nonsense in connection
with his system. For him, as for Hegel, the final formula, obtained
it is true by a very different way, is the thesis: The actual is
rational.

But just this reference to Hegel's system makes clear to us the
opposition which Herbert Spencer's system found in Germany,
first of all, but also in wide circles in England and in America. If
it could be objected against Hegel that the activity of the individual,
in contrast to the might of the developing process of the logical idea,
is reduced to insignificance, this consideration returns with doubled
force in contrast to the concept of the thought of development, which
is found in the modern theory of evolution of Spencer. For here
it is not teleological necessities which prevail, but causal. To have
proved evolution by the laws of nature is precisely his system's title
to fame. The question must then be raised whether an obligation
to any definite practical action can be deduced from the proof of the
necessity of any event. If the development is necessary, it will be
completed whether I cooperate with it or not. If it needs my coopera-
tion, it need not be regarded as a law of nature. It is exactly the
same difficulty which beset the Stoics, when they tried to harmon-
ize the determinism of world events with the demands which their
ethics put upon the moral resolves of the individual. It is absurd
to will any necessary event of the laws of nature; I can suspend my
action so that I count upon the occurrence of such an incident, but
I cannot make this incident the object of my will. I can decide that

408 ETHICS

I will observe an eclipse of the moon, but I cannot will the occurrence
of this eclipse of the moon, or not will it.

If we reduce the difficulty to the simplest formula, it would be as
follows: the theory of evolution did not distinguish between two
completely different kinds of attitudes on the part of human mental
activity ; between the knowledge of the necessity of what exists and
its judgment by standards of value. But it is precisely with the
latter that ethics has to do. It is, like logic and aesthetics, a science
of values; the interest in the question how something has come to be,
is quite different from the interest in determining its value. Every-
thing has come to be, the valueless as well as the valued, with the
same necessity; that is a self-evident presupposition of all explana-
tory science. The bungling drawing of a school-toy and the Sistine
Madonna, the hallucinations of a lunatic and the thought of a
Herbert Spencer, a demonic crime and a deed of the purest ethical
fulfillment of duty, are, in the same sense, necessary; but with the
knowledge of this necessity we have not come a single step nearer
to the task of their valuation.

The difference between these two kinds of attitudes has perhaps
never been more clearly sketched than in Fichte's book On the
Calling of Man. If we assume that I have a fully adequate scientific
knowledge of the course of nature, I might discern that this grain
of sand which the storm has set in motion could not drift a hair's
breadth farther, unless the whole previous course of nature had been
quite different; what then would be gained for my own moral action?
The answer must be: Nothing. More than that, if this point of view
were the only possible for man, then this action would have no
longer, as a moral action, any significance, and could have none;
since as a part of the world event alike in value to all other parts it
would remain like in value, and it would be meaningless to select and
emphasize out of this continuum of facts and environments, alike in
value, single elements as especially valuable and significant. The
man who could not resign himself to this knowledge, who could
not be satisfied to continue, in cool content, at the point of view of
the silent contemplation of causes, must fall into conflicts similar
to those which Carlyle so vividly described in Sartor Resartus. We
must then, in order to an understanding for this new problem,
provisionally disregard, above all else, whatever the theory of evo-
lution has accomplished by way of scientific explanation, and reserve
for a later investigation the ethical valuation of this sequence of
development. The question which is now to occupy us is directed,
first of all, to the subject of our moral valuation. What do we call
good or bad?

This is the main question of all normative ethics in general, and its
answer by Kant will always remain a brilliant feat in this field. He

PROBLEMS OF ETHICS 409

proved, in the first place, that this predicate can be properly applied
to no action whatever, that we can speak of a good action in figur-
ative language only, when we believe that we can make from this
action an inference with regard to something else, — the disposition
of the actor; and that the same action which we do not hesitate
to describe as good, on the supposition of the correctness of this
inference, loses directly this character as soon as doubt of the cor-
rectness of the inference arises. This disposition, which we distin-
guish in this way, which forms the substrate of our moral valuation,
we call the good will, and the Magna Charta of the Kantian ethics
consists in the celebrated thesis: Nothing can possibly be good
except a good will. This reasoning appears to be as self-evident
as its result is important.

The whole ethical process is removed within the soul. While the
theory of evolution and, still more, utilitarianism could still hope
to obtain, with the character of the work, at the same time an ex-
pression with regard to the ethical value of the action; while, in this
combination of ideas, the ethical goodness of the disposition could
be judged by the usefulness or value to civilization of the performance
done, so that both these systems would have essentially the character
of an ethics of results, we have in Kant and his successors, most
decidedly, an ethics of dispositions. It has rightly been pointed
out that this ethics could grow only upon Protestant soil, that here
the same contradiction prevails which Luther once summed up in the
words: "Good works do not make the good man, but the good man
creates good works." All the excellences, but all the weaknesses
also, of Protestantism, cling to Kant's ethics.

First, let us follow the further stages of Kant's thought. How
must a good will be constituted, so that we may count it as ethically
good? All our acts happen in order to fulfill a purpose. The character
of the action depends upon the character of the purpose, which the
actor proposes for himself, which he affirms with his will, which he
makes his own. But if the purpose be no longer willed, then all the
actions cease, which hitherto had had to be accomplished for its
fulfillment. All those purposes, which under the circumstances
cannot be willed, cannot therefore produce that lasting constitution
of the will which we understand under the term the good will. But
among the different motivations of the will, there are some which
for the observer become separated. They have not a character such
that they could, under any circumstances, cease to motivate the
will; they are necessary and universal determinations of the will.
The imperative which they contain and with which they demand
action has not the hypothetical form: "If thou wilt obtain this or
that, you must; " but the absolute: " Thou shalt." It is a categorical
imperative, to which the will is here subordinated, which determines

410 ETHICS

my actions; and such a categorical imperative we term duty. Only
the dutiful will is good. It is clear that this determination shows
an exact analogy to the other norms of judgment in the logical and
the sesthetical field. The principle of contradiction states nothing
at all with regard to the single thoughts, it only asserts that our think-
ing can then alone make a claim upon a logical valuation while it
fills the condition which the principle of contradiction states. Like-
wise, the impulses of our wills can be morally valued only when they
refer to an absolute "Thou shalt;" if this is not the case, they are
excluded from the range of valuation, just as the play of our fancy,
which does not recognize the principle of contradiction, is excluded
from the realm of the norm of scientific thinking.

Here again the normal action of ethics is represented as a selective
process. While the evolutionist ethicist can estimate every single
content of human consciousness with reference to the point whether
it is preservative of the species or not, and thus give it ethical value,
the realm of the Kantian ethics is much more confined. Only those
impulses of the will occur with conscious subordination under the
command of duty, or in conscious opposition to it fall within the
realm of moral valuation. All others — and their name is legion -
must be termed unmoral. Not as if they become thereby actually
valueless; they may stand as high as you please in the intellectual,
aesthetic, or religious scale of values. But to bring them under just
the moral norms of judgment would be an attempt at an unappli-
able object. This is the point, perhaps, where the Kantian ethics
gives the hardest shock to the healthy human understanding. It
will always seem a paradox that we have a moral act when a man
with strong desires for theft, after a severe inner struggle, does not
put a silver spoon into his pocket, while the man who omits all this
quite as a matter of course may have no claim upon moral desert.
And yet each one of us would feel it as an insult, if he should be
praised for such omission. The solution of this difficulty lies in the
distinction of the value of the single resolve and that of the whole
moral personality. The man who is still led into temptation by silver
spoons stands morally upon the same plane upon which the scholar
stands who struggles with extreme mental effort to calculate a simple
example in multiplication. In the case of the more advanced person
our moral approval is not aroused because he no longer needs, in
this simple case, to appeal to the law of duty, but because we be-
lieve that we may conclude that his moral personality is attacking
other more difficult problems with full force, and that he is here in
himself feeling the full weight of the contest. If we were deceived
in this, if it prove true that he, content with what had been attained,
had withdrawn to the position of the ethical capitalist, our ethical
interest in him would likewise cease, just as our intellectual interest

PROBLEMS OF ETHICS 411

ceases in the scholar for whom there are no more problems in his
science. From this point of view the result is necessary that the
category of duties, to speak with Hegel, is absolutely infinite; and
in this perhaps lies the considerable difference between modern and
ancient ethics. For ancient ethics the ideal of the wise man was
a distinctly finitely determined amount. However difficult it might
be to fulfill the conditions for it, it could still be fulfilled in a human
life; and a further advance beyond this fulfilled ideal would have
been to the Greeks an absurdity: it is the "nothing too much"
transferred to the ethical point of view. It is otherwise in modern
ethics, and with this is connected the change in that the concept of
the infinite has become a concept of value. It is as Carlyle says:
" Fulfill the next duty which presents itself to thee, and when thou
hast fulfilled it, wait for ten, twenty, a hundred to be fulfilled." But
we recognize the degree of ethical development which a man has
attained by noting that it is no longer duty to him.

If the limits of the moral valuation have been much restricted
by the introduction of the concept of unmoral actions, it has been
extended in the other direction by the insight that now every action
which happens in fulfillment of a command of duty is to be valued
as the result of a moral disposition. We come thus to the problem
which, since the time of the ancient sophists, has not ceased to occupy
minds, and which may most simply be termed the anthropological
problem. What in the world is there that is not by individuals and
by people deemed to be moral! With what strange contents the
formal " Thou shalt " of morality is filled ! In face of these contradic-
tions, is there any sense at all in speaking of ethical commands? All
skeptical attacks upon ethics find in such considerations their strong-
est support; and here again the answer is easy when we reflect upon
the analogy with science, art, and religion. Aristotle and Democritus,
Hegel and Hobbes, have taught very differently, and yet all have been
busy with science. Raphael and Menzel are surely to be valued as
artists; Mahomet and Buddha were both religious geniuses of the first
magnitude. Why should it be different in the field of ethics? What
other men have held to be moral, how they have acted, this can be
valuable to me, in order for me to become clear with regard to my
own moral determination, just as the artist sees the works of other
masters, just as the scientific man must know the theorems of others.
But all this cannot be the standard for the formation of my own life.
I am, once for all, placed in this world, to be active there; I am
responsible to myself for what I wish to accomplish with this life.
And so it can, it is true, be an encouragement to me that other men
have felt in themselves the same motive to moral activity; I can
give them my hand as striving for the same with me through the
separating centuries and across the estranging seas. But their way

412 ETHICS

of solving the great problems of life cannot be the standard for me
save in the sense that I receive them into my will, recognize them as
valid for my own life.

So, then, the whole weight of the distinction, the whole moral
process, is transferred to the individual. He is the point of depart-
ure and the goal of the struggle for a content in life. Is this now
egoism? This much-discussed question also suffers, as I believe, by
a defect in the statement of the problem. If it is intended that that
action is meant by egoism, the motive for which is one's own welfare
or happiness, by altruism, however, the action which aims at the
happiness of others, it is quite clear that these two contrasts have as
little meaning for the ethics of disposition as the complementary
contrast of beautiful and ugly. Moral action is completely indifferent
with regard to these contrasts. Moral actions can be characterized
as altruistic as well as egoistic, and the same is the case for unmoral
or bad actions. By knowing that distinct advantages have resulted
to the doer from an action, or that "the greatest happiness of the
greatest number" has resulted from it, I have not gained one step
for the moral valuation of this action. I should surely act immorally
if I omitted an action acknowledged as moral by me because it
would involve pain for others and thus would have an anti-altruistic
character. Whence this confusion of the altruistic with the moral
arose is easy to see. Long before the child could himself act morally,
it must be accustomed to feel that its beloved self cannot be the sole
standard for its action; and to the end that it keep peace and content
with its brothers and playmates, it is properly accustomed to regard
in its action the welfare of the human beings about it. That is a
preparatory step to moral action; but, strictly speaking, it can be
counted as moral by those only who are determined not to recognize
the limits between psychological motivation and normative deter-
mination.

It would be an interesting task to trace the relations into which
the autonomous moral individual enters with the great moral
institutions which dominate the community and have combined in
usage, society, and state, and which Hegel described in a happy
expression as " objective morality." Here it is no longer the regard
for the weal or woe of fellow men which strives to gain influence
over my action; here the ethical will of past generations of my own
ancestors accosts and asks me whether I can bring my action into
harmony with that which they willed and for which they strove.
It is a slight disadvantage to the ethically directed man that, in
order to protect these moral institutions from injury, an arsenal
of punishments, of social influences, of boycotts, and of whatever
finer or coarser means of compulsion there may be, are set up. This
arsenal is necessary to sustain the social structure which alone

PROBLEMS OF ETHICS 413

affords the chance for moral action; and he who calculates with
pleasure and pain, who tries to arrange his life as happily as possible,
will be restrained by shrewd calculation from injuring the prevailing
moral institutions. The moral man has nothing to do with such
considerations. When he affirms the objective morality, he does so
because he recognizes his moral will as identical with that of previous
generations which have made these forms. But the time can come
when he discovers that a moral life within these forms is no longer
possible for him, when with deep regret he sees the bond of continuity
break which knit him in affection with the past, when he must
resolve to enter new untrodden paths, just as Copernicus was forced
to resolve to substitute a new knowledge for those which had satisfied
centuries. Such a man will endure calmly and patiently the con-
sequences which result from such a course; he will not expect to
be justified, through the purity of his intentions, in the eyes of his
fellows, if he undertakes to lay hands on the institutions which the
moral consciousness of his contemporaries recognizes as valid. But
he will also know that these same institutions owe what sacredness
they possess to the blood of previous martyrs, that these shadows
of a past can only then speak to a living generation when they have
tasted the sacred blood of sacrifice.

So then we see two great movements in our time struggling about
the ethical questions. The one has on its side the whole apparatus
of scientific conceptions, the presupposition of necessary events
without exceptions, the knowledge that the single individual is an
infinitely small element in a necessary sequence of development. It
can explain everything, deduce everything from its conditions. At
one point only its power breaks down : it cannot make the individual
comprehend why he should raise a finger to keep in motion this
machine which goes of itself.

And, opposed to this, is the other movement, which rests upon the
one fact that the point of view of its opponent, the scientific, is also
a relation of reality to values, and that man alone introduces these
values into reality, measures and tests it by these values. According
to this movement, every new human life has the question put to it,
what it can accomplish with these values, whether it is capable of
making something out of reality, out of itself, which has in itself a
value such as to raise it above the flux of appearances as the bearer
of these values. Everything previous as well as everything subsequent
vanishes before these thoughts that it is now day, that the night is
soon coming when no man can work, that at the day's end the day's
work must be done. But what each recognizes as his day's work, he
must himself find within himself. This decision is his destiny.

I cannot better close than with the words of the man whose life
had little joy, but who grappled with these questions in the solitude

414 ETHICS

of Craigenputtock, in the supreme solitude of the human wilderness
of London, with a seriousness which still to-day proves to be soul-
wooing and soul- winning: "Centuries have passed that thou might-
est be born, and centuries are waiting in dumb expectation of what
thou wilt accomplish with this life, now that it has begun." And
what this life can offer Carlyle, by combining the thoughts of Fichte
and of Goethe, has united in the call:

" Work and despair not."

SECTION F — AESTHETICS

SECTION F — .ESTHETICS

(Hall 4, September 23, 3 p. TO.)

CHAIRMAN: PROFESSOR JAMES H. TUFTS, University of Chicago.
SPEAKERS: DR. HENRY RUTGERS MARSHALL, New York City.

PROFESSOR MAX DESSOIR, University of Berlin.
SECRETARY: PROFESSOR MAX MEYER, University of Missouri.

THE RELATION OF AESTHETICS TO PSYCHOLOGY AND

PHILOSOPHY

BY HENRY RUTGERS MARSHALL

[Henry Rutgers Marshall, Practicing Architect, President of the New York
Chapter, American Institute of Architects, Member of Art Commission,
City of New York. b. July 22, 1852, New York City. B.A. Columbia Uni-
versity, 1873; M.A. ibid. 1875; L.H.D. Rutgers College, 1903. Member
American Psychological Association, Society of American Naturalists, Fel-
low American Institute of Architects, Honorary Member National Society of
Mural Painters, Member American Philosophical Association. Author of
Pain, Pleasure, and ^Esthetics; ^Esthetic Principles; Instinct and Reason.]

IF conventional divisions of time are to serve as means by which we
may mark the movement of thought as it develops, we may well
say that the nineteenth century saw a real awakening in relation
to aesthetics among those who concern themselves with accurate
thinking, — a coming to consciousness, as it were, of the importance
to the philosophy of life of the existence of beauty in the world, and
of the sense of beauty in man.

And with this awakening came a marked breadth of inquiry; an
attempt to throw the light given by psychological analysis upon the
broad field of aesthetics, and an effort to grasp the relations within
the realm in which beauty holds sway to philosophy as a whole.

That the questions thus presented to us have been answered, I
imagine few, if any, would claim; rather may we say that the nine-
teenth century set the problems which it concerns the sesthetician
of the twentieth century to solve; and this without underestimating
the value of the work of the masters in esthetics who lived and
wrote in the century so lately closed, some of whom are fortunately
with us still.

Of these present problems M. Dessoir will treat in his address to
follow mine; in the regretted absence of Professor Lipps the privilege
has been granted to me to consider with you briefly the relations
of aesthetics to psychology, and to philosophy, which must in the

418 AESTHETICS

end determine the nature of the problems to be studied by the sesthe-
tician, and the import of the solutions of these problems which they
present for our consideration.

I. The Relation of Esthetics to Psychology

We live in what may well be called the era of psychological develop-
ment, an era marked by the recognition of the truth that no philo-
sophical view of life can be adequate which does not take full account
of the experience of the individual human spirit which interprets this
life. And so quite naturally for ourselves, and in all probability
quite in accord with the habit of thought of the immediate future,
we begin our study by the consideration of the relation of aesthetics
to psychology.

In turning for light to psychology, the sesthetician finds himself of
course asking what is the nature of the states of mind related to his
inquiry; and here at once he finds himself confronted with a distinc-
tion which must be made if a correct aesthetic doctrine is to become
established. He notes that there is a sharp difference between (1)
the mental attitude of an artist who produces works of beauty; and
(2) the mental attitude of a man at the moment when he appreciates
beauty in his experience.1 The failure to note this distinction has in
my view led to much confusion of thought among the sestheticians
of the past, and to the defense of dogmas which otherwise would
not have been maintained.

That this distinction is an important one becomes clear in the
fact that the sense of beauty is aroused in us by objects in nature
which bear no relation to what men call fine art. The mental state
of the appreciator of beauty has therefore a breadth which does not
belong to the mental state which accompanies, or leads to, the pro-
duction of works of beauty by the artist.

And yet it should not surprise us that this distinction has so
often been overlooked; for the theorists first follow the trend of
thought of the uncritical man, and this uncritical man does not
naturally make the distinction referred to.

For, on the one hand, even the least talented of men has some
little tendency to give part of his strength to artistic creation in one
form or another; the creative artist is guided by what is a truly racial
instinct, which under favorable conditions will appear in any man
who is not defective: each of us thus in the appreciation of beauty
throws himself to some degree into the attitude of the creative artist.

And, on the other hand, the artist, when not in creative mood, falls
back into the ranks of men who keenly appreciate beauty but who

1 Cf . my Esthetic Principles, chap, i, " The Observer's Standpoint," and
chap, in, " The Artist's Standpoint."

THE RELATIONS OF AESTHETICS 419

are not productive artists; he thus alternately creates and appre-
ciates, and with difficulty separates his diverse moods.

We may well consider these two distinguishable mental attitudes
separately.

a

In asking what is the nature of the experience which we call the
sense of beauty, we are stating what may well be held to be the most
important problem in aesthetics that is presented to the psychologist.

Man is practical before he deals with theory, and his first theo-
retical questionings are aroused by practical demands in connection
with his failures to reach the goal toward wrhich he strives. The de-
velopment of modern aesthetic theory has in the main quite naively
followed this course, and we may properly consider first the psycho-
logical inquiries which seem to have the most direct bearing upon
practical questions.

The artist asks why his efforts so often fail, and he is led to inquire
what are the qualities in his work which he so often misses, but now
and again gains with the resulting attainment of beauty.

It is thus that we naturally find the aesthetician appealing to the
psychologist, asking him what special types of impression yield
beauty, what special characteristics of our mental states involve the
fullest aesthetic experience.

The psychologist is naturally first led to consider certain striking
relations found within the beautiful object which impresses us, and
to inquire into the nature of the psychic functioning which is in-
volved with the impressions thus given. He thus comes to consider
the relations of the lineal parts of pleasing plane-surface figures; and
the study of these relations has given to us such investigations as
the notable ones of Fechner in respect to the "Golden Section,"
which have been supplemented by the more rigid tests of Dr. Witmer
and Doctors Haines and Davies in our own day. In similar manner
the basis of the beauty found in symmetry and in order, and the
problems related to rhythm, have been closely studied, especially
in late years by Lipps; and the fundamental principles of tonal
relation, and of melodic succession, by Helmholtz, Stumpf, and
later writers.

But all these studies of the striking characteristics found in the
object are for the psychologist necessarily involved with the study
of the distinctly subjective accompaniments in the sense of beauty
aroused by the objective forms thus brought to our attention, and
he is led to dwell upon the active part the mind takes in connection
with aesthetic appreciation. We see this tendency in Berenson's
emphasis, and perhaps on the whole over-emphasis, of the import-
ance of the interpretation of works of art, in the group of what I
would call the arts of sight, in terms of the tactile sensibilities. But

420 .ESTHETICS

we see it much more markedly in the important studies of Lipps,
who shows us how far our appreciation of beauty in nature, and in
artistic products, is due to the sympathetic introjection of ourselves
as it were into the object, — to what he calls Einfuhlung.

But, broad as he shows the applicability of this principle to be, it
is clear that we have not in it the solution of the fundamental aesthetic
problem with which the psychologist must deal when appealed to by
the aesthetician. For no one would claim that all of this sympathetic
introjection — this Einfuhlung — is aesthetic : the aesthetic Einfuhl-
ung is of a special type. Nor to my mind does it seem clearly
shown that there are no sources of beauty which do not involve this
introjection, as would be the case if we had reached in this principle
the solution of the fundamental aesthetico-psychologic problem. For
instance, the sense of beauty experienced when I look at some one
bright star in the deep blue of the heaven seems to me to be inex-
plicable in terms of such introjection.

All this work, howrever, brings help to the practical artist and to
the critic. They do not acknowledge it fully to-day, but year by year,
more and more will the influence of the results of these studies be
felt as they gain the attention of thinking men.

Nevertheless, we cannot but face the fact that the practical benefit
to be gained from them is of a negative sort. There is no royal road
to the attainment of beauty; but the psychologist is able to point
out, by the methods here considered, the inner nature of certain
sources of beauty; thus teaching the artist how he may avoid ugliness,
and even indicating to him the main direction in which he may best
travel toward the attainment of his goal.

But, after all, the relations thus discovered in the beautiful object,
and the related special analyses of mental functioning which are
involved with our appreciation of beauty, tell us of but relatively
isolated bits of the broad realm of beauty. The objects which arouse
within us the sense of beauty are most diverse, and equally diverse
are the modes of mental functioning connected with the appreciation
of their beauty.1

And this has led to the formulation of such principles as that of
the " unity of manifoldness " of which Fechner makes so much, and
that of the monarchischen Unterordnung which Lipps has more lately
enunciated.

It is indeed of great interest to inquire why it is that the processes
which lead to the recognition of these principles are so clearly defined
in many cases where the sense of beauty is aroused. But very evi-
dently these general principles, important though they be in them-

Nothing has shown this more clearly than the investigations of Haines and
•9 in reference to the Golden Section of which we have spoken above. See
Psychological Review, xi, 415.

THE RELATIONS OF .ESTHETICS 421

selves, are not ones upon which we can afford to rest: for clearly
they apply in very many cases where beauty does not claim sway.

Our whole mental life exemplifies the unification of the manifold,
and monarchic subordination, whether the processes be sesthetic or
not. It does not suffice us to show, what is thus shown, that the
sesthetic states conform with conditions of our mental life that
have a broad significance, although it is of great importance to
demonstrate the fact: for our mental functioning in the apprecia-
tion of beauty appears thus as in truth an important type, but
for all that but a special and peculiar type of the functioning which
we thus bring into prominence.

The problem then remains, what is the special nature of this
functioning which yields to us the sense of beauty?

And here in my view we have the problem which is of prime
importance to aesthetics to-day, and which psychology alone can
answer; namely, what is the characteristic that differentiates the
sense of beauty from all other of our mental states? Until this
question is answered, all else must seem of secondary importance
from the standpoint of theoretical psychology, however important
other forms of inquiry may be from a practical point of view.

When the psychologist turns his attention to this problem, he
at once perceives that he is unable to limit his inquiry to the experi-
ence of the technically trained artist, or even to that of the man of
culture who gives close attention to sesthetic appreciation.

Beauty is experienced by all men. But beauty is very clearly of
varied types, and the sense of beauty is evidently called out by
impressions of most varied nature; but the fields of what is considered
beautiful by different people so far overlap that we can rest assured
that we all refer to an experience of the same characteristic mental
state when we proclaim the existence of beauty; for when we by
general agreement use a special term as descriptive of an objective
impression, we do so because this impression excites in us certain
more or less specific mental states; and when different people use
the same term in reference to objects of diverse nature, we are wont
to assume, and are in general correct in assuming, that these objects
affect these different people in approximately the same way.

It seems probable, therefore, that if the child, who has learned how
to apply words from his elders, speaks of having a beautiful time at
his birthday party; and if the grown man speaks of a beautiful day;
and if the pathologist speaks of his preparation of morbid tissue as
beautiful; and if the artist or connoisseur speaks of the beauty of
a picture, a statue, a work of architecture, a poem, a symphony;
then the word beauty must be used to describe a certain special mental
state which is aroused in different people by very diverse objective
impressions.

422 AESTHETICS

This view is strengthened when we consider that the application
of the term by individuals changes as they develop naturally or by
processes of education; and that the standards of beauty alter in
like manner in a race from generation to generation as it advances
in its development.

We must then look for the essence of beauty in some quality of
our mental states which is called up by different objective impres-
sions in different people, and under diverse conditions by different
objects at different times in the same individual.

Search for such a quality has led not a few psychologists to look
to pleasure as the quality of our mental states which is most likely
to meet our demand. It is true that the consideration of pleasure
as of the essence of the sense of beauty has not often been seriously
carried out ; apparently because so many of what we speak of as our
most vivid pleasures appear as non-aesthetic; and because pleasure
is recognized to be markedly evanescent, while beauty is thought of
as at least relatively permanent.

It is true, also, that there is a hesitancy in using the word pleasure
in this connection; many writers preferring the less definite word
"feeling" in English, and "gefiihl" in German. But by a large
number of psychologists the words pleasure and feeling are used as
synonyms; and those who, with me, agree that what we loosely call
feeling is broader than mere pleasure, must note that it is the pleas-
urable aspect alone of what is called "feeling" that is essentially
related to our experience of the sense of beauty.

All of us agree, in any event, that the sense of beauty is highly
pleasant; and, in fact, most of our sestheticians have come to assume
tacitly in their writings that the field of aesthetics must be treated
as a field of pleasure-getting; and this whether or not they attempt
to indicate the relation of pleasure-getting to the sense of beauty.

The suggestion that pleasure of a certain type is of the essence of
beauty seems the more likely to prove to be satisfactory when we
consider that pleasure is universally acknowledged to be the con-
tradictory opposite of pain; and that we have in ugliness, which is
always unpleasant, a contradictory opposite of beauty.1

Clearly then it behooves the psychologist to give to the aesthetician
an account of the nature of pleasure which shall be compatible with
the pleasurable nature of the sense of beauty; and which shall either
explain the nature of this sense of beauty in terms of pleasure, or
explain the nature of pleasure in a manner which shall throw light
upon the nature of this sense of beauty to which pleasure is so indis-
solubly attached.

1 It is of course agreed that beauty and ugliness may be held together in a
complex impression: but in such cases the beauty and the ugliness are inherent
in diverse elements of the complex.

THE RELATIONS OF AESTHETICS 423

The jesthetician thus demands urgently of the psychologist an
analysis of the nature of pleasure; and an analysis of this so-
called "feeling," which shall show the relation between the two
experiences.

Concerning the latter problem I hope some day to have something
to say.

Those of you who happen to be familiar with my published works
will realize that my efforts in this field in the past have been given
largely to the study of the former problem. My own view may be
succinctly stated thus.

While all aesthetic experiences are pleasant, very evidently much
that we call pleasant is not aesthetic. We must look then for some
special differentiation of aesthetic pleasure, and this I find in its
relative permanency.

This view is led up to by a preliminary study of the psychological
nature of pleasure.

Pleasure I find to be one phase of a general quality — Pleasure-
Pain — which, under proper conditions, may inhere in any emphasis
within the field of attention, or, to use more common language, may
belong to any element of attention.

Now pleasure, as we have said, is notably evanescent, but this
does not preclude the existence of pleasurable states of attention
which are relatively permanent. This permanency may be given by
the shifting of attention from one pleasurable element to another;
by the summation of very moderate pleasures, etc., etc.

Any pleasant psychic element may become an element of an
aesthetic complex: and any psychic complex which displays a relative
permanency of pleasure is in that fact aesthetic. Our aesthetic states
are those in which many pleasant elements are combined to produce
a relative permanency of pleasure.

Our "non-aesthetic pleasures," so called, are those states which
have been experienced in the past as vividly pleasant, and to which
the name pleasure has become indissolubly attached: but they are
states which do not produce a relatively permanent pleasure in
revival; and correctly speaking, are not pleasures at the moment
when they are described as such, and at the same time as non-
aesthetic.

I am glad to feel that this view of mine is not discrepant from that
of Dr. Santayana, as given in quite different terms in his book en-
titled The Sense of Beauty. For what is relatively permanent has
the quality which I call realness; and that in experience which has
realness we tend to objectify. Hence it is quite natural to find Dr.
Santayana defining beauty as objectified pleasure.

You will not blame me I believe for thinking that my own defini-
tion cuts down closer to the root of the matter than Dr. Santayana 's.

424 AESTHETICS

But if this theory of mine is found wanting, the aesthetician will
not cease to call upon the psychologist for some other which shall
meet the demands of introspection; and which shall accord with our
experiences of the sense of beauty, which in all their wealth of impres-
sion the acsthetician offers to the psychologist as data for the labor-
ious study asked of him.

Before leaving this subject I may perhaps be allowed to call
attention to the fact that the theoretical view, which places the essence
of the sense of beauty in pleasure-getting, if it prove to be true, is
not without such practical applications as are so properly demanded
in our time. For if this view is correct, it teaches to the critic a lesson
of sympathetic tolerance; for he learns from it that the sources from
which the sense of beauty are derived differ very markedly in people
of diverse types: and it warns him also against the danger of an
artificial limitation of his own aesthetic sense, which will surely
result unless he carefully avoids the narrowing of his interests.

It teaches further that there is no validity in the distinction
between fine art and aesthetics on the one hand, and beauty on the
other, on the ground, commonly accepted by the highly trained
artist and connoisseur, that a work of art may deal with what is not
beautiful.

For it appears that while the sense of beauty is the same for each
of us, the objects which call it out are in some measure different for
each.

Now it happens naturally that the objects which arouse the sense
of beauty in a large proportion of men of culture get the word beauty
firmly attached to them in common speech.

But under the view here maintained, it must be that the highly
trained artist or critic will pass beyond these commoner men, and
find his sense of beauty aroused by objects and objective relations
quite different from those which arouse the sense of beauty in the
commoner man; so that often he may deal with the beauty of
elements in connection with which beauty is unknown to the com-
moner man, and even with elements which arouse a sense of ugliness
in the commoner man; while on the other hand the objects which
the commoner man signalizes as most beautiful, and which are cur-
rently so called, may not arouse in the trained artist or critic the
sense of beauty which is now aroused in him by effects of broader
nature, and of less common experience.

The critic and the skilled artist thus often find their aesthetic sense
aroused no longer by the objects to which the word beauty has by com-
mon consent come to be attached ; although with the commoner man he
still uses the word beauty as descriptive of the object which arouses
the aesthetic thrill in the mass of normally educated men. He may

THE RELATIONS OF .ESTHETICS 425

even find his aesthetic sense aroused by what the common man calls
ugly; although it is for himself really beautiful. And he comes thus
quite improperly to think of the highest art as in a measure inde-
pendent of what he calls "mere beauty." What he has a right to
say, however, is merely this, that the highest art deals with sources
of beauty which are not appreciated by even the generally well-
cultivated man.

I have dwelt, perhaps, too long on the psychological problems
presented when the psychologist attempts to describe to the sesthet-
ician the nature of the experience of one who appreciates beauty;
and have left perhaps too little time for the consideration of the
problems presented when he is asked to consider the nature of the
experience of the artist who creates.

The man who finds strongly developed within him the creative
tendency, is wont, when he turns to theory, to lay emphasis upon
expression as of the essence of beauty.

It is, of course, to be granted that the process of Einfiihlung, -
of introjection, — above referred to, leads us to find a source of
beauty in the vague imagination of ourselves as doing what others
have done; and we may take great aesthetic delight in reading,
through his work, the mind of the man who has created the object
of beauty for us. But evidently, when we lay stress upon this intro-
jection, we are dealing with the appreciation of beauty, and not with
the force which leads to its production.

Just as clearly is it impossible to hold that expression is of the
essence of the making of beauty. For expressiveness is involved in
all of man's creative activity, much of which has no relation what-
ever to the aesthetic. The expression of the character of the genius
of the inventor of a cotton loom, or of the successful leader of an
army in a bloody battle, excites our interest and wonder; but the
expression of his character as read in the result accomplished does
not constitute it a work of beauty.

I speak of this point at this length because in my opinion views
of the nature of that here objected to could not have been upheld
by such men as Bosanquet and V4ron had they kept clear the dis-
tinction referred to above between the experience of one who ap-
preciates beauty, and the experience of the creative artist; and
especially because the teaching of the doctrine thus combated is
wont to lead the artist whose cry is " Art for Art's sake " to excessive
self-satisfaction, and to lack of restraint which leads to failure.1

1 In order to avoid misunderstanding, I may say here that notwithstanding
these remarks I am in full sympathy with the artist who thus expresses himself,
as will presently appear clear.

426 .ESTHETICS

The strong hold which this theory has in many minds has its value,
however, in the emphasis of the fact that {esthetic creation is due
to impulses which are born of innate instincts expressing them-
selves in the production of works of beauty. And if this be so, we see
how true it must be that each of us must have in him some measure
of this instinct; and that the appearance of its appropriate impulses
should not mislead us, and induce us to devote our lives to the
worship of the Muses, unless we become convinced that no other work
can adequately express the best that is in us.

But the true artist is not troubled by such questionings. He finds
himself carried away by what is a true passion; by what is instinct-
ive and not ratiocinative.

The fact that the artist is thus impelled by what may well be called
the " art instinct " is one he could only have learned from the psy-
chologist, or when in introspective mood he became a psychologist
himself; and it carries with it corollaries of great value, which the
psychologist alone can elucidate.

It teaches the artist, for instance, that his success must be deter-
mined by the measure of this instinct that is developed within him;
that he must allow himself to be led by this instinct; that his best
work will be his "spontaneous" work. This, of course, is very far
from saying that he cannot gain by training; but it does mean that
he must learn to treat this training as his tool; that he must riot
trust overmuch to his ratiocinative work, the result of which must
be assimilated by, and become part of, his impulsive nature, if he is
to be a master.

An artist is one in whom is highly developed the instinct which
leads him to create objects that arouse the sense of beauty. The
expression of this instinct marks his appropriate functioning. He
may incidentally do many useful things, and produce results apart
from his special aptitude; but as an artist his work is solely and
completely bound up in the production of works of beauty.

We naturally ask here what may be the function in life of the
expressions of such an instinct as we have been studying, and this
leads us to consider a point of more than psychological interest,
and turns our thought to our second division.

II. The Relation of Esthetics to Philosophy

For while the science of psychology must guide, it can never dom-
inate the thought of the philosopher who strives to gain a broad view
of the world of experience; and, as will appear below, the aesthetician
calls upon the philosopher for aid which the psychologist as such
cannot give.

THE RELATIONS OF AESTHETICS 427

a

In approaching this subject we may take at the start what we may
call the broadly philosophical view, and may consider the question
raised immediately above, where we ask what may be the function
in life of the art instinct, and what the significance of the aesthetic
production to which its expression leads.

We, in our day, are still strongly influenced by the awakening of
interest in the problems of organic development with which Darwin's
name is identified, and thus naturally look upon this problem from
a genetic point of view; from which, to my mind, artistic expression
appears, as I have elsewhere argued at length, as one of nature's
means to enforce social consolidation. But it is possible that we
are led, by the present-day interest above spoken of, to over-
emphasize the importance of the processes of the unfolding of our
capacities, and it is not improbable that those who follow us, less
blinded by the brilliancy of the achievement of the evolutionists,
may be able to look deeper than we can into the essence of the
teleological problem thus raised.

That art is worthy for art's sake is the conviction of a large body
of artists, who labor in their chosen work often with a truly martyr-
like self-abnegation; and as an artist I find myself heartily in sym-
pathy with this attitude. But aesthetics looks to philosophy for
some account of this artistic re'Aos, which shall harmonize the artist's
effort with that of mankind in general, from whom the artist all too
often feels himself cut off by an impassable gulf.

The study of aesthetics by the philosopher from the genetic stand-
point has, however, already brought to our attention some facts
which are both significant and helpful.

It has shown us how slow and hesitant have been the steps in the
development of aesthetic accomplishment and appreciation in the
past, and how dependent these steps have been upon economic con-
ditions. This on the one hand arouses in us a demand for a fuller
study of the relations of the artistic to the other activities of men;
and on the other hand is a source of encouragement to critic and
artist alike, each of whom in every age is apt to over-emphasize the
artistic failures of his time, and to minimize the importance of its
artistic accomplishment.

This genetic study has a further value in the guidance of our
critical judgment, in that it shows us that the artistic tendencies
of our time are but steps in what is a continuous process of develop-
ment. It shows us arts which have differentiated in the past, and
teaches us to look for further artistic differentiations of the arts in
the future; thus leading us to critical conclusions of no little im-
portance. This consideration seems to me to be of sufficient interest
to warrant our dwelling upon it a little at length.

428 .ESTHETICS

The arts of greatest importance in our time may well be divided
into the arts of hearing (that is, literature, poetry, music), and the
arts of sight (that is, architecture, sculpture, painting, and the
graphic arts).

These diverse groups of arts were differentiated long before any
age of which we have a shadow of record. But many animals display
what seem to be rudimentary art instincts, in which rhythmical move-
ment (which is to be classed as an art of sight) and tonal accompani-
ment are invariably combined — as they are also in the dance and
song of the savage; and this fact would seem to indicate that in the
earliest times of man's rise from savagery the differentiation between
the arts of sight and the arts of hearing was at least very incom-
plete.

But leaving such surmises, we may consider the arts of sight and
the arts of hearing in themselves. We see them still in a measure
bound together; for many an artist, for instance, devotes his life
to the making of paintings which "tell a story," and many a poet
to the production of "word-pictures."

In general, however, it may be said that the arts of hearing and
the arts of sight express themselves in totally different languages,
so to speak, and they have thus differentiated because each can give
a special form of aesthetic delight.

Turning to the consideration of each great group, we note that
the arts of sight have become clearly differentiated on lines which
enable us to group them broadly as the graphic arts, painting,
sculpture, and architecture. Each of these latter has become im-
portant in itself, and has separated itself from the others, just so
far as it has shown that it can arouse the sense of beauty in a man-
ner which its kindred arts of sight cannot approach. It is true that
all the arts of sight hold together more closely than do the arts of
sight, as such, with the arts of hearing, as such. But it is equally
clear that the bond between the several arts of sight was closer
in earlier times than it is to-day, in the fact that modeled paint-
ing, and colored sculpture, were common media of artistic expres-
sion among the ancients, the latter being still conventional even so
late as in the times of the greatest development of art among the
Greeks.

But the modern has learned that in painting and graphics the
artist can gain a special source of beauty of color and line which he
is able to gain with less distinctness when he models the surface upon
which he works : and the experience of the ages has gradually taught
the sculptor once for all that he in his own special medium is able
to gain a special source of beauty of pure form which no other arts
can reach, and that this special type of beauty cannot be brought
into as great emphasis when he colors his modeled forms.

THE RELATIONS OF ESTHETICS 429

In my view we may well state, as a valid critical principle, that,
other things being equal, in any art the artist does best who presents
in his chosen medium a source of beauty which cannot be as well
presented by any other art. That this principle is appreciated and
widely accepted (although implicitly rather than explicitly) is
indicated by the unrationalized objection of the cultivated critic in
our day to colored sculpture or to modeled painting, and in a more
special direction to the use of body-color in aquarelle work. The
objection in all cases is apparently to the fact that the artist fails to
bring into prominence that type of beauty which his medium can
present as no other medium can.

Personally I have no objection to raise to a recombination of the
arts of sight, provided a fuller sense of beauty can thereby be
reached. But it is clear that this recombination becomes more and
more difficult as the ages of development pass; and I believe the
principle of critical judgment above enunciated is valid, based as
it is upon the inner sense of cultivated men.

Better than attempts to recombine the already differentiated
arts of sight are attempts to use them in conjunction, so that our
shiftings of attention from one type of beauty to another may carry
with them more permanent and fuller aesthetic effects ; and such
attempts we see common to-day in the conjunction of architecture
and of sculpture and of painting, in our private and public galleries,
in which are collected together works of the arts of sight.

Now if we turn to the consideration of the arts of hearing, we find
a correspondence which leads to certain suggestions of no little
importance to the critical analyst in our day.

The arts of hearing have become differentiated on lines which
enable us to group them broadly as rhetoric, poetry and literature,
and music. Each has become important in itself, and has gradually
separated itself from the others; — and this just so far as it has
shown that it can arouse in men, in a special and peculiar manner,
the sense of beauty.

It is true, as with the arts of sight, that the special arts of hearing
still hold well together.

But in relatively very modern times music, having discovered a
written language of its own, has differentiated very distinctly from
the other arts of hearing. Men have discovered that pure music
can arouse in a special manner the sense of beauty, and can bring to
us a form of esthetic delight which no other art can as well give.

Poetry has long been written which is not to be sung; and it has
gained much in freedom of development in that fact.

Music in our modern times is composed by the greatest masters
for its own intrinsic worth, and not as of old as a mere accompani-

430 .ESTHETICS

ment of the spoken word of the poet; the existence of the works
of Bach, to mention no others, tells of the value of this differentiation.

And here I think we may apply with justice the principle of criticism
above presented. The poet and the musician each do their best work,
other things being equal, when they emphasize the forms of beauty
which their several arts alone can give. We have here in my view
a rational ground for the repulsion many of us feel for the so-called
" programme music " of our day.

Music and literature of the highest types nowadays present
sources of beauty of very diverse character, and any effort to make
one subsidiary to the other is likely to lessen the aesthetic worth of
each, and of the combination.

Here again I may say that I have no objection to raise to a recom-
bination of the arts of hearing, provided a fuller sense of beauty
can thereby be reached. But this recombination becomes year by
year more difficult as the several arts become more clearly differen-
tiated, and must in my view soon reach its limit.

The opera of to-day attempts such a recombination, but does so
either to the detriment of the musical or of the literary constituent ;
that is clear when we consider the musical ineptitude of such operas
as deal with a finely developed drama, and the literary crudeness of
the plot-interest in Wagner's very best works. Such a consideration
makes very clear to us how much each of the great divisions of the
arts of hearing has gained by their differentiation, and by their inde-
pendent development.

Here as with the arts of sight we may, in my view, hope for
better aesthetic results from the development of each of the differ-
entiated arts in conjunction; rather from the persistent attempt to
recombine them, with the almost certain result that the aesthetic
value of each will be reduced.

6

But aesthetics demands more of philosophy than an account of the
genesis of art, with all the valuable lessons that this involves. It de-
mands, rightly, that it be given a place of honor in any system which
claims to give us a rationalized scheme of the universe of experience.

The aesthetician tells the philosopher that he cannot but ask
himself what significance aesthetic facts have for his pluralism, or
for his monism. He claims that this question is too often overlooked
entirely or too lightly considered; but that it must be satisfactorily
answered if the system-maker is to find acceptance of his view.
And in the attempt to answer this and kindred questions, the aesthet-
ician is not without hope that no inconsiderable light may be thrown
by the philosopher upon the solution of the problems of aesthetics
itself.

THE RELATIONS OF ESTHETICS 431

;

Nor are the problems of aesthetics without relation to pure meta-
physic. The existence of aesthetic standards must be considered by
the metaphysician, and these standards, with those of logic and ethics,
must be treated by him as data for the study of ontological
problems.

But beyond this, aesthetics cries out for special aid from the
ontologist. What, he asks, is the significance of our standards of
aesthetic appreciation? What the inner nature of that which we call
the real of beauty? What its relation with the real of goodness
and the real of truth?

From a practical standpoint this last-mentioned question is of
special import at this time. For the world of art has for centuries
been torn asunder by the contention of the aesthetic realists and their
opponents.

That, in its real essence, beauty is truth, and truth beauty, is
a claim which has often been, and is still heard; and it is a claim
which must finally be adjudicated by the metaphysician who deals
with the nature of the real.

The practical importance of the solution of this problem is brought
home forcibly to those who, like myself, seem to see marked aesthetic
deterioration in the work of those artists who have been led to listen
to the claims of aesthetic realism; who learn to strive for the expres-
sion of truth, thinking thus certainly to gain beauty.

That many great artists have announced themselves as aesthetic
realists shows how powerfully the claims of the doctrine appeal to
them. But one who studies the artistic work of Leonardo, for in-
stance, cannot but believe that he was a great artist notwithstanding
his theoretical belief, and cannot but believe that all others of his
way of thinking, so far as they are artists, are such because in them
genius has overridden their dogmatic thought.

It is clearly not without significance that the world of values is
by common consent held to be covered by the categories of the True,
the Good, and the Beautiful. This common consent seems surely
to imply that each of the three is independent of the other two,
although all are bound together in one group. And if this is true, then
the claim of the aesthetic realist can surely not be correct.

But this claim will not be overthrown by any reference to such
a generalization as that above mentioned. The claim of the aesthetic
realist is based upon what he feels to be clear evidence founded upon
experience; and he cannot be answered unless we are able to show
him what is the basis for his ready conviction that truth and beauty
are one and identical; and what is the true relation between the
True, the Good, and the Beautiful. And these problems, which are
in our day of vital importance to the artist, the philosopher alone
can answer.

432 .ESTHETICS

In my view some aid in the solution of this problem may be gained
from the examination of the meaning of our terms. From this study
I feel convinced that we must hold that when we speak of the True,
and the Good, and the Beautiful, as mutually exclusive as above,
we use the term "true" in a narrow sense. On the other hand, the
True is often used in a broader sense, as equivalent to the Real.

This being so we may say

That the Beautiful is the Real as discovered in the world of im-
pression; the relatively permanent pleasure which gives us the sense
of beauty being the most stable characteristic of those parts of the
field of impression which interest us we may also assent

That the Good is the Real as discovered in the world of expression,
that is, of impulse, which is due to the inhibited capacity for expres-
sion, and the reaction of the self in its efforts to break down the
inhibition. And in the same way we may conclude

That the True (using the term in the narrow sense) is the Real
as discovered in the realm of experience exclusive of impression or
expression.

a. The Real of Impression — The Beautiful
THE REAL

J. The Real of Expression — The Good

(. The Real in realms — The True

exclusive of a and ft (in the narrower

sense of the term)

or

THE TRUE
(in the broad sense
of the term)

That the Beautiful is part of the REAL, that is, is always the
TRUE, using the term true in the broader sense, is not questioned: and
that, in my view, is the theoretical truth recognized by the aesthetic
realists. But in practice the aesthetic realist maintains that the
beautiful is always the true, using the term true in the narrow sense,
and in this, in my view, lies his error.

And if the relation of the beautiful to the true demands the
attention of the philosopher, equally so does the relation of the
beautiful to the good. As I look upon it, all of the true (using the
term as above explained in the narrow sense) and all of the good,
so far as either involve relatively permanent pleasure of impression,
are possible elements of beauty. But, on the other hand, it seems clear
that neither the true (still using the term in the narrower sense), nor
the good, is necessarily pleasing, but may be unpleasant, and there-
fore either of them may be an element of ugliness, and as such must
lose all possibility of becoming an element in the beautiful.

One further word, in closing, upon the closely allied question as
to the nature of worth-values. There is a worth-value involved
in the Good, and a worth-value involved in the True, and a worth-

THE RELATIONS OF AESTHETICS 433

value involved in the Beautiful: and each of these worth-values
in itself seems to be involved with pleasure-getting. Now if this is
the case, then, under the theory I uphold, any worth-value should be
a possible aesthetic element, and this I think it will be granted is
true. But the distinctions between these worth-values are on differ-
ent planes, as it were. In the case of the worth-value of the Good,
we appreciate the worth-pleasure within the realm of the Real of
Expression, that is, of impulse. In the case of the worth-value of the
True (in the narrow sense) , we appreciate the worth-pleasure within
the realm of the Real in other fields than that of expression or that
of impression. In the case of the worth-value of the Beautiful, we
appreciate the worth-pleasure within the realm of the Real of Im-
pression; that is, we appreciate, with pleasure, the significance for
life of the existence of relatively permanent pleasure in and for
itself.

THE FUNDAMENTAL QUESTIONS OF CONTEMPORARY

AESTHETICS

BY MAX DESSOIB
(Translated from the German by Miss Ethel D. Puffer, Cambridge, Mass.)

[Max Dessoir, Professor of Philosophy, University of Berlin, since 1897. b.
1867, Berlin, Germany. Ph.D. Berlin, 1889; M.D. Wiirzburg, 1892. Pri-
vat-docent, University of Berlin, 1892-97. Member German Psychological
Society, Society for Psychical Research, London. Author of The Double Ego;
History of the New German Psychology ; Philosophical Reader; JEslhetik u. A II-
gemeine Kunstwissenschaft ; and many other works and papers on philosophy.]

IN the development which our science has undergone, from its
inception up to the present day, one thought has held a central
place, — that aesthetic enjoyment and production, beauty and art,
are inseparably allied. The subject-matter of this science is held to
be, though varied, of a unitary character. Art is considered as the
representation of the beautiful, which comes to pass out of an aes-
thetic state or condition, and is experienced in a similar attitude; the
science which deals with these two psychical states, with the beau-
tiful and its modifications, and with art in its varieties, is, inasmuch
as it constitutes a unity, designated by the single name of aesthetics.

The critical thought of the present day is, however, beginning to
question whether the beautiful, the aesthetic, and art stand to one
another in a relation that can be termed almost an identity. The
undivided sway of the beautiful has already been assailed. Since
art includes the tragic and the comic, the graceful and the sublime,
and even the ugly, and since aesthetic pleasure can attach itself to
all these categories, it is clear that by "the beautiful" something
narrower must be meant than the artistically and aesthetically
valuable. Yet beauty might still constitute the end and aim and
central point of art, and it might be that the other categories but
denote the way to beauty — beauty in a state of becoming, as it
were.

But even this view, which sees in beauty the real content of art,
and the central object of aesthetic experiences, is open to serious
question. It is confronted with the fact, above all, that the beauty
enjoyed in life and that enjoyed yi art are not the same. The artist's
copy of the beauty of nature takes on a quite new character. Solid
objects in space become in painting flat pictures, the existent is in
poetry transformed into matter of speech; and in every realm is a

THE FUNDAMENTAL QUESTIONS OF ESTHETICS 435

like metamorphosis. The subjective impression might indeed be sup-
posed to remain the same, in spite of objective differentiations. But
even that is not the case. Living human beauty — an acknowledged
passport for its possessor — speaks to all our senses; it often stirs
sex-feeling in however delicate and scarce conscious a way; it
involuntarily influences our actions. On the other hand, there hangs
about the marble statue of a naked human being an atmosphere
of coolness in which we do not consider whether we are looking
upon man or woman: even the most beauteous body is enjoyed as
sexless shape, like the beauty of a landscape or a melody. To
the aesthetic impression of the forest belongs its aromatic fragrance,
to the impression of tropical vegetation its glowing heat, while
from the enjoyment of art the sensations of the lower senses are
barred. In return for what is lost, as it were, art-enjoyment involves
pleasure in the personality of the artist, and in his power to over-
come difficulties, and in the same way many other elements of pleas-
ure, which are never produced by natural beauty. Accordingly,
what we call beautiful in art must be distinguished from what goes
by that name in life, both as regards the object and the subjective
impression.

Another point, too, appears from our examples. Assuming that
we may call the pure, pleasurable contemplation of actual things
and events aesthetic, — and what reason against it could be adduced
from common usage ? — it is thus clear that the circle of the aesthetic
is wider than the field of art. Our admiring and adoring self-abandon-
ment to nature-beauties bears all the marks of the aesthetic attitude,
and needs for all that no connection with art. Further: in all in-
tellectual and social spheres a part of the productive energy expresses
itself in aesthetic forms; these products, which are not works of art,
are yet aesthetically enjoyed. As numberless facts of daily experience
show us that taste can develop and become effective independently
of art, we must then concede to the sphere of the aesthetic a wider
circumference than that of art.

This is not to maintain that the circle of art is a narrow section of
a large field. On the contrary, the aesthetic principle does not by
any means exhaust the content and purpose of that realm of human
production which taken together we call "art." Every true work of
art is extraordinarily complex in its motives and its effects; it arises
not alone from the free play of aesthetic impulse, and aims at more
than pure beauty — at more than aesthetic pleasure. The desires
and energies in which art is grounded are in no way fulfilled by
the serene satisfaction which is the traditional criterion of the aes-
thetic impression, as of the aesthetic object. In reality the arts
have a function in intellectual and social life, through which they are
closely bound up with all our knowing and willing.

436 AESTHETICS

It is, therefore, the duty of a general science of art to take account
of the broad facts of art in all its relations. ^Esthetics is not capable
of this task, if it is to have a determined, self-complete, and clearly
bounded content. We may no longer obliterate the differences
between the two disciplines, but must rather so sharply separate
them by ever finer distinctions that the really existent connections
become clear. The first step thereto has been taken by Hugo Spitzer.
The relation of earlier to current views is comparable to that between
materialism and positivism. While materialism ventured on a pretty
crude resolution of the spiritual into the corporeal, positivism set
up a hierarchy of forces of nature, whose order was determined
by the relation of dependence. Thus mechanical forces, physico-
chemical processes, the biological and the social-historical groups
of facts, are not traced back each to the preceding by an inner con-
nection, but are so linked that the higher orders appear as dependent
on the lower. In the same way is it now sought to link art methodo-
logically with the aesthetic. Perhaps even more closely, indeed, since
already aesthetics and the science of art often play into each other's
hands, like the tunnel- workers who pierce a mountain from opposite
points, to meet at its centre.

Often it so happens, but not invariably. In many cases research is
carried to an end, quite irrespectively of what is going on in other
quarters. The field is too great, and the interests are too various.
Artists recount their experiences in the process of creation, con-
noisseurs enlighten us as to the technique of the special arts; socio-
logists investigate the social function, ethnologists the origin, of art;
psychologists explore the aesthetic impression, partly by experiment,
partly through conceptual analysis; philosophers expound aesthetic
methods and principles; the historians of literature, music, and
pictorial art have collected a vast deal of material — and the sum
total of these scientific inquiries constitutes the most substantial
though not the greatest part of the published discussions, which,
written from every possible point of view, abound in newspapers and
magazines. " There is left, then, for the serious student, naught but
to resolve to fix a central point somewhere, and thence to find out
a way to deal with all the rest as outlying territory " (Goethe).

Only by the mutual setting of bounds can a united effect be pos-
sible from the busy whirl of efforts. Contradictory and heterogeneous
facts are still very numerous. He who should undertake to construct
thereof a clear intelligible unity of concepts, would destroy the
energy which now proves itself in the encounters, crossing of swords,
and lively controversies of scholars, and would mutilate the fullness
of experience which now expresses itself in the manifold special
researches. System and method signify for us: to be free from one
system and one method.

THE FUNDAMENTAL QUESTIONS OF ESTHETICS 437

II

If we are to consider how we answer to-day the questions put for
scientific consideration as to the facts of aesthetic life and of art, first
of all we must examine the now prevailing theories of aesthetics.
They fall in general into aesthetic objectivism and subjectivism.
By the first collective name we denote the aggregate of all theories
which find the characteristic of their field of inquiry essentially in
the quality and conformation of the object, not in the attitude of
the enjoying subject. This quality of the aesthetically valuable is
most easily characterized by setting it off against reality. Of such
theories, which explain "the beautiful" and art from their relation
to what is given in nature, naturalism stands for the identity of real-
ity and art, while the various types of idealism set forth art as more
than reality, and vice versa, formalism, illusionism, sensualism make
it less than reality.

Inasmuch as naturalism is still defended only by a handful of
artists who write, it would appear almost superfluous to consider it.
But the refutations of it which are still appearing indicate that it
must have some life. And in fact it still exists, partly as a present-
day phenomenon in literature and art, partly as the permanent
conviction of many artists. The naturalistic style testifies to revolt
against forms and notions which are dying out; it therefore only
attains a pure aesthetic interest through the theoretic ground which
is furnished to it. And this rests above all on the testimony of the
artists, who are never weary of assuring us that they immediately
reproduce what is given in perception. Some philosophical concep-
tions also play therein a certain role. The adherents of the doctrine
that only the sense-world is real come as a matter of course to the
demand that art shall hold itself strictly to the given. And what
optimist, who explains the real world as the best of all possible
worlds, can, without a logical weakening, admit a play of imagination
different from the reality.

^Esthetic idealism, too, is borne on general philosophical premises.
However various these are, in this they all agree, that the world is
not exhausted by appearances, but has an ideal content and import,
which finds in the aesthetic and in the field of. art its expression to
sense. Even H. Taine sets to art the task of showing the " dominant
character" of things. The beautiful is therefore something higher
than the chance reality, — the typical as over against the anomalous
natural objects or events. It can then be objectively determined
with reference to its typical and generic quality and in its various
kinds.

Somewhat different is the case of formalism, which to-day scarcely
anywhere sets up to be a complete system of aesthetics, but points

438 ESTHETICS

the way for many special investigations. It seeks the aesthetically
effective in the form, that is, in the relation of parts, which has
in principle nothing to do with the content of the object. Every
clearly perceptible unity in manifoldness is pleasing. As "this ar-
rangement is independent of the material, the aesthetic represents
only a part of reality.

In contrast thereto, illusionism sets the world of art as a whole over
against the whole of reality. Art, we are taught, presents neither a
new aspect of the given nor hidden truth, nor pure form; it is, on the
contrary, a world of appearance only, and is to be enjoyed without
regard to connections in life or any consequences. While we other-
wise consider objects as to how they serve our interests and as to their
place in the actual connection of all things, in the aesthetic experi-
ence this twofold relation is disregarded. Neither what things do
for us, nor what they do for each other, comes in question. Their
reality disappears, and the beautiful semblance comes to its own.
Konrad Lange has given to this theory — especially in the line of
a subjective side, to be later mentioned — its modern form.

Of the nearly-related sensualism, the connoisseur Fiedler and the
sculptor Hildebrand are the recent exponents; Rutgers Marshall
and certain French scholars also lean that way. It is the arts which
fix the transitory element of the sense-image, hold fast the fleeting,
make immortal the perishable, and lend stability and permanence
to all pleasure that is bound up with perception. What does painting
accomplish? Arisen, as it has, out of the demands of the eye, its
sole task is to gain for the undefined form- and color-impressions
of reality a complete and stable existence. The same thing is true
of the other arts, for their respective sense-impressions.

To sum up: If the transformation of reality is acknowledged as
a fundamental principle of art, it is also to be granted that this takes
place in two directions: — art is something at once more and less
than nature. Inasmuch as art pushes on to the vraie vtrite, and at
the same time disregards all that is not of the nature of semblance
or image, we take from it ideas whose quality enthralls and stimu-
lates us quite independently of their meaning. Art shows us the
hidden essence of the world and of life and at the same time the
outsides of things created for our pleasure; that is, the objects'
pure psychical value in the field of sense. It involves a lifting above
nature, and at the same time the rounding out and fulfillment of
sense. Through making of the object an image, it frees us from our
surrounding, yet leaves us at rest in it.

We turn now to aesthetic subjectivism. Under this name we com-
prehend the essence of those theories which seek to solve the riddle
of the beautiful by a general characterization of the aesthetic atti-
tude. Many of these are near akin to the objectivistic theories; some,

THE FUNDAMENTAL QUESTIONS OF AESTHETICS 439

however, like the Einfiihlung-iheory, take an independent place.
For the former, therefore, a mere indication will suffice. The prin-
ciple of "semblance" or illusion, for instance, takes very easily a
subjectivistic turn. The question then runs: Wherein consists the
peculiarity of the conscious processes which are set up by the
semblance? The answer as given by Meinong and Witasek starts
from the fact that the totality of psychical processes falls into two
divisions. Every process in one division has its counterpart in the
other. To perception corresponds imagination, to judgment assump-
tion, to real emotion ideal emotion, to earnest desire fancied desire.
The aesthetic emotions attached to assumptions, the semblance-emo-
tions, that is, are held to be scarcely distinguished, so far as feeling
goes, from other emotions, at most, perhaps, by less intensity. The
chief difference lies rather in the premise or basis of emotion; and
this is but a mere assumption or fiction.

A critical treatment of the foregoing cannot be given here; nor
of that view which explains the psychical condition in receiving an
aesthetic impression as a conscious self-deception, a continued and
intentional confusion of reality and semblance. The aesthetic pleas-
ure, according to this, is a free and conscious hovering between
reality and unreality; or, otherwise expressed, the never successful
seeking for fusion of original and copy. The enjoyment of a good
graphic representation of a globe would then depend on the specta-
tor's now thinking he sees a real globe, now being sure he views a flat
drawing.

While this theory has found but small acceptance, comparatively
many modern ajstheticians admit the doctrine of Einfilhlung. Its
leading exponent, Theodor Lipps, sees the decisive characteristic
of aesthetic enjoyment in the fusion of an alien experience with one's
own: as soon as something objectively given furnishes us the pos-
sibility of freely living ourselves into it, we feel aesthetic pleasure.
In the example of the Doric column, rearing itself and gathering
itself up to our view, Lipps has sought to show how given space-
forms are interpreted first dynamically, then anthropomorphically.
We read into the geometrical figure not only the expression of energy,
but also free purposiveness. In so far as we look at it in the light of
our own activity, and sympathize with it accordingly, in so far do
we feel it as beautiful.

Could we enter upon a critical discussion at this point, it would
appear that the Einfiihlung-iheory, like its fellows, is open to well-
founded objections. The belief in an all-explaining formula is a
delusion. In truth, every one of the enumerated principles is rela-
tively justified. And as they all have points of similarity with one
another, it is not hard for the past-master of terminology and the
technique of concepts to epitomize the common element in a single

440 AESTHETICS

phrase or thesis. Still, nothing is gained by such a general formula
in presence of the richness of the reality; and just as little — as an
exhaustive treatment would have to prove — by the concise ex-
positron of a single method for our science.

The specially approved method of procedure at the present day
is that of psychological description and explanation. It seems,
indeed, very natural to see in psychical processes the real subject-
matter of aesthetics, and in psychology, accordingly, the science to
which it is subordinate. Some philosophers, however, — among
whom I may instance Jonas Cohn, — wish to make of aesthetics a
science of values, and demand that on the basis of this pretension
the mutually contradictory judgments of taste and types of art be
tried and tested. They will have no mere descriptive and explan-
atory aesthetics, but a normative, precept-giving science. In truth,
the opposition of the schools is complete at every point; in the
writings of Volkelt and Groos we have the proof of it.

Ill

The special research in the narrower field of aesthetics is at present
almost entirely of the psychological type. Our survey can touch
upon only the salient points.

The aim of the extended and highly detailed study consists in
fixating by means of psychological analysis the course of develop-
ment, the effective elements, and the various sub-species of the
aesthetic experience. Certain philosophers seek a point of departure
for this undertaking in the aesthetic object. Thus Volkelt's system
of aesthetics finds, for the chief elements of the aesthetic enjoyment,
corresponding features in the object; in the special field of poetry
Dilthey has undertaken an analysis along the same lines. For the
most part, however, such dissection is limited to the subjective side.
In Wundt's psychology, for instance, the aesthetic state of mind is
shown to be built up of sense-feelings, feelings from perceptions, in-
tellectual and emotional excitements; the most important, that is
to say, the pivotal feelings, which are bound up with space- and
time-relations, become in turn the condition and support of the
higher emotions, because they lead over from the field of sense to
that of the logical and emotional.

If we limit ourselves to the psychological, we must first ask in what
order the elements of the aesthetic impression are wont to follow
each other. The phases of this development, however, are as yet not
completely studied, although they are of great significance for the
differences in enjoyment. The second problem concerns the con-
stitution (taken as timeless) of the experience. All formulas which
attempt to fix in two words the totality of the impression fail com-

THE FUNDAMENTAL QUESTIONS OF AESTHETICS 441

pletely, — so extraordinarily various and manifold are the factors
which enter here. What these are and how they are bound together
is the question which is for the moment occupying the scholars with
a leaning toward psychology.

The aesthetic impression is an emotion. According to the well-
known sensualistic theory of the emotions, it must therefore, in so
far as it is more than mere perception or idea, be composed of organic
sensations. G. Sergi and Karl Lange see, in fact, the peculiar mark
of the aesthetic experience in the general sensations which appear
with changes in the circulation, breathing, etc. Unprejudiced ob-
servation must satisfy every one that much in all this is true. On the
other hand, it is to be recalled that we do not reckon the organic
sensations to the objective qualities of aesthetic things, and that we
cannot explain in this way every artistic enjoyment. — In regard to
the sensations of taste, smell, and touch, it is generally granted that
they play a certain role, even if but as reproduced ideas and only
corresponding to natural beauty. Among the most important are
the attitudes and imitative movements, finely investigated by Karl
Groos. — To this must be added the sensuous pleasantness of visual
and auditory perceptions. Yet attempts to construct the aesthetic
enjoyment in its entirety out of such pleasure-factors have so far
failed. The undertaking is already wrecked by the fact that elements
displeasing to sense are demonstrably present, not only as negligible
admixtures, but also as necessary factors. The relations of similarity
between the contents of a sense-field, and the spatial and temporal
connections between them, are in any case incomparably more
important; we devote to them, therefore, a closer consideration.
Finally, alongside all these ideas and the emotions immediately
attaching to them, there must be arrayed the great multitude of as-
sociated ideas and connecting judgments. While scientific interest
in the associations is now greatly diminished, explanations of the
part played by the element of really active thought are many. A
universally satisfactory theory is still to appear, for the reason,
above all, that here the higher principles referred to in the second
section enter into the problems.

Elementary aesthetics, therefore, willingly turns aside from the
shore of the very complex emotions, of association, Einfuhlung and
illusion in aesthetic experience, in order to become independent of
general philosophical fundamental conceptions. Its own field lies
in the general province of the perception-feelings determined im-
mediately by the object: more exactly, of the feelings which are
induced partly by the relations of similarity, partly by the outer
connections of the content, partly by the linking of inner and outer
reference. The qualitative relation of tones and colors arouses the
so-called feelings of harmony; the arrangement in space and time

442 .ESTHETICS

awakes the so-called proportion-feelings; and from the cooperation
of these two arise the so-called aesthetic complication-feelings.

As to the pleasurable tone- and color-combinations, the first are
better known than the second, but even their theoretical interpre-
tation is not well settled. More diligent and successful at the present
time is the research into the proportion-feelings. So far as these
bear upon space-relations, they attach either to the outlines or to the
structure of the forms. The bounding-lines are then pleasing, One
theory holds, when they correspond to the easiest eye-movements, and
in general meet our desire for easy, effortless orientation. Another
doctrine, already referred to, explains their aesthetic value from a
cooperation of general bodily feelings, especially sensations of
breathing and equilibrium. Accurate experiments have not succeeded
in finding a real conformity to law in either the first or the second
direction. In the matter of the structure of forms, symmetry in
the horizontal position, and the proportion of the golden section
in the vertical position, receive especial attention. All those space-
shapes may be called symmetrical, whose halves are of equal value
aesthetically. How these must be constituted, has been studied
from the simplest examples by Miinsterberg and his pupils. The
explanation of the pleasing quality rests on the fact that the spec-
tator feels the contents of the two halves — lines or colors — as light
or heavy, according to the energy expended in the necessary eye-
movements. In the vertical position a proportion pleases (as does also
equality) which is only approximately that of the golden section.
The numerical proportion is, therefore, not the ground of pleasure,
for otherwise those forms which are thus divided would have to be
the absolutely beautiful ones, and the more a division varies from
the exact fraction, the more would it sacrifice in beauty. The ground
of pleasure is rather descried in the fact that in the case of the pleas-
ing divisions the two parts stand out as distinct and clearly character-
ized, while yet unified effect is secured through the larger division.

The temporal ordering of an aesthetic character is that of rhythm.
Concerning the aesthetic object as such — that is, concerning the
metrical forms in music and poetry, the views are still widely at
variance; this is true to a startling degree of poetry, because here
the element, that is to say, the word, is made up of accented and
unaccented syllables, and because the tendency of the logical con-
nections of the content to create unities cannot be done away with.
This state of confusion is so much the more to be regretted as it is
just to the art-forms that the most vivid rhythmical feelings attach.
The psychological investigations of Neumann, Bolton, and others have
nevertheless much advanced our scientific understanding of this
subject. A new point of view has taken its rise from Souriau and
Biicher: the connection of the art-rhythm with work and other

THE FUNDAMENTAL QUESTIONS OF ESTHETICS 443

aspects of life. But the collections of data do not yet render it pos-
sible to settle the question in what manner the rhythm of work,
which runs on automatically, and is controlled by the idea of an end,
goes over into aesthetic rhythm.

The aesthetic complication-feelings are bound up with the products
of the fusion of rhythm and harmony, form and color, rhythm and
form (in the dance). So long as all elements of association are
neglected, three characteristics remain to be noted: an increasing
valuation of the absolute quantity, the building-up of definite
form-qualities (Gestaltqualitateri) , and a reconciliation or harmony
of differences, wherein the quantitative element is wont to be the
unifying, the qualitative element the separating factor. I need not,
however, go any further into investigations so subtle, and even now
merely in their beginnings.

This entire fabric of experience, from which but a few threads
have been drawn out 'to view, can now take on various shadings.
These we refer to as the aesthetic moods, or by a less psychological
name, as the aesthetic categories. The ideally beautiful and the
sublime, the tragic and the ugly, the comic and the graceful, are
the best known among them. Modern science has shown most
interest in the study of the comic and the tragic. According to Lipps
the specific emotion of the comic arises in the disappointing of a
psychical preparation for a strong impression, by the appearance of
a weak one. The pleasurable character of the experience would be
explained by the fact that the surplus of psychical impulse, like every
excess of inner energy, is felt as agreeable. The tragic mood is under-
stood no longer as arising in fear and pity, but in pathos and wonder.
Its objective correlate should not be forced to the standard of a nar-
row ethics. The demand for guilt and expiation is being given up
by progressive thinkers in aesthetics; but the constituents of tragedy
remain fast bound to the realm of harshness, cruelty, and dissonance.

IV

From a period more or less remote there have existed poetics,
musical theory, and the science of art. To examine the presupposi-
tious methods and aims of these disciplines from the epistemological
point of view, and to sum up and compare their most important results ,
is the task of a general science of art; this has besides, in the pro-
blems of artistic creation and the origin of art, and of the classification
of the arts and their social function, certain fields of inquiry that
would otherwise have no definite place. They are worked, indeed,
with remarkable diligence and productiveness. Most to be regretted,
on the other hand, is that so little energy is applied to laying the
epistemological foundation.

The theory of the development of art deals with it both in its

444 .ESTHETICS

individual and its generally human aspect. Concerning the genesis of
the child's understanding of art and impulse to produce it, we learn
most from the studies of his drawings at an early age. Here are to be
noted well-established results of observation, even though as yet
they are few in number. On the other hand, the unfolding of primi-
tive feeling (and of the aesthetic sensibility in general) during the
historical period can be only approximately reconstructed. The
case is somewhat more favorable for our information in regard to
the beginnings of art, especially since it has been systematically
assembled by Ernst Grosse and Yrjo Him. If the conditions of the
most primitive of the races now living in a state of nature can be
taken as identical with those at the beginnings of civilization, the
entire vast material of ethnology can be made use of. We gather
therefrom how close-linked with the useful and the necessary beauty
is, and see clearly that primitive art is thoroughly penetrated by
the purpose of a common enjoyment, and is effective in a social
way; but beyond such general principles one can go only with
hesitation, inasmuch as it seems scarcely possible to us, creatures of
civilization, to fix the boundaries of what is really art there.

There are three conjectures as to objective origin of art. It may
be that the separate arts have developed through variation from one
embryonic state. Or the main arts may have been separate from the
very first, having arisen independently of each other. Finally, there
are middle views, like that of Spencer, according to which poetry,
music, and the dance on the one hand, and writing, painting, and
sculpture on the other, have a common root ; Mobius recognizes
three primitive arts, to which the others are to be traced back. The
solution of this question would be especially important, could one
hope to find Darwin's maxim for all jetiological investigations valid
for our field also — that is, the dictum: What is of like origin is of
like character.

As psychological conditions, from which the artistic activity is
likely first to have arisen, the following functions have been suggested
and maintained, — the play-instinct, imitation, the need for expres-
sion and communication, the sense for order and arrangement, the
impulse to attract others and the opposed impulse to startle others.
Each of these theories of conditions must clearly connect itself with
one or the other of the just-named three theories of art's origin; for
had music, taken in our sense and independently, existed as the orig-
inal art, one could hardly regard imitation as the psychological root
of art. All in all, art and the play-instinct seem most closely linked;
that is also true, moreover, of its development with the child.

I come now to the fundamental problems of artistic creation. It
is they which present the most obstinate difficulties to a thorough
and exact investigation, for experiment and the questionnaire —

THE FUNDAMENTAL QUESTIONS OF AESTHETICS 445

which aims at least at objectivity — are but crude means to the end in
view. At the present day, as earlier, there is no lack of very refined,
penetrating, — nay, brilliant analyses. They have a very superior
value; but this has no special significance for the present status of
the science of aesthetics, and for this reason our survey may omit
much which yet has an interest for individuals.

The influence of heredity and environment on the artist's talent
offers rich material for research. It is conceded, though, that how
the most material and the most spiritual of influences, inherited
disposition and fortune, the chances of descent and of intercourse
with one's fellows, — how all this is fused into a unified personality,
can be established only in individual cases by the biographer. A
second very productive source of material in this field has appeared
in Lombroso's teaching. The days of the most violent controversies
lie behind us. It is the general view that genius and madness are near
allied in their expression, that greatness often breaks forth in ques-
tionable forms; yet the majority perceive an essential difference ; the
genius points onward, the mind diseased harks back; the one has
purposive significance, the other not. After these more introductory
inquiries, the real work begins. It has to show in what points every
gift for art coincides with generally disseminated abilities, and just
where the specific power sets in, which the inartistic person lacks.
Take, for example, the memory. We retain this or that fact without,
in principle, any selection; the remembrance of the artist, on the
contrary, is dissociative — it favors what is needful for its own ends.
The memory of the painter battens on forms and colors, the conscious-
ness of the musician is filled with melodies, the fancy of the poet lives
in verbal images. Also there is, especially with the poet, a peculiar
understanding for human experience. In truth, the fanciful products
of the imagination are but the starting-point for the soul-know-
ledge of the poet. Without going into details we may say that by
such penetrating and delimiting analyses the superficial theory of
inspiration is refuted. Out of date, too, is the notion that the artist
creates by putting things together; on the contrary his fancy has
the whole before the parts, it gives to the world an organism, within
which the members gradually emerge. Finally, the old theory is no
longer held, according to which the work of art is already complete
in the inner man, and afterwards merely brought to light. More
definite explanation is given by the doctrine of the way in which the
artistic creation runs its course,, which Eduard v. Hartmann has
skillfully portrayed.

The distinction, differentiation, and comparison of the special
arts offers opportunity and material for numberless special studies.
Music is here the least fully represented, since it is only exceptionally
that art-philosophers feel a drawing to it. So much the more, how-

446 .ESTHETICS

ever, are they inclined to the study of poetry. They are even begin-
ning to make use, for poetics, of the studies in the modern psychology
of language, since it is acknowledged that language is the essential
element, and thus more than the mere form of expression, of the
poetic art. Th. A. Meyer has thrown an apple of discord into the
question whether the poet's words must, in order to arouse pleasure,
also awake an image. As a matter of fact, the aesthetic value does not
depend on the chance-aroused sense-images, but on the language
itself and the images which belong to it alone; for the most part the
understanding of the words alone is enough to give the reader pleas-
ure in the poetic treatment. In the general theory of the visu-
ally representative arts there are two opposed doctrines. The one
emphasizes the common element, and believes to have found it in
the so-called Fernbild, or distant image; the other seeks salvation
in complete separations — as, for instance, of the so-called Griff el-
kunst. or graphic art, from painting. Only the future can decide
between them.

The existence of the total field of art as an essential factor of hu-
man endeavors involves difficulties which must be removed partly
in the philosophical consideration, partly in law and governmental
practice. The last factor must also be taken account of in theory;
for so long as we do not live in an ideal world, the state will claim
regulation of all activities expressing themselves in it, and so also
of art. In first line it is concerned for art's relation to morality.
Secondly, the social problems arise: does art bind men together,
or part them? does it reconcile or intensify oppositions? is it demo-
cratic or aristocratic? is it a necessity or a luxury? does it further or
reject patriotic, ethical, pedagogical ends? The artistic education of
youth and the race has become a burning question. Ruskin and
Morris have developed from art-critics to critics of the social order,
and Tolstoi has contracted the democratic point of view to the
most extreme degree. With the desire to transform art from the
privilege of the few to the possession of all is, finally, bound up the
wish that art shall emerge from another seclusion — that it shall not
be throned in museums and libraries, in theatres and concert-halls,
but shall mingle with our daily domestic life, and direct and color
every act of the scholar as of the peasant.

A satisfactory decision can be reached only by him who keeps in
view that art presents something extremely complex, and by no
means mere aesthetic form; that, on the other hand, the aesthetic
life is not banished to the sacred circle of the independent arts.
With this conclusion we return to the first words of our reflec-
tions herein presented.

SPECIAL BIBLIOGRAPHY PREPARED BY PROFESSOR
DESSOIR FOR HIS ADDRESS

Thaddeus L. Bolton, Rhythm. Americ. Joum. of Psychol. 1894. vr, 145-238.

Karl Biicher, Arbeit und Rhythmus. 3 Aufl. Leipzig, 1902.

Jonas Cohn, Allgemeine ^Esthetik, Leipzig, 1901.

Wilhelm Dilthey, Die Einbildungshraft des Dichters. Bausterne zu einer Poetik.

In den Zeller gewidmeten Philos. Aufsatzen, Leipzig, 1887.
Konrad Fiedler, Schriften iiber Kunst, Leipzig, 1896.
Karl Groos, Der sesthetische Genuss. Giessen, 1902.
Ernst Grosse, Die Anfange der Kunst, Freiburg und Leipzig, 1894.
Eduard von Hartmann, ^Esthetik, Bd. n, Leipzig, 1887.
Adolf Hildebrand, Das Problem der Form in der bildenden Kunst. 3 Aufl.

Strassburg, 1901

Yrjo Hirn, The Origins of Art, London, 1900. Deutsch, Leipzig, 1904.
Karl Lange, Sinnesgeniisse und Kunstgenuss, Wiesbaden, 1903.
Konrad Lange, Das Wesen der Kunst, 2 Bde., Berlin, 1901.
Theodor Lipps, Raumsesthetik, Leipzig, 1897. — Komik und Humor, Hamburg

und Leipzig, 1898. — Grundlegung der ^Esthetik, Hamburg und Leipzig, 1903.
Cesare Lombroso, L'uomo di genio in rapporto alia psichiatria. Torino, 1889

und ofter, Deutsch, Hamburg, 1890.

H. Rutgers Marshall, ^Esthetic Principles, New York, 1895.
A. Meinong, Ueber Annahmen, Leipzig, 1902.
Th. A. Meyer, Das Stilgesetz der Poesie, Leipzig, 1901.
P. J. Mobius, Ueber Kunst und Kiinstler, Leipzig, 1901.
William Morris, Hopes and Fears for Art, London, 1882. Deutsch, Bd. i, Die

niederen Ktinste. n, Die Kunst des Volkes. Leipzig, 1891.

Hugo Munsterberg, Harvard Psychological Studies. Bd. i, Lancaster, Pa. 1903.
Ernst Neumann, Untersuchungen fur Psychologic und ^Esthetik des Rhythmus

Philos. Studien, herau«g. von W. Wundt, 1894, Bd. x.
John Ruskin, Ausgewahlte Werke. Deutsch, Leipzig, 1900,
G. Sergi, Dolore e Piacere, Milano, 1897.
Paul Souriau, L'esthe'tique du mouvement, Paris, 1889.
Herbert Spencer, Principles of Psychology, Bd. u, London, 1855 und ofter.

Deutsch, Leipzig, 1875, ff.

Hugo Spitzer, Hermann Hettners Kunstphilosophische Anfange, Graz, 1903.
H. Taine, Philosophic de 1'Art. 2 Bde. 7 Aufl. Paris, 1895.
Leo Tolstoj, Was ist Kunst? Deutsch, Berlin, 1892.
Johannes Volkelt, /Esthetische Zeitfragen, Muenchen, 1895. Deutch, Leipzig,

1902-03. — jEsthetik des Tragischen, Munchen, 1897. — System der ^Esthetik,

Bd. i, Munchen, 1905.

Stephan Witasek, Grundziige der allgemeinen ^Esthetik, Leipzig.
Wilheim Wundt, Grundziige der physiologsichen Psychologic, 1904. Bd. in. 5

Aufl. Leipzig, 1903.

SHORT PAPERS

A short paper was contributed by Professor A. D. F. Hamlin, of Columbia
University, on the "Sources of Savage Conventional Patterns." The speaker
said that, in the exhibit of the Department of the Interior, two glass cases displayed
side by side the handiwork of the American Indian of one hundred years ago and
of to-day. In the Fine Arts palace the blankets and basketry of the Navahoes
were shown beside the leather work and other handicrafts of white Americans.
In both instances the contrast between the savage and the civilized work em-
phasizes the fact that civilization tends to stifle or destroy the decorative instinct.
The savage art is spontaneous, instructive, unpremeditated. The work of the
civilized artist is thoughtful, carefully elaborated, intellectual. Among these
peoples both the crafts and the patterns are traditional, and there is little or no
ambition to innovate. The forms and combinations we admire in their work are
the result of long-continued processes of evolution and elimination in which, as in
the world of living organisms, the fittest have survived. The structure of savage
patterns is almost always extremely simple. There are three theories advanced to
account for them: that they were invented out of hand; that they were evolved
out of the technical processes, tools, and materials of primitive industry; that
they are descended from fetish or animistic representations of natural forms.
The first is the common view of laymen; the second was first expressed (though
chiefly with reference to civilized art) by Semper; and the third is widely enter-
tained by anthropologists.

Ths savage instinct for decoration has probably developed from primitive
animism — from that fear of the powers of nature, and that confounding of the
animate and inanimate world which is universally recognized as a primitive
trait. But once awakened in even the slightest degree, it has found exercise in
the operations of primitive industry, and given existence to a long series of repeti-
tive forms produced in weaving, basketry, string-lashing, and carving. The two
classes of patterns thus originated — those derived from the imitation of nature
under fetish ideas, and those derived from technical processes — have invariably
converged, overlapping at last in many forms of decorative art, so that the real
origin of a given pattern may be dual. Myths have invariably arisen to explain
the origin of the technical patterns, which have received magic significance and
names, in accordance with savage tendency to assign magical powers to all visible
or at least to all valued objects: all savage art is talismanic. One ought to be
cautious about dogmatizing as to origins in dealing with savage art, because both
the phenomenon of what I call convergence in ornament evolution, and that of
the myths, poetic faculty, and habit among savages, tend to confuse and obscure
the real origin of the patterns with which they deal. And finally, for the artist
as distinguished from the archaeologist and the theorist, the real lesson of savage
art is not in its origins, but in its products; in the strength, simplicity, admirable
distribution, and high decorative effects of poor and despised peoples. Savage
all-over patterns and Greek carved ornament and decorative sculpture represent
the opposed poles of decorative design, and both are of fundamental value as
objects of study for the designer.

BIBLIOGRAPHY: DEPARTMENT OF PHILOSOPHY

PREPARED THROUGH THE COURTESY OF DR. RALPH BARTON PERRY,
OF HARVARD UNIVERSITY

HISTORY OF PHILOSOPHY

BOUILLIER, F. , Philosophie Cartesienne.
BURNET, J., Early Greek Philosophy.
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EUCKEN, R., Lebensanschauungen der grossen Denker.
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FISCHER, K. , Geschichte der neueren Philosophie.
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HOFFDING, H., Geschichte der neueren Philosophie.
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ROYCE, J. , Spirit of Modern Philosophy.
SIDGWICK, H., History of Ethics.
TURNER, W., History of Philosophy.
UEBERWEG, F., Geschichte der Philosophie.
WEBER, A., Histoire de la philosophic europeenne.
WINDELBAND, W., Geschichte der Philosophie.

Geschichte der alten Philosophie.
ZELLER, E., Geschichte der griechischen Philosophie.

PHILOSOPHICAL CLASSICS

ABELARD, Dialectic.
ANSELM, Monologium.
ARISTOTLE, Metaphysics.
De Anirna.
Physics.

Nicomachean Ethics.
BACON, F., Novum Organum.

BERKELEY, G., The Principles of Human Knowledge.
BRUNO, G., Dialogi, De la Causa Principio et Uno, etc.
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ANAXAGORAS, etc.
DESCARTES, R. , Discours de la Me"thode.

Meditationes de Prima Philosophia.
DUNS SCOTUS, Opus Oxoniense.
FICHTE, J. G., Wissenschaftslehre.
HEGEL, G. W. F., Wissenschaft der Logik.

Encyklopadie.
HOBBES, T., Leviathan.
HUME, D., Enquiry Concerning the Human Understanding.

Enquiry Concerning the Principles of Morals.
KANT, I., Kritik der reinen Vernunft.

Kritik der praktischen Vernunft.
Kritik der Urteilskraft.

450 BIBLIOGRAPHY: DEPARTMENT OF PHILOSOPHY

LEIBNIZ, G. W., Monadologie.

Theodicee.

LOCKE, J., An Essay Concerning Human Understanding.
LOTZE, R. H., Metaphysik.
LUCRETIUS, De Rerum Natura.
PLATO, Republic. Phaedo. Theaetetus. Symposium. Phaedrus. Protagoras

(and other dialogues).
PLOTINUS, Enneades.
ST. AUGUSTINE, De Civitate Dei.
SCHELLING, Philosophie der Natur.
SCHOPENHAUER, A., Die Welt als Wille und Vorstellung.
SPENCER, H., Synthetic Philosophy.
SPINOZA, B., Ethica.
THOMAS AQUINAS, Summa Theologiae.

INTRODUCTION TO PHILOSOPHY

BALDWIN, J. M., Dictionary of Philosophy.

HIBBEN, J. G., Problems of Philosophy.

KULPE, O., Einleitung in die Philosophie.

MARVIN, W. T., Introduction to Philosophy.

PAULSEN, F., Einleitung in die Philosophie.

PERRY, R. B., Approach to Philosophy.

SIDGWICK, H., Philosophy, its Scope and Relations.

STUCKENBERG, J. H. W., Introduction to the Study of Philosophy.

WATSON, J., Outline of Philosophy.

WINDELBAND, W., Praludien.

METAPHYSICS

AVENARIUS, R., Kritik der reinen Erfahrung.

BERGSON, H., Matiere et memoire.

BRADLEY, F. H., Appearance and Reality.

DEUSSEN, P., Elements of Metaphysics.

EUCKEN, R., Der Kampf um einen geistigen Lebensinhalt.

FULLERTON, G. S., System of Metaphysics.

HODGSON, S., Metaphysics of Experience.

HOWISON, G. H., The Limits of Evolution.

JAMES, W., The Will to Believe.

LIEBMANN, Analysis der Wirklichkeit.

ORMOND, A. T., Foundations of Knowledge.

PETZOLDT, J., Philosophie der reinen Erfahrung.

RENOUVIER, C., Les Dilemmes de la me"taphysique pure.

RICKERT, H. , Der Gegenstand der Erkeuntniss.

RIEHL, A., Philosophische Kriticismus.

ROYCE, J., The World and the Individual.

SCHILLER, F. C. S., Humanism.

SETH, A., Man and the Cosmos.

STURT, H. (editor), Personal Idealism.

TAYLOR, A. E., Elements of Metaphysics.

VOLKELT, J., Erfahrung und Denken.

WINDELBAND, W., Praludien.

WUNDT, W., System der Philosophie.

BIBLIOGRAPHY: DEPARTMENT OF PHILOSOPHY 451

PHILOSOPHY OF RELIGION

BOUSSET, W., Das Wesen der Religion, dargestellt in ihrer Geschichte.

CAIRO, E., The Evolution of Religion.

DORNER, A., Religionsphilosophie.

EUCKEN, R., Der Wahrheitsgehalt der Religion.

EVERETT, C. C., The Psychological Elements of Religious Faith.

HARTMANN, VON, E., Religionsphilosophie.

HOFFDING, H., Religionsphilosophie.

JAMES, W., Varieties of Religious Experience.

MARTINEAU, J., A Study of Religion, its Sources and Contents.

MULLER, M., Einleitung in die vergleichende Religionswissenschaft.

PFLEIDERER, O., Religionsphilosophie auf geschichtelichen Grundlage.

RAUENHOFF, Religionsphilosophie.

ROYCE, J., The Religious Aspect of Philosophy.

SABATIER, A., Religionsphilosophie auf psychologischen und geschichtlichen

Grundlage.

SAUSSAYE, Lehrhuch der Religionsgeschichte.
SEYDEL, R., Religionsphilosophie.
TEICHMULLER, G., Religionsphilosophie.
TIELE, C. P., Grundzuge der Religionswissenschaft.

LOGIC

BRADLEY, F. H., The Principles of Logic.

BOSANQUET, B., Logic.

COHEN, H., Die Logik der reinen Erkenntniss.

DEWEY, J., Studies in Logical Theory.

ERDMANN, B., Logik.

HIBBEN, J. G., Logic.

HOBHOUSE, L. T., Theory of Knowledge.

HUSSF.RL, Logische Untersuchungen.

LOTZE, R. H., Grundziige der Logik.

SCHUPPE, W., Erkenntnisstheoretische Logik.

SIGWART, C., Logik.

WUNDT, W., Logik.

METHODOLOGY OF SCIENCE

CANTOR, G. , Grundlagen einer allgemeinen Mannigfaltigkeitslehre.

DEDEKIND, R., Was sind und was sollen die Zahlen?

HERTZ, H., Die Principien der Mechanik.

JEVONS, W. S., Principles of Science.

MACH, E., Die Analyse der Empfmdung.

MUNSTERBERG, H., Grundzuge der Psychologie.

NATORP, P., Einleitung in die Psychologie.

OSTWALD, W., Vorlesungen iiber Naturphilosophie.

PEARSON, K., Grammar of Science.

POINCARE, H., La Science et l'Hypothvse.

RICKERT, H., Die Grenzen der naturwissenschaftlichen Begriffsbildung.

ROYCE, J., The World and the Individual, Second Series.

RUSSELL, B., The Principles of Mathematics.

WARD, J., Naturalism and Agnosticism.

WINDELBAND, W., Geschichte und Naturwissenschaft.

452 BIBLIOGRAPHY: DEPARTMENT OF PHILOSOPHY

ETHICS

ALEXANDER, S., Moral Order and Progress.

BRADLEY, F. H., Ethical Studies.

COHEN, H., Ethik des reinen Willens.

GIZYCKI, G., Grundziige der Moral.

GREEN, T. H., Prolegomena to Ethics.

GUYAU, M. J., Esquisse d'une morale sans obligation ni sanction.

LADD, G. T., Philosophy of Conduct.

MARTINEAU, J., Types of Ethical Theory.

MEZES, S. E., Ethics, Descriptive and Explanatory.

MOORE, G. E., Principia Ethica.

PALMER, G. H., The Nature of Goo«lness.

PAULSEN, F., System der Ethik.

ROYCE, J., Studies of Good and Evil.

SETH, J., Principles of Ethics.

SIDGWICK, H., Methods of Ethics.

SIMMEL, G., Einleitung in die Moralwissenschaft.

SORLEY, W. R., Ethics of Naturalism.

SPENCER, H., Principles of Ethics.

STEPHEN, L., Science of Ethics.

TAYLOR, A. E., The Problem of Conduct.

WUNDT, W., Ethik.

AESTHETICS

COHN, Allgemeine JEsthetik.

GUYAU, M. J., Les Problemes de 1'esth^tique contemporaine.

HIRN, YRJO, The Origins of Art.

LANGE, K., Das Wesen der Kunst.

LIPPS, T., ^Esthetik.

PUFFER, E., Psychology of Beauty.

SOURIAU, P., La Beaute" Rationelle.

VOLKELT, J., System der ^Esthetik.

WITASEK, S., Grundziige der allgemeinen sesthetik.

CHARLES EMILE PI CARD, LL.D.,
Professor of Algebra and Analysis, University of Paris (.Sorbonne).

HEINRICH MASCHKE, Ph.D.,
Associate Professor of Mathematics, University of Chicago.

E. H. MOORE, Ph.D., LL.D.,
Professor of Mathematics. University of Chicago.

DEPARTMENT II — MATHEMATICS

DEPARTMENT II — MATHEMATICS

(Hall 7, September 20, 11.15 a. m.)

CHAIRMAN: PROFESSOR HENRY S. WHITE, Northwestern University.
SPEAKERS: PROFESSOR MAXIME BOCHER, Harvard University.
PROFESSOR JAMES P. PIERPONT, Yale University.

THE Chairman of the Department of Mathematics was Professor
Henry S. White, of Northwestern University. In opening the pro-
ceedings Professor White said:

" Influenced by patriotism and by pride in material progress, cities
and whole nations meet and celebrate the building of bridges, the
opening of long railways, the tunneling of difficult mountain passes,
the acquisition of new territories, or commemorate with festivity the
discovery of a continent. These things are real and significant to us
all.

" In the realm of ideas also there are events of no less moment,
discoveries and conquests that greatly enlarge the empire of human
reason. In the lapse of a century there may be many such notable
achievements, even in the domain of a single science.

" Mathematics is a science continually expanding; and its growth,
unlike some political and industrial events, is attended by universal
acclamation. We are wont to-day, as devotees of this noble and
useful science, to pass in review the newest phases of her expansion,
— I say newest, for in retrospect a century is but brief, — and to
rejoice in the deeds of the past. At the same time, also, we turn
an eye of aspiration and resolution towards the mountains, rivers,
deserts, and the obstructing seas that are to test the mathematicians
of the future."

THE FUNDAMENTAL CONCEPTIONS AND METHODS OF

MATHEMATICS

BY PROFESSOR MAXIME BQCHER

[Maxima Bocher, Professor of Mathematics, Harvard University, b. August 28,
1867, Boston, Mass. A.B. Harvard, 1888; Ph.D. Gottingen, 1891. In-
structor, Assistant Professor and Professor, Harvard University, 1891-.
Fellow of the American Academy. Author of Ueber die Reihenentwickel-
ungen der Potentialtheorie; and various papers on mathematics.]

I. Old and New Definitions of Mathematics

I AM going to ask you to spend a few minutes with me in consider-
ing the question: what is mathematics? In doing this I do not propose
to lay down dogmatically a precise definition ; but rather, after hav-
ing pointed out the inadequacy of traditional views, to determine
what characteristics are common to the most varied parts of mathe-
matics but are not shared by other sciences, and to show how this
opens the way to two or three definitions of mathematics, any one of
which is fairly satisfactory. Although this is, after all, merely a dis-
cussion of the meaning to be attached to a name, I do not think that
it is unfruitful, since its aim is to bring unity into the fundamental
conceptions of the science with which we are concerned. If any of
you, however, should regard such a discussion of the meaning of words
as devoid of any deeper significance, I will ask you to regard this
question as merely a bond by means of which I have found it con-
venient to unite what I have to say on the fundamental conceptions
and methods of what, with or without definition, we all of us agree
to call mathematics.

The old idea that mathematics is the science of quantity, or that
it is the science of space and number, or indeed that it can be charac-
terized by any enumeration of several more or less heterogeneous
objects of study, has pretty well passed away among those mathe-
maticians who have given any thought to the question of what
mathematics really is. Such definitions, which might have been
intelligently defended at the beginning of the nineteenth century,
became obviously inadequate as subjects like projective geometry,
the algebra of logic, and the theory of abstract groups were de-
veloped; for none of these has any necessary relation to quantity
(at least in any ordinary understanding of that word), and the last
two have no relation to space. It is true that such examples have
had little effect on the more or less orthodox followers of Kant,
who regard mathematics as concerned with those conceptions which

CONCEPTIONS AND METHODS OF MATHEMATICS 457

are obtained by direct intuition of time and space without the aid of
empirical observation. This view seems to have been held by such
eminent mathematicians as Hamilton and DeMorgan; and it is a
very difficult position to refute, resting as it does on a purely meta-
physical foundation which regards it as certain that we can evolve
out of our inner consciousness the properties of time and space.
According to this view the idea of quantity is to be deduced from
these intuitions; but one of the facts most vividly brought home to
pure mathematicians during the last half-century is the fatal weak-
ness of intuition when taken as the logical source of our knowledge
of number and quantity.1

The objects of mathematical study, even when we confine our
attention to what is ordinarily regarded as pure mathematics are,
then, of the most varied description; so that, in order to reach a
-satisfactory conclusion as to what really characterizes mathematics,
one of two methods is open to us. On the one hand we may seek
some hidden resemblance in the various objects of mathematical
investigation, and having found an aspect common to them all we
may fix on this as the one true object of mathematical study. Or,
on the other hand, we may abandon the attempt to characterize
mathematics by means of its objects of study, and seek in its methods
its distinguishing characteristic. Finally, there is the possibility of
our combining these two points of view. The first of these methods is
that of Kempe, the second will lead us to the definition of Benjamin
Peirce, while the third has recently been elaborated at great length
by Russell. Other mathematicians have naturally followed out more
or less consistently the same ideas, but I shall nevertheless take the
liberty of using the names Kempe, Peirce, and Russell as convenient
designations for these three points of view. These different methods
of approaching the question lead finally to results which, without
being identical, still stand in the most intimate relation to one an-
other, as we shall now see. Let us begin with the second method.

II. Peirce' s Definition

More than a third of a century ago Benjamin Peirce wrote : 2
Mathematics is the science which draws necessary conclusions. Accord-
ing to this view there is a mathematical element involved in every
inquiry in which exact reasoning is used. Thus, for instance,8 a
jury listening to the attempt of the counsel for the prisoner to prove
an alibi in a criminal case might reason as follows: "If the witnesses

1 I refer here to such facts as that there exist continuous functions without
derivatives, whereas the direct untutored intuition of space would lead any one
to believe that every continuous curve has tangents.

2 Linear Associative Algebra. Lithographed 1870. Reprinted in the American
Journal of Mathematics, vol. iv.

3 This illustration was suggested by the remarks by J. Richard, Sur la philoso-
phie des mathmcatiques. Paris, Gauthier-Villars, 1903, p. 50.

458 MATHEMATICS

are telling the truth when they say that the prisoner was in St. Louis
at the moment the crime was committed in Chicago, and if it is
true that a person cannot be in two places at the same time, it follows
that the prisoner was not in Chicago when the crime was committed."
This, according to Peirce, is a bit of mathematics; while the further
reasoning by which the jury would decide whether or not to believe
the witnesses, and the reasoning (if they thought any necessary)
by which they would satisfy themselves that a person cannot be
in two places at once, would be inductive reasoning, which can give
merely a high degree of probability to the conclusion, but never
certainty. This mathematical element may be, as the example
just given shows, so slight as not to be worth noticing from a prac-
tical point of view. This is almost always the case in the transac-
tions of daily life and in the observational sciences. If, however, we
turn to such subjects as chemistry and mineralogy, we find the
mathematical element of considerable importance, though still
subordinate. In physics and astronomy its importance is much
greater. Finally in geometry, to mention only one other science, the
mathematical element predominates to such an extent that this
science has been commonly rated a branch of pure mathematics,
whereas, according to Peirce, it is as much a branch of applied
mathematics as is, for instance, mathematical physics.

It is clear from what has just been said that, from Peirce's point
of view, mathematics does not necessarily concern itself with quanti-
tative relations, and that any subject becomes capable of mathe-
matical treatment as soon as it has secured data from which import-
ant consequences can be drawn by exact reasoning. Thus, for
example, even though psychologists be right when they assure us
that sensations and the other objects with which they have to deal
cannot be measured, we need still not necessarily despair of one day
seeing a mathematical psychology, just as we already have a math-
ematical logic.

I have said enough, I think, to show what relation Peirce's con-
ception of mathematics has to the applications. Let us then turn
to the definition itself and examine it a little more closely. You
have doubtless already noticed that the phrase, " the science which
draws necessary conclusions, " contains a word which is very much
in need of elucidation. What is a necessary conclusion? Some of
you will perhaps think that the conception here involved is one
about which, in a concrete case at least, there can be no practical
diversity of opinion among men with well-trained minds; and in
fact when I spoke a few minutes ago about the reasoning of the
jurymen when listening to the lawyer trying to prove an alibi, I
assumed tacitly that this is so. If this really were the case, no further
discussion would be necessary, for it is not my purpose to enter into

CONCEPTIONS AND METHODS OF MATHEMATICS 459

any purely philosophical speculations. But unfortunately we can-
not dismiss the matter in this way; for it has happened not infre-
quently that the most eminent men, including mathematicians,
have differed as to whether a given piece of reasoning was exact or
not; and, what is worse, modes of reasoning which seem absolutely
conclusive to one generation no longer satisfy the next, as is shown
by the way in which the greatest mathematicians of the eighteenth
century used geometric intuition as a means of drawing what they
regarded as necessary conclusions.1

I do not wish here to raise the question whether there is such a
thing as absolute logical rigor, or whether this whole conception of
logical rigor is a purely psychological one bound to change with
changes in the human mind. I content myself with expressing the
belief, which I will try to justify a little more fully in a moment,
that as we never have found an immutable standard of logical rigor
in the past, so we are not likely to find it in the future. However
this may be, so much we can say with tolerable confidence, as past
experience shows, that no reasoning which claims to be exact can
make any use of intuition, but that it must proceed from definitely
and completely stated premises according to certain principles of
formal logic. It is right here that modern mathematicians break
sharply with the tradition of a priori synthetic judgments (that is,
conclusions drawn from intuition) which, according to Kant, form an
essential part of mathematical reasoning.

If then we agree that " necessary conclusions " must, in the present
state of human knowledge, mean conclusions drawn according to
certain logical principles from definitely and completely stated
premises, we must face the question as to what these principles
shall be. Here, fortunately, the mathematical logicians from Boole
down to C. S. Peirce, Schroder, and Peano have prepared the field
so well that of late years Peano and his followers 2 have been able
to make a rather short list of logical conceptions and principles upon
which it would seem that all exact reasoning depends.3 We must
remember, however, when we are tempted to put implicit confidence
in certain fundamental logical principles, that, owing to their extreme
generality and abstractness, no very great weight can be attached
to the mere fact that these principles appeal to us as obviously

1 All writers on elementary geometry from Euclid down almost to the close
of the nineteenth century use intuition freely, though usually unconsciously, in
obtaining results which they are unable to deduce from their axioms. The first
few demonstrations of Euclid are criticised from this point of view by Russell in
his Principles of Mathematics, vol. i, 404-407. Gauss's first proof (1799) that
every algebraic equation has a root gives a striking example of the use of intuition
in what was intended as an absolutely rigorous proof by one of the greatest and at
the same time most critical mathematical minds the world has ever seen.

2 And, independently, Frege.

3 It is not intended to assert that a single list has been fixed upon. Different
writers naturally use different lists.

460 MATHEMATICS

true; for, as I have said, other modes of reasoning which are now
universally recognized as faulty have appealed in just this way to
the greatest minds of the past. Such confidence as we feel must,
I think, come from the fact that those modes of reasoning which
we trust have withstood the test of use in an immense number of
cases and in very many fields. This is the severest test to which any
theory can be put, and if it does not break down under it we may
feel the greatest confidence that, at least in cognate fields, it will
prove serviceable. But we can never be sure. The accepted modes
of exact reasoning may any day lead to a contradiction which would
show that what we regard as universally applicable principles are
in reality applicable only under certain restrictions.1

To show that the danger which I here point out is not a purely
fanciful one, it is sufficient to refer to a very recent example. Inde-
pendently of one another, Frege and Russell have built up the theory
of arithmetic from its logical foundations. Each starts with a definite
list of apparently self-evident logical principles, and builds up a
seemingly flawless theory. Then Russell discovers that his logical
principles when applied to a very general kind of logical cZass lead
to an absurdity; and both Frege and Russell have to admit that
something is wrong with the foundations which looked so secure.
Now there is no doubt that these logical foundations will be somehow
recast to meet this difficulty, and that they will then be stronger
than ever before.2 But who shall say that the same thing will not
happen again?

It is commonly considered that mathematics owes its certainty
to its reliance on the immutable principles of formal logic. This,
as we have seen, is only half the truth imperfectly expressed. The
other half would be that the principles of formal logic owe such
degree of permanence as they have largely to the fact that they
have been tempered by long and varied use by mathematicians.
"A vicious circle!" you will perhaps say. I should rather describe
it as an example of the process known to mathematicians as the
method of successive approximations. Let us hope that in this
case it is really a convergent process, as it has every appearance of
being.

But to return to Peirce's definition. From what are these neces-

1 If the view which I here maintain is correct, it follows that if the term " abso-
lute logical rigor" has a meaning, and if we should some time arrive at this abso-
lute standard, the only indication we should ever have of the fact would be that
for a long period, several thousand years let us say, the logical principles in ques-
tion had stood the test of use. But this state of affairs might equally well mean
that during that time the human mind had degenerated, at least with regard to
some of its functions. Consider, for instance, the twenty centuries following Euclid
when, without doubt, the high tide of exact thinking attained during Euclid's gen-
eration had receded.

2 Cf. Poincar<§'s view in La Science et VHypothese, p. 179, according to which
a theory never renders a greater service to science than when it breaks down.

sary conclusions to be drawn? The answer clearly implied is, from
any premises sufficiently precise to make it possible to draw neces-
sary conclusions from them. In geometry, for instance, we have a
large number of intuitions and fixed beliefs concerning the nature
of space: it is homogeneous and isotropic, infinite in extent in every
direction, etc.; but none of these ideas, however clearly defined
they may at first sight seem to be, gives any hold for exact reasoning.
This was clearly perceived by Euclid, who therefore proceeded to
lay down a list of axioms and postulates, that is, specific facts which
he assumes to be true, and from which it was his object to deduce all
geometric propositions. That his success here was not complete
is now well known, for he frequently assumes unconsciously further
data which he derives from intuition; but his attempt was a monu-
mental one.

III. The Abstract Nature of Mathematics

Now a further self-evident point, but one to which attention seems
to have been drawn only during the last few years, is this : since we
are to make no use of intuition, but only of a certain number of
explicitly stated premises, it is not necessary that we should have
any idea what the nature of the objects and relations involved in
these premises is.1 I will try to make this clear by a simple example.
In plane geometry we have to consider, among other things, points and
straight lines. A point may have a peculiar relation to a straight
line which we express by the words, the point lies on the line. Now
one of the fundamental facts of plane geometry is that two points
determine a line, that is, if two points are given, there exists one and
only one line on which both points lie. All the facts that I have just
stated correspond to clear intuitions. Let us, however, eliminate our
intuition of what is meant by a point, a line, a point lying on a line.
A slight change of language will make it easy for us to do this. In-
stead of points and lines, let us speak of two different kinds of objects,
say ^-objects and -S-objects; and instead of saying that a point
lies on a line we will simply say that an .A -object bears a certain
relation R to a 5-object. Then the fact that two points determine
a line will be expressed by saying: If any two J. -objects are given,
there exists one and only one 5-object to which they both bear the
relation R. This statement, while it does not force on us any specific
intuitions, will serve as a basis for mathematical reasoning 2 just as
well as the more familiar statement where the terms points and lines

1 This was essentially Kempe's point of view in the papers to be referred to

Eresently. In the geometric example which follows it was clearly brought out
y H. Wiener: Jahresbericht d. deutschen Mathematiker-Vereinigung, vol. i (1891),
p. 45.

2 In conjunction, of course, with further postulates with which we need not
here concern ourselves.

462 MATHEMATICS

are used. But more than this. Our A. -objects, our B-objects, and our
relation R may be given an interpretation, if we choose, very different
from that we had at first intended.

We may, for instance, regard the A-objects as the straight lines in
a plane, the .^-objects as the points in the same plane (either finite
or at infinity), and when an ^.-object stands in the relation R to a
.B-object, this may be taken to mean that the line passes through the
point. Our statement would then become: Any two lines being given,
there exists one and only one point through which they both pass.
Or we may regard the A-objects as the men in a certain community,
the 5-objects as the women, and the relation of an A-object to a
B-object as friendship. Then our statement would be: In this com-
munity any two men have one, and only one, woman friend in com-
mon.

These examples are, I think, sufficient to show what is meant
when I say that we are not concerned in mathematics with the
nature of the objects and relations involved in our premises, except
in so far as their nature is exhibited in the premises themselves.
Accordingly mathematicians of a critical turn of mind, during the
last few years, have adopted more and more a purely nominalistic
attitude towards the objects and relations involved in mathematical
investigation. This is, of course, not the crude mixture of nominalism
and empiricism of the philosopher Hobbes, whose claim to mathe-
matical fame, it may be said in passing, is that of a circle-squarer.1
The nominalism of the present-day mathematician consists in treating
the objects of his investigation and the relations between them as
mere symbols. He then states his propositions, in effect, in the fol-
lowing form: If there exist any objects in the physical or mental
world with relations among themselves which satisfy the conditions
which I have laid down for my symbols, then such and such facts
will be true concerning them.

It will be seen that, according to Peirce's view, the mathematician
as such is in no wise concerned with the source of his premises or with
their harmony or lack of harmony with any part of the external
world. He does not even assert that any objects really exist which
correspond to his symbols. Mathematics may therefore be truly
said to be the most abstract of all sciences, since it does not deal
directly with reality.2

This, then, is Peirce's definition of mathematics. Its advantages
in the direction of unifying our conception of mathematics and of
assigning to it a definite place among the other sciences are clear.

1 Hobbes practically obtains as the ratio of a circumference to its diameter
the value vTO. Cf. for instance Molesworth's edition of Hobbes's English Works,
vol. vn, p. 431.

2 Cf. the very interesting remarks along this line of C. S. Peirce in The Monist,
vol. vii, pp. 23-24.

CONCEPTIONS AND METHODS OF MATHEMATICS 463

What are its disadvantages? I can see only two. First that, as has
been already remarked, the idea of drawing necessary conclusions
is a slightly vague and shifting one. Secondly, that it lays exclusive
stress on the rigorous logical element in mathematics and ignores
the intuitional and other non-rigorous tendencies which form an
important element in the great bulk of mathematical work concern-
ing which I shall speak in greater detail later.

IV. Geometry an Experimental Science

Some of you will also regard it as an objection that there are
subjects which have almost universally been regarded as branches
of mathematics but are excluded by this definition. A striking
example of this is geometry, I mean the science of the actual space
we live in; for though geometry is, according to Peirce's definition,
preeminently a mathematical science, it is not exclusively so. Until
a system of axioms is established mathematics cannot begin its work.
Moreover, the actual perception of spatial relations, not merely
in simple cases but in the appreciation of complicated theorems, is
an essential element in geometry which has no relation to mathe-
matics as Peirce understands the term. The same is true, to a con-
siderable extent, of such subjects as mechanical drawing and model-
making, which involve, besides small amounts of physics and math-
ematics, mainly non-mathematical geometry. Moreover, although the
mathematical method is the traditional one for arriving at the truth
concerning geometric facts, it is not the only one. Direct appeal to
the intuition is often a short and fairly safe cut to geometric results;
and on the other hand experiments may be used in geometry, just
as they are used every day in physics, to test the truth of a proposi-
tion or to determine the value of some geometric magnitude.1

We must, then, admit, if we hold to Peirce's view, 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 math-
ematical 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. This view, while it has not yet
gained universal recognition, should, I believe, prevail, and geo-
metry be recognized as a science independent of mathematics, just
as psychology is gradually being recognized as an independent
science and not as a branch of philosophy.

The view here set forth, according to which geometry is an ex-
perimental science like physics or chemistry, has been held ever

1 I am thinking of measurements and observations made on accurately con-
structed drawings and models. A famous example is Galileo's determination of
the area of a cycloid by cutting out a cycloid from a metallic sheet and weighing it.

464 MATHEMATICS

since Gauss's time by almost all the leading mathematicians who
have been conversant with non-Euclidean geometry.1 Recently,
however, Poincare1 has thrown the weight of his great authority
against this view,2 claiming that the experiments by which it is
sought to test the truth of geometric axioms are really not geometrical
experiments at all but physical ones, and that any failure of these
experiments to agree with the ordinary geometrical axioms could
be explained by the inaccuracy of the physical laws ordinarily as-
sumed. There is undoubtedly an important element of truth here.
Every experiment depends for its results not merely on the law we
wish to test, but also on other laws which for the moment we assume
to be true. But, if we prefer, we may, in many cases, assume as
true the law we were before testing and our experiment will then
serve to test some of the remaining laws. If, then, we choose to stick
to the ordinary Euclidean axioms of geometry in spite of what any
future experiments may possibly show, we can do so, but at the cost,
perhaps, of our present simple physical laws, not merely in one
branch of physics but in several. Poincare^s view 3 is that it will
always be expedient to preserve simple geometric laws at all costs.,
an opinion for which I fail to see sufficient reason.

V. Kempe's Definition

Let us now turn from Peirce's method of defining mathematics to
Kempe's, which, however, I shall present to you in a somewhat
modified form.4 The point of view adopted here is to try to define
mathematics, as other sciences are defined, by describing the objects
with which it deals. The diversity of the objects with which mathe-
matics is ordinarily supposed to deal being so great, the first stcjp
must be to divest them of what is unessential for the mathematical
treatment, and to try in this way to discover their common and
characteristic element.

The first point on which Kempe insists is that the objects of mathe-
matical discussion, whether they be the points and lines of geometry,
the numbers real or complex of algebra or analysis, the elements of
groups or anything else, are always individuals, infinite in number
perhaps, but still distinct individuals. In a particular mathematical
investigation we may, and usually do, have several different kinds of
individuals; as for instance, in elementary plane geometry, points,
straight lines, and circles. Furthermore, we have to deal with certain
relations of these objects to one another. For instance, in the example

1 Gauss, Riemann, Helmholtz are the names which will carry perhaps the
greatest weight.

2 Cf. La Science et THypothese. Paris, 1903.
* L. c., chapter v. In particular, p. 93.

« Kempe has set forth his ideas in rather popular form in the Proceedings of
the London Mathematical Society, vol. xxvi (1894), p. 5; and in Nature, vol. XLIII
1890), p. 156, where references to his more technical writings will be found.

CONCEPTIONS AND METHODS OF MATHEMATICS 465

just cited, a given point may or may not lie on a given line; a given
line may or may not touch a given circle; three or more points may
or may not be collinear, etc. This example shows how in a single
mathematical problem a large number of relations may be involved,
relations some of which connect two objects, others three, etc.
Moreover these relations may connect like or they may connect
unlike objects; and finally the order in which the objects are taken
is not by any means immaterial in general, as is shown by the relation
between three points which states that the third is collinear with and
lies between the first two.

But even this is not all; for, besides these objects and relations
of various kinds, we often have operations by which objects can be
combined to yield another object, as, for instance, addition or multi-
plication of numbers. Here the objects combined and the resulting
object are all of the same kind, but this is by no means necessary.
We may, for instance, consider the operation of combining two
points and getting the perpendicular bisector of the line connecting
them; or we may combine a point and a line and get the perpen-
dicular dropped from the point on the line.

These few examples show how diverse the relations and operations,
as well as the objects of mathematics, seem at first sight to be. Out
of this apparent diversity it is not difficult to obtain a very great
uniformity by simply restating the facts in a little different language.
We shall find it convenient to indicate that the objects a, b, c, . . . ,
taken in the order named, satisfy a relation R by simply writing
R(a. b, c, . . . ), where it should be understood that among the
objects a, b. c, . . . the same object may occur a number of times.
On the other hand, if two objects a and b are combined to yield
a third object c, we may write a o b = c,1 where the symbol o is
characteristic of the special operation with which we are concerned.

Let us first notice that the equation aob=c denotes merely
that the three objects a, b, c bear a certain relation to one another,
say R(a, b, c). In other words the idea of an operation or law of
combination between the objects we deal with, however convenient
and useful it may be as a matter of notation, is essentially merely
a way of expressing the fact that the objects combined bear a certain
relation to the object resulting from their combination. Accordingly,
in a purely abstract discussion like the present, where questions of
practical convenience are not involved, we need not consider such
rules of combination.2

1 I speak here merely of dyadic operations, — i. e., of operations by which
two objects are combined to yield a third, — these being by far the most import-
ant as well as the simplest. What is said, however, obviously applies to opera-
tions by which any number of objects are combined.

2 Even from the point of view of the technical mathematician it may some-
times be desirable to adopt the point of view of a relation rather than that of an
operation. This is seen, for instance, in laying down a system of postulates for the

466 MATHEMATICS

Furthermore, it is easy to see that when we speak of objects of
different kinds, as, for instance, the points and lines of geometry, we
are introducing a notion which can very readily be expressed in our
relational notation. For this purpose we need merely to introduce
a further relation which is satisfied by two or more objects when and
only when they are of the same "kind."

Let us turn finally to the relations themselves. It is customary
to distinguish here between dyadic relations, triadic relations, etc.,
according as the relation in question connects two objects, three
objects, etc. There are, however, relations which may connect any
number of objects, as, for instance, the relation of collinearity which
may hold between any number of points. Any relation holds for
certain ordered groups of objects but not for others, and it is in no
way necessary for us to fix our attention on the fact, if it be true,
that the number of objects in all the groups for which a particular
relation holds is the same. This is the point of view we shall adopt,
and we shall relegate the property that a relation is dyadic, triadic,
etc., to the background along with the various other properties
relations may have,1 all of which must be taken account of in the
proper place.

We are thus concerned in any mathematical investigation, from
our present point of view, with just two conceptions: first a set, or
as the logicians say, a class of objects a, b, c, . . .; and secondly a
class of relations R, S, T , . . . . We may suppose these objects
divested of any qualitative, quantitative, spatial, or other attributes
which they may have had, and regard them merely as satisfying or not
satisfying the relations in question, where, again, we are wholly
indifferent to the nature which these relations originally had. And
now we are in a position to state what I conceive to be really the
essential point in Kempe's definition of mathematics; although I
have omitted one of the points on which he insists most strongly,2
by saying:

If we have a certain class of objects and a certain class of relations,
and if the only questions which we investigate are whether ordered
groups of these objects do or do not satisfy the relations, the results
of the investigation are called mathematics.

theory of abstract groups (cf., for example, Huntington, Bulletin of the Ameri-
can Mathematical Society, June, 1902), where the postulate:
If a and b belong to the class, a o b belongs to the class,

which in this form looks indecomposable, immediately breaks up, when stated in
the relational form, into the following two:

1. If a and b belong to the class, there exists an element c of the class such that
R(a, b, c).

2. If a, b, c, d belong to the class, and if R(a, b, c) and R(a, b, d), then c = d.

1 For instance, the property of symmetry. A relation is said to be symmetrical
if it holds or fails to hold independently of the order in which the objects are taken.

2 Namely, that the only relation that need be considered is that of being " in-
distinguishable," t. e., a symmetrical and transitive relation between two groups
of objects.

CONCEPTIONS AND METHODS OF MATHEMATICS 467

It is convenient to have a term to designate a class of objects
associated with a class of relations between these objects. Such an
aggregate we will speak of as a mathematical system. If now we have
two different mathematical systems, and if a one-to-one correspond-
ence can be set up between the two classes of objects, and also
between the two classes of relations in such a way that whenever
a certain ordered set of objects of the first system satisfies a relation
of that system, the set consisting of the corresponding objects of the
second system satisfies the corresponding relation of that system,
and vice versa, then it is clear that the two systems are, from our
present point of view, mathematically equivalent, however different
the nature of the objects and relations may be in the two cases.1 To
use a technical term, the two systems are simply isomorphic.2

It will be noticed that in the definition of mathematics just given
nothing is said as to the method by which we are to ascertain whether
or not a given relation holds between the objects of a given set. The
method used may be a purely empirical one, or it may be partly or
wholly deductive. Thus, to take a very simple case, suppose our class
of objects to consist of a large number of points in a plane and sup-
pose the only relation between them with which we are concerned
is that of collinearity. Then, if the points are given us by being
marked in ink on a piece of white paper, we can begin by taking three
pins, sticking them into the paper at three of the points; then, by
sighting along them, we can determine whether or not these points
are collinear. We can do the same with other groups of three
points, then with all groups of four points, etc. The same result
can be obtained with much less labor if we make use of certain
simple properties which the relation of collinearity satisfies, pro-
perties which are expressed by such propositions as:

R(a, b, c) implies R(b, a, c),

R(a, b, c, d) implies R(a, b, c),

R(a, b, c) and R(a, b, d) together imply R(a, b, c, d), etc.

By means of a small number of propositions of this sort it is easy
to show that no empirical observations as to the collinearity of
groups of more than three points need be made, and that it may
not be necessary to examine even all groups of three points. Having

1 The point of view here brought out, including the term isomorphism, was
first developed in a special case, — the theory of groups.

2 Inasmuch as the relations in a mathematical system are themselves objects,
we may, if we choose, take our class of objects so as to include these relations as
well as what we called objects before, some of which, we may remark in passing,
may themselves be relations. Looked at from this point of view, we need one
additional relation which is now the only one which we explicitly call a relation.
If we denote this relation by inclosing the objects which satisfy it in parentheses,
then if the relation denoted before by R(a, b} is satisfied, we should now write
(R, a, 6), whereas we should not have (a, R, b) (S, R, a, b), etc. Thus we see that
any mathematical system may be regarded as consisting of a class of objects and
a single relation between them.

468 MATHEMATICS

made this relatively small number of observations, the remaining
results would be obtained deductively. Finally, we may suppose
the points given by their coordinates, in which case the complete
answer to our question may be obtained by the purely deductive
method of analytic geometry.

According to the modified form of Kempe's definition which I
have just stated, mathematics is not necessarily a deductive science.
This view, while not in accord with the prevailing ideas of mathe-
maticians, undoubtedly has its advantages as well as its dangers.
The non-deductive processes, of which I shall have more to say
presently, play too important a part in the life of mathematics to
be ignored, and the definition just given has the merit of not exclud-
ing them. It would seem, however, that the definition in the form
just given is too broad. It would include, for instance, the deter-
mination by experimental methods of what pairs of chemical com-
pounds of the known elements react on one another when mixed
under given conditions.

VI. Axioms and Postulates. Existence Theorems

If, however, we restrict ourselves to exact or deductive mathe-
matics, it will be seen that Kempe's definition becomes coextensive
with Peirce's. Here, in order to have a starting-point for deductive
reasoning, we must assume a certain number of facts or primitive
propositions concerning any mathematical system we wish to study,
of which all other propositions will be necessary consequences.1
We touch here on a subject whose origin goes back to Euclid and
which has of late years received great development, primarily at
the hands of Italian mathematicians.2

It is important for us to notice at this point that not merely these
primitive propositions but all the propositions of mathematics may
be divided into two great classes. On the one hand, we have pro-
positions which state that certain specified objects satisfy certain
specified relations. On the other hand are the existence theorems.
which state that there exist objects satisfying, along with certain
specified objects, certain specified relations.3 These two classes of
propositions are well known to logicians and are designated by them

1 These primitive propositions may be spoken of as axioms or postulates, ac-
cording to the point of view we wish to take concerning their source, the word
axiom, which has been much misused of late, indicating an intuitional or empirical
source.

2 Peano, Fieri, Padoa, Burali-Forti. We may mention here also Hilbert, who,
apparently without knowing of the important work of his Italian predecessors,
has also done valuable work along these lines.

* Or we might conceivably have existence theorems which state that there
exist relations which are satisfied by certain specified objects; or these two kinds
of existence theorems might be combined. If we take the point of view explained
in the second footnote on p. 467, all existence theorems will be of the type men-
tioned in the text.

CONCEPTIONS AND METHODS OF MATHEMATICS 460

universal and particular propositions respectively.1 It is only during
the last fifty years or so that mathematicians have become conscious
of the fundamental importance in their science of existence theorems,
which until then they had frequently assumed tacitly as they needed
them, without always being conscious of what they were doing.

It is sometimes held by non-mathematicians that if mathematics
were really a purely deductive science, it could not have gained
anything like the extent which it has without losing itself in trivial-
ities and becoming, as Poincare puts it, a vast tautology.2 This
view would doubtless be correct if all primitive propositions were
universal propositions. One of the most characteristic features of
mathematical reasoning, however, is the use which it makes of aux-
iliary elements. I refer to the auxiliary points and lines in proofs
by elementary geometry, the quantities formed by combining in
various ways the numbers which enter into the theorems to be
proved in algebra, etc. Without the use of such auxiliary elements
mathematicians would be incapable of advancing a step; and
whenever we make use of such an element in a proof, we are in reality
using an existence theorem.3 These existence theorems need not,
to be sure, be among the primitive propositions; but if not, they must
be deduced from primitive propositions some of which are existence
theorems, for it is clear that an existence theorem cannot be deduced
from universal propositions alone.4 Thus it may fairly be said that
existence theorems form the vital principle of mathematics, but these
in turn, it must be remembered, would be impotent without the
material basis of universal propositions to work upon.

VII. Russell's Definition

We have so far arrived at the view that exact mathematics is
the study by deductive methods of what we have called a mathe-
matical system, that is, a class of objects and a class of relations
between them. If we elaborate this position in two directions we
shall reach the standpoint of Russell.5

In the first place Russell makes precise the term deductive method

"All men are mortals" is a standard example of a universal proposition;
while as an illustration of a particular proposition is often given: "Some men are
Greeks. " That this is really an existence theorem is seen more clearly when we
state it in the form: "There exists at least one man who is a Greek."

2 Cf. La Science et I'Hypothese, p. 10.

5 Even when in algebra we consider the sum of two numbers a + b, we are using
the existence theorem which says that, any two numbers a and b being given,
there exists a number c which stands to them in the relation which we indicate in
ordinary language by saying that c is the sum of a and b.

4 The power which resides in the method of mathematical induction, so called,
comes from the fact that this method depends on an existence theorem. It is,
however, not the only fertile principle in mathematics as Poincare' would have
us believe (cf. La Science et I'Hypothese). In fact there are great branches of
mathematics, like elementary geometry, in which it takes little or no part.

8 The Principles of Mathematics, Cambridge, England, 1903.

470 MATHEMATICS

by laying down explicitly a list of logical conceptions and prin-
ciples which alone are to be used; and, secondly, he insists,1 on the
contrary, that no mathematical system, to use again the technical
term introduced above, be studied in pure mathematics whose exist-
ence cannot be established solely from the logical principles on which
all mathematics is based. Inasmuch as the development of mathemat-
ics during the last fifty years has shown that the existence of most,
if not all the mathematical systems which have proved to be im-
portant can be deduced when once the existence of positive integers
is granted, the point about which interest must centre here is the
proof, which Russell attempts, of the existence of this latter sys-
tem.2 This proof will necessarily require that, among the logical
principles assumed, existence theorems be found. Such theorems
do not seem to be explicitly stated by Russell, the existence theorems
which make their appearance further on being evolved out of some-
what vague philosophical reasoning. There are also other reasons,
into which I cannot enter here, why I am not able to regard the
attempt made in this direction by Russell as completely successful.3
Nevertheless, in view of the fact that the system of finite positive
integers is necessary in almost all branches of mathematics (we
cannot speak of a triangle or a hexagon without having the numbers
three and six at our disposal), it seems extremely desirable that the
system of logical principles which we lay at the foundation of all
mathematics be assumed, if possible, broad enough so that the
existence of positive integers — at least finite integers — follows from
it; and there seems little doubt that this can be done in a satisfactory
manner. When this has been done we shall perhaps be able to regard,
with Russell, pure mathematics as consisting exclusively of deduc-
tions "by logical principles from logical principles."

VIII. The Non-Deductive Elements in Mathematics

I fear that many of you will think that what I have been saying
is of an extremely one-sided character, for I have insisted merely on
the rigidly deductive form of reasoning used and the purely abstract
character of the objects considered in mathematics. These, to the
great majority of mathematicians, are only the dry bones of the
science. Or, to chang3 the simile, it may perhaps be said that instead
of inviting you to a feast I have merely shown you the empty dishes

' In the formal definition of mathematics at the beginning of the book this is
not stated or in any way implied; and yet it comes out so clearly throughout
the book that this is a point of view which the author regards as essential, that
I have not hesitated to include it as a part of his definition.

1 Cf. also Burali-Forti, Congres Internationale de philosophic. Paris, vol. in,
p. 289.

* Russell's unequivocal repudiation of nominalism in mathematics seems to
me a serious if not in insurmountable barrier to progress.

CONCEPTIONS AND METHODS OF MATHEMATICS 471

and explained how the feast would be served if only the dishes were
filled.1 I fully agree with this opinion, and can only plead in excuse
that my subject was the fundamental conceptions and methods of
mathematics, not the infinite variety of detail and application
which give our science its real vitality. In fact I should like to
subscribe most heartily to the view that in mathematics, as else-
where, the discussion of such fundamental matters derives its interest
mainly from the importance of the theory of which they are the
so-called foundations.2 I like to look at mathematics almost more
as an art than as a science; for the activity of the mathematician,
constantly creating as he is, guided though not controlled by the
external world of the senses, bears a resemblance, not fanciful I
believe but real, to the activity of an artist, of a painter let us say.
Rigorous deductive reasoning on the part of the mathematician
may be likened here to technical skill in drawing on the part of the
painter. Just as no one can become a good painter without a certain
amount of this skill, so no one can become a mathematician without
the power to reason accurately up to a certain point. Yet these
qualities, fundamental though they are, do not make a painter or
a mathematician worthy of the name, nor indeed are they the most
important factors in the case. Other qualities of a far more subtle
sort, chief among which in both cases is imagination, go to the
making of the good artist or good mathematician. I must content
myself merely by recalling to you this somewhat vague and difficult
though interesting field of speculation which arises when we attempt
to attach value to mathematical work, a field which is familiar
enough to us all in the analogous case of artistic or literary criticism.
We are in the habit of speaking of logical rigor and the considera-
tion of axioms and postulates as the foundations on which the superb
structure of modern mathematics rests; and it is often a matter of
wonder how such a great edifice can rest securely on such a small
foundation. Moreover, these foundations have not always seemed so
secure as they do at present. During the first half of the nineteenth
century certain mathematicians of a critical turn of mind — Cauchy,
Abel, Weierstrass, to mention the greatest of them — perceived to
their dismay that these foundations were not sound, and some of the
best efforts of their lives were devoted to strengthening and improv-
ing them. And yet I doubt whether the great results of mathematics

1 Notice that just as the empty dishes could be filled by a great variety of
viands, so the empty symbols of mathematics can be given meanings of the most
varied sorts.

2 Cf. the following remark by Study, Jahrcsbericht der deutschcn Mathematiker-
Vereinigung, vol. xi (1902), p. 313:

" So wertvoll auch Untersuchungen liber die systematische Stellung der math-
ematischen Grundbegriffe sind . . . wertvoller ist doch noch der materielle Inhalt
der einzelnen Disciplinen, um dessentwillen allein ja derartige Untersuchungen
liberhaupt Zweck haben. ..."

472 MATHEMATICS

seemed less certain to any of them because of the weakness they
perceived in the foundations on which these results are built up.
The fact is that what we call mathematical rigor is merely one of
the foundation stones of the science; an important and essential
one surely, yet not the only thing upon which we can rely. A science
which has developed along such broad lines as mathematics, with
such numerous relations of its parts both to one another and to other
sciences, could not long contain serious error without detection.
This explains how, again and again, it has come about, that the
most important mathematical developments have taken place by
methods which cannot be wholly justified by our present canons of
mathematical rigor, the logical "foundation" having been supplied
only long after the superstructure had been raised. A discussion
and analysis of the non-deductive methods which the creative
mathematician really uses would be both interesting and instructive.
Here I must content myself with the enumeration of a few of them.

First and foremost there is the use of intuition, whether geometrical,
mechanical, or physical. The great service which this method has
rendered and is still rendering to mathematics both pure and applied
is so well known that a mere mention is sufficient.

Then there is the method of experiment; not merely the physical
experiments of the laboratory or the geometrical experiments I
had occasion to speak of a few minutes ago, but also arithmetical
experiments, numerous examples of which are found in the theory
of numbers and in analysis. The mathematicians of the past fre-
quently used this method in their printed works. That this is now
seldom done must not be taken to indicate that the method itself is
not used as much as ever.

Closely allied to this method of experiment is the method of
analogy, which assumes that something true of a considerable num-
ber of cases will probably be true in analogous cases. This is, of
course, nothing but the ordinary method of induction. But in mathe-
matics induction may be employed not merely in connection with
the experimental method, but also to extend results won by deduct-
ive methods to other analogous cases. This use of induction has
often been unconscious and sometimes overbold, as, for instance,
when the operations of ordinary algebra were extended without
scruple to infinite series.

Finally there is what may perhaps be called the method of optim-
ism, which leads us either willfully or instinctively to shut our eyes
to the possibility of evil. Thus the optimist who treats a problem in
algebra or analytic geometry will say, if he stops to reflect on what
he is doing: "1 know that 1 have no right to divide by zero; but
there are so many other values which the expression by which I am
dividing might have that I will assume that the Evil One has not

CONCEPTIONS AND METHODS OF MATHEMATICS 473

thrown a zero in my denominator this time." This method, if a pro-
ceeding often unconscious can be called a method, has been of great
service in the rapid development of many branches of mathematics,
though it may well be doubted whether in a subject as highly devel-
oped as is ordinary algebra it has not now survived its usefulness.1
While no one of these methods can in any way compare with
that of rigorous deductive reasoning as a method upon which to
base mathematical results, it would be merely shutting one's eyes
to the facts to deny them their place in the life of the mathematical
world, not merely of the past but of to-day. There is now. and there
always will be room in the world for good mathematicians of every
grade of logical precision. It is almost equally important that the
small band whose chief interest lies in accuracy and rigor should
not make the mistake of despising the broader though less accurate
work of the great mass of their colleagues; as that the latter should
not attempt to shake themselves wholly free from the restraint the
former would put upon them. The union of these two tendencies
in the same individuals, as it was found, for instance, in Gauss and
Cauchy, seems the only sure way of avoiding complete estrangement
between mathematicians of these two types.

1 Cf. the very suggestive remarks by Study, Jahresbericht d. Deutschen Math-
ematiker-Vereinigung, vol. xi (1902), p. 100, footnote, in which it is pointed out
how rigor, in cases of this sort, may not merely serve to increase the correctness of
the result, but actually to suggest new fields for mathematical investigation.

THE HISTORY OF MATHEMATICS IN THE NINETEENTH

CENTURY

BY PROFESSOR JAMES P. PIERPONT OF YALE UNIVERSITY

THE extraordinary development of mathematics in the last century
is quite unparalleled in the long history of this most ancient of
sciences. Not only have those branches of mathematics which were
taken over from the eighteenth century steadily grown, but entirely
new ones have sprung up in almost bewildering profusion, and
many of these have promptly assumed proportions of vast extent.

As it is obviously impossible to trace in the short time allotted to
me the history of mathematics in the nineteenth century even in
merest outline, I shall restrict myself to the consideration of some
of its leading theories.

Theory of Functions of a Complex Variable

Without doubt one of the most characteristic features of mathe-
matics in the last century is the systematic and universal use of the
complex variable. Most of its great theories received invaluable aid
from it, and many owe their very existence to it. What would the
theory of differential equations or elliptic functions be to-day without
it, and is it probable that Poncelet, Steiner, Chasles, and von Staudt
would have developed synthetic geometry with such elegance and
perfection without its powerful stimulus?

The necessities of elementary algebra kept complex numbers
persistently before the eyes of every mathematician. In the eight-
eenth century the more daring, as Euler and Lagrange, used them
sparingly; in general one avoided them when possible. Three events,
however, early in the nineteenth century changed the attitude of
mathematicians toward this mysterious guest. In 1813 Argand
published his geometric interpretation of complex numbers. In
1824 came the discovery by Abel of the imaginary period of the
elliptic function. Finally Gauss in his second memoir on biquadratic
residues (1832) proclaims them a legitimate and necessary element
of analysis.

The theory of function of a complex variable may be said to have
had its birth when Cauchy discovered his integral theorem

ff(x)dx=Q

published in 1825. In a long series of publications beginning with
the Cows d> Analyse (1821), Cauchy gradually developed his theory
of functions and applied it to problems of the most diverse nature;

MATHEMATICS IN THE NINETEENTH CENTURY 475

for example, existence theorems for implicit functions and the solu-
tions of certain differential equations, the development of functions
in infinite series and products, and the periods of integrals of one
and many valued functions.

Meanwhile Germany is not idle; Weierstrass and Riemann de-
velop Cauchy's theory along two distinct and original paths. Weier-
strass starts with an explicit analytical expression, a power series,
and defines his function as the totality of its analytical continua-
tions. No appeal is made to geometric intuition, his entire theory
is strictly arithmetical. Riemann growing up under Gauss and
Dirichlet not only relies largely on geometric intuition, but he also
does not hesitate to impress mathematical physics into his service.
Two noteworthy features of his theory, are the many leaved surfaces
named after him, and the extensive use of conformal representation.

The history of functions as first developed is largely a theory of
algebraic functions and their integrals. A general theory of func-
tions is only slowly evolved. For a long time the methods of Cauchy,
Riemann, and Weierstrass were cultivated along distinct lines by
their respective pupils. The schools of Cauchy and Riemann were
the first to coalesce. The entire rigor which has recently been im-
parted to their methods has removed all reason for founding, as
Weierstrass and his school have urged, the theory of functions on
a single algorithm, namely, the power series. We may therefore say
that at the close of the century there is only one theory of functions
in which the ideas of its three great creators are harmoniously united.

Let us note briefly some of its lines of advance. Weierstrass early
observed that an analytic expression might represent different
analytic functions in different regions. Associated with this is the
phenomenon of natural boundaries. The question therefore arose,
What is the most general domain of definition of an analytic function?
Runge has shown that any connected region may serve this purpose.
An important line of investigation relates to the analytic expression
of a function by means of infinite series, products, and fractions.
Here may be mentioned Weierstrass *s discovery of prime factors;
the theorems of Mittag-Leffler and Hilbert; Poincar4's uniform-
ization of algebraic and analytic functions by means of a third
variable, and the work of Stieljes, Fade", and Van Vleck on infinite
fractions. Since an analytic function is determined by a single
power series, which in general has a finite circle of convergence, two
problems present themselves: determine, first, the singular points of
the analytic function so defined, and, second, an analytic expression
valid for its whole domain of definition. The celebrated memoir of
Hadamard inaugurated a long series of investigations on the first
problem; while Mittag-Leffler 's star theorem is the most important
result yet obtained relating to the second.

476 MATHEMATICS

Another line of investigation relates to the work of Poincare",
Borel, Fade", et al., on divergent series. It is, indeed, a strange vicissi-
tude of our science that these series which early in the century
were supposed to be banished once and for all from rigorous mathe-
matics should at its close be knocking at the door for readmission.

Let us finally note an important series of memoirs on integral
transcendental functions, beginning with Weierstrass, Laguerre, and
Poincare*.

Algebraic Functions and their Integrals

A branch of the theory of functions has been developed to such
an extent that it may be regarded as an independent theory; we
mean the theory of algebraic functions and their integrals. The
brilliant discoveries of Abel and Jacobi in the elliptic functions from
1824 to 1829 prepared the way for a similar treatment of the hyper-
elliptic case. Here a difficulty of gravest nature was met. The cor-
responding integrals have 2p linearly independent periods; but as
Jacobi had shown, a one valued function having more than two
periods admits a period as small as we choose. It therefore looked
as if the elliptic functions admitted no further generalization.
Guided by Abel's theorem, Jacobi at last discovered the solution to
the difficulty (1832); to get functions analogous to the elliptic func-
tions we must consider functions not of one but of p independent
variables, namely, the p independent integrals of the first species.
The great problem now before mathematicians, known as Jacobi 's
Problem of Inversion, was to extend this apergu to the case of any
algebraic configuration and develop the consequences. The first to
take up this immense task were Weierstrass and Riemann, whose
results belong to the most brilliant achievements of the century.
Among the important notions hereby introduced we note the fol-
lowing: the birational transformation, rank of an algebraic con-
figuration, class invariants, prime functions, the theta and multiply
periodic functions in several variables. Of great importance is
Riemann 's method of proving existence theorems, as also his repre-
sentation of algebraic functions by means of integrals of the second
species.

A new direction was given to research in this field by Clebsch, who
considered the fundamental algebraic configuration as defining a
curve. His aim was to bring about a union of Riemann 's ideas and
the theory of algebraic curves for their mutual benefit. Clebsch's
labors were continued by Brill and Nother; in their work the tran-
scendental methods of Riemann are placed quite in the background.
More recently Klein and his school have sought to unite the tran-
scendental methods of Riemann with the geometric direction begun
by Clebsch, making systematic use of homogeneous coordinates and

MATHEMATICS IN THE NINETEENTH CENTURY 477

the invariant theory. Noteworthy, also, is his use of normal curves
in (p — 1) way space, to represent the given algebraic configuration.
Dedekind and Weber, Hensel and Landsberg, have made use of the
ideal theory with marked success. Many of the difficulties of the
older theory, e. g., the resolution of singularities of the algebraic
configuration, are treated with a truly remarkable ease and generality.
In the theory of multiply periodic functions and the general &
functions we mention, besides Weierstrass, the researches of Prym,
Krazer, Frobenius, Poincare, and Wirtinger.

Automorphic Functions

Closely connected with the elliptic functions is a class of functions
which has come into great prominence in the last quarter of a cen-
tury, namely, the elliptic modular and automorphic functions. Let
us consider first the modular functions of which the modulus K and
the absolute invariant J are the simplest types.

The transformation theory of Jacobi gave algebraic relations be-
tween such functions in endless number. Hermite, Fuchs, Dedekind,
and Schwarz are forerunners, but the theory of modular functions as
it stands to-day is principally due to Klein and his school. Its goal
is briefly stated thus : Determine all sub-groups of the linear group

(1) xl

where a, /?, 7-, d are integers and ad — /?^ = 1; determine for each
such group associate modular functions and investigate their rela-
tion to one another and especially to J. Important features in this
theory are the congruence groups of (1); the fundamental polygon
belonging to a given sub-group, and its use as substitute for a Rie-
mann surface; the principle of reflection over a circle, the modular
forms.

The theory of automorphic functions is due to Klein and Poincare.
It is a generalization of the modular functions; the coefficients in
(1) being any real or imaginary numbers, with non- vanishing de-
terminant, such that the group is discontinuous. Both authors have
recourse to non- Euclidean geometry to interpret the substitutions (1).
Their manner of showing the existence of functions belonging to
a given group is quite different. Poincare by a brilliant stroke of
genius actually writes down their arithmetic expressions in terms
of his celebrated 8 series. Klein employs the existence methods of
Riemann. The relation of automorphic functions to differential
equations is studied by Poincare in detail. In particular, he shows that
both variables of a linear differential equation with algebraic coeffi-
cients can be expressed uniformly by their means.

478 MATHEMATICS

Differential Equations

Let us turn now to another great field of mathematical activity,
the theory of differential equations. The introduction of the theory
of functions has completely revolutionized this subject. At the
beginning of the nineteenth century many important results had
indeed been established, particularly by Euler and Lagrange; but
the methods employed were artificial, and broad comprehensive
principles were lacking. By various devices one tried to express
the solution in terms of the elementary functions and quadratures
— a vain attempt; for as we know now, the goal they strove so
laboriously to reach was in general unattainable.

A new epoch began with Cauchy, who by means of his new theory
of functions first rigorously established the existence of the solution
of certain classes of equations in the vicinity of regular points. He
also showed that many of the properties of the elliptic functions
might be deduced directly from their differential equations. Ere
long, the problem of integrating a differential equation changed
its base. Instead of seeking to express its solution in terms of the
elementary functions and quadratures, one asked what is the nature
of the functions defined by a given equation. To answer this ques-
tion we must first know what are the singular points of the integral
function and how does it behave in their vicinity. The number of
memoirs on this fundamental and often difficult question is enormous;
but this is not strange if we consider the great variety of interesting
and important classes of equations which have to be studied.

One of the first to open up this new path was Fuchs, whose classic
memoirs (1866-68) gave the theory of linear differential equations
its birth. These equations enjoy a property which renders them
particularly accessible, namely, the absence of movable singular
points. They may, however, possess points of indetermination, to
use Fuchs's terminology, and little progress has been made in this
case. Noteworthy in this connection is the introduction by v. Koch
of infinite determinants, first considered by our distinguished coun-
tryman Hill ; also the use of divergent series — that invention of
the Devil, as Abel called them — by Poincare". A particular class
of linear differential equations of great importance is the hyper-
geometric equation; the results obtained by Gauss, Kummer,
Riemann, and Schwarz relating to this equation have had the great-
est influence on the development of the general theory. The vast
extent and importance of the theory of linear differential equations
may be estimated when we recall that within its borders it embraces
not only almost all the elementary functions, but also the modular
and automorphic functions.

Too important to pass over in silence is the subject of algebraic

MATHEMATICS IN THE NINETEENTH CENTURY 479

differential equations with uniform solutions. The brilliant researches
of Poinleve" deserve especial mention.

Another field of great importance, especially in mathematical
physics, relates to the determination of the solution of differential
equations with assigned boundary conditions. The literature of this
subject is enormous; we may therefore be pardoned if mention is
made only of the investigation of our countrymen Bocher, Van
Vleck, and Porter.

Since 1870 the theory of differential equations has been greatly
advanced by Lie's theory of groups. Assuming that an equation or a
system of equations admits one or more infinitesimal transformations,
Lie has shown how they may be employed to simplify the problem
of integration. In many cases they give us exact information how
to conduct the solution and upon what system of auxiliary equations
the solution depends. One of the most striking illustrations of this
is the theory of ordinary linear differential equations which Picard
and Vessiot have developed, analogous to Galois 's theory for algebraic
equations. An interesting result of this theory is a criterion for the
solution of such equations by quadratures. As an application, we
find that Ricatti's equation cannot be solved by quadratures. The
attempts to effect such a solution of this celebrated equation in the
century before were therefore necessarily in vain.

A characteristic feature of Lie's theories is the prominence given
to the geometrical aspects of the questions involved. Lie thinks in
geometrical images, the analytical formulation comes afterwards.
Already Morge had shown how much might be gained in geometrizing
the problem of integration. Lie has gone much farther in this direc-
tion. Besides employing all the geometrical notions of his predeces-
sors extended to n-way space, he has introduced a variety of new
conceptions, chief of which are his surface element and contact
transformations.

He has also used with great effect Pliicker's line geometry, and his
own sphere geometry in the study of certain types of partial differential
equations of the first and second orders which are of great geometrical
interest, for example, equations whose characteristic curves are lines
of curvature, geodesies, etc. Let us close by remarking that Lie's
theories not only afford new and valuable points of view for attack-
ing old problems, but also give rise to a host of new ones of great
interest and importance.

Groups

We turn now to the second dominant idea of the century, the
group concept.

Groups first became objects of study in algebra when Lagrange
(1770), Ruffini (1799), and Abel (1826) employed substitution groups

480 MATHEMATICS

with great advantage in their work on the quintic. The enormous
importance of groups in algebra was. however, first made clear by
Galois, whose theory of the solution of algebraic equations is one
of the great achievements of the century. Its influence has stretched
far beyond the narrow bounds of algebra.

With an arbitrary but fixed domain of rationality, Galois observed
that every algebraic equation has attached to it a certain group of
substitutions. The nature of the auxiliary equations required to
solve the given equation is completely revealed by an inspection of
this group.

Galois 's theory showed the importance of determining the sub-
groups of a given substitution group, and this problem was studied
by Cauchy, Serret, Matthieu, Kirkmann, and others. The publica-
tion of Jordan's great treatise in 1870 is a noteworthy event. It
collects and unifies the results of his predecessors and contains an
immense amount of new matter.

A new direction was given to the theory of groups by the introduc-
tion by Cayley of abstract groups (1854, 1878). The work of Sylow,
Holder and Frobenius, Burnside and Miller, deserve especial notice.

Another line of research relates to the determination of the finite
groups in the linear group of any number of variables. These groups
are important in the theory of linear differential equations with
algebraic solutions, in the study of certain geometrical problems
as the points of inflection of a cubic, the twenty-seven lines on a
surface of the third order, in crystallography, etc. They also enter
prominently into Klein's Formen-problem. An especially important
class of finite linear groups are the congruence groups first considered
by Galois. Among the laborers in the field of linear groups, we note
Jordan, Klein, Moore, Maschke, Dickson. Frobenius, and Wiman.

Up to the present we have considered only groups of finite order.
About 1870 entirely new ideas coming from geometry and differential
equations give the theory of groups an unexpected development.
Foremost in this field are Lie and Klein.

Lie discovers and gradually perfects his theory of continuous
transformation groups and shows their relations to many different
branches of mathematics. In 1872 Klein publishes his Erlanger
Programme and in 1877 begins his investigations on elliptic modular
functions, in which infinite discontinuous groups are of primary im-
portance, as we have already seen. In the now famous Programme,
Klein asks what is the principle which underlies and unifies the
heterogeneous geometrical methods then in vogue, as, for example,
the geometry of the ancients, whose figures are rigid and invariable;
the modern protective geometry, whose figures are in ceaseless
flux passing from one form to another; the geometries of Plucker
and Lie, in which the elements of space are no longer points, but line

MATHEMATICS IN THE NINETEENTH CENTURY 481

spheres, or other configurations at pleasure, the geometry of birational
transformation, the analysis situs, etc., etc. Klein finds this answer:
In each geometry we have a system of objects and a group which
transforms these objects one into another. We seek the invariants
of this group. In each case it is the abstract group and not the con-
crete objects which is essential. The fundamental r61e of a group in
geometrical research is thus made obvious. Its importance is the
solution of algebraic equation, in the theory of differential equations
in the automorphic functions we have already seen. The immense
theory of algebraic invariants developed by Cayley and Sylvester,
Aronhold. Clebsch, Gordan, Hermite, Brioschi, and a host of zealous
workers in the middle of the century, also finds its place in the far
more general invariant theory of Lie's theory of groups. The same is
true of the theory of surfaces, so far as it rests on the theory of differ-
ential forms. In the theory of numbers, groups have many important
applications, for example, in the composition of quadratic forms and
the cyclotomic bodies. Finally, let us note the relation between hyper-
complex numbers and continuous groups discovered by Poincare.

In re'sume', we may thus say that the group concept, hardly not-
iceable at the beginning of the century, has at its close become one
of the fundamental and most fruitful notions in the whole range of
our science.

Infinite Aggregates

Leaving the subject of groups, we consider now briefly another
fundamental concept, namely, infinite aggregates. In the most
diverse mathematical investigations we are confronted with such
aggregates. In geometry the conceptions of curves, surface, region,
frontier, etc., when examined carefully, load us to a rich variety of
aggregates. In analysis they also appear, for example, the domain
of definition of an analytic function, the points where a function of
a real variable ceases to be continuous or to have a differential coeffi-
cient, the points where a series of functions ceases to be uniformly
convergent, etc.

To say an aggregate (not necessarily a point aggregate) is infinite
is often an important step; but often again only the first step. To
penetrate farther into the problem may require us to state how
infinite. This requires us to make distinctions in infinite aggregates,
to discover fruitful principles of classification, and to investigate the
properties of such classes.

The honor of having done this belongs to George Cantor. The
theory of aggregates is for the most part his creation; it has en-
riched mathematical science with fundamental and far-reaching
notions and results.

The theory falls into two parts; a theory of aggregates in general,

482 MATHEMATICS

and a theory of point aggregates. In the theory of point aggregates
the notion of limiting points gives rise to important classes of aggre-
gates as discrete, dense, everywhere dense, complete, perfect, con-
nected, etc., which are so important in the function theory.

In the general theory two notions are especially important,
namely, the one to one correspondence of the elements of two ag-
gregates, and well-ordered aggregates. The first leads to cardinal
numbers and the idea of enumerable aggregates, the second to trans-
finite or ordinal numbers.

Two striking results of Cantor's theory are these: the algebraic
and therefore the rational numbers, although everywhere dense, are
enumerable; and secondly, one-way and n-way space have the
same cardinal number.

Cantor's theory has already found many applications, especially
in the function theory, where it is to-day an indispensable instrument
of research.

Functions of Real Variables — The Critical Movement

One of the most conspicuous and distinctive features of mathe-
matical thought in the nineteenth century is its critical spirit. Be-
ginning with the calculus, it soon permeates all analysis, and toward
the close of the century it overhauls and recasts the foundation of
geometry and aspires to further conquests in mechanics and in the
immense domains of mathematical physics.

Ushered in with Lagrange and Gauss just at the close of the
eighteenth century, the critical movement receives its first decisive
impulse from the teachings of Cauchy, who in particular introduces
our modern definition of limit and makes it the foundation of the
calculus. We must also mention in this connection Abel, Bolzano,
and Dirichlet. Especially Abel adopted the reform ideas of Cauchy
with enthusiasm, and made important contributions in infinite series.

The figure, however, which towers above all others in this move-
ment, whose name has become an epithet of rigor, is Weierstrass.
Beginning at the very foundations, he creates an arithmetic of real
and complex numbers, assuming the theory of positive integers to be
given. The necessity of this is manifest when we recall that until
then the simplest properties of radicals and logarithms were utterly
devoid of a rigorous foundation; so, for example,

V2 N/5=ViO log 2+log 5=log 10

Characteristic of the pre-Weierstrassean era is the loose way in
which geometrical and other intuitional ideas were employed in
the demonstration of analytical theorems. Even Gauss is open to
this criticism. The mathematical world received a great shock
when Weierstrass showed them an example of a continuous function

MATHEMATICS IN THE NINETEENTH CENTURY 483

without a derivative, and Hankel and Cantor, by means of their
principle of condensation of singularities, could construct analytic
expressions for functions having in any interval however small an
infinity of points of oscillation, an infinity of points in which the
differential coefficient is altogether indeterminate, or an infinity of
points of discontinuity. Another rude surprise was Cantor's dis-
covery of the one to one correspondence between the points of a
unit segment and a unit square, followed up by Peano's example
of a space-filling curve.

These examples and many others made it very clear that the
ideas of a curve, a surface region, motion, etc., instead of being clear
and simple, were extremely vague and complex. Until these notions
had been cleared up, their admission in the demonstration of an
analytical theorem was therefore not to be tolerated. On a purely
arithmetical basis, with no appeal to our intuition, Weierstrass
develops his stately theory of functions which culminates in the
theory of Abelian and multiply periodic functions.

But the notion of rigor is relative and depends on what we are
willing to admit either tacitly or explicitly. As we observed, Gauss,
whose rigor was the admiration of his contemporaries, freely ad-
mitted geometrical notions. This Weierstrass would criticise. On
the other hand, Weierstrass has made a grave oversight: he no-
where shows that his definitions relative to the number he introduces
do not involve mutual contradictions. If he replied that such con-
tradictions would involve contradictions in the theory of positive
integers, one might ask what assurance have we that such contradic-
tions may not actually exist. A flourishing young school of mathe-
matical logic has recently grown up under the influence of Peano.
They have investigated with marked success the foundations of
analysis and geometry, and in particular have attempted to show
the non-contradictoriness of the axioms of our number-system by
making them depend on the axioms of logic, which axioms we must
admit, to reason at all.

The critical spirit, which in the first half of the century was to
be found in the writings of only a few of the foremost mathematicians,
has in the last quarter of the century become almost universal, at
least in analysis. A searching examination of the foundation of
arithmetic and the calculus has brought to light the insufficiency of
much of the reasoning formerly considered as conclusive. It became
necessary to build up these subjects anew. The theory of irrational
numbers invented by Weierstrass has been supplanted by the more
flexible theories of Dedekind and Cantor. Stolz has given us a sys-
tematic and rigorous treatment of arithmetic. The calculus has
been completely overhauled and arithmetized by Thomae, Harnack,
Peano, Stolz, Jordan, and Valle'e-Poussin.

484 MATHEMATICS

Leaving the calculus, let us notice briefly the theory of functions
of real variables. The line of demarcation between these two sub-
jects is extremely arbitrary. We might properly place in the latter
all those finer and deeper questions relating to the number-system;
the study of our curve, surface, and other geometrical notions, the
peculiarities that functions present with reference to discontinuity,
oscillation, differentiation, and integration; as well as a very exten-
sive class of investigations whose object is the greatest possible
extension of the processes, concepts, and results of the calculus.
Among the many not yet mentioned who have made important
contributions to this subject we note: Fourier, Riemann, Stokes,
Dini, Tannery, Pringsheim, Arzela, Osgood, Broden, Ascoli, Borel,
Baire, Kopke, Holder, Volterra, and Lebesgue.

Closely related with the differential calculus is the calculus of
variations; in the former the variables are given infinitesimal varia-
tions, in the latter the functions. Developed in a purely formal
manner by Jacobi, Hamilton, Clebsch, and others in the first part
of the century, a new epoch began with Weierstrass, who, having
subjected the labors of his predecessors to an annihilating criticism,
placed the theory on a new and secure foundation and so opened the
path for further research by Schwarz, A. Mayer, Scheffers, v. Esche-
rich, Kneser, Osgood, Bolza, Kobb, Zermelo, and others. At the
very close of the century Hilbert has given the theory a fresh im-
pulse by the introduction of new and powerful methods, which
enable us in certain cases to neglect the second variation and sim-
plifies the consideration of the first. As application he gives the
first direct and yet simple demonstration of Dirichlet's celebrated
Principle.

Theory of Numbers — Algebraic Bodies

The theory of numbers as left by Fermat, Euler, and Legendre
was for the most part concerned with the solution of Diophantine
equations, that is, given an equation f(x, y, z, . . . ) =0 whose
coefficients are integers, find all rational, and especially all integral
solutions. In this problem Lagrange had shown the importance
of considering the theory of forms. A new era begins with the ap-
I>earance of Gauss's Disquisitiones arithmeticae in 1801. This great
work is remarkable for three things: (1) The notion of divisibility
in the form of congruences is shown to be an instrument of wonder-
ful power; (2) the Diophantine problem is thrown in the back-
ground and the theory of forms is given a dominant r61e; (3) the
introduction of algebraic numbers, namely, the roots of unity.

The theory of forms has been further developed along the lines
of the Disquisitiones by Dirichlet, Eisenstein, Hermite, H. Smith, and
Minkowski.

MATHEMATICS IN THE NINETEENTH CENTURY 485

Another part of the theory of numbers also goes back to Gauss,
namely, algebraic numerical bodies. The Law of Reciprocity of
Quadratic Residues, one of the gems of the higher arithmetic, was
first rigorously proved by Gauss. His attempts to extend this
theorem to cubic and biquadratic residues showed that the elegant
simplicity which prevailed in quadratic residues was altogether
missing in these higher residues, until one passed from the domain
of real integers to the domain formed of the third and fourth roots of
unity. In these domains, as Gauss remarked, algebraic integers have
essentially the same properties as ordinary integers. Further explor-
ation in this new and promising field by Jacobi, Eisenstein, and
others soon brought to light the fact that already in the domain
formed of the twenty-third roots of unity the laws of divisibility were
altogether different from those of ordinary integers; in particular,
a number could be expressed as the product of prime factors in more
than one way. Further progress in this direction was therefore
apparently impossible.

It is Rummer's immortal achievement to make further progress
possible by the invention of his ideals. These he applied to Fermat's
celebrated Last Theorem and the Law of Reciprocity of Higher
Residues.

The next step in this direction was taken by Dedekind and Kro-
necker, who developed the ideal theory for any algebraic domain.
So arose the theory of algebraic numerical bodies, which has come
into such prominence in the last decades of the century through
the researches of Hensel, Hurwitz, Minkowski, Weber, and, above
all, Hilbert.

Kronecker has gone farther, and in his classic Grundziige he has
shown that similar ideas and methods enable us to develop a theory
of algebraic bodies in any number of variables. The notion of divis-
ibility so important in the preceding theories is generalized by Kro-
necker still farther in the shape of his system of moduli.

Another noteworthy field of research opened up by Kronecker is
the relation between quadratic forms with negative determinant
and complex multiplication of elliptic functions. H. Smith, Gierster,
Hurwitz, and especially Weber have made important contributions.
A method of great power in certain investigations has been created
by Minkowski, which he called the Geometric der Zahlen. Introduc-
ing a generalization of the distance function, he is led to the concep-
tion of a fundamental body (Aichkorper) . Minkowski shows that
every fundamental body is nowhere concave, and conversely to
each such body belongs a distance function. A theorem of great
importance is now the following: The minimum value which each
distance function has at the lattice points is not greater than a certain
number depending on the function chosen.

486 MATHEMATICS

We wish finally to mention a line of investigation which makes
use of the infinitesimal calculus arid even the theory of functions.
Here belong the brilliant researches of Dirichlet relating to the num-
ber of classes of binary forms for a given determinant, the number
of primes in a given arithmetic progression; and Riemann's remark-
able memoir on the number of primes in a given interval.

In this analytical side of the theory of numbers we notice also the
researches of Mertens. Weber, and Hadamard.

Protective Geometry

The tendencies of the eighteenth century were predominantly
analytical. Mathematicians were absorbed for the most part in
developing the wonderful instrument of the calculus with its countless
applications. Geometry made relatively little progress. A new era
begins with Monge. His numerous and valuable contributions to
analytical descriptive and differential geometry, and especially his
brilliant and inspiring lectures at the Ecole Poly technique (1795,
1809), put fresh life into geometry and prepared it for a new and
glorious development in the nineteenth century.

When one passes in review the great achievements which have
made the nineteenth century memorable in the annals of our science,
certainly projective geometry will occupy a foremost place. Pascal,
De la Hire, Monge, and Carnot are forerunners, but Poncelet, a pupil
of Monge, is its real creator. The appearance of his Traite des pro-
prietes projectiles des figures, in 1822, gives modern geometry its
birth. In it we find the line at infinity, the introduction of imagin-
aries, the circular points at infinity, polar reciprocation, a discus-
sion of homology, the systematic use of projection, section, and
anharmonic ratio.

While the countrymen of Poncelet, especially Chasles, do not fail
to make numerous and valuable contributions to the new geometry,
the next great steps in advance are made on German soil. In 1827
Mobius publishes the Barycentrische Calcul; Pliicker's Analytisch-
geometrische Entwickelungen appears in 1828-31 and Steiner's Ent-
wickelung der Abhdngigkeit geometrischer Gestalten von einander in
1832. In the ten years which embrace the publication of these
immortal works of Poncelet, Pliicker, and Steiner, geometry has
made more real progress than in the two thousand years which had
elapsed since the time of Appolonius. The ideas which had been
slowly taking shape since the time of Descartes suddenly crystallized
and almost overwhelmed geometry with an abundance of new ideas
and principles.

To Mobius we owe the introduction of homogeneous coordinates,
and the far-reaching conception of geometric transformation, includ-
ing collineation and duality as special cases. To Pliicker we owe the

MATHEMATICS IN THE NINETEENTH CENTURY 487

use of the abbreviate notation which permits us to study the proper-
ties of geometric figures without the intervention of the coordinates,
the introduction of line and plane coordinates, and the notion of
generalized space elements. Steiner, who has been called the greatest
geometer since Appolonius, besides enriching geometry in countless
ways, was the first to employ systematically the method of generating
geometrical figures by means of protective pencils.

Other noteworthy works belonging to this period are Pliicker's
System der analytischen Geometric (1835), and Chasles's classic Apercu
(1837).

Already at this stage we notice a bifurcation in geometrical
methods. Steiner and Cha'sles become eloquent champions of the
synthetic school of geometry, while Pliicker, and later Hesse and
Cayley, are leaders in the analytical movement. The astonishing
fruitfulness and beauty of synthetic methods threatened for a short
time to drive the analytic school out of existence. The tendency
of the synthetic school was to banish more and more metrical methods.
In effecting this the anharmonic ratio became constantly more promi-
nent. To define this fundamental ratio without reference to measure-
ment, and so free projective geometry from the galling bondage
of metric relations, was thus a problem of fundamental importance.
The glory of this achievement, which has, as we shall see, a far
wider significance, belongs to v. Staudt. Another equally important
contribution of v. Staudt to synthetic geometry is his theory of
imaginaries. Poncelet, Steiner, Chasles operate with imaginary
elements as if they were real. Their only justification is recourse to
the so-called principles of continuity or to some other equally vague
principle. V. Staudt gives this theory a rigorous foundation, defining
the imaginary points, lines, and planes by means of involutions
without ordinal elements.

The next great advance made is the advent of the theory of alge-
braic invariants. Since projective geometry is the study of those
properties of geometric figures which remain unaltered by projective
transformations, and since the theory of invariants is the study of
those forms which remain unaltered (except possibly for a numerical
factor) by the group of linear substitutions, these two subjects are
inseparably related and in many respects only different aspects of the
same thing. It is no wonder, then, that geometers speedily applied
the new theory of invariants to geometrical problems. Among the
pioneers in this direction were Cayley, Salmon. Aronhold, Hesse,
and especially Clebsch.

Finally we must mention the introduction of the line as a space
element. Forerunners are Grassmann (1844) and Cayley (1859), but
Pliicker in his memoirs of 1865, and his work Neue Geometric des
Raumes (1868-69), was the first to show its great value by studying

488 MATHEMATICS

complexes of the first and second order and calling attention to
their application to mechanics and optics.

The most important advance over Pliieker has been made by
Klein, who takes as coordinates six-line complexes in involution.
Klein also observed that line geometry may be regarded as a point
geometry on a quadric in five-way space. Other laborers in this
field are Clebsch, Reye, Segre, Sturm, and Konigs.

Differential Geometry

During the first quarter of the century this important branch of
geometry was cultivated chiefly by the French. Monge and his
school study with great success the generation of surfaces in vari-
ous ways, the properties of envelopes, evolutes, lines of curvature,
asymptotic lines, skew curves, orthogonal systems, and especially the
relation between the surface theory and partial differential equations.

The appearance of Gauss's Disquisitiones generates circa super-
ficies curvas, in 1828, marks a new epoch. Its wealth of new ideas
has furnished material for countless memoirs, and given geometry
a new direction. We find here the parametric representation of a
surface, the introduction of curvilinear coordinates, the notion of
spherical image, the Gaussian measure of curvature, and a study of
geodesies. But by far the most important contributions that Gauss
makes in this work is the consideration of a surface as a flexible,
inextensible film or membrane, and the importance given quadratic
differential forms.

We consider now some of the lines along which differential geometry
has advanced. The most important is perhaps the theory of differen-
tial quadratic forms with their associate invariants and parameters.
We mention here Lame", Beltrami, Menardi, Codazzi, Christoffel,
and Weingarten.

An especially beautiful application of this theory is the immense
subject of applicability and deformation of surfaces, in which Mind-
ing, Bauer, Beltrami, Weingarten, and Voss have made important
contributions.

Intimately related with the theory of applicability of two surfaces
is the theory of surfaces of constant curvature which play so import-
ant a part in non-Euclidean geometry. We mention here the work
of Minding, Beltrami, Dini, Backlund, and Lie.

The theory of rectilinear congruences has also been the subject
of important researches from the standpoint of differential geometry.
First studied by Monge as a system of normals to a surface and then
in connection with optics by Mains, Dupin, and Hamilton, the gen-
eral theory has since been developed by Kummer, Ribaucour,
Guichard, Darboux, Voss, and Weingarten. An important applica-
tion of this theorv is the infinitesimal deformation of a surface.

MATHEMATICS IN THE NINETEENTH CENTURY 489

Minimum surfaces have been studied by Monge, Bonnet, and
Enneper. The subject owes its present extensive development prin-
cipally to Weierstrass, Riemann, Schwarz, and Lie. In it we find
harmoniously united the theory of surfaces, the theory of functions,
the calculus of variations, the theory of groups, and mathematical
physics.

Another extensive division of differential geometry is the theory of
orthogonal systems, of such importance in physics. We note espe-
cially the investigations of Dupin, Jacobi, Darboux, Combescure,
and Bianchi.

Under this head we group a number of subjects too important
to pass oVer in silence, yet which cannot be considered at length for
lack of time.

In the first place is the immense subject of algebraic curves and
surfaces. To develop adequately all the important and elegant
properties of curves and surfaces of the second order alone would
require a bulky volume. In this line of ideas would follow curves
and surfaces of higher order and class. Their theory is far less
complete, but this lack it amply makes good by offering an almost
bewildering variety of configurations to classify and explore. No
single geometer has contributed more to this subject than Cayley.

A theory of great importance is the geometry on a curve or sur-
face inaugurated by Clebsch in 1863.

Expressing the coordinates of a plane cubic by means of elliptic
functions and employing their addition theorems, he deduced with
hardly any calculation Steiner's theorem relating to the inscribed
polygons and various theorems concerning conies touching the curve.
Encouraged by such successes, Clebsch proposed to make use of
Riemann 's theory of Abelian functions in the study of algebraic
curves of any order. The most important result was a new classifica-
tion of such curves. Instead of the linear transformation, Clebsch
in harmony with Riemann 's ideas employs the birational transforma-
tion as a principle of classification. From this standpoint we ask
what arc the properties of algebraic curves which remain invariant
for such transformation.

Brill and Nother follow Clebsch. Their method is, however, alge-
braical, and rests on their celebrated Residual theorem which in
their hands takes the place of Abel's theorem. We mention further
the investigation of Castelnuovo, Weber, Krause, and Segre. An
important division of this subject is the theory of correspondences.
First studied by Chasles for curves of deficiency 0 in 1864, Cayley,
and, immediately after, Brill extended the theory to the case of any
•p. The most important advance made in later years has been made

490 MATHEMATICS

by Hurwitz, who considers the totality of possible correspondences
on an algebraic curve, making use of the corresponding integrals of
the first species.

Alongside the geometry on a curve is the vastly more difficult and
complicated geometry on a surface, or more generally, on any algebraic
spread in n-way space. Starting from a remark of Clebsch (1868).
Nother made the first great step in his famous memoir of 1868-
74. Further progress has been due to the French and Italian mathe-
maticians. Picard, Poincare", and Humbert make use of transcend-
ental methods, in which figure prominently double integrals which
remain finite on the surface and single integrals of total differentials.
On the other hand, Enriques and Castelnuovo have attacked the
subject from a more algebraic-geometric standpoint by means of
linear systems of algebraic curves on the surface.

The first invariants of a surface were discovered by Clebsch and
Nother; still others have been found by Castelnuovo and Enriques
in connection with irregular surfaces.

Leaving this subject, let us consider briefly the geometry of n
dimensions. A characteristic of nineteenth-century mathematics
is the generality of its methods and results. When such has been
impossible with the elements in hand, fresh ones have been invented;
witness the introduction of imaginary numbers in algebra and the
function theory, the ideals of Kummer in the theory of numbers,
the line and plane at infinity in projective geometry. The benefit
that analysis derived from geometry was too great not to tempt
mathematicians to free the latter from the narrow limits of three
dimensions, and so give it the generality that the former has long
enjoyed. The first pioneer in this abstract field was Grassmann (1844) ;
we must, however, consider Cayley as the real founder of n-dimen-
sional geometry (1869). Notable contributions have been made by
the Italian school, Veronese, Segre, etc.

Non-Euclidean Geometry

Each century takes over as a heritage from its predecessor a
number of problems whose solution previous generations of mathe-
maticians have arduously but vainly sought. It is a signal achieve-
ment of the nineteenth century to have triumphed over some of the
most celebrated of these problems.

The most ancient of them is the Quadrature of the Circle, which
already appears in our oldest mathematical document, the Papyrus
Rhind, B.C. 2000. Its impossibility was finally shown by Lindemann
(1882).

Another famous problem relates to the solution of the quintic,
which had engaged the attention of mathematicians since the middle
of the sixteenth century. The impossibility of expressing its roots by

MATHEMATICS IN THE NINETEENTH CENTURY 491

radicals was finally shown by the youthful Abel (1824), while Hermite
and Kroneker (1858) showed how they might be expressed by the
elliptic modular functions, and Klein (1875) by means of the icosa-
hedral irrationality.

But of all problems which have come down from the past, by far
the most celebrated and important relates to Euclid's parallel
axiom. Its solution has profoundly affected our views of space,
and given rise to questions even deeper and more far-reaching which
embrace the entire foundation of geometry and our space conception.
Let us pass in rapid review the principal events of this great move-
ment. Wallis in the seventeenth, Seccheri, Lambert, and Legendre
in the eighteenth, are the first to make any noteworthy progress
before the nineteenth century. The really profound investigations
of Seccheri and Lambert, strangely enough, were entirely over-
looked by later writers and have only recently come to light.

In the nineteenth century non-Euclidean geometry develops along
four directions, which roughly follow each other chronologically.
Let us consider them in order.

The naive-synthetic direction. — The methods employed are similar to
those of Euclid. His axioms are assumed with the exception of the
parallel axiom; the resulting geometry is what is now called hyper-
bolic or Lobatschewski 's geometry. Its principal properties are de-
duced, in particular its trigonometry, which is shown to be that of a
sphere with imaginary radius as Lambert had divined. As a specific
result of these investigations the long-debated question relating to
the independence of the parallel axiom was finally settled. The great
names in this group are Lobatschewski, Bolyai, and Gauss. The first
publications of Lobatschewski are his Exposition succinct des prin-
cipesde la geometric (1829), and the Geometrische Untersuchungen, in
1840. Bolyai's Appendix was published in 1832. As to the extent
of Gauss's investigations, we can only judge from scattered remarks
in private letters and his reviews of books relating to the parallel
axioms. His dread of the Geschrei der Sootier, that is, the followers
of Kant, prevented him from publishing his extensive speculations.

The metric-differential direction. — This is inaugurated by three great
memoirs by Riemann, Helmholtz, and Beltrami, all published in the
same year, 1868.

Beltrami, making use of results of Gauss and Minding relating to
the applicability of two surfaces, shows that the hyperbolic geometry
of a plane may be interpreted on a surface of constant negative
curvature, the pseudosphere. By means of this discovery the purely
logical and hypothetical system of Lobatschewski and Bolyai takes
on a form as concrete and tangible as the geometry of a plane.

The work of Riemann is as original as profound. He considers
space as an n-dimensional continuous numerical multiplicity, which

492 MATHEMATICS

is distinguished from the infinity of other such multiplicities by
certain well-defined characters. Chief of them are (1) the quadratic
differential expression which defines the length of an elementary arc,
and (2) a property relative to the displacements of this multiplicity
about a point. There are an infinity of space multiplicities which
satisfy Riemann's axioms. By extending Gauss's definition of a
curvature &, of a surface at a point to curvature of space at a point,
by considering the geodesic surfaces passing through that point,
Riemann finds that all these spaces fall into three classes according
as k is equal to, greater, or less than 0. For ?i=3 and fc=0 we have
Euclidean space; when fc<0 we have the space found by Gauss,
Lobatschewski, and Bolyai; when fc>0 we have the space first
considered in the long-forgotten writings of Seccheri and Lambert,
in which the right line is finite.

Helmholtz, like Riemann, considers space as a numerical multiplic-
ity. To characterize it further, Helmholtz makes use of the notions
of rigid bodies and free mobility. His work has been revised and ma-
terially extended by Lie from the standpoint of the theory of groups.

In the present category also belong important papers by New-
comb and Killing.

The protective direction. — We have already noticed the efforts of
the synthetic school to express metric properties by means of project-
ive relations. In this the circular points at infinity were especially
serviceable. An immense step in this direction was taken by Laguerre,
who showed, in 1853, that all angles might be expressed as an anhar-
monic ratio with reference to these points, that is, with reference to
a certain fixed conic. The next advance is made by Cay ley in his
famous sixth memoir on quantics, in 1859. Taking any fixed conic
(or quadric, for space) which he calls the absolute, Cayley introduces
two expressions depending on the anharmonic ratio with reference
to the absolute. When this degenerates into the circular points
at infinity, these expressions go over into the ordinary expressions
for the distance between two points and the angle between two
lines. Thus all metric relations may be considered as protective
relations with respect to the absolute. Cayley does not seem to be
aware of the relation of his work to non-Euclidean geometry. This
was discovered by Klein, in 1871. In fact, according to the nature of
the absolute, three geometries are possible; these are precisely the
three already mentioned. Klein has made many important contri-
butions to non-Euclidean geometry. We mention his modification
of v. Staudt's definition of anharmonic ratio so as to be independ-
ent of the parallel axiom, his discovery of the two forms of Rie-
mann's space, and finally his contributions to a class of geometries
first noticed by Clifford and which are characterized by the fact that
only certain of its motions affect space as a whole.

MATHEMATICS IN THE NINETEENTH CENTURY 493

As a result of all these investigations, both in the protective as
also in the metric differential direction, we are led irresistibly to the
same conclusion, namely: The facts of experience can be explained
by all three geometries when the constant k is taken small enough.
It is, therefore) merely a question of convenience whether we adopt
the parabolic, hyperbolic, or elliptic geometry.

The critical synthetic direction represents a return to the old syn-
thetic methods of Euclid, Lobatschewski, and Bolyai, with the added
feature of a refined and exacting logic. Its principal object is no
longer a study of non-Euclidean but of Euclidean geometry. Its
aim is to establish a system of axioms for our ordinary space which
is complete, compatible, and irreducible. The fundamental terms
point, line, plane, between, congruent, etc., are introduced as ab-
stract marks whose properties are determined by inter-relations in
the form of axioms. Geometric intuition has no place in this order
of ideas which regards geometry as a mere division of pure logic.
The efforts of this school have already been crowned with eminent
success, and much may be expected from it in the future. Its leaders
are Peano, Veronese, Fieri, Padoa, Burali-Forti, and Levi-Civitta, in
Italy, Pasch and Hilbert in Germany, and Moore in America.

Closing at this point our hasty and imperfect survey of mathe-
matics in the last century, let us endeavor to sum up its main charac-
teristics. What strikes us at once is its colossal proportions and rapid
growth in nearly all directions, the great variety of its branches, the
generality and complexity of its methods; an inexhaustible creative
imagination, the fearless introduction and employment of ideal
elements, and an appreciation for a refined and logical development
of all its parts.

We who stand on the threshold of a new century can look back on
an era of unparalleled progress. Looking into the future, an equally
bright prospect greets our eyes; on all sides fruitful fields of re-
search invite our labor and promise easy and rich returns.

Surely this is the golden age of mathematics.

GEBMAN UNIVERSITY STUDENTS
Photogravure from the Painting by Carl Heyden.

SECTION A— ALGEBRA AND ANALYSIS

SECTION A — ALGEBRA AND ANALYSIS

(Hall 9, September 22, 10 a. m.)

CHAIRMAN: PROFESSOR E. H. MOORE, University of Chicago.

SPEAKERS: PROFESSOR CHARLES EMILE PICARD, The Sorbonne; Member of the

Institute of France.

PROFESSOR HEINRICH MASCHKE, University of Chicago.
SECRETARY: PROFESSOR A. G. BLISS, University of Chicago.

BY CHARLES EMILE PICARD
(Translated from the French by Professor George Bruce Hoisted, Kenyan College)

[Charles Emile Picard, Professor of Higher Algebra and Higher Analysis, Uni-
versity of Paris; also Professor of General Mechanics, 1'Ecole Centrale des
Arts et Manufactures, Paris, b. Paris, France, July 24, 1856. LL.D. Clark
University, Glasgow University, University of Christiania. Member of In-
stitute of France; Academy of Science, Berlin, St. Petersburg, Bologna,
Boston, Turin, Copenhagen, Washington, and many others; Mathematical
Society of London. Former President of Mathematical Society of France,
Mathematical Societies of London and Kharkow, and many other math-
ematical societies. Author and editor of Memoirs, Traits and Discussions
of Mathematics; Theory of Algebraic Functions of Two Variables.]

IT is one of the objects of a congress such as this which now
brings us together, to show the bonds between the diverse parts of
science taken in its most extended acceptation. So the organizers
of this meeting have insisted that the relations between different
sections should be put in evidence.

To undertake a study of this sort, somewhat indeterminate in
character, it is necessary to forget that all is in all; in what con-
cerns algebra and analysis, a Pythagorean would be dismayed at the
extent of his task, remembering the celebrated formula of the school:
" Things are numbers." From this point of view my subject would
be inexhaustible.

But I, for the best of reasons, will make no such pretensions.

In casting merely a glance over the development of our science
through the ages, and particularly in the last century, I hope to be
able to characterize sufficiently the role of mathematical analysis in
its relations to certain other sciences.

It would appear natural to commence by speaking of the concept
itself of whole number; but this subject is not alone of logical order,

498 ALGEBRA AND ANALYSIS

it is also of order historic and psychologic, and would draw us away
into too many discussions.

Since the concept of number has been sifted, in it have been found
unfathomable depths; thus, it is a question still pending to know,
between the two forms, the cardinal number and the ordinal number,
under which the idea of number presents itself, which of the two is
anterior to the other, that is to say, whether the idea of number
properly so called is anterior to that of order, or if it is the inverse.

It seems that the geometer-logician neglects too much in these
questions psychology and the lessons uncivilized races give us; it
would seem to result from these studies that the priority is with the
cardinal number.

It may also be there is no general response to the question, the
response varying according to races and according to mentalities.

I have sometimes thought, on this subject, of the distinction be-
tween auditives and visuals, auditives favoring the ordinal theory,
visuals the cardinal.

But I will not linger on this ground full of snares; I fear that our
modern school of logicians with difficulty comes to agreement with
the ethnologists and biologists; these latter in questions of origin
are always dominated by the evolution idea, and, for more than one
of them, logic is only the resume of ancestral experience. Mathe-
maticians are even reproached with postulating in principle that
there is a human mind in some way exterior to things, and that it
has its logic. We must, however, submit to this, on pain of con-
structing nothing. We need this point of departure, and certainly,
supposing it to have evolved during the course of prehistoric time,
this logic of the human mind was perfectly fixed at the time of the
oldest geometric schools, those of Greece; their works appear to
have been its first code, as is expressed by the story of Plato writing
over the door of his school, " Let no one not a "geometer enter
here."

Long before the bizarre word algebra was derived from the Arabic,
expressing, it would seem, the operation by which equalities are
reduced to a certain canonic form, the Greeks had made algebra
without knowing it; relations more intimate could not be imagined
than those binding together their algebra and their geometry, or
rather, one would be embarrassed to classify, if there were occasion,
their geometric algebra, in which they reason not on numbers but on
magnitudes.

Among the Greeks also we find a geometric arithmetic, and one of
the most interesting phases of its development is the conflict which,
among the Pythagoreans, arose in this subject between number and
magnitude, apropos of irrationals.

Though the Greeks cultivated the abstract study of numbers, called

DEVELOPMENT OF MATHEMATICAL ANALYSIS 499

by them arithmetic, their speculative spirit showed little taste for
practical calculation, which they called logistic.

In remote antiquity, the Egyptians and the Chaldeans, and later
the Hindus and the Arabs, carried far the science of calculation.

They were led on by practical needs; logistic preceded arithmetic,
as land-surveying and geodesy opened the way to geometry; in the
same way trigonometry developed under the influence of the in-
creasing needs of astronomy.

The history of science at its beginnings shows a close relation
between pure and applied mathematics; this we shall meet again
constantly in the course of this study.

We have remained up to this point in the domain which ordinary
language calls elementary algebra and arithmetic.

In fact, from the time that the incommensurability of certain
magnitudes had been recognized, the infinite had made its appearance,
and, from the time of the sophisms of Zeno on the impossibility of
motion, the summation of geometric progressions must have been
considered.

The procedures of exhaustion which are found in Eudoxus and in
Euclid appertain already to the integral calculus, and Archimedes
calculates definite integrals.

Mechanics also appeared in his treatise on the quadrature of the
parabola, since he first finds the surface of the segment bounded by
an arc of a parabola and its chord with the help of the theorem of
moments; this is the first example of the relations between me-
chanics and analysis, which since have not ceased developing.,

The infinitesimal method of the Greek geometers for the measure
of volumes raised questions whose interest is even to-day not ex-
hausted.

In plane geometry, two polygons of the same area are either
equivalent or equivalent-by-completion, that is to say, can be de-
composed into a finite number of triangles congruent in pairs, or
may be regarded as differences of polygons susceptible of such a
partition.

It is not the same for the geometry of space, and we have lately
learned that stereometry cannot, like planimetry, get on without
recourse to procedures of exhaustion or of limit, which require the
axiom of continuity or the Archimedes assumption.

Without insisting further, this hasty glance at antiquity shows
how completely then were amalgamated algebra, arithmetic, geo-
metry, and the first endeavors at integral calculus and mechanics, to
the point of its being impossible to recall separately their history.

In the Middle Ages and the Renaissance, the geometric algebra of
the ancients separated from geometry. Little by little algebra
properly so called arrived at independence, with its symbolism and

500 ALGEBRA AND ANALYSIS

its notation more and more perfected; thus was created this lan-
guage so admirably clear, which brings about for thought a veritable
economy and renders further progress possible.

This is also the moment when distinct divisions are organized.

Trigonometry, which, in antiquity, had been only an auxiliary of
astronomy, is developed independently; toward the same time the
logarithm appears, and essential elements are thus put in evidence.

II

In the seventeenth century, the analytic geometry of Descartes,
distinct from what I have just called the geometric algebra of the
Greeks by the general and systematic ideas which are at its base,
and the new-born dynamic were the origin of the greatest progress of
analysis.

When Galileo, starting from the hypothesis that the velocity of
heavy bodies in their fall is proportional to the time, from this
deduced the law of the distances passed over, to verify it afterward
by experiment, he took up again the road upon which Archimedes
had formerly entered and on which would follow after him Cavalieri,
Fermat, and others still, even to Newton and Leibnitz. The integral
calculus of the Greek geometers was born again in the kinematic of
the great Florentine physicist.

As to the calculus of derivatives or of differentials, it was founded
with precision apropos of the drawing of tangents.

In reality, the origin of the notion of derivative is in the confused
sense of the mobility of things and of the rapidity more or less great
with which phenomena happen; this is well expressed by the words
fluents and fluxions, which Newton used, and which one might
suppose borrowed from old Heraclitus.

The points of view taken by the founders of the science of motion,
Galileo, Huygens, and Newton, had an enormous influence on the
orientation of mathematical analysis.

It was with Galileo an intuition of genius to discover that, in
natural phenomena, the determining circumstances of the motion
produce accelerations: this must have conducted to the statement
of the principle that the rapidity with which the dynamic state of
a system changes depends in a determinate manner on its static state
alone. In a more general way we reach the postulate that the in-
finitesimal changes, of whatever nature they may be, occurring in
a system of bodies, depend uniquely on the actual state of this
system.

In what degree are the exceptions apparent or real? This is a ques-
tion which was raised only later and which I put aside for the
moment.

From the principles enunciated becomes clear a point of capital

DEVELOPMENT OF MATHEMATICAL ANALYSIS 501

importance for the analyst: Phenomena are ruled by differential
equations which can be formed when observation and experiment
have made known for each category of phenomena certain physical
laws.

We understand the unlimited hopes conceived from these results.
As Bertrand says in the preface of his treatise, "The early successes
were at first such that one might suppose all the difficulties of science
surmounted in advance, and believe that the geometers, without
being longer distracted by the elaboration of pure mathematics,
could turn their meditations exclusively toward the study of the
natural laws."

This was to admit gratuitously that the problems of analysis, to
which one was led, would not present very grave difficulties.

Despite the disillusions the future was to bring, this capital point
remained, that the problems had taken a precise form, and that a
classification could be established in the difficulties to be surmounted.

There was, therefore, an immense advance, one of the greatest
ever made by the human mind. We understand also why the theory
of differential equations acquired a considerable importance.

I have anticipated somewhat, in presenting things under a form
so analytic. Geometry was intermingled in all this progress. Huy-
gens, for example, followed always by preference the ancients, and
his Horologium oscillatorium rests at the same time on infinitesi-
mal geometry and mechanics; in the same way, in the Principia
of Newton, the methods followed are synthetic.

It is, above all, with Leibnitz that science takes the paths which
were to lead to what we call mathematical analysis; it is he who,
for the first time, in the latter years of the seventeenth century,
pronounces the word function.

By his systematic spirit, by the numerous problems he treated,
even as his disciples James and John Bernoulli, he established in a
final way the power of the doctrines to the edification of which had
successively contributed a long series of thinkers from the distant
times of Eudoxus and of Archimedes.

The eighteenth century showed the extreme fecundity of the new
methods. That was a strange time, the era cf mathematical duels
where geometers hurled defiance, combats not always without
acrimony, when Leibnitzians and Newtonians encountered in the
lists.

From the purely analytic point of view, the classification and study
of simple functions is particularly interesting; the function idea, on
which analysis rests, is thus developed little by little.

The celebrated works of Euler hold then a considerable place.
However, the numerous problems which present themselves to the
mathematicians leave no time for a scrutiny of principles; the

502 ALGEBRA AND ANALYSIS

foundations themselves of the doctrine are elucidated slowly, and
the mot attributed to d'Alembert, "Allez en avant et la foi vous
viendra, " is very characteristic of this epoch.

Of all the problems started at the end of the seventeenth century
or during the first half of the eighteenth, it will suffice for me to recall
those isoperimetric problems which gave birth to the calculus of
variations.

I prefer to insist on the interpenetration still more intimate
between analysis and mechanics when, after the inductive period of
the first age of dynamics, the deductive period was reached where one
strove to give a final form to the principles. The mathematical and
formal development played then the essential role, and the analytic
language was indispensable to the greatest extension of these prin-
ciples.

There are moments in the history of the sciences and, perhaps, of
society, when the spirit is sustained and carried forward by the words
and the symbols it has created, and when generalizations present
themselves with the least effort. Such was particularly the rdle of
analysis in the formal development of mechanics.

Allow me a remark just here. It is often said an equation contains
only what one has put into it. It is easy to answer, first, that the
new form under which one finds the things constitutes often of itself
an important discovery.

But sometimes there is more; analysis, by the simple play of
its symbols, may suggest generalizations far surpassing the primitive
outline. Is it not so with the principle of virtual velocities, of which
the first idea comes from the simplest mechanisms; the analytic
form which translates it will suggest extensions leading far from the
point of departure.

In the same sense, it is not just to say analysis has created nothing,
since these more general conceptions are its work. Still another
example is furnished us by Lagrange's system of equations; here
calculus transformations have given the type of differential equations
to which one tends to carry back to-day the notion of mechanical
explanation.

There are in science few examples comparable to this, of the
importance of the form of an analytic relation and of the power of
generalization of which it may be capable.

It is very clear that, in each case, the generalizations suggested
should be made precise by an appeal to observation and experiment,
then it is still the calculus which searches out distant consequences
for checks, but this is an order of ideas which I need not broach here.

Under the impulse of the problems set by geometry, mechanics,
and physics, we see develop or take birth almost all the great divisions
of analysis. First were met equations with a single independent vari-

able. Soon appear partial differential equations, with vibrating cords,
the mechanics of fluids and the infinitesimal geometry of surfaces.

This was a wholly new analytic world; the origin itself of the
problems treated was an aid which from the first steps permits no
wandering, and in the hands of Monge geometry rendered useful
services to the new-born theories.

But of all the applications of analysis, none had then more renown
than the problems of celestial mechanics set by the knowledge of the
law of gravitation and to which the greatest geometers gave their
names.

Theory never had a more beautiful triumph; perhaps one might
add that it was too complete, because it was at this moment above
all that were conceived for natural philosophy the hopes at least
premature of which I spoke above.

In all this period, especially in the second half of the eighteenth
century, what strikes us with admiration and is also somewhat
confusing, is the extreme importance of the applications realized,
while the pure theory appeared still so ill assured. One perceives it
when certain questions are raised like the degree of arbitrariness in
the integral of vibrating cords, which gives place to an interminable
and inconclusive discussion.

Lagrange appreciated these insufficiencies when he published his
theory of analytic functions, where he strove to give a precise foun-
dation to analysis.

One cannot too much admire the marvelous presentiment he had
of the role which the functions, which with him we call analytic,
were to play; but we may confess that we stand astonished before
the demonstration he believed to have given of the possibility of the
development of a function in Taylor's series.

The exigencies in questions of pure analysis were less at this
epoch. Confiding in intuition, one was content with certain probabil-
ities, and agreed implicitly about certain hypotheses that it seemed
useless to formulate in an explicit way; in reality, one had con-
fidence in the ideas which so many times had shown themselves
fecund, which is very nearly the mot of d'Alembert.

The demand for rigor in mathematics has had its successive
approximations, and in this regard our sciences have not the absolute
character so many people attribute to them.

Ill

We have now reached the first years of the nineteenth century.
As we have explained, the great majority of the analytic researches
had, in the eighteenth century, for occasion a problem of geometry,
and especially of mechanics and of physics, and we have scarcely
found the logical and sesthetic preoccupations which are to give a

504 ALGEBRA AND ANALYSIS

physiognomy so different to so many mathematical works, above all
in the latter two thirds of the nineteenth century.

Not to anticipate, however, after so many examples of the in-
fluences of physics on the developments of analysis, we meet still a
new one, and one of the most memorable, in Fourier's theory of heat.
He commences by forming the partial differential equations which
govern temperature.

What are for a partial differential equation the conditions at the
limits permitting the determination of a solution?

For Fourier, the conditions are suggested by the physical problem,
and the methods that he followed have served as models to the
physicist-geometers of the first half of the last century.

One of these consists in forming a series with certain simple solu-
tions. Fourier thus obtained the first types of developments more
general than the trigonometric developments, as in the problem of
the cooling of a sphere, where he applies his theory to the terrestrial
globe, and investigates the law which governs the variations of
temperature in the ground, trying to go even as far as numerical
applications.

In the face of so many beautiful results, we understand the enthu-
siasm of Fourier which scintillates from every line of his preliminary
discourse. Speaking of mathematical analysis, he says, " There could
not be a language more universal, more simple, more exempt from
errors and from obscurities, that is to say, more worthy to express
the invariable relations of natural things. Considered under this
point of view, it is as extended as nature herself; it defines all sen-
sible relations, measures times, spaces; forces, temperatures. This
difficult science forms slowly, but it retains all the principles once
acquired. It grows and strengthens without cease in the midst of
so many errors of the human mind."

The eulogy is magnificent, but permeating it we see the tendency
which makes all analysis uniquely an auxiliary, however incom-
parable, of the natural sciences, a tendency, in conformity, as we
have seen, with the development of science during the preceding two
centuries; but we reach just here an epoch where new tendencies
appear.

Poisson having in a report on the Fundamenta recalled the re-
proach made by Fourier to Abel and Jacobi of not having occupied
themselves preferably with the movement of heat, Jacobi wrote to
Legendre: "It is true that Monsieur Fourier held the view that
the principal aim of mathematics was public utility, and the ex-
planation of natural phenomena; but a philosopher such as he
should have known that the unique aim of science is the honor of
the human spirit, and that from this point of view a question about
numbers is as important as a question about the system of the

DEVELOPMENT OF MATHEMATICAL ANALYSIS 505

world." This was without doubt also the opinion of the grand geo-
meter of Goettingen, who called mathematics the queen of the sciences,
and arithmetic the queen of mathematics.

It would be ridiculous to oppose one to the other these two
tendencies; the harmony of our science is in their synthesis.

The time was about to arrive when one would feel the need of
inspecting the foundations of the edifice, and of making the inventory
of accumulated wealth, using more of the critical spirit. Mathematical
thought was about to gather more force by retiring into itself; the
problems were exhausted for a time, and it is not well for all seekers
to stay on the same road. Moreover, difficulties and paradoxes
remaining unexplained made necessary the progress of pure theory.

The path on which this should move was traced in its large outlines,
and there it could move with independence without necessarily losing
contact with the problems set by geometry, mechanics, and physics.

At the same time more interest was to attach to the philosophic
and artistic side of mathematics, confiding in a sort of pree'stab-
lished harmony between our logical and aesthetic satisfactions and the
necessities of future applications.

Let us recall rapidly certain points in the history of the revision
of principles where Gauss, Cauchy, and Abel likewise were laborers
of the first hour. Precise definitions of continuous functions, and their
most immediate properties, simple rules on the convergence of series,
were formulated; and soon was established, under very general
conditions, the possibility of trigonometric developments, legiti-
matizing thus the boldness of Fourier.

Certain geometric intuitions relative to areas and. to arcs give
place to rigorous demonstration. The geometers of the eighteenth
century had necessarily sought to give account of the degree of the
generality of the solution of ordinary differential equations. Their
likeness to equations of finite differences led easily to the result; but
the demonstration so conducted must not be pressed very close.

Lagrange, in his lessons on the calculus of functions, had intro-
duced greater precision, and starting from Taylor's series, he saw
that the equation of order m leaves indeterminate the function,
and its m — 1 first derivatives for the initial value of the variable;
we are not surprised that Lagrange did not set himself the question
of convergence.

In twenty or thirty years the exigencies in the rigor of proofs had
grown. One knew that the two preceding modes of demonstration
are susceptible of all the precision necessary.

For the first, there was need of no new principle; for the second
it was necessary that the theory should develop in a new way. Up
to this point, the functions and the variables had remained real.
The consideration of complex variables comes to extend the field of

506 ALGEBRA AND ANALYSIS

analysis. The functions of a complex variable with unique derivative
are necessarily developable in Taylor's series; we come back thus
to the mode of development of which the author of the theory of
analytic functions had understood the interest, but of which the
importance could not be put fully in evidence in limiting one's self
to real variables. They also owe the grand r61e that they have not
ceased to play to the facility with which we can manage them, and
to their convenience in calculation.

The general theorems of the theory of analytic functions permitted
to reply with precision to questions remaining up to that time un-
decided, such as the degree of generality of the integrals of differential
equations. It became possible to push even to the end the demon-
stration sketched by Lagrange for an ordinary differential equa-
tion. For a partial differential equation or a system of such equations,
precise theorems were established. It is not that on this latter point
the results obtained, however important they may be, resolve
completely the diverse questions that may be «et ; because in mathe-
matical physics, and often in geometry, the conditions at the limits
are susceptible of forms so varied that the problem called Cauchy's
appears often under very severe form. I will shortly return to this
capital point.

IV

Without restricting ourselves to the historic order, we will follow
the development of mathematical physics during the last century,
in so far as it interests analysis.

The problems of calorific equilibrium lead to the equation already
encountered by Laplace in the study of attraction. Few equations
have been the object of so many works as this celebrated equation.
The conditions at the limits may be of divers forms. The simplest
case is that of the calorific equilibrium of a body of which we main-
tain the elements of the surface at given temperatures; from the
physical point of view, it may be regarded as evident that the tem-
perature, continuous within the interior since no source of heat is
there, is determined when it is given at the surface.

A more general case is that where, the state remaining permanent,
there might be radiation toward the outside with an emissive power
varying on the surface in accordance with a given law; in particular
the temperature may be given on one portion, while there is radiation
on another portion.

These questions, which are not yet resolved in their greatest gen-
erality, have greatly contributed to the orientation of the theory of
partial differential equations. They have called attention to types of
determinations of integrals, which would not have presented them-
selves in remaining at a point of view purely abstract.

DEVELOPMENT OF MATHEMATICAL ANALYSIS 507

Laplace's equation had been met already in hydrodynamics and
in the study of attraction inversely as the square of the distance.
This latter theory has led to putting in evidence the most essential
elements, such as the potentials of simple strata and of double
strata. Analytic combinations of the highest importance were there
met, which since have been notably generalized, such as Green's
formula.

The fundamental problems of static electricity belong to the
same order of ideas, and that was surely a beautiful triumph for
theory, the discovery of the celebrated theorem on electric phe-
nomena in the interior of hollow conductors, which later Faraday
rediscovered experimentally, without having known of Green's
memoir.

All this magnificent ensemble has remained the type of the theories
already old of mathematical physics, which seem to us almost to
have attained perfection, and which exercise still so happy an in-
fluence on the progress of pure analysis in suggesting to it the most
beautiful problems. The theory of functions offers us another mem-
orable affiliation.

There the analytic transformations which come into play are not
distinct from those we have met in the permanent movement of
heat. Certain fundamental problems of the theory of functions of
a complex variable lost then their abstract enunciation to take a
physical form, such as that of the distribution of temperature on
a closed surface of any connection and not radiating, in calorific
equilibrium with two sources of heat which necessarily correspond
to flows equal and of contrary signs. Transposing, we face a ques-
tion relative to Abelian integrals of the third species in the theory of
algebraic curves.

The examples which precede, where we have envisaged only the
equations of heat and of attraction, show that the influence of
physical theories has been exercised not only on the general nature
of the problems to be solved, but even in the details of the analytic
transformations. Thus is currently designated in recent memoirs on
partial differential equations under the name of Green's formula,
a formula inspired by the primitive formula of the English physicist.
The theory of dynamic electricity and that of magnetism, with
Ampere and Gauss, have been the origin of important progress; the
study of curvilinear integrals and that of the integrals of surfaces
have taken thence all their developments, and formulas, such as
that of Stokes which might also be called Ampere's formula, have
appeared for the first time in memoirs on physics. The equations
of the propagation of electricity, to which are attached the names of
Ohm and Kirchoff, while presenting a great analogy with those of
heat, offer often conditions at the limits a little different; we know

508 ALGEBRA AND ANALYSIS

all that telegraphy by cables owes to the profound discussion of a
Fourier's equation carried over into electricity.

The equations long ago written of hydrodynamics, the equations
of the theory of electricity, those of Maxwell and of Hertz in electro-
magnetism, have offered problems analogous to those recalled above,
but under conditions still more varied. Many unsurmounted diffi-
culties are there met with; but how many beautiful results we owe
to the study of particular cases, whose number one would wish to
see increase. To be noted also as interesting at once to analysis and
physics are the profound differences which the propagation may
present according to the phenomena studied. With equations such
as those of sound, we have propagation by waves; with the equa-
tion of heat, each variation is felt instantly at every distance, but
very little at a very great distance, and we cannot then speak of
velocity of propagation.

In other cases of which Kirchoff 's equation relative to the propa-
gation of electricity with induction and capacity offers the simplest
type, there is a wave front with a velocity determined but with a
remainder behind which does not vanish.

These diverse circumstances reveal very different properties of
integrals; their study has been delved into only in a few particular
cases, and it raises questions into which enter the most profound
notions of modern analysis.

I will enter into certain analytic details especially interesting for
mathematical physics.

The question of the generality of the solution of a partial differential
equation has presented some apparent paradoxes. For the same
equation, the number of arbitrary functions figuring in the general
integral was not always the same, following the form of the integral
envisaged. Thus Fourier, studying the equation of heat in an indefin-
ite medium, considers as evident that a solution will be determined
if its value for ^=0 is given, that is to say one arbitrary function of
the three coordinates x, y, z ; from the point of view of Cauchy , we
may consider, on the contrary, that in the general solution there are
two arbitrary functions of the three variables. In reality, the ques-
tion, as it has long been stated, has not a precise signification.

In the first place, when it is a question only of analytic functions,
any finite number of functions of any number of independent vari-
ables presents, from the arithmetical point of view, no greater gen-
erality than a single function of a single variable, since in the one
case and in the other the ensemble of coefficients of the development
forms an enumerable series. But there is something more. In reality,
beyond the conditions which are translated by given functions, an

DEVELOPMENT OF MATHEMATICAL ANALYSIS 509

integral is subjected to conditions of continuity, or is to become in-
finite in a determined manner for certain elements; one may so be
led to regard as equivalent to an arbitrary function the condition
of continuity in a given space, and then we clearly see how badly
formulated is the question of giving the number of the arbitrary
functions. It is at times a delicate matter to demonstrate that con-
ditions determine in a unique manner a solution, when we do not
wish to be contented with probabilities; it is then necessary to make
precise the manner in which the function and certain of. its deriva-
tives conduct themselves.

Thus in Fourier's problem relative to an indefinite medium cer-
tain hypotheses must be made about the function and its first
derivatives at infinity, if we wish to establish that the solution is
unique.

Formulas analogous to Green's render great services, but the
demonstrations one deduces from them are not always entirely
rigorous, implicitly supposing fulfilled for the limits conditions
which are not, a priori at least, necessary. This is, after so many
others, a new example of the evolution of exigencies in the rigor of
proofs.

We remark, moreover, that the new study, rendered necessary,
has often led to a better account of the nature of integrals.

True rigor is fecund, thus distinguishing itself from another purely
formal and tedious, which spreads a shadow over the problems it
touches.

The difficulties in the demonstration of the unity of a solution
may be very different according as it is question of equations of
which all the integrals are or are not analytic. This is an important
point, and shows that even when we wish to put them aside, it is
necessary sometimes to consider non-analytic functions.

Thus we cannot affirm that Cauchy's problem determines in a
unique manner one solution, the data of the problem being general,
that is to say not being characteristic.

This is surely the case, if we envisage only analytic integrals,
but with non-analytic integrals there may be contacts of order
infinite. And theory here does not outstrip applications; on the
contrary, as the following example shows:

Does the celebrated theorem of Lagrange on the potentials of
velocity in a perfect fluid hold good in a viscid fluid? Examples have
been given where the coordinates of different points of a viscous
fluid starting from rest are not expressible as analytic functions of
the time starting from the initial instant of the motion, and where
the nul rotations as well as all their derivatives with respect to the
time at this instant are, however, not identically nul; Lagrange 's
theorem, therefore, does not hold true.

510 ALGEBRA AND ANALYSIS

These considerations sufficiently show the interest it may have
to be assured that all the integrals of a system of partial differential
equations continuous as well as all their derivatives up to a deter-
mined order in a certain field of real variables are analytic functions;
it is understood, we suppose, there are in the equations only analytic
elements. We have for linear equations precise theorems, all the
integrals being analytic, if the characteristics are imaginary, and
very general propositions have also been obtained in other cases.

The conditions at the limits that one is led to assume are very
different according as it is question of an equation of which the
integrals are or are not analytic. A type of the first case is given
by the problem generalized by Dirichlet; conditions of continuity
there play an essential part, and, in general, the solution cannot
be prolonged from the two sides of the continuum which serves as
support to the data; it is no longer the same in the second case,
where the disposition of this support in relation to the characteris-
tics plays the principal r61e, and where the field of existence of the
solution presents itself under wholly different conditions.

All these notions, difficult to make precise in ordinary language
and fundamental for mathematical physics, are not of less interest
for infinitesimal geometry.

It will suffice to recall that ah1 the surfaces of constant positive
curvature are analytic, while there exist surfaces of constant 'nega-
tive curvature not analytic.

From antiquity has been felt the confused sentiment of a certain
economy in natural phenomena; one of the first precise examples
is furnished by Format's principle relative to the economy of time
in the transmission of light.

Then we came to recognize that the general equations of mechanics
correspond to a problem of minimum, or more exactly of variation,
and thus we obtained the principle of virtual velocities, then Ham-
ilton's principle, and that of least action. A great number of problems
appeared then as corresponding to minima of certain definite in-
tegrals.

This was a very important advance, because the existence of
a minimum could in many cases be regarded as evident, and con-
sequently the demonstration of the existence of the solution was
effected.

This reasoning has rendered immense services; the greatest geo-
meters, Gauss in the problem of the distribution of an attracting
mass corresponding to a given potential, Riemann in his theory of
Abelian functions, have been satisfied with it. To-day our attention
has been called to the dangers of this sort of demonstration; it is
possible for the minima to be simply limits and not to be actually
attained by veritable functions possessing the necessary properties

of continuity. We are, therefore, no longer content with the prob-
abilities offered by the reasoning long classic.

Whether we proceed indirectly or whether we seek to give a rigor-
ous proof of the existence of a function corresponding to the mini-
mum, the route is long and arduous.

Further, not the less will it be always useful to connect a ques-
tion of mechanics or of mathematical physics with a problem of
minimum; in this first of all is a source of fecund analytic trans-
formations, and besides in the very calculations of the investigation
of variations useful indications may appear, relative to the condi-
tions at the limits; a beautiful example of it was given by Kirchoff
in the delicate investigation of the conditions at the limits of the
equilibrium of flexure of plates.

VI

I have been led to expand particularly on partial differential
equations.

Examples chosen in rational mechanics and in celestial mechanics
would readily show the role which ordinary differential equations
play in the progress of these sciences whose history, as we have seen,
has been so narrowly bound to that of analysis.

When the hope of integrating with simple functions was lost, one
strove to find developments permitting to follow a phenomenon as long
as possible, or at least to obtain information of its qualitative bearing.

For practice, the methods of approximation form an extremely
important part of mathematics, and it is thus that the highest parts
of theoretic arithmetic find themselves connected with the. applied
sciences. As to series, the demonstrations themselves of the exist-
ence of integrals furnish them from the very first; thus Cauchy's
first method gives developments convergent as long as the integrals
and the differential coefficients remain continuous.

When any circumstance permits our foreseeing that such is always
the case, we obtain developments always convergent. In the pro-
blem of n bodies, we can in this way obtain developments valid so
long as there are no shocks.

If the bodies, instead of attracting, repel each other, this contin-
gency need not be feared and we should obtain developments valid
indefinitely; unhappily, as Fresnel said one day to Laplace, nature
is not concerned about analytic difficulties and the celestial bodies
attract instead of repelling each other.

One would even be tempted at times to go further than the great
physicist and say that nature has sown difficulties in the paths of
the analysts.

Thus, to take another example, we can generally decide, given a
system of differential equations of the first order, whether the gen-

512 ALGEBRA AND ANALYSIS

eral solution is stable or not about a point, and to find developments
in series valid for stable solutions it is only necessary that certain
inequalities be verified.

But if we apply these results to the equations of dynamics to dis-
cuss stability, we find ourselves exactly in the particular case which
is unfavorable. Nay, in general, here it is not possible to decide on
the stability; in the case of a function of forces having a maximum,
reasoning classic, but indirect, establishes the stability which cannot
be deduced from any development valid for every value of the time.

Do not lament these difficulties; they will be the source of future
progress.

Such are also the difficulties which still present to us, in spite of
so many works, the equations of celestial mechanics; the astro-
nomers have almost drawn from them, since Newton, by means of
series practically convergent and approximations happily con-
ducted, all that is necessary for the foretelling of the motions of the
heavenly bodies.

The analysts would ask more, but they no longer hope to attain
the integration by means of simple functions or developments al-
ways convergent.

What admirable recent researches have best taught them is the
immense difficulty of the problem; a new way has, however, been
opened by the study of particular solutions, such as the periodic
solutions and the asymptotic solutions which have already been
utilized. It is not perhaps so much because of the needs of practice
as in order not to avow itself vanquished, that analysis will never
resign itself to abandon, without a decisive victory, a subject where
it has met so many brilliant triumphs; and again, what more beau-
tiful field could the theories new-born or rejuvenated of the modern
doctrine of functions find, to essay their forces, than this classic
problem of n bodies?

It is a joy for the analyst to encounter in applications equations
that he can integrate with known functions, with transcendents
already classed.

Such encounters are unhapily rare; the problem of the pendulum,
the classic cases of the motion of a solid body around a fixed point,
are examples where the elliptic functions have permitted us to effect
the integration.

It would also be extremely interesting to encounter a question
of mechanics which might be the origin of the discovery of a new
transcendent possessing some remarkable property; I should be
embarrassed to give an example of it unless in carrying back to the
pendulum the de"but of the theory of elliptic functions.

The interpenetration between theory and applications is here
much less than in the questions of mathematical physics. Thus

DEVELOPMENT OF MATHEMATICAL ANALYSIS 513

is explained that, since forty years, the works on ordinary differ-
ential equations attached to analytic functions have had in great
part a theoretic character altogether abstract.

The pure theory has notably taken the advance; we have had
occasion to say that it was well it should be so, but evidently there
is here a question of measure, and we may hope to see the old pro-
blems profit by the progress accomplished.

It would not be over-difficult to give some examples, and I will re-
call only those linear differential equations, where figure arbitrary
parameters whose singular values are roots of entire transcendent
functions ; which in particular makes the successive harmonics of
a vibrating membrane correspond to the poles of a meromorphic
function.

It happens also that the theory may be an element of classifica-
tion in leading to seek conditions for which the solution falls under
a determined type, as for example that the integral may be uniform.
There have been and there yet will be many interesting discoveries
in this way, the case of the motion of a solid heavy body treated
by Madame de Kovalevski and where the Abelian functions were
utilized is a remarkable example.

VII

In studying the reciprocal relations of analysis with mechanics
and mathematical physics, we have on our way more than once
encountered the infinitesimal geometry, which has proposed so
many celebrated problems; in many difficult questions, the happy
combination of calculus and synthetic reasonings has realized con-
siderable progress, as is shown by the theories of applicable surfaces
and systems triply orthogonal.

It is another part of geometry which plays a grand role in certain
analytic researches, I mean the geometry of situation or analysis
situs. We know that Riemann made from this point of view a com-
plete study of the continuum of two dimensions, on which rests his
theory of algebraic functions and their integrals.

When this number of dimensions augments, the questions of
analysis situs become necessarily complicated; the geometric intui-
tion ceases, and the study becomes purely analytic, the mind being
guided solely by analogies which may be misleading and need to be
discussed very closely. The theory of algebraic functions of two
variables, which transports us into a space of four dimensions,
without getting from analysis situs an aid so fruitful as does the
theory of functions of one variable, owes it, however, useful orient-
ations.

There is also another order of questions where the geometry of
situation intervenes; in the study of curves traced on a surface and

514 ALGEBRA AND ANALYSIS

defined by differential equations, the connection of this surface plays
an important r6le; this happens for geodesic lines.

The notion of connexity, moreover, presented itself long ago in
analysis, when the study of electric currents and magnetism led
to non-uniform potentials; in a more general manner certain multi-
form integrals of some partial differential equations are met in
difficult theories, such as that of diffraction, and varied researches
must continue in this direction.

From a different point of view, I must yet recall the relations of
algebraic analysis with geometry, which manifest themselves so
elegantly in the theory of groups of finite order.

A regular polyhedron, say an icosahedron, is on the one hand the
solid that all the world knows; it is also, for the analyst, a group of
finite order, corresponding to the divers ways of making the poly-
hedron coincide with itself.

The investigation of all the types of groups of motion of finite
order interests not alone the geometers, but also the crystallo-
graphers; it goes back essentially to the study of groups of ternary
linear substitutions of determinant +1, and leads to the thirty-
two systems of symmetry of the crystallographers for the particular
complex.

The grouping in systems of polyhedra corresponding so as to fill
space exhausts all the possibilities in the investigation of the struc-
ture of crystals.

Since the epoch when the notion of group was introduced into
algebra by Galois, it has taken, in divers ways, considerable devel-
opment, so that to-day it is met in all parts of mathematics. In the
applications, it appears to us above all as an admirable instrument
of classification. Whether it is a question of substitution groups
or of Sophus Lie's transformation groups, whether it is a question
of algebraic equations or of differential equations, this comprehen-
sive doctrine permits explanation of the degree of difficulty of the
problems treated and teaches how to utilize the special circumstances
which present themselves; thus it should interest as well mechanics
and mathematical physics as pure analysis.

The degree of development of mechanics and physics has per-
mitted giving to almost all their theories a mathematical form;
certain hypotheses, the knowledge of elementary laws, have led
to differential relations which constitute the last form under which
these theories settle down, at least for a time. These latter have
seen little by little their field enlarge with the principles of thermo-
dynamics; to-day chemistry tends to take in its turn a mathemat-
ical form.

I will take as witness of it only the celebrated memoir of Gibbs
on the equilibrium of chemical systems, so analytic in character,

DEVELOPMENT OF MATHEMATICAL ANALYSIS 515

and where it needed some effort on the part of the chemists to
recognize, under their algebraic mantle, laws of high importance.

It seems that chemistry has to-day gotten out of the premathe-
matic period, by which every science begins, and that a day must
come when will be systematized grand theories, analogous to those
of our present mathematical physics, but far more vast, and com-
prising the ensemble of physicochemic phenomena.

It would be premature to ask if analysis will find in their develop-
ments the source of new progress; we do not even know before-
hand what analytic types one might find.

I have constantly spoken of differential equations ruling phe-
nomena; will this always be the final form which condenses a theory?
Of this I know nothing certain, but we should, however, remember
that many hypotheses have been made of more or less experimental
nature ; among them, one is what has been called the principle of
non-heredity, which postulates that the future of a system depends
only on its present state and its state at an instant infinitely near,
or, more briefly, that accelerations depend only on positions and
velocities.

We know that in certain cases this hypothesis is not admissible,
at least with the magnitudes directly envisaged; one has sometimes
misemployed on this subject the memory of matter, which recalls
its past, and has spoken in affected terms of the life of a morsel of
steel. Different attempts have been made to give a theory of these
phenomena, where a distant past seems to interfere; of them I need
not speak here. An analyst may think that in cases so complex it
is necessary to abandon the form of differential equations, and resign
one's self to envisage functional equations, where figure definite
integrals which will be the witness of a sort of heredity.

To see the interest which is attached at this moment to functional
equations, one might believe in a presentiment of the future needs
of science.

VIII

After having spoken of non-heredity, I scarcely dare touch the
question of the applications of analysis to biology.

It will be some time, no doubt, before one forms the functional
equations of biologic phenomena; the attempts so far made are
in a very modest order of ideas; yet efforts are being made to get
out of the purely qualitative field, to introduce quantitative meas-
ures. In the question of the variation of certain characteristics,
mensuration has been engaged in, and statistic measures which are
translated by curves of frequency. The modifications of these curves
with successive generations, their decompositions into distinct curves,
may give the measure of the stability of species or of the rapidity

516 ALGEBRA AND ANALYSIS

of mutations, and we know what interest attaches itself to these
questions in recent botanic researches. In all this so great is the
number of parameters that one questions whether the infinitesimal
method itself could be of any service. Some laws of a simple arith-
metic character like those of Mendel come occasionally to give
renewed confidence in the old aphorism which I cited in the begin-
ning, that all things are explained by numbers; but, in spite of
legitimate hopes, it is clear that, in its totality, biology is still far
from entering upon a period truly mathematical.

It is not so, according to certain economists, with potential econ-
omy. After Cournot, the Lausanne school made an effort extremely
interesting to introduce mathematical analysis into political econ-
omy.

Under certain hypotheses, which fit at least limiting cases, we
find in learned treatises an equation between the quantities of
merchandise and their prices, which recalls the equation of virtual
velocities in mechanics: this is the equation of economic equilib-
rium. A function of quantities plays in this theory an essential role
recalling that of the potential function. Moreover, the best author-
ized representatives of the school insist on the analogy of economic
phenomena with mechanical phenomena. "As rational mechanics,"
says one of them, " considers material points, pure economy con-
siders the homo oeconomicus."

Naturally, we find there also the analogues of Lagrange's equa-
tions, indispensable matrix of all mechanics.

While admiring these bold works, we fear lest the authors have
neglected certain hidden masses, as Helmholtz and Hertz would
have said. But although that may happen, there is in these doctrines
a curious application of mathematics, which, at least, in well-circum-
scribed cases, has already rendered great services.

I have terminated, messieurs, this summary history of some of
the applications of analysis, with the reflections which it has at
moments suggested to me. It is far from being complete; thus I have
omitted to speak of the calculus of probabilities, which demands
so much subtlety of mind, and of which Pascal refused to explain the
niceties to the Chevalier de Me"r6 because he was not a geometer.

Its practical utility is of the first rank, its theoretic interest has
always been great; it is further augmented to-day, thanks to the
importance taken by the researches that Maxwell called statistical
and which tend to envisage mechanics under a wholly new light.

I hope, however, to have shown, in this sketch, the origin and
the reason of the bonds. so profound which unite analysis to geometry
and physics, more generally to every science bearing on quantities
numerically measurable.

DEVELOPMENT OF MATHEMATICAL ANALYSIS 517

The reciprocal influence of analysis and physical theories has been
in this regard particularly instructive.

What does the future hold?

Problems more difficult, corresponding to an approximation of
higher order, will introduce complications which we can only vaguely
forecast, in speaking, as I have just done, of functional equations
replacing systematically our actual differential equations, or further
of integrations of equations infinite in number with an infinity of
unknown functions. But even though that happens, mathematical
analysis will alwrays remain that language which, according to the
mot of Fourier, has no symbols to express confused notions, a lan-
guage endowed with an admirable power of transformation and
capable of condensing in its formulas an immense number of results.

ON PRESENT PROBLEMS OF ALGEBRA AND ANALYSIS

BY HEINRICH MASCHKE

[Heinrich Maschke, Associate Professor of Mathematics, University of
Chicago, b. Breslau, Germany, October 24, 1853. A.B. Magdalenen Gym-
nasium, Breslau, 1872; Ph.D. Gottingen, 1880. Post-graduate Heidelberg,
Breslau, Berlin, and Gottingen. Professor Mathematics Lvisenstadt. Gym-
nasium, Berlin, 1880-90; Electric Engineer at Weston Electric Company,
Newark, New Jersey, 1890-92; Assistant Professor of Mathematics, Uni-
versity of Chicago, 1892-96.]

As set forth by the Committee directing the affairs of this Interna-
tional Congress, the address which I have the distinguished privilege
of delivering to-day shall be on "Present Problems in Algebra and
Analysis," - but it is not provided by the Committee how many
of these problems shall be treated.

The different branches of algebra and analysis which have been
investigated are so numerous that it would be quite impossible to
give an approximately exhaustive representation even only of the
most important problems, within the limits of the time allowed to
me. I, therefore, have confined myself to the minimum admissible
number, namely one, or rather one group of problems.

Of this one problem, however, this Section of Algebra and Analysis
has the right to expect that it is neither purely algebraic nor purely
analytic, but one which touches both fields; and at least in this
respect I hope that my selection has been fortunate.

I purpose to speak to-day on the Theory of Invariants of Quad-
ratic Differential Quantics. Invariants suggest at once algebra,
differential quantics: analysis. At the same time the subject also
leads into geometry, — it contains, for instance, a great part of
differential geometry and of geometry of hyperspace. But is there,
indeed, any algebraic or analytic problem which does not allow
geometrical interpretation in some way or other? And when it comes
to geometry of hyperspace, — it is then only geometrical language
that we are using, — what we are actually considering are analytic
or algebraic forms. Moreover, rigorous definitions and discussions
of geometrical propositions of an invariant character in particular
can only be given by tracing them back to their analytic origin.

In the following exposition I shall first speak on the various in-
variant expressions of differential quadratics as they occur in geo-
metry of two and more dimensions, and then take up the purely
analytic representation in the second part of the paper.

This corresponds also to the historical development of the sub-

PROBLEMS OF ALGEBRA AND ANALYSIS 519

ject: geometry has here as well as in many other branches of mathe-
matics indicated the problems which in their later development
turned out to be of paramount interest in pure analysis.

A few preliminary remarks concerning the nomenclature of the
different types of invariant expressions will be necessary.

To a given differential quadratic form

where the a^'s are functions of the n independent variables Xi,x2, . . .
xn, we apply a general point transformation of the variables x,

Xi = xi(yl,y,, . . .yn).

We observe that the differentials dx are then transformed into
linear expressions of the differentials dy with the Jacobian of the
x's with respect to the y's as the substitution-determinant which
we shall call r.

By this transformation A goes into

A' =2a'ikdyidyk.

Let now 0 be a function

(a) of the coefficients aik and their first, second, . . . derivatives,

(b) of U, V, . . , and their derivatives, where U, V, . . . are any
arbitrary functions of xlf x2, . . . xn.

If then 0 remains the same whether formed for the new or for
the old quantities, that is, if

dyX dyX dxX dxX

. . . V, . . . )
we say that 0 is an invariant (in the wider sense) of A.

If 0 contains only the a^'s and their derivatives, we call it an
invariant proper, and its order the order of the highest derivative
occurring in it. If <? contains also one or more arbitrary functions
U,V, . . . we call it a differential parameter, the definition of order
being the same as before.

If more than one differential quadratic is given it is easily under-
stood what is meant by simultaneous invariants and simultaneous
differential parameters.

In strict analogy with the algebraic theory of invariants we call
covariants expressions 0 of the above invariantive nature, provided
that we also allow the differentials dx to enter into <?.

The first and the most important example of a differential quad-
ratic quantic is the square of the arc-element on a surface

ds2 = Edu2 + 2Fdudv+Gdv2.

It was Gauss who made (1827), in his Disquisitiones generates
circa superficies curvas, this expression the fundamental object of

520

ALGEBRA AND ANALYSIS

investigation. He also gave, in what has been called after him the
Gaussian Curvature

the first example of an invariant. Gauss defines this curvature
geometrically and finds for it the analytic expression

LN-M2
EG-F2

which is a simultaneous invariant of two differential quantics,

ds2
namely, of ds2 and of — =Ldu2+2Mdudv+Ndv2.

This shows that K is independent of the w,v-system on the
surface. And now Gauss expresses K in terms of E, F, G and the
first and second derivatives of these quantities alone. A direct
demonstration that K is an invariant proper of the differential
quantic ds2 alone, — that is, without passing through the second

ds2

differential quantic — , — is of course desirable.1 Each one of the
P

general methods of treating the theory of invariants, which will be
discussed in the latter part of this paper, furnishes such a direct
proof. In particular, the aspect of the formula for K, on p. 528,
deduced by the symbolic method, shows immediately the invariant
character of K.

Differential parameters were introduced into differential geometry
by Beltrami in 1863. These are the well-known expressions

dv dv

EG-F2

+ Q

du dv dv duj du du

V)

' I-

EG-1

~<&-F*f
du dv

pa
d

do —,&V~\
E -- F—

dv du

VEG-F2\du

_VEG-F2_

dv

_ VEG-F2]}

where <p and <f> are the arbitrary functions which take the place of
U, V in our general definition of differential parameters. Beltrami
adopted the name "differential parameters" and also the notation

> Cf . on this subject the interesting paper by Knoblauch : " Der Gauss'sche Satz
vom Krummungsmass," Sitzungsbericnte der Berliner Mathem. Gesellschaft. April
27, 1904.

PROBLEMS OF ALGEBRA AND ANALYSIS 521

J from Lame, who, in his Lecons sur les coordonnees curvilignes,
defined in 1859 his differential parameters

,oy/ \oz

02(£> 0(0 0 <p

dx2 3y2 3z2

for the three-dimensional case where the arc-element is of the form

ds2=dx2+dy2+dz2.

Lame recognized the fundamental importance of these quantities
and made a systematical use of them on account of their invariance
with respect to any point-transformation preserving the form ds2.

The general theory of invariants defines the differential parameters
J, and J2 for the case of n variables. From these general expressions
Beltrami's differential parameters are directly obtained for n = 2,
Lame's quantities (Ji)2 and J2 for the special form of ds2 in the case
n = 3.

The number of differential parameters is of course infinite, but
Darboux in his Lecons sur la theorie generate des surfaces has proved
that all of them are expressible by means of Jt, J2, p and the evident
differential parameter

du ~3v~ 3v 3u

VEG-F2

(by forming, for instance, Ji(J2<^) etc.) — an important theorem
which has later been extended by Staeckel to an analogous theorem
for the case of n variables.

The expression J^ occurs already in Gauss's Disquisitiones.
By taking as parameter curves a singly infinite system of geodesies
and its orthogonal trajectories he transforms the arc-element into
the form

ds2 =dr2 +m2d<p2

and shows that r satisfies the differential equation

J,r = l.

An important differential parameter is the geodesic curvature.
Its expression was thrown by Bonnet into a form which is easily
recognized as a differential parameter (of the second order). Its
numerator =0 represents the differential equation of geodesic lines
in an invariant form.

Since a transformation of the two independent variables u, v which
preserves the same value of ds2 can also be considered as a transfor-
mation of two surfaces which are applicable to each other, it follows
that all invariants of ds2 are also invariants of a surface with respect
to the process of bending. From this reason these invariants have

522 ALGEBRA AND ANALYSIS

been called by Weingarten and Knoblauch, who were among the first
writers emphasizing and developing to a certain extent the invari-
antive side of differential geometry, in the case of invariants proper,
" Biegungsinvarianten," in the case of differential parameters, " Bie-
gungscovarianten," and this notation has been more or less generally
adopted. The notation " Biegungscovarianten " does not agree with
the definition of a covariant given above, but a differential para-
meter of ds2 can easily be modified into a covariantive form by
replacing according to the differential equation of the curve

U(u,v) = const.

3U dU

the derivatives — — and — - by udv and — udu.
Su dv

A surface is completely defined, apart from its location in space,
when in addition to the quadratic form ds2 also

ds2

- =Ldu2+2Mdudv+Ndv2
P

is given, where p denotes the radius of curvature along ds, — a the-
orem which was proved (1867) by Bonnet.

With these two differential quantics given, we can now at once
form simultaneous invariants and differential parameters. The six
coefficients, E, F, G, L, M, N are, however, not independent; they
are related by three partial differential equations, — the Gaussian
relation and the two Codazzi-Mainardi equations. These three
relations are expressible in an invariantive form. The Gaussian re-
lation is

W-M* dE

while the two Codazzi formulas are given by the identical vanishing
of one simultaneous linear covariant.

As examples of simultaneous differential parameters and covariants
I mention the expressions which, when set equal to zero, represent
the differential equations of conjugate lines, asymptotic lines, and
lines of curvature. The differential equation of lines of curvature, for
instance, if written in terms of du, dv represents a linear simultaneous
covariant; if written as a partial differential equation derived from

U(u,v) = const.

it represents a simultaneous differential parameter involving the
arbitrary function U. The differential equation of conjugate lines,
if written in two sets of differentials du, dv and du, 8v represents
a bilinear simultaneous covariant; if written as a partial differential
equation it represents a differential parameter involving two arbi-
trary functions U and V.

The theory of invariants of the above two differential quadratics,

PROBLEMS OF ALGEBRA AND ANALYSIS 523

together with the condition of the vanishing of one simultaneous
invariant proper and one simultaneous covariant, dominates then,
in a certain sense, the whole of differential geometry.

Passing now to the case of n variables we may consider the differ-
ential quadratic form

^aikdxfdxk =ds2

i,A-=l

as the square of the arc in a hyperspace of n dimensions.

The fundamental role which the Gaussian curvature plays in the
case n = 2 is here represented by -an invariant expresson of ds2 which
- in a certain sense — might be regarded as a generalization of the
Gaussian curvature, namely, the Riemann curvature of the hyper-
space. Riemann 's investigations on this subject are found in his
paper, Ueber die Hypothesen, welche der Geometric zu Grunde liegen,
and in the mathematical supplement to it Commentatio mathematica,
etc. in the prize-problem of the Parisian Academy, 1861.

The geometrical definition of the Riemann curvature is briefly the
following: Starting from any point P with the coordinates x^ we
consider two linear directions defined by the increments dxi and dxi.
If we remain in the vicinity of P these two directions define a plane
of two dimensions and the determinants

may be considered as the coordinates of this plane. If now we
draw geodesic lines from the point P whose initial arc-elements
lie all in this plane, then these geodesies define a surface of two di-
mensions and the Gaussian curvature of this geodesic surface at the
point P is the Riemann curvature. The analytic expression for it is

2 (ikrs) (dx{dxs — dxsdx{) (dxkdxr — dxrdxk)

R = — $ — •

2(aikars — airaks)(dxioxs—dxsoxi)(dxkoxr — dxrdxlc,

where the sum is to be taken over all values of i, k, r, s from 1 to n
with the exception of those for which i =k or r =s.

The coefficients (ikrs) are certain quantities depending on the
coefficients aik, their first and second derivatives; they occur in the
literature mostly under the name of the " Christoffel quadruple index
symbols." A better, certainly shorter, notation would be the one
used by Ricci, namely, " Riemann symbols."

The Riemann curvature R is an invariant expression, and as its
form shows it is a covariant of two sets of differentials. For n = 2
it is identical with the Gaussian curvature. For greater numbers
n the value of R depends, at a given point, on the plane-direction
at that point and in general varies with the plane. If it should be
constant for all plane-directions through one point, and if this is
so for all the points, then R is, as Schur has shown, altogether con-
stant that is, for every point.

524 ALGEBRA AND ANALYSIS

Spaces of constant Riemann curvature have been the object of
numerous interesting investigations, but these are more or less of
a specific geometric character.

If in particular R is zero, then all the Riemann symbols vanish
and it can easily be shown that ds2 can be transformed into the
sum of n squares

The converse is true. In this case the hyperspace of n dimensions
is called a flat or also Euclidean space.
In every case the quadratic

can be transformed into

n+r

where r has the maximum value n^~)' We might say then that the
given hyperspace of n dimensions is always contained in an
Euclidean space of n+r dimensions, where r is one of the numbers,

n 1 n(n-l)

v) *•> 2

The number r is evidently characteristic for the hyperspace the
square of the arc-element of which is the given quadratic. This
number r has been called by Ricci the class of the given differential
quadratic quantic. It is evident that this class is an invariant num-
ber, and the condition that a given differential quadratic be of class
r must certainly be an invariantive condition. For r=0 we have
just seen that the condition is R = 0. For higher values of r no at-
tempt has yet been made, so far as I know, to establish this invari-
antive condition though this problem is certainly one of fundamental
interest.

Beltrami, in his paper, Teoria generale dei parametri differenziali,
has extended the definition of his differential parameters to the
case of n variables. The definition, for instance, of the first differ-
ential parameters is

! " dtp d<p

A&=a 2 A'*-F- T-
a 0=1 dxi dxk

where Aik denotes the minor of the element a^ in the determinant

I flf* I =«•

Beltrami shows that by means of the geodesies emanating from one
point and of the hypersurfaces orthogonal to them he can choose
his parameters such that ds2 is transformed into

»,*-=!

where r satisfies the equation A^r = 1, and that thus Gauss's theorems

PROBLEMS OF ALGEBRA AND ANALYSIS 525

on geodesic polar coordinates for n=2 admit a perfect analogon
in hyperspace. Also in hyperspace then the determination of systems
of geodesies amounts to the integration of the partial differential
equation

J^ = l.

This leads now to the application of differential quadratics to
analytic mechanics. If we write down the expression of th'e vis
viva of a (holonomous) material system in terms of generalized
coordinates qv q2, • • -qn-

T = \Zaikd^ ***
dt dt

we have at once in

2Tdt*=ds2
a differential quadratic before us.

If no external forces act on the system, then a geodesic line of ds2
represents at once, as also Beltrami has shown, a path of the sys-
tem. Thus the mechanical problem is practically reduced to the
integration of the equation Ajp = \.

In the case of the existence of external forces having a potential
U, the above differential quantic has to be replaced by

I '(U +h)aikdqidqk

and the mechanical problem is equivalent to the integration of the
equation

Ai<p = U+h

where A\<p is the differential parameter of the quadratic form de-
noted before by ds2.

A detailed exposition of the above-mentioned researches of Bel-
trami, as well as this application to mechanics, is given in the second
volume of Darboux's Lecons sur la theorie des surfaces.

Passing now to the second part of my address, the purely ana-
lytic theory of invariants of differential quadratics, I have first
to discuss that paper which forms the foundation of almost all
later literature on the subject: Christoffel's article in Crelle's Jour-
nal, vol. LXX (1870), "Ueber die Transformation der homogenen
Differentialausdriicke des zweiten Grades."

Christoffel puts his problem in this form: Given two differential
quadratics

A —la^dxidxk and A' =Ia' ikdyidy^,

what are the necessary and sufficient conditions for the equivalence
of the two quadratics, that is, for the existence of a transformation
of one quantic into the other; and if these conditions are established
how can the required transformation be determined? (I should men-
tion that Lame" in his already quoted work, Lecons sur les coor-

526 ALGEBRA AND ANALYSIS

danntes curvilignes, treats and solves the analogous problem for the
case A =dx2+dy2+dz2).

Since the differentials dx are substituted linearly in terms of the
dy there exists one and only one algebraic condition for the trans-
formation, namely,

This condition would be sufficient if the coefficients a^ and the
elements of the determinant r were constants. In our case, how-
ever, other conditions must be satisfied, namely, the conditions of
integrability in order that the expressions for the dx's are com-
plete differentials. This is the way in which Christoffel introduces
his problem to the reader.

The difficulty lies in the fact that the integrability conditions
lead at once to a great number of partial differential equations of
an apparently highly complex character. But Christoffel succeeds
in substituting for all these partial differential equations a purely
algebraic problem: The equivalence of two finite systems of alge-
braic forms in the sense of the algebraic theory of invariants. If
this equivalence is satisfied, — which is merely a question of algebra,

- no further discussion of the integrability conditions is required;
they are all taken care of by the equivalence of the two systems.

For the following it will be necessary to sketch briefly the char-
acter of these forms.

The first is the quadratic form A itself. The next form is a quad-
rilinear covariant GI in four sets of differentials dx1, dx2, dx3, dx*,
the coefficients of which are precisely the quantities (i k r s ) — the
"Christoffel quadruple index symbols" or the "Riemann symbols"

- which occur in the expression for the Riemann curvature:

G =

It is highly interesting to observe how the quantities ( i k r s )
have entered into the theory from two so apparently different stand-
points. Christoffel found these expressions quite independently.
Though Riemann's paper was written in 1861, that is, before Chris-
toffePs article which appeared in 1870, it was only published in 1876,
ten years after Riemann's death, by Weber-Dedekind.

For the deduction of the following forms Gb, G9, — ... — these
forms are covariants linear in resp. 5, «, . . . sets of differentials -
Christoffel uses a certain reduction process. The coefficients (Xikr s)
for instance of (?5 are obtained from (i k r s) first by differentiating
(i k r s) with respect to zx and then by the addition of a sum of 5n
terms which are linear in the different symbols ( i k r s ) with co-
efficients depending on the so-called Christoffel triple index sym-
bols of the second kind — expressions involving the quantities
aac and their first derivatives.

PROBLEMS OF ALGEBRA AND ANALYSIS 527

Continuing in this way Christoffel obtains a well-defined set of
co variants G&, Ge, . . . , and this is his final result: the necessary
and sufficient condition for the equivalence of the two differential
quadratics is the algebraic equivalence — in the sense of the alge-
braic theory of invariants — of the forms A, G4, Gs, . . . G^and A',
G'4, G'5 . . . G'M, where /< is a certain finite number.

In several papers covering the period from about 1884 up to the
present time Ricci has worked out in a systematical way the funda-
mental principles of ChristoffeFs investigation, and has applied his
theory to many problems in analysis, geometry, mechanics, and
mathematical physics. He recognized in particular the importance
of ChristoffeFs deduction of the co variants GA+1 from GA. He found
that this process of deduction can be applied with a proper modifi-
cation to any functions of the x's and the az-&'s and that whenever
invariantive relations with respect to the fundamental differential
quadratic A come into question, this process is always of vital im-
portance. He calls this process covariantive differentiation with
respect to the fundamental quadratic A. On the systematical use
of this covariantive differentiation Ricci based a calculus which he
called Calcolo differenziale assoluto.

A collection of all his various investigations is given in two places:

(1) In a paper published, together with Levi-Civitta in the Math.
Annalcn, vol. LIV.

(2) In his Lezioni sulla teoria delle super-fide, Verona, Padua, 1898.
In the introduction of these autographed lectures he presents a

complete exposition of his absolute differential calculus. Charac-
teristic is the way in which he treats in his Lezioni the differential
geometry. He divides it into two parts:

(1) Properties of surfaces depending on the one differential quad-
ratic ds2.

(2) Properties of surfaces depending on the two quadratics

ds2
ds2 and — .

P
We are here chiefly interested in his applications to the theory of

differential invariants. This is the result in his language: In order
to obtain all invariants proper and differential parameters of order //.
it is sufficient to determine the algebraic invariants of the system
of the following forms:

(1) The fundamental differential quantic A.

(2) The covariantive derivatives of the arbitrary functions

t/, V, ... up to the order /*.

(3) (for j«>l) the quadrilinear covariant (74 and its covariantivo
derivatives up to the order « — 2.

Another treatment of the invariant theory of differential quan<

528 ALGEBRA AND ANALYSIS

tics was given by myself. I applied a symbolic method to the theory
which consists chiefly in identifying the fundamental quadratic

with the square of a linear expression

Widxy

by setting fifk=dik- This is strictly analogous to the introduction of
symbols in the algebraic theory. The difference, of course, comes
in at once when we have to consider also the derivatives of a,*.

A systematic development leads to expressions and formulas
which with respect to simplicity and shortness are as superior to
the formulas of the ordinary notation as the formulas of the so-
called symbolic notation in the algebraic theory are superior to the
non-symbolic expressions.

As examples I give the most important invariant expressions
for the case n=2.

Let us introduce the abbreviation

(Pi Q* -P, Qi ) =(PQ), where P*=— etc.;

— a2i2

let further /, <p, $ . . .be symbols of .A, so that

and let f/, F ... be arbitrary functions of xl} xr
Then we have

=Ai U,

= 2K (Gaussian curvature),
<p) (<pU) ((/£/)£/) : (Ji t/)3 =Geodesic curvature of curve U=const.
To give also some examples of simultaneous invariant expressions
let /, <p, . . . be as before symbols of

Edu2+2Fdudv+Gdv2
and F, d> . . . symbols of

Lduz+2Mdudv +Ndv2,
Then:

(/F)2 = mean curvature.
The differential equations

of asymptotic curves U =c are (Ft/)2=0,
of conjugate curves U =c, V=c: (FU)(FV)=Q,
of lines of curvature U = c : (fF) (fU) (FU) = 0.
The equation (fo) (<pF) ( (/^) C7) =0 gives the two Cadazzi formulas
by setting the coefficients of C7, and U? separately equal to zero.

In these examples the invariant expressions always appear as
products of factors of the type (P.R). The general theorem holds
that any product of factors of this type represents always an in-

PROBLEMS OF ALGEBRA AND ANALYSIS 529

variant expression provided that the symbols/, <p, . . . , F, <D, . . .
occur in such a connection as to permit actual meaning.

The symbolic representation of invariant expressions suggested
by the case n=2 can without essential difficulty be extended to the
general case of n variables. In this treatment of the subject all the
essential quantities entering into the theory present themselves quite
naturally; they lie, so to say, on the surface; so, for instance, all the
Christoffel symbols of the different kinds including the Riemann
symbols and in particular also the process of covariantive differ-
entiation.

The results of my investigation are chiefly laid down in the paper
"A symbolic treatment of the theory of invariants of quadratic
differential quantics of n variables," Transactions of the American
Mathematical Society, vol. iv.

A third method of investigation of our theory of invariants is
based on Lie's theory of continuous groups. The general point
transformation by which A is transformed into A' defines a so-
called "infinite" continuous group. In order to obtain the invari-
ants of A, this group must first be "extended" in Lie's sense to
include the coefficients a^ of A and also the arbitrary functions
involved in the differential parameters.

Lie himself developed a short outline of the determination of
invariants in the second volume of the Mathematische Annalen for
the case n=2, and indicated in particular how the Gaussian curv-
ature and the parameter Ji^> could be found. The general plan of
investigation was taken up in the sixteenth volume of the Acta
Mathematica by Zorowski, who studied the case n = 2 in detail, adding
the complete computation of the Gaussian curvature and the most
important differential parameters.

An extension of Lie's methods to the general case of n variables
as far as the actual determination of invariants is concerned has,
so far as I know, not yet been made; only the problem of deter-
mining the number of functionally independent invariants of a given
order has been taken up. It seems that Lie's method is especially
well adapted to this particular problem. In a paper in the Atti del
Reale Institute Veneto (1897), Levi-Civitta found a lower limit for the
number of invariants of a given order. The actual number was
determined by Haskins in the Transactions of the American Mathe-
matical Society, vol. in, for the case of invariants proper (including
also simultaneous invariants) and in vol. v, of differential parameters.

I am at the end of my paper. I have attempted to show, in a
compendious way, what has been done in this attractive field of
research which is so closely connected with various interesting parts

530 ALGEBRA AND ANALYSIS

of pure and applied mathematics. The number of problems that re-
main to be solved are numerous. Excepting the lowest cases as to
the number of variables and the order of the invariants, not much
more than the mere existence of the invariants is known, so that
we have hardly the right to speak of a theory of these invariants.

When it comes to the question which of the different methods
will be best adapted to a further systematical study of the subject,
it seems probable that a combination of two or more of them will
be the most promising one. But here, as always, it is the man, not
the method, that solves the problem.

SHORT PAPERS

The Section of Algebra and Analysis attracted wide interest and caused many
supplementary papers on various topics to be submitted. It is impossible to give
a resume of these, as their analytical nature demands that they be printed in full
or not at all.

The first paper was presented by Professor G. A. Miller, of Leland Stanford Jr.
University, on the " Bearing of Several Recent Theorems on Group Theory."

The second paper was read by Professor James Birney Shaw, of Milliken
University, on " Linear Associative Algebra."

The third paper was presented by Professor M. W. Haskell, of the University
of California, on "The Reduction of any Collineation to a Product of Perspect-
ive Collineations."

The fourth paper was presented by Professor M. B. Porter, of the University
of Texas, "On Functions defined by an Infinite Series of Analytic Functions
of a Complex Variable."

The fifth paper was presented by Professor Edward V. Huntington, of Harvard
University, on "A Set of Postulates for Real Algebra comprising Postulates for
a One Dimensional Continuum and for the Theory of Groups."

The sixth paper was presented by Professor J. I. Hutchinson, of Cornell Uni-
versity, on " Uniformizing of Algebraic Functions."

The seventh paper was read by Professor E. R. Hedrick, of the University of
Missouri, on " Generalization of the Analytic Functions of a Complex Variable."

SECTION B — GEOMETRY

SECTION B — GEOMETRY

(Hall 9, September 24, 10 a. TO.)

CHAIRMAN: PROFESSOR M. W. HASKELL, University of California.
SPEAKERS: M. JEAN GASTON DARBOTJX, Perpetual Secretary of the Academy of

Sciences, Paris.

DR. EDWARD KASNER, Columbia University.
SECRETARY: PROFESSOR THOMAS J. HOLGATE, Northwestern University.

A STUDY OF THE DEVELOPMENT OF GEOMETRIC

METHODS

BY M. JEAN GASTON DARBOUX

(Translated from the French by Professor George Bruce Hoisted, Kenyon College)

[JeanGastonDarboux, Perpetual Secretary Academy of Sciences, Paris; Doyen
Honorary, Professor of Higher Geometry of the Faculty of Sciences, Paris,
b. August 13, 1842, Nimes, France. Dr.Sc., LL.D., University of Cambridge,
University of Christiania, University of Heidelberg, et al. Professor of
Special Mathematics, Lyce"e Louis le Grand, 1867—73; Master of Confer-
ences in Superior Normal Schools, Paris, 1873-81; Professor Suppleant of
Rational Mechanics and Higher Geometry, The Sorbonne, 1873-81; since
1881, Professor Titulaire of the Faculty of Sciences, and Doyen of the Fac-
ulty of Sciences since 1889; also Professor in Higher Normal School for
Schools of Science ; Member of Bureau des Longitudes; President of the
First General Assembly of the International Association of Academies; and
Honorary Vice-President for France of the Congress of Arts and Science;
Member of Institute of France, Royal Society of London; Academies of
Berlin, St. Petersburg, Rome, Amsterdam, Munich, Stockholm; American
Philosophical Society, et al. Author of many publications and addresses
on Mathematics, and editor of the Bulletin of Science of Mathematics.}

To appreciate the progress geometry has made during the cen-
tury just ended, it is of advantage to cast a rapid glance over the
state of mathematical science at the beginning of the nineteenth
century.

We know that, in the last period of his life, Lagrange, fatigued by
the researches in analysis and mechanics, which assured him, however,
an immortal glory, neglected mathematics for chemistry (which,
according to him, was easy as algebra), for physics, for philosophic
speculations.

This mood of Lagrange we almost always find at certain moments
of the life of the greatest savants. The new ideas which came to
them in the fecund period of youth and which they introduced into
the common domain have given them all they could have expected;
they have fulfilled their task and feel the need of turning their

536 GEOMETRY

mental activity towards wholly new subjects. This need, as we
recognize, manifested itself with particular force at the epoch of La-
grange. At this moment, in fact, the programme of researches opened
to geometers by the discovery of the infinitesimal calculus appeared
very nearly finished up. Some differential equations more or less
complicated to integrate, some chapters to add to the integral
calculus, and one seemed about to touch the very outmost bounds
of science.

Laplace had achieved the explanation of the system of the world
and laid the foundations of molecular physics. New ways opened
before the experimental sciences and prepared the astonishing
development they received in the course of the century just ended.
Ampere, Poisson, Fourier, and Cauchy himself, the creator of the
theory of imaginaries, were occupied above all in studying the appli-
cation of the analytic methods to molecular physics, and seemed to
believe that outside this new domain, which they hastened to cover,
the outlines of theory and science were finally fixed.

Modern geometry, a glory we must claim for it, came, after the
end of the eighteenth century, to contribute in large measure to the
renewing of all mathematical science, by offering to research a way
new and fertile, and above all in showing us, by brilliant successes,
that general methods are not everything in science, and that even
in the simplest subject there is much for an ingenious and inventive
mind to do.

The beautiful geometric demonstrations of Huygens, of Newton,
and of Clairaut were forgotten or neglected. The fine ideas introduced
by Desargues and Pascal had remained without development and
appeared to have fallen on sterile ground.

Carnot, by his Essai sur les transversales and his Geometric de.
position, above all Monge, by the creation of descriptive geometry
and by his beautiful theories on the generation of surfaces, came to
renew a chain which seemed broken. Thanks to them, the conceptions
of the inventors of analytic geometry, Descartes and Fermat, retook
alongside the infinitesimal calculus of Leibnitz and Newton the place
they had lost, yet should never have ceased to occupy. With his
geometry, said Lagrange, speaking of Monge, this demon of a man
will make himself immortal.

Arid, in fact, not only has descriptive geometry made it possible
to coordinate and perfect the procedures employed in all the arts
where precision of form is a condition of success and of excellence for
the work and its products; but it appeared as the graphic translation
of a geometry, general and purely rational, of which numerous and
important researches have demonstrated the happy. fertility.

Moreover, beside the Geometric descriptive we must not forget
to place that other masterpiece, the Application de I'analyse d la

DEVELOPMENT OF GEOMETRIC METHODS 537

geometric ; nor should we forget that to Monge are due the notion
of lines of curvature and the elegant integration of the differential
equation of these lines for the case of the ellipsoid, which, it is said,
Lagrange envied him. To be stressed is this character of unity of the
work of Monge.

The renewer of modern geometry has shown us from the beginning,
what his successors have perhaps forgotten, that the alliance of
geometry and analysis is useful and fruitful, that this alliance is
perhaps for each a condition of success.

II

In the school of Monge were formed many geometers: Hachette,
Brianchon, Chappuis, Binet, Lancret, Dupin, Malus, Gaultier de
Tours, Poncelet, Chasles, et al. Among these Poncelet takes first
rank. Neglecting, in the works of Monge, everything pertaining to
the analysis of Descartes or concerning infinitesimal geometry, he
devoted himself exclusively to developing the germs contained in
the purely geometric researches of his illustrious predecessor.

Made prisoner by the Russians in 1813 at the passage of the Dnieper
and incarcerated at Saratoff, Poncelet employed the leisure captivity
left him in the demonstration of the principles which he has developed
in the Traite des proprietes projectives des figures, issued in 1822,
and in the great memoirs on reciprocal polars and on harmonic
means, which go back nearly to the same epoch. So we may say the
modern geometry was born at Saratoff.

Renewing the chain broken since Pascal and Desargues, Poncelet
introduced at the same time homology and reciprocal polars, putting
thus in evidence, from the beginning, the fruitful ideas on which the
science has evolved during fifty years.

Presented in opposition to analytic geometr}', the methods of Ponce-
let were not favorably received by the French analysts. But such
were their importance and their novelty, that without delay they
aroused, from divers sides, the most profound researches.

Poncelet had been alone in discovering the principles; on the
contrary, many geometers appeared almost simultaneously to study
them on all sides and to deduce from them the essential results which
they implicitly contained.

At this epoch, Gergonne was brilliantly editing a periodical which
has to-day for the history of geometry an inestimable value. The
Annales de Mathematiqucs, published at Nimes from 1810 to 1831.
was during more than fifteen years the only journal in the entire
world devoted exclusively to mathematical researches.

Gergonne, who, in many regards, was a model editor for a scienti-
fic journal, had the defects of his qualities; he collaborated, often

538 GEOMETRY

against their will, with the authors of the memoirs sent him, rewrote
them, and sometimes made them say more or less than they would
have wished. Be that as it may, he was greatly struck by the origin-
ality and range of Poncelet's discoveries.

In geometry some simple methods of transformation of figures
were already known; homology even had been employed in the plane,
but without extending it to space, as did Poncelet, and especially
without recognizing its power and fruitfulness. Moreover, all these
transformations were punctual ; that is to say, they made correspond
a point to a point.

In introducing polar reciprocals, Poncelet was in the highest
degree creative, because he gave the first example of a transformation
in which to a point corresponded something other than a point.

Every method of transformation enables us to multiply the num-
ber of theorems, but that of polar reciprocals had the advantage of
making correspond to a proposition another proposition of wholly
different aspect. This was a fact essentially new. To put it in evi-
dence, Gergonne invented the system, which since has had so much
success, of memoirs printed in double columns with correlative
propositions in juxtaposition; and he had the idea of substituting
for Poncelet's demonstrations, which required an intermediary
curve or surface of the second degree, the famous "principle of
duality," of which the signification, a little vague at first, was suffi-
ciently cleared up by the discussions which took place on this subject
between Gergonne, Poncelet, and Pluecker.

Bobillier, Chasles. Steiner, Lame", Sturm, and many others whose
names escape me, were, at the same time as Pluecker and Poncelet,
assiduous collaborators of the Annales de Mathematiques. Gergonne,
having become rector of the Academy of Montpellier, was forced to
suspend in 1831 the publication of his journal. But the success it had
obtained, the taste for research it had contributed to develop, had
commenced to bear their fruit. Que"telet had established in Belgium
the Correspondance mathematique et physique. Crelle, from 1826,
brought out at Berlin the first sheets of his celebrated journal, where
he published the memoirs of Abel, of Jacobi, of Steiner.

A great number of separate works began also to appear, wherein
the principles of modern geometry were powerfully expounded and
developed.

First came in 1827 the Barycentrische Calcul of Moebius, a work
truly original, remarkable for the profundity of its conceptions, the
elegance and the rigor of its exposition; then in 1828 the Analytisch-
geometrische Entwickelungen of Pluecker, of which the second part
appeared in 1831, and which was soon followed by the System der
analytischen Geometrie of the same author, published at Berlin in
1835.

DEVELOPMENT OF GEOMETRIC METHODS. 539

In 1832 Steiner brought out at Berlin his great work: Systemat-
ische Entwickelung der Abhaengigkeit der geometrischen Gestalten von
einander, and, the following year, Die geometrischen Konstruktionen
ausgefuehrt mittels der geraden Linie und eines festen Kreises, where
was confirmed by the most elegant examples a proposition of Pon-
celet's relative to the employment of a single circle for the geometric
constructions.

Finally, in 1830, Chasles sent to the Academy of Brussels, which
happily inspired had offered a prize for a study of the principles of
modern geometry, his celebrated Apergu historique sur I'origine et
le developpement des methodes en geometric, followed by Memoire
sur deux principes generaux de la science : la dualite et I'homographie,
which was published only in 1837.

Time would fail us to give a worthy appreciation of these beautiful
works and to apportion the share of each. Moreover, to what would
such a study conduct us, but to a new verification of the general laws
of the development of science ? When the times are ripe, when the
fundamental principles have been recognized and enunciated, nothing
stops the march of ideas ; the same discoveries, or discoveries almost
equivalent, appear at nearly the same instant, and in places the most
diverse. Without undertaking a discussion of this sort, which, besides,
might appear useless or become irritating, it is, however, of import-
ance to bring out a fundamental difference between the tendencies
of the great geometers, who, about 1830, gave to geometry a scope
before unknown.

Ill

Some, like Chasles and Steiner, who consecrated their entire lives
to research in pure geometry, opposed what they called synthesis to
analysis, and, adopting in the ensemble if not in detail the tendencies
of Poncelet, proposed to constitute an independent doctrine, rival of
Descartes's analysis.

Poncelet could not content himself with the insufficient resources
furnished by the method of projections; to attain imaginaries he
created that famous principle of continuity which gave birth to such
long discussions between him and Cauchy.

Suitably enunciated, this principle is excellent and can render
great service. Poncelet was wrong in refusing to present it as a simple
consequence of analysis; and Cauchy, on the other hand, was not
willing to recognize that his own objections, applicable without
doubt to certain transcendent figures, were without force in the
applications made by the author of the Traite des proprietes pro-
jectives.

Whatever be the opinion of such a discussion, it showed at least
in the clearest manner that the geometric system of Poncelet rested

540 GEOMETRY

on an analytic foundation, and besides we know, by the untoward
publication of the manuscripts of Saratoff, that by the aid of
Descartes's analysis were established the principles which serve as
foundation for the TraiU des proprietes projectiles.

Younger than Poncelet, who besides abandoned geometry for
mechanics where his works had a preponderant influence, Chasles,
for whom was created in 1847 a chair of Geometrie superieure in the
Faculty of Science of Paris, endeavored to constitute a geometric
doctrine entirely independent and autonomous. He has expounded
it in two works of high importance, the Traite de geometric supe-
rieure, which dates from 1852, and the Traite des sections conigues,
unhappily unfinished and of which the first part alone appeared in
1865.

In the preface of the first of these works he indicates very clearly
the three fundamental points which permit the new doctrine to share
the advantages of analysis and which to him appear to mark an
advance in the cultivation of the science. These are: (1) The intro-
duction of the principle of signs, which simplifies at once the enuncia-
tions and the demonstrations, and gives to Carnot's analysis of trans-
versals all the scope of which it is susceptible; (2) the introduction of
imaginaries, which supplies the place of the principle of continuity
and furnishes demonstrations as general as those of analytic geo-
metry; (3) the simultaneous demonstration of propositions which are
correlative, that is to say, which correspond in virtue of the principle
of duality.

Chasles studies indeed in his work homography and correlation;
but he avoids systematically in his exposition the employment of
transformations of figures, which, he thinks, cannot take the place of
direct demonstrations since they mask the origin and the true nature
of the properties obtained by their means.

There is truth in this judgment, but the advance itself of the science
permits us to declare it too severe. If it happens often that, em-
ployed without discernment, transformations multiply uselessly the
number of theorems, it must be recognized that they often aid us to
better understand the nature of the propositions even to which they
have been applied. Is it not the employment of Poncelet's projection
which has led to the so fruitful distinction between projective proper-
ties and metric properties, which has taught us also the high import-
ance of that cross-ratio whose essential property is found already
in Pappus, and of which the fundamental r61e has begun to appear
after fifteen centuries only in the researches of modern geometry?

The introduction of the principle of signs was not so new as Chasles
supposed at the time he wrote his TraiU de Geometric superieure.

Moebius, in his Barycentrische Calcul, had already given issue to
a desideratum of Carnot, and employed the signs in a way the largest

DEVELOPMENT OF GEOMETRIC METHODS 541

and most precise, defining for the first time the sign of a segment
and even that of an area.

Later he succeeded in extending the use of signs to lengths not
laid off on the same straight line and to angles not formed about the
same point.

Besides Grassmann, whose mind has so much analogy to that of
Moebius, had necessarily employed the principle of signs in the defini-
tions which serve as basis for his methods, so original, of studying
the properties of space.

The second characteristic which Chasles assigns to his system of
geometry is the employment of imaginaries. Here, his method was
really new, and he illustrates it by examples of high interest. One will
always admire the beautiful theories he has left us on homofocal
surfaces of the second degree, where all the known properties and
others new, as varied as elegant, flow from the general principle that
they are inscribed in the same developable circumscribed to the
circle at infinity.

But Chasles introduced imaginaries only by their symmetric func-
tions, and consequently would not have been able to define the cross-
ratio of four elements when these ceased to be real in whole or in
part. If Chasles had been able to establish the notion of the cross-
ratio of imaginary elements, a formula he gives in the Geometric
sup£rieure (p. 118 of the new edition) would have immediately
furnished him that beautiful definition of angle as logarithm of a
cross-ratio which enabled Laguerre, our regretted confrere, to give
the complete solution, sought so long, of the problem of the trans-
formation of relations which contain at the same time angles and
segments in homography and correlation.

Like Chasles, Steiner, the great and profound geometer, followed
the way of pure geometry; but he has neglected to give us a complete
exposition of the methods upon which he depended. However, they
may be characterized by saying that they rest upon the introduction
of those elementary geometric forms which Desargues had already
considered, on the development he was able to give to Bobillier's
theory of polars, and finally on the construction of curves and sur-
faces of higher degrees by the aid of sheaves or nets of curves of
lower orders. In default of recent researches, analysis would suffice
to show that the field thus embraced has just the extent of that into
which the analysis of Descartes introduces us without effort.

IV

While Chasles, Steiner, and, later, as we shall see, von Staudt, were
intent on constituting a rival doctrine to analysis and set in some
sort altar against altar, Gergonne, Bobillier, Sturm, and above all
Pluecker, perfected the geometry of Descartes and constituted an

542 GEOMETRY

analytic system in a manner adequate to the discoveries of the
geometers. It is to Bobillier and to Pluecker that we owe the method
called abridged notation. Bobillier consecrated to it some pages truly
new in the last volumes of the Annales of Gergonne.

Pluecker commenced to develop it in his first work, soon followed
by a series of works where are established in a fully conscious manner
the foundations of the modern analytic geometry. It is to him that
we owe tangential coordinates, trilinear coordinates, employed with
homogeneous equations, and finally the employment of canonical
forms whose validity was recognized by the method, so deceptive
sometimes, but so fruitful, called the enumeration of constants.

All these happy acquisitions infused new blood into Descartes's
analysis and put it in condition to give their full signification to the
conceptions of which the geometry called synthetic had been unable
to make itself completely mistress.

Pluecker, to whom it is without doubt just to adjoin Bobillier,
carried off by a premature death, should be regarded as the veritable
initiator of those methods of modern analysis where the employment
of homogeneous coordinates permits treating simultaneously and,
so to say, without the reader perceiving it, together with one figure
all those deducible from it by homography and correlation.

Parting from this moment, a period opens brilliant for geometric
researches of every nature.

The analysts interpret all their results and are occupied in trans-
lating them by constructions.

The geometers are intent on discovering in every question some
general principle, usually undemonstrable without the aid of ana-
lysis, in order to make flow from it without effort a crowd of particu-
lar consequences, solidly bound to one another and to the principle
whence they are derived. Otto Hesse, brilliant disciple of Jacobi,
develops in an admirable manner that method of homogeneous
coordinates to which Pluecker perhaps had not attached its full
value. Boole discovers in the polars of Bobillier the first notion of
a covariant; the theory of forms is created by the labors of Cayley,
Sylvester, Hermite, Brioschi. Later Aronhold, Clebsch and Gordan,
and other geometers still living, gave to it its final notation, estab-
lished the fundamental theorem relative to the limitation of the
number of covariant forms and so gave it all its amplitude.

The theory of surfaces of the second order, built up principally
by the school of Monge, was enriched by a multitude of elegant
properties, established principally by O. Hesse, who found later in
Paul Serret a worthy emulator and continues

DEVELOPMENT OF GEOMETRIC METHODS 543

The properties of the polars of algebraic curves are developed by
Pluecker and above all by Steiner. The study, already old, of curves
of the third order is rejuvenated and enriched by a crowd of new
elements. Steiner, the first, studies by pure geometry the double
tangents of curves of the fourth order, and Hesse, after him, applies
the methods of algebra to this beautiful question, as well as to that
of points of inflection of curves of the third order.

The notion of class introduced by Gergonne, the study of a para-
dox in part elucidated by Poncelet and relative to the respective
degrees of two curves reciprocal polars one of the other, give birth
to the researches of Pluecker relative to the singularities called ordi-
nary of algebraic plane curves. The celebrated formulas to which
Pluecker is thus conducted are later extended by Cayley and by
other geometers to algebraic skew curves, by Cayley again and by
Salmon to algebraic surfaces.

The singularities of higher order are in their turn taken up by
the geometers; contrary to an opinion then very widespread, Hal-
phen demonstrates that each of these singularities cannot be con-
sidered as equivalent to a certain group of ordinary singularities, and
his researches close for a time this difficult and important question.

Analysis and geometry, Steiner, Cayley, Salmon, Cremona, meet in
the study of surfaces of the third order, and, in conformity with
the anticipations of Steiner, this theory becomes as simple and as
easy as that of surfaces of the second order.

The algebraic ruled surfaces, so important for applications, are
studied by Chasles, by Cayley, of whom we find the influence and the
mark in all mathematical researches, by Cremona, Salmon, La Gour-
nerie; so they will be later by Pluecker in a work to which we must
return.

The study of the general surface of the fourth order would seem
to be still too difficult; but that of the particular surfaces of this order
with multiple points or multiple lines is commenced, by Pluecker for
the surface of waves, by Steiner, Kummer, Cayley, Moutard, Laguerre,
Cremona, and many other investigators.

As for the theory of algebraic skew curves, grown rich in its ele-
mentary parts, it receives finally, by the labors of Halphen and of
Noether, whom it is impossible for us here to separate, the most
notable extensions.

A new theory with a great future is born by the labors of Chasles,
of Clebsch, and of Cremona; it concerns the study of all the algebraic
curves which can be traced on a determined surface.

Homography and correlation, those two methods of transformation
which have been the distant origin of all the preceding researches,
receive from them in their turn an unexpected extension; they are
not the only methods which make a single element correspond to a

544 GEOMETRY

single element, as might have shown a particular transformation
briefly indicated by Poncelet in the Traite des proprtites projectives.

Pluecker defines the transformation by reciprocal radii vectores or
inversion, of which Sir W. Thomson and Liouville hasten to show all
the importance, as well for mathematical physics as for geometry.

A contemporary of Moebius and Pluecker, Magnus believed he had
found the most general transformation which makes a point corre-
spond to a point, but the researches of Cremona show us that the
transformation of Magnus is only the first term of a series of bira-
tional transformations which the great Italian geometer teaches us to
determine methodically, at least for the figures of plane geometry.

The Cremona transformations long retained a great interest,
though later researches have shown us that they reduce always to
a series of successive applications of the transformation of Magnus.

VI

All the works we have enumerated, others to which we shall return
later, find their origin and, in some sort, their first motive in the con-
ceptions of modern geometry; but the moment has come to indicate
rapidly another source of great advances for geometric studies.
Legendre's theory of elliptic functions, too much neglected by the
French geometers, is developed and extended by Abel and Jacobi.
With these great geometers, soon followed by Riemann and Weier-
strass, the theory of Abelian functions which, later, algebra would
try to follow solely with its own resources, brought to the geometry
of curves and surfaces a contribution whose importance will continue
to grow.

Already, Jacobi had employed the analysis of elliptic functions
in the demonstration of Poncelet's celebrated theorems on inscribed
and circumscribed polygons, inaugurating thus a chapter since en-
riched by a multitude of elegant results; he had obtained also, by
methods pertaining to geometry, the integration of Abelian equa-
tions.

But it was Clebsch who first showed in a long series of works all
the importance of the notion of deficiency (Geschlecht, genre) of a
curve, due to Abel and Riemann, in developing a crowd of results
and elegant solutions that the employment of Abelian integrals would
seem, so simple was it, to connect with their veritable point of
departure.

The study of points of inflection of curves of the third order, that
of double tangents of curves of the fourth order, and, in general, the
theory of osculation on which the ancients and the moderns had so
often practiced, were connected with the beautiful problem of the
division of elliptic functions and Abelian functions.

In one of his memoirs, Clebsch had studied the curves which are

545

rational or of deficiency zero; this led him, toward the end of his
too short life, to envisage what may be called also rational surfaces,
those which can be simply represented by a plane. This was a vast
field for research, opened already for the elementary cases by Chasles,
and in which Clebsch was followed by Cremona and many other
savants. It was on this occasion that Cremona, generalizing his re-
searches on plane geometry, made known not indeed the totality of
birational transformations of space, but certain of the most interest-
ing among these transformations.

The extension of the notion of deficiency to algebraic surfaces is
already commenced; already also works of high value have shown
that the theory of integrals, simple or multiple, of algebraic differ-
entials will find, in the study of surfaces as in that of curves, an ample
field of important applications; but it is not proper for the reporter
on geometry to dilate on this subject .

VII

While thus were constituted the mixed methods whose principal
applications we have just indicated, the pure geometers were not
inactive. Poinsot, the creator of the theory of couples, developed,
by a method purely geometric, "that, where one never for a mo-
ment loses from view the object of the research," the theory of the
rotation of a solid body that the researches of d'Alembert, Euler, and
Lagrange seemed to have exhausted; Chasles made a precious con-
tribution to kinematic by his beautiful theorems on the displacement
of a solid body, which have since been extended by other elegant
methods to the case where the motion has divers degrees of freedom.
He made known those beautiful propositions on attraction in gen-
eral, which figure without disadvantage beside those of Green and
Gauss. Chasles and Steiner met in the study of the attraction of
ellipsoids and showed thus once more that geometry has its desig-
nated place in the highest questions of the integral calculus.

Steiner did not disdain at the same time to occupy himself with
the elementary parts of geometry. His researches on the contacts of
circles and conies, on isoperimetric problems, on parallel surfaces, on
the centre of gravity of curvature, excited the admiration of all by
their simplicity and their depth.

Chasles introduced his principle of correspondence between two
variable objects which has given birth to so many applications; but
here anatysis retook its place to study the principle in its essence,
make it precise and generalize it.

It was the same concerning the famous theory of characteristics
and the numerous researches of de Jonquieres, Chasles, Cremona,
and still others, which gave the foundations of a new branch of the
science, Enumerative Geometry.

546 GEOMETRY

During many years, the celebrated postulate of Chasles was ad-
mitted without any objection: a crowd of geometers believed they
had established it in a manner irrefutable.

But, as Zeuthen then said, it is very difficult to recognize whether,
in demonstrations of this sort, there does not exist always some weak
point that their author has not perceived; and, in fact, Halphen,
after fruitless efforts, crowned finally all these researches by clearly
indicating in what cases the postulate of Chasles may be admitted
and in what cases it must be rejected.

VIII

Such are the principal works which restored geometric synthesis
to honor and assured to it, in the course of the last century, the place
belonging to it in mathematical research. Numerous and illustrious
workers took part in this great geometric movement, but we must
recognize that its chiefs and leaders were Chasles and Steiner. So
brilliant were their marvelous discoveries that they threw into the
shade, at least momentarily, the publications of other modest geo-
meters, less preoccupied perhaps in finding brilliant applications,
fitted to evoke love for geometry than to establish this science itself
on an absolutely solid foundation. Their works have received per-
haps a recompense more tardy, but their influence grows each day;
it will assuredly increase still more. To pass them over in silence
would be without doubt to neglect one of the principal factors which
will enter into future researches. We allude at this moment above
all to von Staudt. His geometric works were published in two books
of great interest: the Geometric der Lage, issued in 1847, and the
Beitrage zur Geometric der Lage, published in 1856, that is to say,
four years after the Geometric superieure. Chasles, as we have seen,
had devoted himself to constituting a body of doctrine independent
of Descartes's analysis and had not completely succeeded. We have
already indicated one of the criticisms that can be made upon this
system: the imaginary elements are there defined only by their sym-
metric functions, which necessarily exclude them from a multitude
of researches. On the other hand, the constant employment of cross-
ratio, of transversals, and of involution, which requires frequent
analytic transformations, gives to the Geometrie superieure a char-
acter almost exclusively metric which removes it notably from the
methods of Poncelet. Returning to these methods, von Staudt
devoted himself to constituting a geometry freed from all metric
relation and resting exclusively on relations of situation.

This is the spirit in which was conceived his first work, the Geo-
metrie der Lage of 1847. The author there takes as point of departure
the harmonic properties of the complete quadrilateral and those
of homologic triangles, demonstrated uniquely by considerations

DEVELOPMENT OF GEOMETRIC METHODS 547

of geometry of three dimensions, analogous to those of which the
school of Monge made such frequent use.

In this first part of his work, von Staudt neglected entirely im-
aginary elements. It is only in the Beitrage, his second work, that
he succeeds, by a very original extension of the method of Chasles,
in defining geometrically an isolated imaginary element and dis-
tinguishing it from its conjugate.

This extension, although rigorous, is difficult and very abstract.
It may be defined in substance as follows: Two conjugate imaginary
points may always be considered as the double points of an involu-
tion on a real straight; and just as one passes from an imaginary to
its conjugate by changing i into — i, so one may distinguish the two
imaginary points by making correspond to each of them one of the
two different senses which may be attributed to the straight. In this
there is something a little artificial; the development of the theory
erected on such foundations is necessarily complicated. By methods
purely projective, von Staudt establishes a calculus of cross-ratios of
the most general imaginary elements. Like all geometry, the pro-
jective geometry employs the notion of order and order engenders
number; we are not astonished therefore that von Staudt has been
able to constitute his calculus; but we must admire the ingenuity
displayed in attaining it. In spite of the efforts of distinguished
geometers who have essayed to simplify its exposition, we fear that
this part of the geometry of von Staudt, like the geometry otherwise
so interesting of the profound thinker Grassmann, cannot prevail
against the analytical methods which have won to-day favor almost
universal. Life is short; geometers know and also practice the
principle of least action. Despite these fears, which should discour-
age no one, it seems to us that under the first form given it by von
Staudt, projective geometry must become the necessary companion
of descriptive geometry, that it is called to renovate this geometry
in its spirit, its procedures, and its applications.

This has already been comprehended in many countries, and
notably in Italy, where the great geometer Cremona did not disdain
to write for the schools an elementary treatise on projective geometry.

IX

In the preceding articles, we have essayed to follow and bring out
clearly the most remote consequences of the methods of Monge and
Poncelet. In creating tangential coordinates and homogeneous coor-
dinates, Pluecker seemed to have exhausted all that the method of
projections and that of reciprocal polars give to analysis.

It remained for him, toward the end of his life, to return to his
first researches to give them an extension enlarging to an unexpected
degree the domain of geometry.

548 GEOMETRY

Preceded by innumerable researches on systems of straight lines,
due to Poinsot, Moebius, Chasles, Dupin, Malus. Hamilton, Krummer,
Transon. above all to Cayley, who first introduced the notion of the
coordinates of the straight, researches originating perhaps in statics
and kinematics, perhaps in geometrical optics, Pluecker's geometry of
the straight line will always be regarded as the part of his work where
are met the newest and most interesting ideas.

Pluecker first set up a methodic study of the straight line, which
already is important, but that is nothing beside what he discov
ered. It is sometimes said that the principle of duality shows that
the plane an well as the point may be considered as a space element.
That is true; but in adding the straight line to the plane and point
as possible space element, Plueckcr was led to recognize that any
curve, any surface, may also be considered as space element, and so
was born a new geometry which already has inspired a great number
of works, which will raise up still more in the future.

A beautiful discovery, of which we shall speak further on, has
already connected the geometry of spheres with that of straight lines
and permits the introduction of the notion of coordinates of a sphere.

The theory of systems of circles is already commenced; it will
be developed without doubt when one wishes to study the representa-
tion, which we owe to Laguerre, of an imaginary point in space by an
oriented circle.

But before expounding the development of these new ideas which
have vivified the infinitesimal methods of Monge, it is necessary to go
back to take up the history of branches of geometry that we have
neglected until now.

X

Among the works of the school of Monge, we have hitherto con-
fined ourselves to the consideration of those connected with finite
geometry; but certain of the disciples of Monge devoted themselves
above all to developing the new notions of infinitesimal geometry
applied by their master to curves of double curvature, to lines of curv-
ature, to the generation of surfaces, notions expounded at least in
part in the Application de V Analyse a la Geometrie. Among these
we must cite Lancret, author of beautiful works on skew curves, and
above all Charles Dupin, the only one perhaps who -followed all the
paths opened by Monge.

Among other works, we owe to Dupin two volumes Monge would
not have hesitated to sign: Les Dcveloppements de Geometrie pure,
issued in 1813, and Les Applications de Geometrie et de Mecanique,
dating from 1822.

There we find the notion of indicatrix, which was to renovate,
after Euler and Meunier, all the theory of curvature, that of conjugate

DEVELOPMENT OF GEOMETRIC METHODS 549

tangents, of asymptotic lines which have taken so important a place
in recent researches. Nor should we forget the determination of the
surface of which all the lines of curvature are circles, nor above all
the memoir on triple systems of orthogonal surfaces where is found,
together with the discovery of the triple system formed by surfaces
of the second degree, the celebrated theorem to which the name of
Dupin will remain attached.

Under the influence of these works and of the renaissance of syn-
thetic methods, the geometry of infinitesimals retook in all researches
the place Lagrange had wished to take away from it forever.

Singular thing, the geometric methods thus restored were to receive
the most vivid impulse in consequence of the publication of a memoir
which, at least at first blush, would appear connected with the purest
analysis; we mean the celebrated paper of Gauss, Disquisitiones
generates circa superficies curvas, which was presented in 1827 to the
Gottingen Society, and whose appearance marked, one may say.
a decisive date in the history of infinitesimal geometry.

From this moment, the infinitesimal method took in France a free
scope before unknown.

Frenet, Bertrand, Molins, J. A. Serret, Bouquet, Puiseux, Ossian
Bonnet, Paul Serret, develop the theory of skew curves. Liouville,
Chasles, Minding, join them to pursue the methodic study of the
memoir of Gauss.

The integration made by Jacobi of the differential equation of the
geodesic lines of the ellipsoid started a great number of researches.
At the same time the problems studied in the Application de I' Analyse
of Monge were greatly developed.

The determination of all the surfaces having their lines of curvature
plane or spheric completed in the happiest manner certain partial
results already obtained by Monge.

At this moment, one of the most penetrating of geometers, ac-
cording to the judgment of Jacobi, Gabriel Lam4, who, like Charles
Sturm, had commenced with pure geometry and had already made to
this science contributions the most interesting by a little book pub-
lished in 1817 and by memoirs inserted in the Annales of Gergonne,
utilized the results obtained by Dupin and Binet on the system of
confocal surfaces of the second degree, and, rising to the idea of
curvilinear coordinates in space, became the creator of a wholly new
theory destined to receive in mathematical physics the most varied
applications.

XI

Here again, in this infinitesimal branch of geometry are found the
two tendencies we have pointed out a propos of the geometry of finite
quantities.

550 GEOMETRY

Some, among whom must be placed J. Bertrand and O. Bonnet,
wish to constitute an independent method resting directly on the
employment of infinitesimals. The grand TraiU de Calcul differentiel,
of Bertrand, contains many chapters on the theory of curves and
of surfaces, which are, in some sort, the illustration of this con-
ception.

Others follow the usual analytic ways, being only intent to clearly
recognize and put in evidence the elements which figure in the first
plan. Thus did Lame1 in introducing his theory of differential para-
meters. Thus did Beltrami in extending with great ingenuity the
employment of these differential invariants to the case of two inde-
pendent variables, that is to say, to the study of surfaces.

It seems that to-day is accepted a mixed method whose origin is
found in the works of Ribaucour, under the name perimorphie. The
rectangular axes of analytic geometry are retained, but made mobile
and attached as seems best to the system to be studied. Thus dis-
appear most of the objections which have been made to the method
of coordinates. The advantages of what is sometimes called intrinsic
geometry are united to those resulting from the use of the regular
analysis. Besides, this analysis is by no means abandoned; the com-
plications of calculation which it almost always carries with it, in its
applications to the study of surfaces and rectilinear coordinates, usu-
ally disappear if one employs the notion on the invariants and the
covariants of quadratic powers of differentials which we owe to the
researches of Lipschitz and Christoffel, inspired by Riemann's studies
on the non-Euclidean geometry.

XII

The results of so many labors were not long in coming. The notion
of geodesic curvature which Gauss already possessed, but without
having published it, was given by Bonnet and Liouville; the theory
of surfaces of which the radii of curvature are functions one of the
other, inaugurated in Germany by two propositions which would
figure without disadvantage in the memoir of Gauss, was enriched
by Ribaucour, Halphen, S. Lie, and others, with a multitude of propo-
sitions, some concerning these surfaces envisaged in a general man-
ner; others applying to particular cases where the relation between
the radii of curvature takes a form particularly simple; to minimal
surfaces for example, and also to surfaces of constant curvature,
positive or negative.

The minimal surfaces were the object of works which make of
their study the most attractive chapter of infinitesimal geometry.
The integration of their partial differential equation constitutes one
of the most beautiful discoveries of Monge; but because of the im-
perfection of the theory of imaginaries, the great geometer could not

DEVELOPMENT OF GEOMETRIC METHODS 551

get from its formulas any mode of generation of these surfaces, nor
even any particular surface. We will not here retrace the detailed
history which we have presented in our Lecons sur la theorie des
surfaces ; but it is proper to recall the fundamental researches of
Bonnet which have given us, in particular, the notion of surfaces
associated with a given surface, the formulas of Weierstrass which
establish a close bond between the minimal surfaces and the functions
of a complex variable, the researches of Lie by which it was estab-
lished that just the formulas of Monge can to-day serve as founda-
tion for a fruitful study of minimal surfaces.

In seeking to determine the minimal surfaces of smallest classes
or degrees, we were led to the notion of double minimal surfaces
which is dependent on analysis situs.

Three problems of unequal importance have been studied in this
theory.

The first, relative to the determination of minimal surfaces in-
scribed along a given contour in a developable equally given, was
solved by celebrated formulas which have led to a great number of
propositions. For example, every straight traced on such a surface
is an axis of symmetry.

The second, set by S. Lie, concerns the determination of all the
algebraic minimal surfaces inscribed, in an algebraic developable,
without the curve of contact being given. It also has been entirely
elucidated.

The third and the most difficult is what the physicists solve experi-
mentally, by plunging a closed contour into a solution of glycerine.
It concerns the determination of the minimal surface passing through
a given contour.

The solution of this problem evidently surpasses the resources of
geometry. Thanks to the resources of the highest analysis, it has
been solved for particular contours in the celebrated memoir of
Riemann and in the profound researches which have followed or
accompanied this memoir.

For the most general contour, its study has been brilliantly begun;
it will be continued by our successors.

After the minimal surfaces, the surfaces of constant curvature at-
tracted the attention of geometers. An ingenious remark of Bonnet
connects with each other the surfaces of which one or the other of the
two curvatures, mean curvature or total curvature, is constant.

Bour announced that the partial differential equation of surfaces
of constant curvature could be completely integrated. This result
has not been secured; it would seem even very doubtful if we con-
sider a research where S. Lie has essayed in vain to apply a general
method of integration of partial differential equations to the particu-
lar equation of surfaces of constant curvature.

552 GEOMETRY

But, if it is impossible to determine in finite terms all these sur-
faces, it has at least been possible to obtain certain of them, char-
acterized by special properties, such as that of having their lines of
curvature plane or spheric; and it has been shown, by employing a
method which succeeds in many other problems, that from every sur-
face of constant curvature may be derived an infinity of other surfaces
of the same nature, by employing operations clearly defined which
require only quadratures.

The theory of the deformation of surfaces in the sense of Gauss
has been also much enriched. We owe to Minding and to Bour the
detailed study of that special deformation of ruled surfaces which
leaves the generators rectilineal. If we have not been able, as has
been said, to determine the surfaces applicable on the sphere, other
surfaces of the second degree have been attacked with more success,
and, in particular, the paraboloid of revolution.

The systematic study of the deformation of general surfaces of the
second degree is already entered upon; it is one of those which will
give shortly the most important results.

The theory of infinitesimal deformation constitutes to-day one of
the most finished chapters of geometry. It is the first somewhat
extended application of a general method which seems to have a great
future.

Being given a system of differential or partial differential equations,
suitable to determine a certain number of unknowns, it is advantage-
ous to associate with it a system of equations which we have called
auxiliary system, and which determines the systems of solutions
infinitely near any given system of solutions. The auxiliary system
being necessarily linear, its employment in all researches gives
precious light on the properties of the proposed system and on the
possibility of obtaining its integration.

The theory of lines of curvature and of asymptotic lines has been
notably extended. Not only have been determined these two series
of lines for particular surfaces such as the tetrahedral surfaces of
Lame1; but also, in developing Moutard's results relative to a par-
ticular class of linear partial differential equations of the second
order, it proved possible to generalize all that had been obtained for
surfaces with lines of curvature plane or spheric, in determining com-
pletely all the classes of surfaces for which could be solved the pro-
blem of spheric representation.

Just so has been solved the correlative problem relative to asymp-
totic lines in making known all the surfaces of which the infinitesimal
deformation can be determined in finite terms. Here is a vast field
for research whose exploration is scarcely begun.

The infinitesimal study of rectilinear congruences, already com-
menced long ago by Dupin, Bertrand, Hamilton, Kummer, has come

DEVELOPMENT OF GEOMETRIC METHODS 553

to intermingle in all these researches. Ribaucour, who has taken in
it a preponderant part, studied particular classes of rectilinear con-
gruences and, in particular, the congruences called isotropes, which
intervene in the happiest way in the study of minimal surfaces.

The triply orthogonal systems which Lame used in mathematical
physics have become the object of systematic researches. Cayley
was the first to form the partial differential equation of the third
order on which the general solution of this problem was made to
depend.

The system of homofocal surfaces of the second degree has been
generalized and has given birth to that theory of general cyclides in
which may be employed at the same time the resources of metric
geometry, of projective geometry, and of infinitesimal geometry.
Many other orthogonal systems have been made known. Among
these it is proper to signalize the cyclic systems of Ribaucour, for
which one of the three families admits circles as orthogonal trajecto-
ries and the more general systems for which these orthogonal trajec-
tories are simply plane curves.

The systematic employment of imaginaries, which we must be
careful not to exclude from geometry, has permitted the connection
of all these determinations with the study of the finite deformation
of a particular surface.

Among the methods which have permitted the establishment of
all these results, it is proper to note the systematic employment of
linear partial differential equations of the second order and of systems
formed of such equations. The most recent researches show that this
employment is destined to renovate most of the theories.

Infinitesimal geometry could not neglect the study of the two
fundamental problems set it by the calculus of variations.

The problem of the shortest path on a surface was the object of
masterly studies by Jacobi and by Ossian Bonnet. The study of
geodesic lines has been followed up; we have learned to determine
them for new surfaces. The theory of ensembles has come to permit
the following of these lines in their course on a given surface.

The solution of a problem relative to the representation of two
surfaces one on the other has greatly increased the interest of dis-
coveries of Jacobi and of Liouville relative to a particular class of
surfaces of which the geodesic lines could be determined. The results
concerning this particular case led to the examination of a new ques-
tion : to investigate all the problems of the calculus of variations of
which the solution is given by curves satisfying a given differential
equation.

Finally, the methods of Jacobi have been extended to space of
three dimensions and applied to the solution of a question which
presented the greatest difficulties: the study of properties of mini-

554 GEOMETRY

mum appertaining to the minimal surface passing through a given
contour.

XIII

Among the inventors who have contributed to the development of
infinitesimal geometry, Sophus Lie distinguishes himself by many
capital discoveries which place him in the first rank.

He was not one of those who show from infancy the most char-
acteristic aptitudes, and at the moment of quitting the University of
Christiania in 1865, he still hesitated between philology and mathe-
matics.

It was the works of Pluecker which gave him for the first time
full consciousness of his true calling.

He published in 1869 a first work on the interpretation of imagin-
aries in geometry, and from 1870 he was in possession of the directing
ideas of his whole career. I had at this time the pleasure of seeing
him often, of entertaining him at Paris, where he had come with his
friend F. Klein.

A course by M. Sylow followed by Lie had revealed to him all the
importance of the theory of substitutions; the two friends studied
this theory in the great treatise of C. Jordan; they were fully con-
scious of the important r61e it was called on to play in so many
branches of mathematical science where it had not yet been applied.

They have both had the good fortune to contribute by their works
to impress upon mathematical studies the direction which to them
appeared the best.

In 1870, Sophus Lie presented to the Academy of Sciences of Paris
a discovery extremely interesting. Nothing bears less resemblance
to a sphere than a straight line, and yet Lie had imagined a singular
transformation which made a sphere correspond to a straight line,
and permitted, consequently, the connecting of every proposition
relative to straight lines with a proposition relating to spheres, and
vice versa.

In this so curious method of transformation, each property relative
to the lines of curvature of a surface furnishes a proposition relative
to the asymptotic lines of the surface attained.

The name of Lie will remain attached to these deep-lying relations
which join to one another the straight line and the sphere, those two
essential and fundamental elements of geometric research. He de-
veloped them in a memoir full of new ideas which appeared in 1872.

The works which followed this brilliant de"but of Lie fully con-
firmed the hopes it had aroused. Pluecker's conception relative to
the generation of space by straight lines, by curves or surfaces
arbitrarily chosen, opens to the theory of algebraic forms a field
which has not yet been explored, which Clebsch scarcely began to
recognize and settle the boundaries of. But, from the side of infini-

DEVELOPMENT OF GEOMETRIC METHODS 555

tesimal geometry, this conception has been given its full value by
Sophus Lie. The great Norwegian geometer was able to find in it
first the notion of congruences and complexes of curves, and after-
ward that of contact transformations of which he had found, for the
case of the plane, the first germ in Pluecker. The study of these
transformations led him to perfect, at the same time with M. Mayer,
the methods of integration which Jacobi had instituted for partial
differential equations of the first order; but above all it threw the
most brilliant light on the most difficult and the most obscure parts
of the theories relative to partial differential equations of higher
order. It permitted Lie, in particular, to indicate all the cases in
which the method of characteristics of Monge is fully applicable to
equations of the second order with two independent variables.

In continuing the study of these special transformations, Lie was
led to construct progressively his masterly theory of continuous
groups of transformations and to put in evidence the very important
role that the notion of group plays in geometry. Among the essential
elements of his researches, it is proper to signalize the infinitesimal
transformations, of which the idea belongs exclusively to him.

Three great books published under his direction by able and de-
voted collaborators contain the essential part of his works and their
applications to the theory of integration, to that of complex units and
to the non-Euclidean geometry.

XIV

By an indirect way I have arrived at that non-Euclidean geometry
the study of which takes in the researches of geometers a place which
grows greater each day.

If I were the only one to talk with you about geometry, I should
take pleasure in recalling to you* all that has been done on this sub-
ject since Euclid or at least from Legendre to our days.

Envisaged successively by the greatest geometers of the last cen-
tury, the question has progressively enlarged.

It commenced with the celebrated postulatum relative to parallels;
it ends with the totality of geometric axioms.

The Elements of Euclid, which have withstood the action of so
many centuries, will have at least the honor before ending of arous-
ing a long series of works admirably enchained which will contrib-
ute, in the most effective way, to the progress of mathematics, at the
same time that they furnish to the philosophers the most precise and
the most solid points of departure for the study of the origin and of
the formation of our cognitions.

I am assured in advance that my distinguished collaborator will
not forget, among the problems of the present time, this one, which is
perhaps the most important, and with which he has occupied himself

556 GEOMETRY

with so much success; and I leave to him the task of developing it
with all the amplitude which it assuredly merits.

Thave just spoken of the elements of geometry. They have received
in the last hundred years extensions which must not be forgotten.
The theory of polyhedrons has been enriched by the beautiful dis-
coveries of Poinsot on the star polyhedrons and those of Moebius
on polyhedrons with a single face. The methods of transformation
have enlarged the exposition. We may say to-day that the first book
contains the theory of translation and of symmetry, that the second
amounts to the theory of rotation and of displacement, that the
third rest on homothety and inversion. But it must be recognized
that it is due to analysis that the Elements have been enriched by
their most beautiful propositions.

It is to the highest analysis that we owe the inscription of regular
polygons of seventeen sides and analogous polygons. To it we owe
the demonstrations, so long sought, of the impossibility of the quad-
rature of the circle, of the impossibility of certain geometric con-
structions with the aid of the ruler and the compasses; and to it finally
we owe the first rigorous demonstrations of the properties of maxi-
mum and of minimum of the sphere. It will belong to geometry to
enter upon this ground where analysis has preceded it.

What will be the elements of geometry in the course of the cen-
tury which has just commenced? Will there be a single elementary
book of geometry? It is perhaps America, writh its schools free from
all programme and from all tradition, which will give us the best solu-
tion of this important and difficult question.

Von Staudt has sometimes been called the Euclid of the nine-
teenth century; I would prefer to call him the Euclid of protective
geometry ; but is projective geometry, interesting though it may be,
destined to furnish the unique foundation of the future elements?

XV

The moment has come to close this over-long recital, and yet there
is a crowd of interesting researches that I have been, so to say, forced
to neglect.

I would have loved to talk with you about those geometries of
any number of dimensions of which the notion goes back to the first
days of algebra, but of which the systematic study was commenced
only sixty years ago by Cayley and by Cauchy. This kind of researches
has found favor in your country and I need not recall that our illus-
trious president, after having shown himself the worthy successor
of Laplace and Le Verrier, in a space which he considers with us as
being endowed with three dimensions, has not disdained to publish,
in the American Journal, considerations of great interest on the
geometries of n dimensions.

DEVELOPMENT OF GEOMETRIC METHODS 557

A single objection can be made to studies of this sort, and was
already formulated by Poisson: the absence of all real foundation, of
all substratum permitting the presentation, under aspects visible and
in some sort palpable, of the results obtained.

The extension of the methods of descriptive geometry, and above
all the employment of Pluecker's conceptions on the generation of
space, will contribute to take away from this objection much of its
force.

I would have liked to speak to you also of the method of equi-
pollences, of which we find the germ in the posthumous works of
Gauss, of Hamilton's quaternions, of Grassmann's methods, and in
general of systems of complex units, of the analysis situs, so inti-
mately connected with the theory of functions, of the geometry
called kinematic, of the theory of abaci, of geometrography, of the
applications of geometry to natural philosophy or to the arts. But
1 fear, if I branched out beyond measure, some analyst, as has hap-
pened before, would accuse geometry of wishing to monopolize
everything.

My admiration for analysis, grown so fruitful and so powerful in
our time, would not permit me to conceive such a thought. But if
some reproach of this sort could be formulated to-day, it is not to
geometry, it is to analysis it would be proper, I believe, to address it.
The circle in which the mathematical studies appeared to be inclosed
at the beginning of the nineteenth century has been broken on all
sides.

The old problems present themselves to us under a new form, new
problems offer themselves, whose study occupies legions of workers.

The number of those who cultivate pure geometry has become
prodigiously restricted. Therein is a danger against which it is im-
portant to provide. We must not forget that, if analysis has acquired
means of investigation which it lacked heretofore, it owes them in
great part to the conceptions introduced by the geometers. Geometry
must not remain in some sort entombed in its triumph. It is in its
school we have learned; our successors must learn never to be blindly
proud of methods too general, to envisage the questions in themselves
and to find, in the conditions particular to each problem, perhaps
a direct way towards a solution, perhaps the means of applying in
an appropriate manner the general procedures which every science
should gather.

As Chasles said at the beginning of the Apercu, historique, "The
doctrines of pure geometry offer often, and in a multitude of ques-
tions, that simple and natural way which, penetrating to the very
source of the truths, lays bare the mysterious chain which binds them
to each other and makes us know them individually in the way most
luminous and most complete."

558 GEOMETRY

Cultivate therefore geometry, which has its own advantages, with-
out wishing, on all points, to make it equal to its rival.

For the rest, if we were tempted to neglect it, it would soon find in
the applications of mathematics, as it did once before, means to rise
up again and develop itself anew. It is like the giant Antaeus who
recovered his strength in touching the earth.

THE PRESENT PROBLEMS OF GEOMETRY

BY DR. EDWARD KASNER

[Edward Kasner, Instructor in Mathematics, Columbia University, b. New York
City, 1877. B.S. College of the City of New York, 1896; A.M. Columbia
University, 1899; Ph.D. ibid. 1899. Post-graduate, Fellow in Mathematics,
Columbia University, 1897-99 ; Student, University of Gottingen, 1899-
1900; Tutor in Mathematics, Columbia University, 1900-05; Instructor,
1905; Member American Mathematical Society; Fellow American Associa-
tion for Advancement of Science. Associate editor, Transactions American
Mathematical Society.]

IN spite of the richness and power of recent geometry, it is notice-
able that the geometer himself has become more modest. It was the
ambition of Descartes and Leibnitz to discover universal methods,
applicable to all conceivable questions; later, the Ausdehnungslehre
of Grassmann and the quaternion theory of Hamilton were believed
by their devotees to be ultimate geometric analyses; and Chasles
attributed to the principles of duality and homography the same
role in the domain of pure space as that of the law of gravitation
in celestial mechanics. To-day, the mathematician admits the ex-
istence and the necessity of many theories, many geometries, each
appealing to certain interests, each to be developed by the most
appropriate methods; and he realizes that, no matter how large his
conceptions and how powerful his methods, they will be replaced
before long by others larger and more powerful.

Aside from the conceivability of other spaces with just as self-
consistent properties as those of the so-called ordinary space, such
diverse theories arise, in the first place, on account of the variety
of objects demanding consideration, — curves, surfaces, congruences
and complexes, correspondences, fields of differential elements, and
so on in endless profusion. The totality of configurations is indeed
not thinkable in the sense of an ordinary assemblage, since the total-
ity itself would have to be admitted as a configuration, that is, an
element of the assemblage.

However, more essential in most respects than the diversity in
the material treated is the diversity in the points of view from which
it may be regarded. Even the simplest figure, a triangle or a circle,
has an infinity of properties — indeed, recalling the unity of the
physical world, the complete study of a single figure would involve
its relations to all other figures and thus not be distinguishable from
the whole of geometry. For the past three decades the ruling thought
in this connection has been the principle (associated with the names
of Klein and Lie) that the properties which are deemed of interest
in the various geometric theories may be classified according to the

560 GEOMETRY

groups of transformations which leave those properties unchanged.
Thus almost all discussions on algebraic curves are connected with
the group of displacements (more properly the so-called principal
group), or the group of projective transformations, or the group of
birational transformations; and the distinction between such theories
is more fundamental than the distinction between the theories of
curves, of surfaces, and of complexes.

Historically, the advance has been, in general, from small to larger
groups of transformations. The change thus produced may be likened
to the varying appearance of a painting, at first viewed closely in all
its details, then at a distance in its significant features. The analogy
also suggests the desirability of viewing an object from several stand-
points, of studying geometric configurations with respect to various
groups. It is indeed true, though in a necessarily somewhat vague
sense, that the more essential properties are those invariant under
the more extensive groups; and it is to be expected that such groups
will play a predominating role in the not far distant future.

The domain of geometry occupies a position, as indicated in the
programme of the Congress, intermediate between the domain of
analysis on the one hand and of mathematical physics on the other;
and in its development it continually encroaches upon these adjacent
fields. The concepts of transformation and invariant, the algebraic
curve, the space of n dimensions, owe their origin primarily to the
suggestions of analysis; while the null-system, the theory of vector
fields, the questions connected with the applicability and deforma-
tion of surfaces, have their source in mechanics. It is true that some
mathematicians regard the discussion of point sets, for example,
as belonging exclusively to the theory of functions, and others look
upon the composition of displacements as a part of mechanics.
While such considerations show the difficulty, if not impossibility,
of drawing strict limits about any science, it is to be observed that
the consequent lack of definiteness, deplored though it be by the
formalist, is more than compensated by the fact that such overlap-
ping is actually the principal means by which the different realms
of knowledge are bound together.

If a mathematician of the past, an Archimedes or even a Descartes,
could view the field of geometry in its present condition, the first
feature to impress him would be its lack of concreteness. There are
whole classes of geometric theories which proceed, not merely with-
out models and diagrams, but without the slightest (apparent) use
of the spatial intuition. In the main this is due, of course, to the
power of the analytic instruments of investigation as compared
with the purely geometric. The formulas move in advance of thought,
while the intuition often lags behind; in the oft-quoted words of
d'Alembert, "L'algebre est ge'ne'reuse, elle donne souvent plus qu'on

PRESENT PROBLEMS OF GEOMETRY 561

lui demande." As the field of research widens, as we proceed from
the simple and definite to the more refined and general, we naturally
cease to picture our processes and even our results. It is often neces-
sary to close our eyes and go forward blindly if we wish to advance
at all. But admitting the inevitableness of such a change in the
spirit of any science, one may still question the attitude of the geo-
meter who rests content with his blindness, who does not at least
strive to intensify and enlarge the intuition. Has not such an inten-
sification and enlargement been the main contribution of geometry
to the race, its very raison d'etre as a separate part of mathematics,
and is there any ground for regarding this service as completed?

From the point of view here referred to, a problem is not to be
regarded as completely solved until we are in position to construct
a model of the solution, or at least to conceive of such a construction.
This requires the interpretation, not merely of the results of a geo-
metric investigation, but also, as far as possible, of the intermediate
processes — an attitude illustrated most strikingly in the works of
Lie. This duty of the geometer, to make the ground won by means
of analysis really geometric, and as far as possible concretely intui-
tive, is the source of many problems of to-day, a few of which will
be referred to in the course of this address.

The tendency to generalization, so characteristic of modern geo-
metry, is counteracted in many cases by this desire for the concrete,
in others by the desire for the exact, the rigorous (not to be con-
fused with the rigid). The great mathematicians have acted on the
principle "Devinez avant de demontrer," and it is certainly true
that almost all important discoveries are made in this fashion. But
while the demonstration comes after the discovery, it cannot there-
fore be disregarded. The spirit of rigor, which tended at first to the
arithmetization of all mathematics and now tends to its exhibition
in terms of pure logic, has always been more prominent in analysis
than in geometry. Absolute rigor may be unattainable, but it can-
not be denied that much remains to be done by the geometers, judg-
ing even by elementary standards. We need refer only to the loose
proofs based upon the invaluable but insufficient enumeration of
constants, the so-called principle of the conservation of number, and
the discussions which confine themselves to the "general case."
Examples abound in every field of geometry. The theorem announced
by Chasles concerning the number of conies satisfying five arbitrary
conditions was proved by such masters as Clebsch and Halphen be-
fore examples invalidating the result were devised. Picard recently
called attention to the need of a new proof of Noether's theorem that
upon the general algebraic surface of degree greater than three every
algebraic curve is a complete intersection with another algebraic
surface. The considerations given by Noether render the result

562 GEOMETRY

highly probable, but do not constitute a complete proof; while the
exact meaning of the term general can be determined only from
the context.

The reaction against such loose methods is represented by Study *
in algebraic geometry, and Hilbert in differential geometry. The
tendency of a considerable portion of recent work is towards the
exhaustive treatment of definite questions, including the considera-
tion of the special or degenerate cases ordinarily passed over as
unimportant. Another aspect of the same tendency is the discussion
of converses of familiar problems, with the object of obtaining con-
ditions at once necessary and sufficient, that is, completely character-
istic results.2

Another set of problems is suggested by the relation of geometry
to physics. It is the duty of the geometer to abstract from the physical
sciences those domains which may be expressed in terms of pure
space, to study the geometric foundations (or, as some would put it,
the skeletons) of the various branches of mechanics and physics.
Most of the actual advance, it is true, has hitherto come from the
physicists themselves, but undoubtedly the time has arrived for
more systematic discussions by the mathematicians. In addition to
the importance which is due to possible applications of such work,
it is to be noticed that we meet, in this way, configurations as inter-
esting and remarkable as those created by the geometer's imagina-
tion. Even in this field, one is tempted to remark, truth is stranger
than fiction.

We have now considered, briefly and inadequately, some of the
leading ideals and influences which are at work towards both the
widening and the deepening of geometry in general; and turn to our
proper topic, a survey of the leading problems or groups of problems
in certain selected (but it is hoped representative) fields of contem-
poraneous investigation.

Foundations

The most striking development of geometry during the past decade
relates to the critical revision of its foundations, more precisely, its
logical foundations. There are, of course, other points of view, for

1 " [Es ist eine] tief eingewurzelte Gewohnheit yieler Geometer, Satze zu formu-
lieren, die 'im allgemeinen ' gelten sollen. d. h. einen klaren Sinn iiberhaupt nicht
haben, zudem noch haufig als allgemein giiltig hingestellt oder mangelhaft be-
griindet werden. [Dies Verfahren wird], trotz etwanigen Verweisungen auf Trager
sehr beriihmter Namen, spateren Geschlechtern sicher als ganz unzulassig erschei-
nen, scheint aber in unserem 'kritischen' Zeitalter von vielen als eine berechtigte
Eigentiimlichkeit der Geometric betrachtet zu werden . . ." Jahr. Deut. Math.-
Ver., vol. xi (1902), p. 100.

2 As an example may be mentioned the theorem of Malus and Dupin, known
for almost a century, that the rays emanating from a point are converted, by any
refraction, into a normal congruence. Quite recently, Levi-Civitta succeeded in
showing that this property is characteristic; that is, any normal congruence may
be refracted into a bundle.

PRESENT PROBLEMS OF GEOMETRY 563

example, the physical, the physiological, the psychological, the meta-
physical, but the interest of mathematicians has been confined to the
purely logical aspect. The main results in this direction are due to
Peano and his co-workers; but the whole field was first brought
prominently to the attention of the mathematical world by the
appearance, five years ago, of Hilbert's elegant Festschrift.

The central problem is to lay down a system of primitive (unde-
fined) concepts or symbols and primitive (unproved) propositions
or postulates, from which the whole body of geometry (that is, the
geometry considered) shall follow by purely deductive processes.
No appeal to intuition is then necessary. " We might put the axioms
into a reasoning apparatus like the logical machine of Stanley Jevons,
and see all geometry come out of it" (Poincare). Such a system of
concepts and postulates may be obtained in a great (indeed end-
less) variety of ways: the main question, at present, concerns the
comparison of various systems, and the possibility of imposing lim-
itations so as to obtain a unique and perhaps simplest basis.

The first requirement of a system is that it shall be consistent.
The postulates must be compatible with one another. No one has yet
deduced contradictory results from the axioms of Euclid, but what
is our guarantee that this will not happen in the future? The only
method of answering this question which has suggested itself is the
exhibition of some object (whose existence is admitted) which fulfills
the conditions imposed by the postulates. Hilbert succeeded in con-
structing such an ideal object out of numbers; but remarks that the
difficulty is merely transferred to the field of arithmetic. The most
far-reaching result is the definition of number in terms of logical
classes as given by Pieri and Russell; but no general agreement is
yet to be expected in these discussions. Will the ultimate conclu-
sion be the impossibility of a direct proof of compatibility?

More accessible is the question concerning the independence of
postulates (and the analogous question of the irreducibility of con-
cepts). Most of the work of the last few years has been concentrated
on this point. In Hilbert's original system the various groups of
axioms (relating respectively to combination, order, parallels, con-
gruence, and continuity) are shown to be independent, but the dis-
cussion is not carried out completely for the individual axioms. In
Dr. Veblen's recently published system of twelve postulates, each
is proved independent of the remaining eleven.1 This marks an ad-
vance, but, of course, it does not terminate the problem. In what
respect does a group of propositions differ from what is termed a
single proposition? Is it possible to define the notion of an absolutely
simple postulate? The statement that any two points determine a
straight line involves an infinity of statements, and its fulfillment for
1 Trans. Amer. Math. Soc., vol. v (1904).

564 GEOMETRY

certain pairs of points may necessitate its fulfillment for all pairs.
If in Euclid's system the postulate of parallels is replaced by the
postulate concerning the sum of the angles of a triangle, a well-known
example of such a reduction is obtained; for it is sufficient to as-
sume the new postulate for a single triangle, the general result being
then deducible. As other examples we may mention Peano's reduc-
tion of the Euclidean definition of the plane; and the definition of
a collineation which demands, instead of the conversion of all straight
lines into straight lines, the existence of four simply infinite systems
of such straight lines.1

These examples illustrate the difficulty, if not the impossibility,
of formulating a really fundamental, that is, absolute standard of
independence and irreducibility. It is probable that the guiding
ideas will be obtained in the discussion of simpler deductive theories,
in particular, the systems for numbers and groups.

Two features are especially prominent in the actual develop-
ment of the body of geometry from its fundamental system. First,
the consideration of what may be termed the collateral geometries,
which arise by replacing one of the original postulates by its opposite,
or otherwise varying the system. Such theories serve to show the
limitation of that point of view which restricts the term general
geometry (pangeometry) to the Euclidean and non-Euclidean geo-
metries. The variety of possible abstract geometries is, of course,
inexhaustible; this is the central fact brought to light by the ex-
hibition of such systems as the non-Archimedean and the non-
arguesian. In the second place, much valuable work is being done in
discussing the various methods by which the same theorem may
be deduced from the postulates, the ideal being to use as few of the
postulates as possible. Here again the question of simplicity (simplest
proof), though it baffles analysis, forces itself upon the attenti9n.

Among the minor problems in this field, it is sufficient to consider
that concerning the relation of the theory of volume to the axiom of
continuity. This axiom need not be used in establishing the theory
of areas of polygons ; but after Dehn and others had proved the exist-
ence of polyhedra having the same volume though not decomposable
into mutually congruent parts (even after the addition of congruent
polyhedra), it was stated by Hilbert, and deemed evident generally,
that reference to continuity could not be avoided in three dimensions.
In a recent announcement 2 of Vahlen's forthcoming Abstrakte
Geometric this conclusion is declared unsound. It seems probable,
however, that the difference is merely one concerning the interpreta-
tion to be given to the term continuity.

1 Together with certain continuity assumptions. Cf. Bull. Amer. Math. Soc.,
vol. ix (1903), p. 545.

2 Jahr. Deut. Math.-Ver., vol. xin (1904), p. 395.

PRESENT PROBLEMS OF GEOMETRY 565

The work on logical foundations has been confined almost entirely
to the Euclidean and projective geometries. It is desirable, however,
that other geometric theories should be treated in a similar deductive
fashion. In particular, it is to be hoped that we shall soon have
a really systematic foundation for the so-called inversion geometry,
dealing with properties invariant under circular transformations.
This theory is of interest, not only for its own sake and for its appli-
cations in function theory, but also because its study serves to free
the mind from what is apt to become, without some check, slavery to
the projective point of view.

The Curve Concept — Analysis Situs

Although curves and surfaces have constituted the almost exclu-
sive material of the geometric investigation of the thirty centu-
ries of which we have record, it can hardly be claimed that the con-
cepts themselves have received their final analysis. Certain vague
notions are suggested by the naive intuition. It is the duty of mathe-
maticians to create perfectly precise concepts which agree more or
less closely with such intuitions, and at the same time, by the reac-
tion of the concepts, to refine the intuition. The problem, evidently, is
not at all determinate. It would be of interest to trace the evolution
which has actually produced several distinct curve concepts defining
more or less extensive classes of curves, agreeing in little beyond the
possession of an infinite number of points.

The more familiar special concepts or classes of curves are defined
in terms of the corresponding equation y =f(x) or function f(x) .
Such are, for example: (1) algebraic curves; (2) analytic curves;
(3) graphs of functions possessing derivatives of all orders; (4) the
curves considered in the usual discussions of infinitesimal geometry,
in which the existence of first and second derivatives is assumed;
(5) the so-called regular curves with a continuously turning tangent
(except for a finite number of corners); (6) the so-called ordinary
curves possessing a tangent and having only a finite number of
oscillations (maxima and minima) in any finite interval; (7) curves
with tangents; (8) the graphs of continuous functions.

How far are such distinctions accessible to the intuition? Of
course there are limitations. For over two centuries, from Descartes
to the publication of Weierstrass's classic example, the intuition of
mathematicians declared the classes (7) and (8) to be identical. Still
later it was found that such extraordinary (pathological or crinkly)
curves may present themselves in class (7). However, even here
partially successful attempts to connect with intuition have been
made by Wiener, Hilbert, Schoenflies, Moore, and others.

Let us consider a simpler extension in the field of ordinary curves.
If the function y (a:) is continuous except for a certain value of x

566 GEOMETRY

where there is an ordinary discontinuity, this is indicated by a break
in the graph; if f is continuous, but the derivative y has such a dis-
continuity, this shows itself by a sharp turn in the curve; if the
discontinuity is only in the second derivative, there is a sudden
change in the radius of curvature, which is, however, relatively
difficult to observe from the figure; finally, if the third derivative
is discontinuous, the effect upon the curve is no longer apparent.
Does this mean that it is impossible to picture it? Does it not rather
indicate a limitation in the usual geometric training which goes
only as far as relations expressible in terms of tangency and curva-
ture? For the interpretation of the third derivative it is necessary
to consider say the osculating parabola at each point of the curve:
in the case referred to, as we pass over the critical point, the
tangent line and osculating circle change continuously, but there is
a sudden change in the osculating parabola. If in fact our intuition
were trained to picture osculating algebraic curves of all orders, it
would detect a discontinuity in a derivative of any order. A partial
equivalent would be the ability to picture the successive evolutes
of a given curve; a complete equivalent would be the picturing of
the successive slope curves y=f'(x), y=f"(x), etc. All this requires,
evidently, only an increase in the intensity of our intuition, not a
change in its nature.

This, however, would not apply to all questions. There are func-
tions which, while possessing derivatives of all orders (then neces-
sarily continuous), are not analytic (that is, not expressible by power
series). What is it that distinguishes the analytic curves among this
larger class? Is it possible to put the distinction in a form capable
of assimilation by an idealized intuition? In short, what is the
really geometric definition of an analytic curve ? 1

Much recent work in function theory has had for its point of de-
parture a more general basis than the theory of curves, namely, the
theory of sets or assemblages of points, with special reference to
the notions of derived set and the various contents or areas. The
geometry of point sets must indeed be regarded as one of the most
important and promising in the whole field of mathematics. It
receives its distinctive character, as compared with the general
abstract theory of assemblages (Mengenlehre), from the fact that it
operates not with all one-to-one correspondences, but with the
group of analysis situs, the group of continuous one-to-one corre-
spondences. From the point of view of the larger group, there is no
distinction between a one-dimensional and a two- or many-dimen-
sional continuum (Cantor). This is still the case if the correspondence

1 One method of attack would be the interpretation of Pringsheim's condi-
tions; this requires not merely the individual derivative curves, but the limit of
the system.

PRESENT PROBLEMS OF GEOMETRY 567

is continuous but not one-to-one (Peano, 1890). In the domain of
continuous one-to-one correspondence, however, spaces of different
dimensions are not equivalent (Jiirgens, 1899).

An important class of curves, much more general than those
referred to above, consists of those point sets which are equivalent
(in the sense of analysis situs) to the straight line or segment of a
straight line. This is Hurwitz's simple and elegant geometric form-
ulation of the concept originally treated analytically by Jordan,
the most fundamental curve concept of to-day. The closed Jordan
curves are defined in analogous fashion as equivalent to the peri-
meter of a square (or the circumference of a circle).

A curve of this kind divides the remaining points of the plane into
two simply connected continua, an inside and an outside. The
necessity for proof of this seemingly obvious result is seen from the
fact that the Jordan class includes such extraordinary types as the
curve with positive content constructed recently by Osgood.1 Such
a separation of the plane may, however, be thought about by other
than Jordan curves: the concept of the boundary of a connected
region gives perhaps the most extensive class of point sets which
deserve to be called curve. Schoenflies proposes a definition for the
idea of a simple closed curve which makes it appear as the natural
extension, in a certain sense, of the polygon: a perfect set of points
P which separates the plane into an exterior region E and an interior
region / such that any E point can be connected with any / point
by a path (Polygonstrecke) having only one point in common with
P. This is in effect a converse of Jordan's theorem, and shows
precisely how the Jordan curve is distinguished from other types
of boundaries of connected regions.

These discussions are mentioned here simply as aspects of a really
fundamental problem: the revision of the concepts and results of
that division of geometry which has been variously termed analysis
situs, theory of connection, topology, geometry of situation — a
revision to be carried out in the light of the theory of assemblages.2

Algebraic Surfaces and Birational Transformations

After the demonstration of the power of the methods based upon
projective transformation, — the chief contribution due to the
geometers of the first half of the nineteenth century, — attempts
were made to introduce other types of one-to-one correspondence or
transformation into algebraic geometry; in particular the inversion
of William Thomson and Liouville, and the quadratic transformation
of Magnus. The general theory of such Cremona transformations
was inaugurated by the Italian geometer in his memoir Sulle tras-

1 Trans. Amer. Math. Soc., vol. iv (1903), p. 107.

2 Cf. Schoenflies, Math. Annalen, vols. LVIII, LIX (1903, 1904).

568 GEOMETRY

jormazioni geometriche delle figure piane, published in 1863. Within
a few years, Clifford, Noether, and Rosanes, working independently,
established the remarkable result that every Cremona transforma-
tion in a plane can be decomposed into a succession of quadratic
transformations, thus bringing to light the fact that there are at
bottom only two types of algebraic one-to-one correspondence, the
homographic and the quadratic.1

The development of a corresponding theory in space has been one
of the chief aims of the geometers of Italy, Germany, and England
for the last thirty years, but the essential question of decomposition
still remains unanswered. Is it possible to reduce the general Cremona
transformation of space to a finite number of fundamental types ?

In its application to the study of the properties of algebraic
curves and surfaces, the theory of the Cremona transformation
is usually merged in the more general theory of the birational trans-
formation. By means of the latter, a correspondence is established
which is one-to-one for the points of the particular figure considered
and the transformed figure, but not for all the points of space. In
the plane theory an important result is that a curve with the most
complicated singularities can, by means of Cremona transformations,
be converted into a curve whose only singularities are multiple
points with distinct tangents (Noether); furthermore, by means of
birational transformations, the singularities may be reduced to the
very simplest type, ordinary double points (Bertini). The known
theory of space curves is also, in this aspect, quite complete. The
analogous problem of the reduction of higher singularities of a sur-
face has been considered by Noether, Del Pezzo, Segre, Kobb, and
others, but no ultimate conclusion has yet been obtained.

One principal source of difficulty is that, while in case of two
birationally equivalent curves the correspondence is one-to-one
without exception, on the other hand, in the case of two surfaces,
there may be isolated points which correspond to curves, and just
such irregular phenomena escape the ordinary methods. Again,
not only singular points require consideration, as is the case in the
plane theory, but also singular lines, and the points may be isolated
or superimposed on the lines. Most success is to be expected from
further application of the method of projection from a higher space
due to Clifford and Veronese. In this direction the most important
result hitherto obtained is the theorem, of Picard and Simart, that
any algebraic surface (in ordinary space) can be regarded as the
projection of a surface free from singularities situated in five-dimen-
sional space.

1 Segre recently called attention to a case where the usual methods of discus-
sion fail to apply; the proof has been completed by Caatelnuovo. Cf. Atti d\
Torino, vol. xxxvi (1901).

PRESENT PROBLEMS OF GEOMETRY 569

A question which awaits solution even in the case of the plane
is that relating to the invariants of the group of Cremona trans-
formations proper. The genus and the moduli of a curve are unaltered
by all birational transformations, but the problem arises: Are there
properties of curves which remain unchanged by Cremona, although
not by other birational transformations ? From the fact that
birationally equivalent curves need not be equivalent under the
Cremona group, it would seem that such invariants — Cremona
invariants proper — do exist, but no actual examples have yet been
obtained. The problem may be restated in the form: What are the
necessary and sufficient conditions which must be fulfilled by two
curves if they are to be equivalent with respect to Cremona trans-
formations? Equality of genera and moduli, as already remarked, is
necessary but not sufficient.

The invariant theory of birational transformations has for its
principal object the study of the linear systems of point groups
on a given algebraic curve, that is, the point groups cut out by
linear systems of curves. Its foundations were implicitly laid by
Riemann in his discussion of the equivalent theory of algebraic func-
tions on a Riemann surface, though the actual application to curves
is due to Clebsch. Most of the later \vork has proceeded along
the algebraic-geometric lines developed by Brill and Noether, the
promising purely geometric treatment inaugurated by Segre being
rather neglected.

The extension of this type of geometry to space, that is, the de-
velopment of a systematic geometry on a fundamental algebraic
surface (especially as regards the linear systems of curves situated
thereon), is one of the main tasks of recent mathematics. The
geometric treatment is given in the memoirs of Enriques and Castel-
nuovo, while the corresponding functional aspect is the subject of
the treatise of Picard and Simart on algebraic functions of two
variables, at present in course of publication.

The most interesting feature of the investigations belonging in
this field is the often unexpected light which they throw on the
inter-relations of distinct fields of mathematics, and the advantage
derived from such relations. For example, Picard (as he himself
relates on presenting the second volume of his treatise to the Paris
Academy a few months ago) l for a long time was unable to prove
directly that the integrals of algebraic total differentials can be
reduced, in general, to algebraic-logarithmic combinations, until
finally a method for deciding the matter was suggested by a theorem
on surfaces which Noether had stated some twenty years earlier.
Again, in the enumeration of the double integrals of the second
species, Picard arrived at a certain result, which was soon noticed
1 Comptes Rendus, February 1, 1904.

570 GEOMETRY

to be essentially equivalent to one obtained by Castelnuovo in his
investigations on linear systems; and thus there was established
a connection between the so-called numerical and linear genera of a
surface, and the number of distinct double integrals. l

A closely related set of investigations, originating with Clebsch's
theorems on intersections and Liouville's on confocal quadrics, may
be termed the "geometry of Abel's theorem." As later applications
we can merely mention Humbert's memoirs on certain metric pro-
perties of curves, and Lie's determination of surfaces of translation.

Investigations in analysis have often suggested the introduc-
tion of new types of configurations into geometry. The field of alge-
braic surfaces is especially fruitful in this respect. Thus, while in the
case of curves (excluding the rational) there always exist integrals
everywhere finite, this holds for only a restricted class of surfaces;
their determination depends on the solution of a partial differential
equation which has been discussed in a few special cases.

In addition to such relations between analysis and geometry,
important relations arise between various fields of geometry. Just
as an algebraic function of one variable is pictured by either a plane
curve or a Riemann surface (according as the independent and de-
pendent variables are taken to be real or complex), so an algebraic
function of two independent variables may be represented by either
a surface in ordinary space or a Riemannian four-dimensional mani-
fold in space of five dimensions. In the case of one variable, the
single invariant number (deficiency or genus p) which arises is
capable of definition in terms of the characteristics of the curve or
the connectivity of the Riemann surface. In passing to two variables,
however, it is necessary to consider several arithmetical invariants
— just how many is an unsettled question. For the algebraic surface
we have, for instance, the geometric genus of Clebsch, the numerical
genus of Cayley, and the so-called second genus, each of which may
be regarded as a generalization, from a certain point of view, of the
single genus of a curve; all are invariant with respect to birational
transformation.

The other geometric interpretation, by means of a Riemannian
manifold, has rendered -necessary the study of the analysis situs of
higher spaces. The connection of such a manifold is no longer ex-
pressed by a single number as in the case of an ordinary surface, but
by a set of two or more, the so-called numbers of Betti and Riemann.
The detailed theory of these connectivities, difficult and delicate
because it must be derived with little aid from the intuition, has been
made the subject of an extensive series of memoirs by Poincare".

From the point of view of analysis, the chief interest in these
investigations is the fact that the connectivities are related to the
1 Comptes Rendus, February 22, 1904.

PRESENT PROBLEMS OF GEOMETRY 571

number of integrals of certain types. The chief problem for the
geometer, however, is the discovery of the precise relations between
the connectivities of the Riemann manifold and the various genera
of the algebraic surface. That relations do exist between such di-
verse geometries — the one operating with all continuous, the other
with the algebraic, one-to-one correspondence — is one of the most
striking results of recent mathematics.

Geometry of Multiple Forms

For some time after its origin, the linear invariant theory of
Boole, Cayley, and Sylvester confined itself to forms containing a
single set of variables. The needs of both analysis and geometry,
however, have emphasized the importance and the necessity of
further development of the theory of forms containing two or more
sets of variables (of the same or different type), so-called multiple
forms.

In the plane we have both point coordinates (x) and line coor-
dinates (u). A form in x corresponds to a point curve (locus), a
form in u to a line curve (envelope), and a form involving both x
and u to a connex. The latter was introduced into geometry, some
thirty years ago, by Clebsch, the suggestion coming from the fact
that, even in the study of a simple form in x, co variants in x and u
present themselves, so that it seemed desirable to deal with such
forms ab initio.

Passing to space, we meet three simple elements, the point (x~),
the plane (u), and the line (p). Forms in a single set of variables
represent, respectively, a surface as point locus, a surface as plane
envelope, and a complex of lines. The compound elements composed
of two simple elements are the point-plane, the point-line, and the
plane-line. The first type, leading to point-plane connexes, has been
studied extensively during the past few years; the second to a more
limited degree; the third is merely the dual of the second. To com-
•plete the series, the case of the point-line-plane as element, or forms
involving x, u, and p, requires investigation.

In the corresponding n-dimensional theory it is necessary to take
account of n simple elements and the various compound elements
formed by their combinations.

The importance of such work is twofold: First, on account of
connection with the algebra of invariants. A fundamental theorem
of Clebsch states that, in the investigation of complete systems of
comitants, it is sufficient to consider forms involving not more than
one set of variables of each type : if in the given forms the types are
involved in any manner, it is possible to find an equivalent reduced
system of the kind described. On the other hand, it is impossible
to reduce the system further, so that the introduction of the n types

572 GEOMETRY

of variables is necessary for the algebraically complete discussion.
Geometry must accordingly extend itself to accommodate the
configurations defined by the new elements.

Second, on account of connection with the theory of differential
equations. The ordinary plane connex in x, u, assigns to each point
of the plane a certain number of directions (represented by the
tangents drawn to the corresponding curve), and thus gives rise to
an (algebraic) differential equation of the first order in two variables;
the point-plane connex in space, associating with each point a single
infinity of incident planes, defines a partial differential equation
of the first order; the point-line connex yields a Monge equation.
The point-line-plane case has not yet been interpreted from this
point of view.

One special problem in this field deserves mention, on account of its
many applications. This is the study of the system composed of a
quadric form in any number of variables and a bilinear form in con-
tragredient variables, that is, a quadric manifold and an arbitrary
(not merely automorphic) collineation in n-space. For n = 6, for
example, this corresponds to the general linear transformation of
line or sphere coordinates.

In addition to forms containing variables of different types, the
forms involving several sets of variables of the same type require
consideration. Forms in two sets of line coordinates present them-
selves in connection with the pfaffian problem of differential systems.
The main interest attaches, however, to forms in sets of point coor-
dinates, since it is these which occur in the theory of contact trans-
formations and of multiple correspondences. For example, while
the ordinary homography on a line is represented by a bilinear form
in binary variables, the trilinear form in similar variables gives rise
to a new geometric variety, the so-called homography of the second
class (associating with any two points a unique third point), which
has applications to the generation of cubic surfaces and to the con-
structions at the basis of photogrammetry. The theory of multilinear
forms in general deserves more attention than it has yet received.

Other important problems, connected with the geometric phases of
linear invariant theory, can merely be mentioned: (1) The general
geometric interpretation of what appears algebraically as the sim-
plest protective relation, namely, apolarity. (2) The invariant dis-
cussion of the simpler discontinuous varieties, for example, the poly-
gon considered as n-point or as n-line.1 (3) The establishment of a
system of forms corresponding to the general space curve. (4) The
study of the properties and the groups of the configurations cor-

1 Cf. F. Morley "On the geometry whose element is the 3-point of a plane,"
Trans. Amer. Math. Soc., voL v (1904). E. Study in his Geometric der Dynamen
develops a new foundation for kinematics by employing as element the Soma or
trirectangular trihedron.

PRESENT PROBLEMS OF GEOMETRY 573

responding in hyperspace to the simpler systems of invariants. (5)
Complete systems of orthogonal or metric invariants for the simpler
curves.1

Transcendental Curves

To reduce to systematic order the chaos of non-algebraic curves
has been the aspiration of many a mathematician; but, despite all
efforts, we have no theory comparable with that of algebraic curves.
The very vagueness and apparent hopelessness of the question is
apt to repel the modern mathematician, to cause him to return to
the more familiar field. The resulting concentration has led to the
powerful methods, already referred to, for studying algebraic varie-
ties. In the transcendental domain, on the other hand, we have a
multitude of interesting but particular geometric forms, — some
suggested by mechanics and physics, others derived from their relation
to algebraic curves, or by the interpretation of analytic results —
a few thousands of which have been considered of sufficient importance
to deserve specific names.2 The problem at issue is then a practical
one (comparable with corresponding discussions in natural history) :
to formulate a principle of classification which will apply, not to all
possible curves, but to as many as possible of the usual important
transcendental curves.

The most fruitful suggestion hitherto applied has come from
the consideration of differential equations : almost all the important
transcendental curves satisfy algebraic differential equations, and
these in the great majority of cases are of the first order. Hence the
need of a systematic discussion of the curves defined by any algebraic
equation F(x, y, y'} =0, the so-called panalgebraic curves of Loria. If
F is of degree n in y' and of degree v in x, y, the curve is said to belong
to a system with the characteristics (n, v}, and we thus have an im-
portant basis for classification. Closely related is the theory of the
Clebsch connex; this figure, it is true, is considered as belonging to
algebraic geometry, but it defines (by means of its principal coinci-
dence) a system of usually transcendental panalgebraic curves.

Both points of view appear to characterize certain systems of
curves rather than individual curves. The following interpretation
may serve as a simple geometric definition of the curves considered.

With any plane curve C we may associate a space curve in this
\vay: at each point of C erect a perpendicular to the plane whose
length represents the slope of the curve at that point; the locus of
the end points of these perpendiculars is the associated space curve

1 Here would belong in particular the theory of algebraic curves based on link-
ages. Little advance has been made beyond the existence theorems of Kempe
and Koenigs. An important unsolved problem is the determination of the link-
age with minimum number of pieces by which a given curve can be described.

2 Cf. Loria, Spezielle Kurven, Leipzig, 1902.

574 GEOMETRY

C". Not every space curve is obtained in this way, but only those
whose tangents belong to a certain linear complex. If C is algebraic
so is C", and then an infinite number of algebraic surfaces may be
passed through the latter. If C is transcendental, so is C', and
usually no algebraic surface can be passed through it. Sometimes,
however, one such algebraic surface F exists. (If there were two,
C' and C would be algebraic.) It is precisely in this case that the
curve C is panalgebraic in the sense of Loria's theory. That such a
curve belongs to a definite system is seen from the fact that while the
surface F is unique, it contains a singly infinite number of curves
whose tangents belong to the linear complex mentioned, and the
orthogonal projections of these curves constitute the required system.
The principal problems in this field which require treatment are:
first, the exhaustive discussion of the simplest systems, correspond-
ing to small values of the characteristics n and v ; second, the study of
the general case in connection with (1) algebraic differential equa-
tions. (2) connexes, and (3) algebraic surfaces and linear complexes.

Natural or Intrinsic Geometry

In spite of the immediate triumph of the Cartesian system at the
time of its introduction into mathematics, rebellion against what
may be termed the tyranny of extraneous coordinates, first expressed
in the Characteristica geometrica of Leibnitz, has been an ever-present
though often subdued influence in the development of geometry.
Why should the properties of a curve be expressed in terms of x's
and y's which are defined not by the curve itself, but by its relation
to certain arbitrary elements of reference? The same curve in differ-
ent positions may have unlike equations, so that it is not a simple
matter to decide whether given equations represent really distinct or
merely congruent curves. The idea of the so-called natural or in-
trinsic coordinates had its birth during the early years of the nine-
teenth century, but it is only the systematic treatment of recent
years which has created a new field of geometry.

For a plane curve there is at each point the arc s measured from
some fixed point on the curve, and the radius of curvature p; these
intrinsic coordinates are connected by a relation p=f(s) which is
precisely characteristic of the curve, that is, the curves corresponding
to the equation differ only in position. There is, however, still
something arbitrary in the point taken as origin. This is eliminated
by taking as coordinates p and its derivative 8 taken with respect
to the arc; so that the final intrinsic equation is of the form 8 =F(p).
There is no difficulty in extending the method to space curves. The
two natural equations necessary are here r=<j)(p), S=t/;(p), where
p and T are the radii of first and second curvature and 8 is the arc
derivative of p.

PRESENT PROBLEMS OF GEOMETRY 575

The application to surfaces is not so evident. Thus, in Cesaro's
standard work, while the discussion of curves is consistently in-
trinsic, this is true to only a slight extent in the treatment of surfaces.
The natural geometry of surfaces is in fact only in process of forma-
tion. Bianchi proposes as intrinsic the familiar representation by
means of the two fundamental quadratic differential forms; but,
although it is true that the surfaces corresponding to a given pair
of forms are necessarily congruent, there is the disadvantage, arising
from the presence of arbitrary parameters, that the same surface
may be represented by distinct pairs of forms. One way of over-
coming this difficulty is to introduce the common feature of all pairs
corresponding to a surface, that is, the invariants of the forms: in
this direction we may cite Ricci's principle of covariant differentia-
tion and Maschke's recent application of symbolic methods.

The basis of natural geometry is, essentially, the theory of differ-
ential invariants. Under the group of motions, a given configuration
assumes oo r positions, where r is in general 6, but may be smaller
in certain cases. The r parameters which thus enter in the analytic
representation may be eliminated by the formation of differential
equations. The aim of natural geometry is to express these differ-
ential equations in terms of the simplest geometric elements of the
given configuration.

The beginning of such a discussion of surfaces was given by Sophus
Lie in 1896 and his work has been somewhat simplified by Scheffers.
As natural coordinates we may take the principal radii of curvature
Ri, R2, at a point of the surface, and their derivatives

dR1 _dR, dRi dR2

ds2 ds2 dst ds2

taken in the principal directions. For a given surface (excluding
the Weingarten class) the radii are independent, and there are four
relations of the form

Conversely, these equations are not satisfied by any surfaces except
those congruent or symmetric to the given surface.

It is to be noticed that four equations thus appear to be necessary
to define a surface, although two are sufficient for a twisted curve.
If a single equation in the above-mentioned natural coordinates is
considered, it is not, as in the case of ordinary coordinates, charac-
teristic: surfaces not congruent or symmetric to the given surface
would satisfy the equation. The apparent inconsistency which arises
is removed, however, by the fact that the four natural equations are

576 GEOMETRY

dependent.1 It is just this that makes the subject difficult as com-
pared with the theory of curves, in which the defining equations are
entirely arbitrary. The questions demanding treatment fall under
these two headings: first, the derivation of the natural equations
of the familiar types of surfaces, and second, the study of the new
types that correspond to equations of simple form. The natural
geometry of the Weingarten class of surfaces requires a distinct basis.

The fact that intrinsic* coordinates are, at bottom, differential
invariants with respect to the group of motions, suggests the exten-
sion of the same idea to the other groups. Thus in the projective
geometry of arbitrary (algebraic or transcendental) curves, coor-
dinates are required which, unlike the distances and angles ordin-
arily used, are invariant under projection. These might, for exam-
ple, be introduced as follows. At each point of the general curve C,
there is a unique osculating cubic and a unique osculating W (self-
projective) curve. Connected with each of these osculating curves
is an absolute projective invariant defined as an anharmonic ratio.
These ratios may then be taken as natural projective coordinates
y and o>, and the natural equation on the curve is of the form
y=f(w}. The principal advantage of such a representation is that
the necessary and sufficient condition for the equivalence of two
curves under projective transformations is simply the identity of the
corresponding equations.

Returning to the theory of surfaces, natural coordinates may
be introduced so as to fit into the so-called geometry of a flexible
but inextensible surface, originated by Gauss, in which the criterion
of equivalence is applicability, or, according to the more accurate
phraseology of Voss, isometry. Intrinsic coordinates must then be
invariant with respect to bending (Biegungsinvariante) . This pro-
perty is fulfilled, for example, by the Gaussian curvature K and the
differential parameters connected with it X=A (K, K), /t=A(/c, X),
^=A(X, X), all capable of simple geometric interpretation. The
intrinsic equations are then of the form p,=(j)(K, X), V=(J>(K, X).

A pair of equations of this kind thus represent, not so much a
single surface S, as the totality of all surfaces applicable on S (or
into which S may be bent) — a totality which is termed a complete
group G, since no additional surfaces are obtained when the same
process is applied to any member of the totality. The discussion of
such groups is ordinarily based on the first fundamental form (repre-
senting the squared element of length), since this is the same for
isometric surfaces; though of course it changes on the introduction
of new parameters.

The simplest example of a complete isometric group is the group

1 The three relations connecting the functions /n, /,„ fn, fn have been worked
out recently by S. Heller, Math. Annalen, vol. LVIII (1904).

PRESENT PROBLEMS OF GEOMETRY 577

typified by the plane, consisting of all the developable surfaces. In
this case the equations of the group may be obtained explicitly, in
terms of eliminations, differentiations, and quadratures. This is,
however, quite exceptional; thus, even in the case of the surfaces
applicable on the unit sphere (surfaces of constant Gaussian curv-
ature + 1), the differential equation of the group has not been
integrated explicitly. In fact, until the year 1866, not a single case
analogous to that of the developable surfaces was discovered. Wein-
garten, by means of his theory -of e volutes, then succeeded in deter-
mining the complete group of the catenoid and of the paraboloid
of revolution, and, some twenty years later, a fourth group defined
in terms of minimal surfaces.

During the past decade, the French geometers have concentrated
their efforts in this field mainly on the arbitrary paraboloid (and to
some extent on the arbitrary quadric). The difficulties even in this
extremely restricted and apparently simple case are great, and are
only gradually being conquered by the use of almost the whole
wealth of modern analysis and the invention of new methods which
undoubtedly have wider fields of application. The results obtained
exhibit, for example, connections with the theories of surfaces of
constant curvature, isometric surfaces, Backlund transformations,
and motions with two degrees of freedom. The principal workers
are Darboux, Goursat, Bianchi, Thybaut, Cosserat, Servant, Gui-
chard, and Raffy.

Geometry im Grossen

The questions we have just been considering, in common with
almost all the developments of general or infinitesimal geometry,
deal with the properties of the figure studied im Kleinen, that is,
in the sufficiently small neighborhood of a given point. Algebraic
geometry, on the other hand, deals with curves and surfaces in their
entirety. This distinction, however, is not inherent in the subject-
matter, but is rather a subjective one due to the limitations of our
analysis: our results being obtained by the use of power series are
valid only in the region of convergence. The properties of a curve
or surface (assumed analytic) considered as a whole are represented
not by means of function elements, but by means of the entire func-
tions obtained say by analytic continuation.

Only the merest traces of such a transcendental geometry im
Grossen are in existence, but the interest of many investigators is
undoubtedly tending in this direction. The difficulty of the problems
which arise (in spite of their simple and natural character) and the
delicacy of method necessary in their treatment may be compared
to the corresponding problems and methods of celestial mechanics.
The calculation of the ephemeris of a planet for a limited time is

578 GEOMETRY

a problem im Kleinen, while the discovery of periodic orbits and the
theory of the stability of the solar system are typical problems im
Grossen.

The principal problems in this field of geometry are connected
with closed curves and surfaces. Of special importance are the inves-
tigations relating to the closed geodesic lines which can be drawn
on a given surface, since these are apt to lead to the invention of
methods applicable to the wider field of dynamics. Geodesies may
in fact be defined dynamically as trajectories of a particle constrained
to the surface and acted upon either by no force or by a force due to
a force function U whose first differential parameter is expressible
in terms of U. The few general theorems known in this connection
are due in the main to Hadamard (Journal de Mathematiques, 1897,
1898). Thus, on a closed surface whose curvature is everywhere
positive, a point describing a geodesic must cross any existing closed
geodesic an infinite number of times, so that, in particular, two
closed geodesies necessarily intersect.1 On a surface of negative
curvature, under certain restrictions, there exist closed geodesies
of various topological types, as well as geodesies which approach
these asymptotically.

As regards surfaces all of whose geodesies are closed, the investi-
gations have been confined entirely to the case of surfaces of revo-
lution, the method employed being that suggested by Darboux in
the Cours de Mecanigue of Despeyrons. Last year Zoll 2 succeeded
in determining such a surface (beyond the obvious sphere) which
differs from the other known solutions in not having any singularities.
Analogous problems in connection with closed lines of curvature
and asymptotic lines will probably soon secure the consideration
they deserve.

A problem of different type is the determination of applicability
criteria valid for entire surfaces. The ordinary conditions (in terms
of differential parameters) assert, for example, the applicability of
any surface of constant positive curvature upon a sphere; but the
bending is actually possible only for a sufficiently small portion of the
surface. A spherical surface as a whole cannot be applied on any
other surface, that is, cannot be bent without extension or tearing.
This result is analogous to the theorem known to Euclid, although
first proved by Cauchy, that a closed convex polyhedral surface is
necessarily rigid. Lagrange, Minding, and Jellet stated the result for
all closed convex surfaces, but the complete discussion is due to
H. Liebmann.8 The theory of the deformation of concave surfaces

1 In a paper read before the St. Louis meeting of the American Mathematical
Society, Pomcare' stated reasons which make very probable the existence of at
least three closed geodesies on a surface of this kind.

2 Math. Anndlen, vol. LVII (1903).

* Gottingen Nachrichten, 1899; Math. Annalen, vols. LIII, LIV.

PRESENT PROBLEMS OF GEOMETRY 579

is far more complicated, and awaits solution even in the case of
polyhedral surfaces.

Beltrami's visualization of Lobachevsky's geometry by pictur-
ing the straight lines of the Lobachevsky plane as geodesies on
a surface of constant negative curvature is well known. However,
since the known surfaces of this kind, like the pseudosphere, have
singular lines, this method really depicts only part of the plane. In
fact Hilbert (Transactions of the American Mathematical Society
for 1900), by very refined considerations, has shown that an analytic
surface of constant negative curvature which is everywhere regular
does not exist, so that the entire Lobachevsky plane cannot be
depicted by any analytic surface.1 There remains undecided the
possibility of a complete representation by means of a non-analytic
surface. The partial differential equation of the surfaces of negative
constant curvature is of the hyperbolic type and hence does admit
non-analytic solutions.2 (This is not true for surfaces of positive
curvature, since the equation is then of elliptic type.) The discussion
of non-analytic curves and surfaces will perhaps be one of the really
new features of future geometry, but it is not yet possible to indicate
the precise direction of such a development.3

Other theories belonging essentially to geometry im Grossen
are the questions of analysis situs, or topology, to which reference has
been made on several occasions, and the properties of the very
general convex surfaces introduced by Minkowski in connection
with his Geometric der Zahlen.

Systems of Curves — Differential Equations

Although projective geometry has for its domain the investigation
of all properties unaltered by collineation, attention has been con-
fined almost exclusively to the algebraic configuration, so that pro-
jective is often confused with algebraic geometry. To the more
general projective geometry belong, for example, the ideas of oscu-
lating conic of an arbitrary curve and the asymptotic lines of an
arbitrary surface, and Mehmke's theorem which asserts that when
two surfaces touch each other, the ratio of their Gaussian curvatures
at the point of contact is an (absolute) projective invariant. The
field for investigation in this direction is of course very extensive,
but we may mention as a problem of special importance the deriva-

1 The entire projective plane, on the other hand, can be so depicted on a sur-
face devised by W. Boy (inaugural Dissertation, Gottingen, 1901).

2 According to Bernstein (Math. AnnaUn, vol. LIX, 1904, p. 72), the proof given
by Liitkemeyer (Inaugural Dissertation, Gottingen, 1902) is not valid, though
the conclusion is correct.

3 Lebesgue (Comptes Rendus, 1900, and Th£se, 1902) has examined the theory
of surfaces applicable on a plane without assuming the existence of derivatives
for the defining functions, and thereby obtains an example of a non-ruled develop-
able.

580 GEOMETRY

tion of the conditions for the protective equivalence of surfaces in
terms of their fundamental quadratic forms.

Coordinate with what has just been stated, that general configura-
tions may be studied from the protective point of view, is the fact
that algebraic configurations may be studied in relation to general
transformation theory. One may object that, with respect to the
group of all (analytic) point transformations, the algebraic con-
figurations do not form a body,1 that is, are not converted into
algebraic configurations; but such a body is obtained by adjoining
to the algebraic all those transcendental configurations which are
equivalent to algebraic. As this appears to have been overlooked,
it seems desirable to give a few concrete instances, of interest in
showing the effect of looking at familiar objects from a new and
more general point of view.

As a first example, consider the idea of a linear system of plane
curves. In algebraic geometry, a linear system is understood to be
one represented by an equation of the form

where the X's are parameters and the F's are polynomials hi x,y. On
the other hand, in general (infinitesimal) geometry, a system is defined
to be linear when it can be reduced (by the introduction of new
parameters) to the same form where the F's are arbitrary functions.
The first definition is invariant under the projective group; the sec-
ond, under the group of all point transformations. If now we apply
the second definition to algebraic curves, the result does not coincide
with that given by the first definition. Thus, every one-parameter
system is linear in the general sense, while only pencils of curves are
linear in the projective sense. The first case of real importance is,
however, the two-parameter system, since here each point of view
gives restricted, though not identical, types. An example in point
is furnished by the vertical parabolas tangent to a fixed line, the
equation of the system being y = (ax+b)2. From the algebraic or
projective point of view, this is a quadratic system since the para-
meters are involved to the second degree; but the system is linear
from the general point of view since its equation may be written
ax+b — V^=0. This suggests the problem: Determine the systems
of algebraic curves which are linear in the general sense.

As a second example, consider, from both points of view, the
equivalence of pencils of straight lines in the plane. By means of
collineations any two pencils may be converted into any other two;

1 The most extensive group for which the algebraic configurations form a body
consists of all algebraic transformations. It is rather remarkable that even this
theory has received no development.

1 Halphen, Laguerre, Forsyth. This theory has been extended to simultaneous
equations and applied geometrically by E. J. Wilczynski (Trans. Amer. Math.
Soc., 1901-1904).

PRESENT PROBLEMS OF GEOMETRY 581

but if three pencils are given, it is necessary to distinguish the case
where the three base points are in a straight line from the case where
they are not so situated. We thus have two protectively distinct
cases, which may be represented canonically by: (1) z=const.,
y=const., x+y=const., and (2) z=const., y = const., y/x=const.
The first type may, however, be converted into the second by the
transcendental transformation xl=ec, ?/1=ey, so that, in the general
group of point transformations, all sets of three pencils are equivalent.
The discussion for four or more pencils yields the rather surprising
result that the protective classification remains valid for the larger
group.

Dropping these special considerations on algebraic systems, let us
pass to the theory of arbitrary systems of curves, or, what is equiva-
lent, the geometry of differential equations. While belonging to the
cycle of theories due primarily to Sophus Lie, it has received little
development in the purely geometric direction. Most attention has
been devoted to special classes of differential equations with respect
to special groups of transformations. Thus there is an extensive
theory of the homogeneous linear equations with respect to the
group xl=$(x), y1=yi](x) which leaves the entire class invariant.1
A special theory which deserves development is that of equations of
the first order with respect to the infinite group of conformal trans-
formations.

As regards the general group of all point transformations, all
equations of the first order are equivalent, so that the first case of
interest is the theory of the two-parameter systems. The invariants
of the differential equation of second order have been discussed
most completely in the prize essay of A. Tresse (submitted to the
Jablonowski Gesellschaft in 1896), with application to the equiva-
lence problem. A specially important class, treated earlier by Lie
and R. Liouville, consists of the equations of cubic type

y" = Ay'3 +By'2 +Cy' +D,

where the coefficients are functions of x, y. It includes, in particular,
the general linear system and all systems capable of representing
the geodesies of any surface. While the analytical conditions which
characterize these subclasses are known, little advance has been
made in their geometric interpretation.

Perhaps the simplest configuration belonging to the field considered,
that is, having properties invariant under all point transformations,
is that composed of three simply infinite systems of curves, which
may be represented analytically by an equation of third degree in
y' with one-valued functions of x, y for coefficients. In the case of
equations of the fourth and higher degree in y', certain invariants

1 The elementary (metric) theory of curve systems has been too much neglected ;
it may be compared in interest and extent with the usual theory of surfaces.

582 GEOMETRY

may be found immediately from the fact that when x and y undergo
an arbitrary transformation, the derivative y' undergoes a fractional
linear transformation (of special type). The invariants found from
this algebraic principle are, however, in a sense, trivial, and the real
problem remains almost untouched: to determine the essential
invariants due to the differential relations connecting the coefficients
in the linear transformation of the derivative.

General Theory of Transformations

Closely connected with the geometry of differential equations
that we have been considering is the geometry of point transform-
ations. In the former theory the transformations enter only as
instruments, in the latter these instruments are made the subject-
matter of the investigation. The distinction is parallel to that which
occurs in protective geometry between the theory of projective
properties of curves and surfaces and the properties of collineations.
(It may be remarked, however, that although a transformation is
generally regarded as dynamic and a configuration as static, the
distinction is not at all essential. Thus a point transformation or
correspondence between the points of a plane may be viewed as
simply a double infinity of point pairs; on the other hand, a curve
in the plane may be regarded as the equivalent of .a correspondence
between the points of two straight lines.1)

We consider first two problems concerning the general (analytic)
point transformation which are of interest and importance from the
theoretic standpoint. The one relates to the discussion of the char-
acter of such a transformation in the neighborhood of a given point.
Transon's theorem states that the effect of any analytjc transform-
ation upon an infinitesimal region is the same as that of a pro-
jective transformation. This is true, however, only in general; it
ceases to hold when the derivatives of the defining functions vanish
at the point considered. What is the character of the transformation
in the neighborhood of such singular points ?

A more fundamental problem relates to the theory of equiva-
lence. Consider a transformation T which puts in correspondence
the points P and Q of a plane. Let the entire plane be subjected to
a transformation S which converts P into Pr and Q into Q'. We thus
obtain a new transformation T' in which P' and Q' are corresponding
points. This is termed the transform of T by means of *S,the relation
being expressed symbolically by T' =S~ *TS. The question then arises
whether all transformations are equivalent, that is, can any one be
converted into any other in the manner defined. The answer de-
pends on certain functional equations which also arise in connection

1 Geometry on a straight line, in its entirety, is as rich as geometry in a plane
or in space of any number of dimensions.

PRESENT PROBLEMS OF GEOMETRY 583

with the question whether an arbitrary transformation belongs to
a continuous group. The problem deserves treatment not merely for
the analytic transformations, but also for the algebraic and for
the continuous transformations.1

Aside from such fundamental questions, further development
is desirable both in the study of the general properties (associated
curve systems and contact relations) of an arbitrary transforma-
tion, and in the introduction of new special types of transformation,
for instance, those which may be regarded as natural extensions of
familiar types.

The main problems in the theory of point transformation are
connected with certain fields of application which we now pass in
review.

1. Cartography. A map may be regarded, abstractly, as the point
by point representation of one surface upon another, the case of
especial practical importance being, of course, the representation of
a spherical or spheroidal surface upon the plane. As it is impossible
to map any but the developable surfaces without distortion upon a
plane, the chief types of available representation are characterized
by the in variance of certain elements, as angles or #reas, or the
simple depiction of certain curves, as of geodesies by straight lines.
Most attention has been devoted to the conformal type, but the
question proposed by Gauss remains unsolved: what is the best
conformal representation of a given surface on the plane, that is,
the one accompanied by the minimum distortion? The answer, of
course, depends on the criterion adopted for measuring the degree
of distortion, and it is in this direction that progress is to be
expected.

2. Mathematical theory of elasticity. As a geometric foundation
for the mechanics of continua, it is necessary to study the most
general deformation of space, defined say by putting xi} yl} Zi equal
to arbitrary functions of x, y, z. The most elegant analytical repre-
sentation, as given for instance in the memoir of E. and F. Cosserat
(Annales de Toulouse, volume 10), is obtained by introducing the
elements of length ds and dsl before and after deformation, and the
related quadratic differential form ds* — ds2 =2eldx2 +2e2dy2 +2e3dz2
+ 2yl dydz + 2y2 dxdz + 2y3dxdy. The theory is thus seen to be ana-
logous to though of course more complicated than the usual theory of
surfaces. The six functions of x, y, z which appear as coefficients
in this form are termed the components of the deformation. Their

1 This problem is not to be confused with the similar (but simpler) question
connected with Lie's division of (analytic) groups into demokratisch and aristo-
kratisch. In those of the first kind all the infinitesimal transformations are
equivalent, in those of the second there exist non-equivalent infinitesimal trans-
formations. Lie shows that all finite groups are aristokratisch, while the groups
of all (analytic) point and contact transformations are demokratisch. Cf. Leip-
ziger Berichte, vol. XLVII (1895), p. 271.

584 GEOMETRY

importance is due to the fact that they vanish only when the trans-
formation is a rigid displacement, so that two deformations have
the same components when, and only when, they differ by a dis-
placement. The case where the components are constants leads to
the homogeneous deformation (or affine transformation of the geo-
meters), the type considered almost exclusively in the usual dis-
cussions of elasticity. It would seem desirable to study in detail
the next case which presents itself, namely, that in which the com-
ponents are linear functions of x, y, z.

In the general deformation, the six components are not inde-
pendent, but are connected by nine differential equations analogous
to those of Codazzi. The fact that a transformation is defined by
three independent functions indicates, however, that there should be
only three distinct relations between the components. This means
that the nine equations of condition which occur in the standard
theory are themselves interdependent; but their relations (analogous
to syzygies among syzygies in the algebra of forms) do not appear
to have been worked out.

3. Vector fields. From its beginning in the Faraday-Maxwell
theory of electricity until the present day, the course which the
discussion of vector fields has followed has been guided almost
entirely by external considerations, namely, the physical applications.
While this is advantageous in many respects, it cannot be denied
that it has led to lack of symmetry and generality. The time seems
to be ripe for a more systematic mathematical development. The
vector field deserves to be introduced as a standard form into geo-
metry.

Abstractly, such a field is equivalent to a point transformation of
space, since each is represented by three scalar relations in six variables.
Instead of taking these variables as the coordinates of corresponding
points, it is more convenient to consider three as the coordinates
x, y, z of a particle and the other three as components u, v, w of its
velocity; we thus picture the set of functional relations by means
of the steady motion of a hypothetical space-filling fluid. This image
should be of service even in abstract analysis; for its role is analogous
to that of the curve in dealing with a single relation between two
variables. The streaming of a material fluid is, of course, not suffi-
ciently general for such a purpose, since, in virtue of the equation of
continuity, it images only a particular class of vector fields.

In addition to the ordinary vector fields, physics makes use of
so-called hypervector fields, which, geometrically, lead to configur-
ations consisting of a triply infinite system of quadric surfaces, one
for each point of space. In the special case of interest in hydro-
dynamics (irrotational motion), the configuration simplifies in that
the quadrics are ellipsoids about the corresponding points as centres.

PRESENT PROBLEMS OF GEOMETRY 585

This is equivalent to the tensor field which arises in studying the
moments of inertia of an arbitrary distribution of mass. The more
general case actually arises in Maxwell's theory of magnetism.

4. As a final domain of application we mention the class of ques-
tions which have received systematic treatment, under the title of
nomography , only during the past few years. This subject deals with
the methods of representing graphically, in a plane, functional
relations containing any number of variables. Thus a function of
two independent variables, z=f(x, y), may be represented by the
system of plane curves /(z, y) =c, each marked with the correspond-
ing value of the. parameter. This " parametered " system is then
a cartesian graphical table, which is the simplest type of abacus or
nomogram.

By means of any point transformation, one nomogram is con-
verted into another which may serve to represent the same functional
relation. The importance of this process of conversion (the so-called
anamorphosis of Lalanne and Massau) depends on the fact that it
may replace a complicated table by a simpler. The problems which
arise (for example, the determination of all relations between three
variables which can be represented by a nomogram composed of
three systems of straight lines1) are of both practical and theoretical
interest. The literature is scattered through the French, Italian,
and German technological journals, but a systematic presentation
of the main results is to be found in the Traite de Nomographie
of d'Ocagne (Paris, 1899).

We return to the abstract theory of transformations. The type
of transformation we have been considering, converting point into
point, is only a special case of more general types. The most im-
portant extension hitherto made depends upon the introduction of
differential elements. Thus the lineal element or directed point
(x, y, y'} leads to transformations which in general convert a point
into a system of elements; when the latter form a curve, every curve
is converted into a curve and the result is termed a contact trans-
formation. Backlund has shown that no extension results from the
elements of second or higher order: osculation transformations are
necessarily contact transformations. The discussion of elements of
infinitely high order, defined by an infinite set of coordinates (x, y,
y') y") • • •)> may perhaps lead to a real extension. The question may
be put in this form: Are there transformations (in addition to or-
dinary contact transformations) which convert analytic curves into
analytic curves in such a way that contact is an invariant relation?
The idea of curve transformation in general will probably be worked

1 The case of three systems of circles has also been discussed. See d'Ocagne,
Journal de I'Ecole Poly technique, 1902.

586 GEOMETRY

out in the near future: what is the most general mode of setting up
a correspondence which associates with every Jordan curve another
Jordan curve? Such discussions are aspects of geometry with an
infinite number of dimensions.

After a review of the kind given in this paper, one is tempted to
ask: What is it which influences the mathematician in selecting
certain (out of an infinite number of equally conceivable) problems
for investigations? It is true, of course, that his subject is ideal,
self-created, and that " Das Wesen der Mathematik liegt in. ihrer
Freiheit." Georg Cantor would indeed replace the term pure mathe-
matics by free mathematics. This freedom, however, is not entirely
caprice. The investigators of each age have always felt it their
duty to deal with the unsolved questions and to generalize the re-
sults and conceptions inherited from the past, to correlate with
other fields of contemporaneous thought, to keep in contact, as far
as possible, with the whole body of truth. This is not all, however.
The influence of aesthetic considerations, though less subject to
analysis, has been, and still is, of at least equal importance in guiding
the course of mathematical development.

SHORT PAPERS

The Section of Geometry was very fully attended and productive of extended
discussion and a number of supplementary papers. For the same reason as in the
Section of Algebra and Analysis it is impossible to give a satisfactory resume of
the short papers on this subject owing to their close technical reasoning.

The first paper was presented by Professor Harris Hancock, of the University
of Cincinnati, on "Algebraic Minimal Surfaces."

The second paper was presented by Professor H. T. Blichfeldt, of Leland Stan-
ford Jr. University, on the subject "Concerning some Geometrical Properties
of Surfaces of Revolution."

The third paper was presented by Professor George Bruce Halsted, of Kenyon
College, on " Non-Euclidean Spherics."

The fourth paper was presented by Professor Arnold Emch, of the University
of Colorado, on "The Configuration of the Points of Inflection of a Plane
Cubic and their Harmonic Polars."

The fifth paper was presented by Professor H. P. Manning, of Brown University,
on " Representation of Complex Variables in Space of Four Dimensions."

The sixth paper was read by Professor G. A. Bliss, of the University of Missouri,
on " Concerning Calcidus of Variations."

The seventh paper was presented by Professor L. W. Dowling, of the University
of Wisconsin, on "Certain Universal Curves."

SECTION C— APPLIED MATHEMATICS

SECTION C— APPLIED MATHEMATICS

(Hall 7, September 24, 3 p. w.)

CHAIRMAN: PROFESSOR ARTHUR G. WEBSTER, Clark University, Worcester,
Mass.

SPEAKERS: PROFESSOR LUDWIG BOLTZMANN, University of Vienna.

PROFESSOR HENRI POINCARE, The Sorbonne; Member of the Insti-
tute of France.

SECRETARY: PROFESSOR HENRY T. EDDY, University of Minnesota.

THE RELATIONS OF APPLIED MATHEMATICS

BY LUDWIG BOLTZMANN
(Translated from the German by Professor S. Epsteen, University of Chicago)

[Ludwig Boltzmann, Professor of Physics, University of Vienna, since 1902.
b. Vienna, Austria, 1840. Studied, Vienna, Heidelberg, and Berlin. Professor
of Physi co-Mathematics, University of Gratz, 1869-73; Professor of Mathe-
matics, University of Vienna, 1873-76; Professor of Experimental Physics,
University of Gratz, 1876-90; Professor of Theoretical Physics, University
of Munich, 1891-95; ibid. University of Vienna, 1895-1900; Professor of
Physics, University of Leipzig, 1900-02. Author of Vorlesungen uber Max-
well's Theorie der Elekt rizitat und des Lichts; Vorlesungen uber Kinetische
Gastheorie; Vorlesungen uber die Prinzipe der Mechanik.]

MY present lecture has been put under the heading of applied
mathematics, while my activity as a teacher and investigator be-
longs to the science of physics. The immense gap which divides
the latter science into two distinct camps has almost nowhere been
so sharply emphasized as in the division of the lecture material
of this scientific congress, which covers such an enormous range of
subjects that one may designate it as a flood, or, to preserve local
coloring, as a Niagara of scientific lectures. I speak of the division
of physics into theoretical and experimental. Although I have
been assigned, as representative of theoretical physics, to "A. —
Normative Science," experimental physics appears much later under
" C. — Physical Science." Between them lie history, science of lan-
guage, literature, art, and science of religion. Over all this, however,
the theoretical physicist must extend his hand to the experimental
physicist. We shall therefore not be able to avoid entirely the ques-
tion of the justification of dividing physics into two parts and, in
particular, into theoretical and experimental.

Let us listen first of all to an investigator of a time when natural
science had not yet grown beyond its first beginnings, to Emmanuel
Kant. Kant requires of each science that it should be developed

592 APPLIED MATHEMATICS

logically from unified principles and firmly established theories.
Natural science seems to him a primary science only in so far as
it rests on a mathematical basis. Thus, he does not reckon the chem-
istry of his day among the sciences, because it rests merely upon
an empirical basis and lacks a unified, regulative principle.

From this point of view theoretical physics is preferred to ex-
perimental physics, and occupies, in a sense, a higher rank. Experi-
mental physics was merely to gather the material, but it remained
for the theoretical physics to form the structure.

But the succession in the order of rank becomes reversed when
we take into account the acquisitions of the last decades as well as
the progress which is to be expected in the immediate future. The
chain of experimental discoveries of the last century received a
fitting completion with the discovery of the Rontgen rays. Con-
nected with these there appear in the present century a multitude
of new rays, with the most enigmatical properties, which have the
profoundest effects upon our conceptions of nature. The more
enigmatical these newly discovered facts are, and the more they
seem at first to contradict our present conceptions, the greater the
successes which they promise for the future. But this is not the occa-
sion for the discussion of these experimental triumphs. I must leave
to the representatives of experimental physics at this Congress the
prolific problem of portraying all of the fruits which have hitherto
been gathered in this domain, one might almost say, daily, and
those which are to be expected.

The representative of theoretical physics scarcely finds himself in
an equally fortunate position. Great activity does indeed prevail
in this domain. One could almost say that it is in process of revolu-
tion. Only how much less tangible are the results here attained in
comparison with those in experimental physics! It appears here
that in a certain sense experimentation deserves precedence over
all theory. An immediate fact is at once comprehensible. Its fruits
may become evident in the shortest time, such as the various appli-
cations of the Rontgen rays and the utilization of the Hertz waves
in wireless telegraphy. The battle which the theories have to fight
is, however, an infinitely wearisome one; indeed, it seems as if cer-
tain disputed questions which existed from the beginning will live
as long as the science.

Every firmly established fact remains forever unchangeable; at
most, it may be generalized, completed, additions may be made,
but it cannot be completely upset. Thus it is explained why the
development of experimental physics is continuously progressive,
never making a sudden jump, and never visited by great tremblings
and revolutions. It occurs only in rare instances that something
which was regarded as a fact turns out afterwards to have been an

RELATIONS OF APPLIED MATHEMATICS 593

error, and in such cases the explanations of the errors follow soon,
and they are not of great influence on the structure of the science as
a whole.

It is, indeed, strongly emphasized that every established and
logically recognized truth must remain incontrovertible. Although
this cannot be doubted, experience teaches that the structure of our
theories is by no means composed entirely of such incontrovertibly
established truths. They are composed rather of many arbitrary
pictures of the connections between phenomena, of so-called hypo-
theses.

Without some departure, however slight, from direct observation,
a theory or even an intelligibly connected practical description for
predicting the facts of nature cannot exist. This is equally true of
the old theories whose foundations have become questionable, and
of the most modern ones, which are resigning themselves to a great
illusion if they regard themselves as free from hypotheses.

The hypotheses may perhaps be indefinite, or may be in the shape
of mathematical formulae, or the thought may be equivalent to the
latter, but expressed in words. In the latter cases the agreement
with given data may be checked step by step; a complete revolu-
tion of that previously constructed is indeed not absolutely impos-
sible, as, for example, if the law of the conservation of energy should
turn out to be incorrect. But such a revolution will be exceedingly
rare and highly improbable.

Such an indefinite, slightly specialized theory might serve as a
guiding thread for experiments whose purpose is a detailed develop-
ment of knowledge previously acquired and which is proceeding in
barren channels ; beyond this its usefulness does not reach.

In contradistinction to these are the hypotheses which give the
imagination room for play and by boldly going beyond the material
at hand afford continual inspiration for new experiments, and are
thus pathfinders for the most unexpected discoveries. Such a theory
will indeed be subject to change, a very complicated mass of inform-
ation will be brought together and will then be replaced by a new
and more comprehensive theory in which the old one will be the pic-
ture of a limited type of phenomena. Examples of this are the theory
of emission in regard to the description of the phenomena of catoptrics
and dioptrics, the hypothesis of an elastic ether in the representation
of the phenomena of interference and refraction of light, and the
notion of the electric fluid in the description of the phenomena of
electrostatics.

Moreover the theories which proudly designate themselves as free
from hypotheses are not exempt from great revolutions; thus, no one
will doubt that the so-called theory of energy will have completely
to alter its form if it desires to remain effective.

594 APPLIED MATHEMATICS

The accusation has been made that physical hypotheses have
sometimes proved injurious and have delayed the progress of the
science. This accusation is based chiefly upon the r61e which the
hypothesis of the electric fluid has played in the development of the
theory of electricity. This hypothesis was brought to a high stage
of perfection by Wilhelm Weber, and the general recognition which
his works found in Germany did indeed stand in the road of the
theory of Maxwell. In a similar manner Newton's emanation theory
stood in the way of the theory of undulations. But such incon-
veniences can scarcely be entirely avoided in the future. It will al-
ways be the tendency to complete as far as possible the prevailing
view, and to make it self-sufficient whenever such a theory is self-
consistent and does not in any way lead to a contradiction, whether
it consist of mechanical models, of geometrical pictures, or of mathe-
matical formulas. It will always be possible that a new theory will
arise which has not yet been tested by experiment and which will
represent a much larger field of phenomena. In such cases the older
theory will count the largest following until this field of phenomena
is brought into the range of experiment, and decisive tests demon-
strate the superiority of the newer one. It is certainly useful, if the
theory of Weber be always held up as a warning example, that one
should bear in , mind the essential progressiveness of the intellect.
The services of Weber are not decreased by this, however; Maxwell
himself speaks of his theory with the greatest wonder. Indeed, this
instance cannot be taken into consideration against the usefulness of
hypotheses, since Maxwell's theory contained as much of the hypo-
thetical as any other. And this was eliminated only after it became
generally known through Hertz, Poynting, and others.

The accusation has also been raised against hypotheses in physics
that the creation and development of mathematical methods for
the computation of the hypothetical molecular motions has been
useless and even harmful. This accusation I cannot recognize as
substantiated. Were it so, the theme selected for my present lecture
would be an unfortunate one; and this fact may excuse me for
having lingered on this much-discussed subject and for having sought
to justify the use of hypotheses in physics.

I have not chosen for the thesis of my present lecture the entire
development of physical theory. Several years ago I treated this
subject at the German Naturforscherversammlung in Munich, and
although some new developments have taken place since then, I
should have to repeat myself a great deal. Moreover, one who has
committed himself to one faction is not in a position to judge the
other factions in a completely objective manner. I do not refer to a
criticism of its value; my lecture shall not criticise, but shall judge.
I am also convinced of the value of the views of my opponents and

RELATIONS OF APPLIED MATHEMATICS 595

only arise to repel them when they attempt to belittle mine. But
one can scarcely give as complete an account according to subject-
matter, and an exposition of the inter-relations of all ideas in the
views of another, as in his own.

I shall therefore select as the goal of my lecture to-day not merely
the kinetic theory of molecules, but, moreover, a highly specialized
branch of it. Far from denying that it contains hypotheses, I must
rather characterize it as a bold advance beyond the facts of observa-
tion. And I nevertheless do not consider it unworthy of this occa-
sion; this much faith do I have in hypotheses which present certain
peculiarities of observation in a new light or which bring forth rela-
tions among them which cannot be reached by other methods. We
must indeed be mindful of the fact that hypotheses require and are
capable of continuous development, and are only then to be aban-
doned when all the relations which they represent can be better
understood in some other manner.

To the above-mentioned problems, which are as old as the science
and still unsolved, belongs the one if matter is continuous, or if it
is to be considered as made up of discrete parts, of very many, but
not in the mathematical sense infinite, individuals. This is one of
the difficult questions which form the boundary of philosophy and
physics.

Even some decades ago, scientists felt very shy of going deeply
into the discussion of such questions. The one before us is too real
to be entirely avoided ; but one cannot discuss it without touching on
some profounder still, such as upon the nature of the law of causation,
of matter, of force, and so forth. The latter are the ones of which it
was said that they did not trouble the scientist, that they belonged
entirely to philosophy. To-day the situation is different, there is
evident a tendency among scientists to consider philosophic questions,
and properly so. One of the first rules of science is never to trust
blindly to the instrument with which one works, but to test it in
all directions. How, then, are we to trust blindly to inherited and
historically developed conceptions, particularly when there are
instances known where they led us into error ? But in the examina-
tion of even the simplest elements, where is the boundary between
science and philosophy at which we should pause ?

I hope that none of the philosophers present will take offense or
perceive an accusation, if I say boldly that by assigning this question
to philosophy the resulting success has been rather meagre. Philo-
sophy has done noticeably little toward the explanation of these
questions, and from her own one-sided point of view she can do so just
as little as natural science can from hers. If real progress is possible,
it is only to be expected by cooperation of both of these sciences.
May I therefore be pardoned if I touch slightly upon these questions

r,96 APPLIED MATHEMATICS

although not a specialist; their connection with the aim of my lec-
ture is very intimate.

Let us consult the famous thinker already quoted, Emmanuel
Kant, on the question if matter is continuous, or if it is composed
of atoms. He treats of this in his Antimonies. Of all the questions
there raised, he shows that both the pro and con can be logically
demonstrated. It can be shown rigorously that there is no limit to
the divisibility of matter while an infinite divisibility contradicts the
laws of logic. Kant shows likewise that a beginning and end of time,
a boundary where space ceases, are as inconceivable as absolutely
endless duration, absolutely endless extension.

This is by no means the sole instance where philosophical thought
becomes tangled in contradictions; indeed, one finds them at every
step. The ordinary things of philosophy are sources of insolvable
riddles; to explain our perceptions it invents the concept of matter
and then finds that it is altogether unsuited to possess perception
itself or to generate perception in a spirit. With consummate acumen
it constructs the concept of space, or of time, and finds that it is
absolutely impossible that things should exist in this space, that
events should occur during this time. It finds insurmountable
difficulties in the relation of cause to effect, of body and soul, in
the possibility of consciousness, in short, everywhere and in every-
thing. Indeed, it finally finds it inexplicable and self-contradictory
that anything can exist at all, that something originated and is cap-
able of continuing, that everything can be classified according to
our categories, nor that there is a quite perfect classification. Such
a classification will always be a variable one and adapted to the
requirements of the moment. Also the breaking up of physics into
theoretical and experimental is merely a consequence of the preval-
ent division of methods and will not last forever.

My present thesis is quite different from the one that certain
questions are beyond the boundary of human comprehension. For
according to the latter, there is a deficiency, an incompleteness in the
human intelligence, while I consider the existence of these questions,
these problems, as an illusion. By superficial consideration it seems
astonishing, after this illusion is recognized, that the impulse to
answer those questions does not cease. Habit of thought is much too
powerful to release us.

It is here as with the ordinary illusion which continues operative
after its cause is recognized. In consequence of this is the feeling of
uncertainty, of want of satisfaction which the scientist feels when he
philosophizes. These illusions will yield but very slowly and gradually,
and I consider it as one of the chief problems of philosophy to set
forth clearly the uselessness of reaching beyond the limits of our
habits of thought and to strive, in the choice and combination of

RELATIONS OF APPLIED MATHEMATICS 597

concepts and words, to give the most useful expression of facts in a
manner which is independent of our inherited habits. Then all these
complications and contradictions must vanish. It must be made
clear what is stone in the structure of our thoughts and what is
mortar, and the oppressive sentiment, that the simplest things are
the most inexplicable and the most trivial are the most mysterious,
becomes mere imagination-change.

To call upon logic seems to me as if one were to put on for a trip
into the mountains a long flowing robe, which always wrapped
itself about the feet so that one fell at the first steps while on the level.
The source of this kind of logic is the immoderate trust in the so-
called laws of thought. It is certain that we could not gather experi-
ence did we not have certain forms of connecting phenomena, that is
to say, of thought, innate. If we wish to call these laws of thought,
they are indeed a priori to the extent that they accompany every
experience in our soul, or if we prefer, in our brain. Only nothing
seems to me less reasonable than the conclusion from the reasoning
in this sense to certainty, to infallibility. These laws of thought
have been developed according to the same laws of evolution as
the optical apparatus of the eye, the acoustic apparatus of the ear,
and the pumping arrangements of the heart. In the course of human
development everything useless was eliminated, and thus a unity
and finish arose which might be mistaken for infallibility. Thus the
perfection of the eye, of the ear, of the arrangement of the heart
excite our admiration, without the absolute perfection of these
organs being emphasized, however. Just so little should the laws of
thought be regarded as absolutely infallible. They are the very ones
which have developed with regard to seizing that which is most
necessary and practically useful in the maintenance of life. With
these, the results of experimental investigation show more relation
than the examination of the mechanism of thought. We should,
therefore, not be surprised that the customary forms of thought
for the abstract are not entirely suited to practical applications
in far removed problems of philosophy, and that they have not
become applicable since the days of Thales. Therefore the simplest
things seem to be the most puzzling to the philosopher. And he
finds everywhere contradictions; these are nothing more, however,
than useless, incorrect facsimiles of that which is given us through
our thoughts. In facts there can be no contradictions. As soon as
contradictions seem unavoidable we must test, extend, and seek
to modify that which we call laws of thought, but which are only
inherited, customary representations, preserved for aeons, for the
description of practical needs. Just as to the inherited discoveries
of the cylinder, the carriage, the plow, numerous artificial ones have
been consciously added, so must we improve, artificially and con-

598 APPLIED MATHEMATICS

sciously, our likewise inherited concepts. Our problem cannot be
to quote facts before the judgment seat of our laws of thought, but
to lit our mental representations and concepts to the facts. Since
we attempt to express with clearness such complicated processes
merely by words, written, spoken, or inwardly thought, it might
also be said that we should combine the words in such wise as to
give the most appropriate expression of the facts, that the relations
indicated by our words should be most adequate for the relations
among the actualities. When the problem is enunciated in this
fashion, its appropriate solution may still offer the greatest difficulties,
but one knows then the end in view and will not stumble on self-
made difficulties.

Much that is useless in the usage and in the bearing of the nature
of life is brought forth by a method of treatment which, being
useful in most cases, becomes through habit a second nature, until
one cannot set it aside when it becomes inapplicable somewhere.
I say that the adaptability goes beyond the point aimed at. This
happens frequently in the commonplaces of thought, and becomes
the source of apparent contradictions between the laws of thought
and the world, as well as between the laws of thought themselves.

Thus, the regularity of the phenomena of nature is the funda-
mental condition for all cognition; thus comes the habit of inquiring
of everything the cause, the non-resisting compulsion, and we
inquire also concerning the cause, why everything must have a cause.
In fact people strove for a long time to determine if cause and effect
is a necessary bond or merely an accidental sequence, and if it did
or did not have a unique meaning to say that a certain particular
phenomenon was connected with, and a necessary consequence of,
a definite group of other phenomena.

Similarly, something is said to be useful, valuable, if it satisfies the
needs of the individual or of humanity; but we go beyond the mark
if we ask concerning the value of life itself, if such it seem to have,
because it has no purpose outside of itself. The same happens when
we strive vainly to explain the simplest concepts, out of which all
others are built, by means of simpler ones still, to explain the simplest
fundamental laws.

We should not attempt to deduce nature from our concepts, but
should adapt the latter to the former. We should not believe our
inherited rules of thought to be conditions preceding our more com-
plicated experiences, for they are not so for the simplest essentials.
They arose slowly in connection with simple experiences and passed,
by heredity, to the more highly organized being. Thus is explained
how synthetic judgments arise which were formed by our ancestors
and were born in us, and are in this sense a priori. Their great
power is also seen in this way, but not their infallibility.

RELATIONS OF APPLIED MATHEMATICS 599

In saying that such judgments as "everything is red or is not red"
are results of experience, I do not mean that every person checks this
empty truth by experience, but that he learns that his parents called
everything either red or not red and that he preserves this nomen-
clature.

It might seem as if we had gone somewhat deeply into philosophical
questions, but I believe that the views we have reached could not
have been attained in a shorter and simpler manner. For we have
reached an impartial judgment how the question of the atomistic
structure of matter is to be viewed. We shall not invoke the law of
thought that there is no limit to the divisibility of matter. This law
is of no more value than if a naive person were to say that no matter
where he went upon the earth the plumb-line directions seemed
always to be parallel and therefore there were no antipodes.

On the one hand we shall start from facts only, and on the other
we shall take nothing into consideration except the effort to attain
to the most adequate expression of these facts.

Regarding the first point, the numerous facts of the theory of
heat, of chemistry, of crystallography, show that bodies which are
apparently continuous do not by any means fill the entire volume
indistinguishably and uniformly with matter. Indeed, it appears
that the space which they occupy is filled with innumerably many
individuals, molecules, and atoms, which are extraordinarily small,
but not infinitely small in the mathematical sense. Their sizes can
be computed in different manners and always with the same result.

The fruitfulness of this line of thought has been verified in the
most recent time. All the phenomena which are observed with the
cathode rays, the Becquerel rays, etc., indicate that we are dealing
with diminutive, moving particles, electrons. After a vigorous
battle, this view vanquished completely the opposing explanation of
these phenomena by the theory of undulations. Not only did the
former theory give a better explanation of the previously known
facts, it inspired new experiments and permitted the prediction of
unknown phenomena, and thus it developed into an atomistic theory
of electricity. If it continue to develop with the same success as
in past years, if phenomena, such as the one observed by Ramsay
on the transmutation of radium into helium, do not remain isolated,
this theory promises deductions concerning the nature and structure
of atoms as yet undreamed of. Computation shows that electrons are
much smaller than the atoms of ponderable matter; and the hypo-
thesis that the atoms are built up of many elements, as well as
various interesting views on the character and structure of this com-
position, is to-day on every tongue. The word atom should not
lead us into error, it comes from a past time; no physicist ascribes
indivisibility to the atoms.

600 APPLIED MATHEMATICS

It is not my intention to confine the thought merely to the above
facts and their resulting consequences; these are not sufficient to
carry through the question as to the finite or infinite divisibility
of matter. If we are going to think of the atoms of chemistry as
made up of electrons, what would hinder us from considering the
electrons as particles filled with rarefied, continuous matter? We
shall adhere faithfully to the previously developed philosophical
principles and shall examine in the most unhampered manner the
concepts themselves in order to express them in a consistent and
most useful form.

It appears now, that we are unable to define the infinite in any other
way except as the limit of continually increasing magnitudes, at
least no one has hitherto been able to set up any other intelligible
conception of the infinite. Should we desire a verbal picture of the
continuum, we must first think of a large finite number of particles
which are endowed with certain properties and study the totality
of these particles.. Certain properties of this totality may approach
a definite limit as the number of particles is increased, and their
size decreased. It can be asserted, concerning these properties, that
they belong to the continuum, and it is my opinion that this is the
only self-consistent definition of a continuum which is endowed
with certain properties.

The question if matter is composed of atoms or is continuous
becomes then the question if the observed properties are accurately
satisfied by the assumption of an exceedingly great number of
such particles or, by increasing number, their limit. We have not
indeed answered the old philosophical question, but we are cured of
the effort to answer it in a senseless and hopeless manner. The
thought-process, that we must investigate the properties of a finite
totality and then let the number of members of this totality increase
greatly, remains the same in both cases. It is nothing other than
the abbreviated expression in algebraic symbols of exactly the same
thought when, as often happens, differential equations are made
the basis of a mathematical-physical theory.

The members of the totality which we select as the picture of the
material body cannot be thought of as absolutely at rest, for there
would then be no motion of any kind, nor can the members be thought
of as relatively at rest in one and the same body, for in this case it
would be impossible to account for the fluids. No effort has been
made to subject them to anything more than to the general laws
of mechanics. In order to explain nature we shall therefore select
a totality of an exceedingly large number of very minute funda-
mental individuals which are constantly in motion, and which are
subject to the laws of mechanics. But an objection is raised that
will be an appropriate introduction to the final considerations of

RELATIONS OF APPLIED MATHEMATICS 601

this lecture. The fundamental equations of mechanics do not alter
their form in the slightest way when the algebraic sign of the time is
changed. All pure mechanical events can therefore occur equally
well in one sense as in its opposite, that is, in the sense of increasing
time or of diminishing time. We remark, however, that in ordinary
life future and past do not coincide as completely as the directions
right and left, but that the two are distinctly different.

This becomes still more definite by means of the second law of the
mechanical theory of heat, which asserts that when an arbitrary
system of bodies is left to itself, uninfluenced by other bodies, the
sense in which changes of condition occur can be assigned. A certain
function of the condition of all the bodies, the entropy, can be
determined, which is such that every change that occurs must be in
the sense of carrying with it an increase of this function; thus,
with increasing time the entropy increases. This law is indeed an
abstraction, just as the principle of Galileo; for it is impossible, in
strict rigor, to isolate a system of bodies from all others. But since
it has given correct results hitherto, in connection with all the other
laws, we assume it to be correct, just as in the case of the principle of
Galileo.

It follows from this law that every closed system of bodies must
tend toward a definite final condition for which the entropy is a
maximum. The outcome of this law, that the universe must come
to a final state in which nothing more can occur, has caused aston-
ishment; but this outcome is only comprehensible on the assump-
tion that the universe is finite and subject to the second law of the
mechanical theory of heat. If one regards the universe as infinite,
the above-mentioned difficulties of thought arise again if one does
not consider the infinite as a mere limit of the finite. Since there is
nothing analogous to the second law in the differential equations
of mechanics, it follows that it can be represented mechanically only
by the initial conditions. In order to find the assumptions suit-
able for this purpose, we must reflect that, to explain the appar-
ent continuity of bodies, we had to assume that every family
of atoms, or more generally, of mechanical individuals, existed in
incredibly many different initial positions. In order to treat this
assumption mathematically, a new science was founded whose pro-
blem is, not the study of the motion of a single mechanical system,
but of the properties of complexes of very many mechanical systems
which begin with a great variety of initial conditions. The task of
systematizing this science, of compiling it into a large book, and of
giving it a characteristic name, was executed by one of the greatest
American scholars, and in regard to abstract thinking, purely theo-
retic investigation, perhaps the greatest, Willard Gibbs, the recently
deceased professor at Yale University. He called this science statis-

602 APPLIED MATHEMATICS

tical mechanics, and it falls naturally into two parts. The first in-
vestigates the conditions under which the outwardly visible proper-
ties of a complex of very many mechanical individuals is not in any
wise altered; this first part I shall call statistical statics. The sec-
ond part investigates the gradual changes of these outwardly visible
properties when those conditions are not fulfilled; it may be called
statistical dynamics. At this point we may allude to the broad view
which is opened by applying this science to the statistics of ani-
mated beings, of human society, of sociology, etc., and not merely
upon mechanical particles. A development of the details of this
science would only be possible in a series of lectures and by means
of mathematical formulas. Apart from mathematical difficulties it is
not free from difficulties of principle. It is based upon the theory
of probabilities. The latter is as exact as any other branch of mathe-
matics if the concept of equal probabilities, which cannot be de-
duced from the other fundamental notions, is assumed. It is here
as in the method of least squares which is only free from objection
when certain definite assumptions are made concerning the equal
probability of elementary errors. The existence of this fundamental
difficulty explains why the simplest result of statistical statics, the
proof of Maxwell's speed law among the molecules of a gas, is still
being disputed.

The theorems of statistical mechanics are rigorous consequences
of the assumptions and will always remain valid, just as all well-
founded mathematical theorems. But its application to the events
of nature is the prototype of a physical hypothesis. Starting from
the simplest fundamental assumption of the equal probabilities, we
find that aggregates of very many individuals behave quite ana-
logously as experience shows of the material world. Progressive or
visible rotary motion must always go over into invisible motion of
the minutest particles, into heat, as Helmholtz characteristically
says: ordered motion tends always to go over into not ordered
motion; the mixture of different substances as well as of different
temperatures, the points of greater or less intense molecular
motion, must always tend toward homogeneity. That this mixture
was not complete from the start, that the universe began in such
an improbable state, belongs to the fundamental hypotheses of the
entire theory; and it may be said that the reason for this is as little
known as the reason why the universe is just so and not otherwise.
But we may take a different point of view. Conditions of great mix-
ture and great differences in temperature are not absolutely impos-
sible according to the theory but are very highly improbable. If the
universe be considered as large enough there will be, according to the
laws of probability, here and there places of the size of fixed stars,
of altogether improbable distributions. The development of such

RELATIONS OF APPLIED MATHEMATICS 603

a spot would be one-sided both in its structure and subsequent dis-
solution. Were there thinking beings at such a spot their impressions
of time would be the same as ours, although the course of events in
the universe as a whole would not be one-sided. The above-developed
theory does indeed go boldly beyond our experience, but it has the
merit which every such theory should have of showing us the facts
of experience in an entirely new light and of inspiring us to new
thought and reflection. In contradistinction to the first fundamental
law, the second one is merely based on probability, as Gibbs pointed
out in the 70 's of the last century.

I have not avoided philosophical questions, in the firm hope that
cooperation between philosophy and natural science will give new
sustenance to both; indeed, that only in this manner a consistent
argument can be carried through. I agree with Schiller when he
says to the scientists and philosophers of his day, " Let there be strife
between you, and the union will come speedily;" I believe that the
time for this union has now arrived.

THE PRINCIPLES OF MATHEMATICAL PHYSICS

BY JULES HENRI POINCARE

(Translated from the French by George Bruce Hoisted, Kenyon College)

[Jules Henri Poincare, Professor University of Paris, and the Polytechnic
School; Member of Bureau of Longitude, b. Nancy, April 29, 1854. D.Sc.
August 3, 1879; D.Sc. Cambridge and Oxford, 1879; Charge of the Course
of the Faculty of Sciences at Caen; Master of Conference of the Faculty of
Sciences of Paris, 1881; Professor of the same Faculty, 1886; Member of the
Institute of France, 1887; Corresponding Member of the National Academy
of Washington; Philosophical Society of Philadelphia; the Academies of
Berlin, London, St. Petersburg, Vienna, Rome, Munich, Gottingen, Bologna,
Turin, Naples, Venice, Amsterdam, Copenhagen, Stockholm, etc. Written
books and numerous articles for reviews and periodicals.]

WHAT is the actual state of mathematical physics? What are the
problems it is led to set itself? What is its future? Is its orientation
on the point of modifying itself?

Will the aim and the methods of this science appear in ten years
to our immediate successors in the same light as to ourselves; or,
on the contrary, are we about to witness a profound transformation?
Such are the questions we are forced to raise in entering to-day upon
our investigation.

If it is easy to propound them, to answer is difficult.

If we feel ourselves tempted to risk a prognostication, we have,
to resist this temptation, only to think of all the stupidities the
most eminent savants of a hundred years ago would have uttered,
if one had asked them what the science of the nineteenth century
would be. They would have believed themselves bold in their pre-
dictions, yet after the event how very timid we should have found
them.

Mathematical physics, we know, was born of celestial mechanics,
which engendered it at the end of the eighteenth century, at the
moment when the latter was attaining its complete development.
During its first years especially, the infant resembled in a striking
way its mother.

The astronomic universe is formed of masses, very great without
doubt, but separated by intervals so immense that they appear to
us only as material points. These points attract each other in the
inverse ratio of the square of the distances, and this attraction is
the sole force which influences their movements. But if our senses
were sufficiently subtle to show us all the details of the bodies which
the physicist studies, the spectacle we should there discover would
scarcely differ from what the astronomer contemplates. There also
we should see material points, separated one from another by inter-

PRINCIPLES OF MATHEMATICAL PHYSICS 605

vals enormous in relation to their dimensions, and describing orbits
following regular laws.

These infinitesimal stars are the atoms. Like the stars properly
so called, they attract or repel each other, and this attraction or this
repulsion directed following the straight line which joins them, de-
pends only on the distance. The law according to which this force
varies as function of the distance is perhaps not the law of Newton,
but it is an analogous law; in place of the exponent — 2, we have
probably a different exponent, and it is from this change of exponent
that springs all the diversity of physical phenomena, the variety of
qualities and of sensations, all the world colored and sonorous which
surrounds us, — in a word, all nature.

Such is the primitive conception in all its purity. It only remains
to seek in the different cases what value should be given to this
exponent in order to explain all the facts. It is on this model that
Laplace, for example, constructed his beautiful theory of capillarity;
he regards it only as a particular case of attraction, or as he says
of universal gravitation, and no one is astonished to find it in the
middle of one of the five volumes of the Mecanique celeste.

More recently Briot believed he had penetrated the final secret
of optics in demonstrating that the atoms of ether attract each other
in the inverse ratio of the sixth power of the distance; and does not
Maxwell himself say somewhere that the atoms of gases repel each
other in the inverse ratio of the fifth power of the distance? We have
the exponent — 6, or — 5 in place of the exponent — 2, but it is
always an exponent.

Among the theories of this period, one alone is an exception, that
of Fourier; in it are indeed atoms, acting at a distance one upon the
other; they mutually transmit heat, but they do not attract, they
never budge. From this point of view, the theory of Fourier must
have appeared to the eyes of his contemporaries, even to Fourier
himself, as imperfect and provisional.

This conception was not without grandeur; it was seductive, and
many among us have not finally renounced it; we know that we
shall attain the ultimate elements of things only by patiently disen-
tangling the complicated skein that our senses give us; that it is
necessary to advance step by step, neglecting no intermediary; that
our fathers were wrong in wishing to skip stations; but we believe
that when we shall have arrived at these ultimate elements, there
again will be found the majestic simplicity of celestial mechanics.

Neither has this conception been useless; it has rendered us an
inestimable service, since it has contributed to make precise in us
the fundamental notion of the physical law.

I will explain myself; how did the ancients understand law? It
was for them an internal harmony, static, so to say, and immutable;

606 APPLIED MATHEMATICS

or it was like a model that nature constrained herself to imitate. A
law for us is not that at all; it is a constant relation between the
phenomenon of to-day and that of to-morrow; in a word, it is a
differential equation.'

The ideal form of physical law is the law of Newton which first
covered it; and then how has one to adapt this form to physics?
by copying as much as possible this law of Newton, that is, in imi-
tating celestial mechanics.

Nevertheless, a day arrived when the conception of central forces
no longer appeared sufficient, and this is the first of those crises of
which I just now spoke.

Then investigators gave up trying to penetrate into the detail
of the structure of the universe, to isolate the pieces of this vast
mechanism, to analyze one by one the forces which put them in
motion, and were content to take as guides certain general prin-
ciples which have precisely for their object the sparing us this minute
study.

How so? Suppose that we have before us any machine; the ini-
tial wheel-work and the final wheel-work alone are visible, but the
transmission, the intermediary wheels by which the movement is
communicated from one to the other are hidden in the interior
and escape our view; we do not know whether the communication
is made by gearing or by belts, by connecting-rods or by other dis-
positives.

Do we say that it is impossible for us to understand anything about
this machine so long as we are not permitted to take it to pieces?
You know well we do not, and that the principle of the conservation
of energy suffices to determine for us the most interesting point. We
easily ascertain that the final wheel turns ten times less quickly than
the initial wheel, since these two wheels are visible; we are able
thence to conclude that a couple applied to the one will be balanced
by a couple ten times greater applied to the other. For that there
is no need to penetrate the mechanism of this equilibrium and to
know how the forces compensate each other in the interior of the
machine; it suffices to be assured that this compensation cannot fail
to occur.

Well, in regard to the universe, the principle of the conservation
of energy is able to render us the same service. This is also a machine,
much more complicated than all those of industry, and of which
almost all the parts are profoundly hidden from us; but in observing
the movement of those that we can see, we are able, by aid of this
principle, to draw conclusions which remain true whatever may be
the details of the invisible mechanism which animates them.

The principle of the conservation of energy, or the principle of
Mayer, is certainly the most important, but it is not the only one;

PRINCIPLES OF MATHEMATICAL PHYSICS 607

there are others from which we are able to draw the same advantage.
These are:

The principle of Carnot, or the principle of the degradation of
energy.

The principle of Newton, or the principle of the equality of action
and reaction.

The principle of relativity, according to which the laws of phys-
ical phenomena should be the same, whether for an observer
fixed, or for an observer carried along in a uniform move-
ment of translation; so that we have not and could not
have any means of discerning whether or not we are carried
along in such a motion.

The principle of the conservation of mass, or principle of

Lavoisier.
I would add the principle of least action.

The application of these five or six general principles to the differ-
ent physical phenomena is sufficient for our learning of them what
we could reasonably hope to know of them.

The most remarkable example of this new mathematical physics
is, beyond contradiction, Maxwell's electro-magnetic theory of light.

We know nothing of the ether, how its molecules are disposed,
whether they attract or repel each other; but we know that this
medium transmits at the same time the optical perturbations and
the electrical perturbations; we know that this transmission should
be made conformably to the general principles of mechanics, and
that suffices us for the establishment of the equations of the electro-
magnetic field.

These principles are results of experiments boldly generalized;
but they seem to derive from their generality itself an eminent
degree of certitude.

In fact the more general they are, the more frequently one has
the occasion to check them, and the verifications, in multiplying
themselves, in taking forms the most varied and the most unex-
pected, finish by no longer leaving place for doubt.

Such is the second phase of the history of mathematical physics,
and we have not yet emerged from it.

Do we say that the first has been useless? that during fifty years
science went the wrong way, and that there is nothing left but to
forget so many accumulated efforts as vicious conceptions condemned
in advance to non-success?

Not the least in the world ; the second phase could not have come
into existence without the first?

The hypothesis of central forces contained all the principles; it
involved them as necessary consequences; it involved both the con-

608 APPLIED MATHEMATICS

servation of energy and that of masses, and the equality of action
and reaction; and the law of least action, which would appear, it
is true, not as experimental verities, but as theorems, and of which
the enunciation would have at the same time a something more pre-
cise and less general than under their actual form.

It is the mathematical physics of our fathers which has familiar-
ized us little by little with these divers principles; which has taught
us to recognize them under the different vestments in which they
disguise themselves. One has to compare them to the data of ex-
perience, to find how it was necessary to modify their enunciation
so as to adapt them to these data; and by these processes they
have been enlarged and consolidated.

So we have been led to regard them as experimental verities;
the conception of central forces became then a useless support, or
rather an embarrassment, since it made the principles partake of its
hypothetical character.

The frames have not therefore broken, because they were elastic;
but they have enlarged; our fathers, who established them, did not
work in vain, and we recognize in the science of to-day the general
traits of the sketch which they traced.

Are we about to enter now upon the eve of a second crisis? Are
these principles on which we have built all about to crumble away
in their turn? For some time, this may well have been asked.

In hearing me speak thus, you think without doubt of radium,
that grand revolutionist of the present time, and in fact I will come
back to it presently; but there is something else.

It is not alone the conservation of energy which is in question;
all the other principles are equally in danger, as we shall see in pass-
ing them successively in review.

Let us commence with the principle of Carnot. This is the only
one which does not present itself as an immediate consequence of
the hypothesis of central forces; more than that, it seems, if not
directly to contradict that hypothesis, at least not to be reconciled
with it without a certain effort.

If physical phenomena were due exclusively to the movements
of atoms whose mutual attraction depended only on the distance,
it seems that all these phenomena should be reversible; if all the
initial velocities were reversed, these atoms, always subjected to
the same forces, ought to go over their trajectories in the contrary
sense, just as the earth would describe in the retrograde sense this
same elliptic orbit which it describes in the direct sense, if the initial
conditions of its movement had been reversed. On this account, if
a physical phenomenon is possible, the inverse phenomenon should
be equally so, and one should be able to reascend the course of
time.

PRINCIPLES OF MATHEMATICAL PHYSICS 609

But it is not so in nature, and this is precisely what the principle
of Carnot teaches us; heat can pass from the warm body to the cold
body; it is impossible afterwards to make it reascend the inverse
way and reestablish differences of temperature which have been
effaced.

Motion can be wholly dissipated and transformed into heat by
friction; the contrary transformation can never be made except in
a partial manner.

We have striven to reconcile this apparent contradiction. If the
world tends toward uniformity, this is not because its ultimate parts,
at first unlike, tend to become less and less different, it is because,
shifting at hazard, they end by blending. For an eye which should
distinguish all the elements, the variety would remain always as
great, each grain of this dust preserves its originality and does not
model itself on its neighbors; but as the blend becomes more and
more intimate, our gross senses perceive no more than the uniform-
ity. Behold why, for example, temperatures tend to a level, without
the possibility of turning backwards.

A drop of wine falls into a glass of water; whatever may be the
law of the internal movements of the liquid, we soon see it colored
to a uniform rosy tint, and from this moment, however well we
may shake the vase, the wine and the water do not seem capable of
further separation. Observe what would be the type of the reversible
physical phenomenon: to hide a grain of barley in a cup of wheat
is easy; afterwards to find it again and get it out is practically im-
possible.

All this Maxwell and Boltzmann have explained; the one who has
seen it most clearly, in a book too little read because it is a little
difficult to read, is Gibbs, in his Elementary Principles of Statistical
Mechanics.

For those who take this point of view, the principle of Carnot is
only an imperfect principle, a sort of concession to the infirmity of
our senses; it is because our eyes are too gross that we do not dis-
tinguish the elements of the blend; it is because our hands are too
gross that we cannot force them to separate; the imaginary demon
of Maxwell, who is able to sort the molecules one by one, could well
constrain the world to return backward. Can it return of itself? That
is not impossible; that is only infinitely improbable.

The chances are that we should long await the concourse of cir-
cumstances which would permit a retrogradation, but soon or late
they would be realized, after years whose number it would take
millions of figures to write.

These reservations, however, all remained theoretic and were not
very disquieting, and the principle of Carnot retained all its practical
value.

610 APPLIED MATHEMATICS

But here the scene changes.

The biologist, armed with his microscope, long ago noticed in his
preparations disorderly movements of little particles in suspension:
this is the Brownian movement; he first thought this was a vital
phenomenon, but he soon saw that the inanimate bodies danced with
no less ardor than the others; then he turned the matter over to the
physicists. Unhappily, the physicists remained long uninterested in
this question; the light is focused to illuminate the microscoprc pre-
paration, thought they; with light goes heat; hence inequalities of
temperature and interior currents produce the movements in the
liquid of which we speak.

M. Gouy, however, looked more closely, and he saw, or thought
he saw, that this explanation is untenable, that the movements
become more brisk as the particles are smaller, but that they are not
influenced by the mode of illumination.

If, then, these movements never cease, or rather are reborn with-
out ceasing, without borrowing anything from an external source
of energy, what ought we to believe? To be sure, we should not
renounce our belief in the conservation of energy, but we see under
our eyes now motion transformed into heat by friction, now heat
changed inversely into motion, and that without loss since the move-
ment lasts forever. This is the contrary of the principle of Carnot.

If this be so, to see the world return backward, we no longer
have need of the infinitely subtle eye of Maxwell's demon; our
microscope suffices us. Bodies too large, those, for example, which
are a tenth of a millimeter, are hit from all sides by moving atoms,
but they do not budge, because these shocks are very numerous and
the law of chance makes them compensate each other: but the
smaller particles receive too few shocks for this compensation to
take place with certainty and are incessantly knocked about. And
thus already one of our principles is in peril.

We come to the principle of relativity : this not only is confirmed
by daily experience, not only is it a necessary consequence of the
hypothesis of central forces, but it is imposed in an irresistible way
upon our good sense, and yet it also is battered.

Consider two electrified bodies; though they seem to us at rest,
they are both carried along by the motion of the earth; an electric
charge in motion, Rowland has taught us, is equivalent to a current;
these two charged bodies are, therefore, equivalent to two parallel
currents of the same sense and these two currents should attract
each other. In measuring this attraction, we measure the velocity
of the earth; not its velocity in relation to the sun or the fixed stars,
but its absolute velocity.

I know it will be said that it is not its absolute velocity that
is measured, but its velocity in relation to the ether. How unsatis-

PRINCIPLES OF MATHEMATICAL PHYSICS 611

factory that is! Is it not evident that from a principle so under-
stood we could no longer get anything? It could no longer tell us
anything just because it would no longer fear any contradiction.

If we succeed in measuring anything, we should always be free
to say that this is not the absolute velocity in relation to the ether,
it might always be the velocity in relation to some new unknown
fluid with which we might fill space.

Indeed, experience has taken on itself to ruin this interpretation
of the principle of relativity; all attempts to measure the velocity
of the earth in relation to the ether have led to negative results.
This time experimental physics has been more faithful to the prin-
ciple than mathematical physics; the theorists, to put in accord
their other general views, would not have spared it; but experiment
has been stubborn in confirming it.

The means have been varied in a thousand ways and finally
Michelson has pushed precision to its last limits; nothing has come
of it. It is precisely to explain this obstinacy that the mathematicians
are forced to-day to employ all their ingenuity.

Their task was not easy, and if Lorentz has gotten through it,
it is only by accumulating hypotheses.

The most ingenious idea has been that of local time.

Imagine two observers who wish to adjust their watches by
optical signals; they exchange signals, but as they know that the
transmission of light is not instantaneous, they take care to cross
them.

When the station B perceives the signal from the station A, its
clock should not mark the same hour as that of the station A at the
moment of sending the signal, but this hour augmented by a con-
stant representing the duration of the transmission. Suppose, for
example, that the station A sends its signal when its clock marks
the hour 0, and that the station B perceives it when its clock marks
the hour t. The clocks are adjusted if the slowness equal to t repre-
sents the duration of the transmission, and to verify it the station B
sends in its turn a signal when its clock marks 0; then the station A
should perceive it when its clock marks t. The time-pieces are then
adjusted. And in fact, they mark the same hour at the same phys-
ical instant, but on one condition, namely, that the two stations are
fixed. In the contrary case the duration of the transmission will not
be the same in the two senses, since the station A, for example,
moves forward to meet the optical perturbation emanating from B,
while the station B flies away before the perturbation emanating
from A. The watches adjusted in that manner do not mark, there-
fore, the true time; they mark what one may call the local time, so
that one of them goes slow on the other. It matters little, since we
have no means of perceiving it. All the phenomena which happen

612 APPLIED MATHEMATICS

at A, for example, will be late, but all will be equally so, and the
observer who ascertains them will not perceive it, since his watch is
slow; so, as the principle of relativity would have it, he will have no
means of knowing whether he is at rest or in absolute motion.

Unhappily, that does not suffice, and complementary hypotheses
are necessary ; it is necessary to admit that bodies in motion undergo
a uniform contraction in the sense of the motion. One of the dia-
meters of the earth, for example, is shrunk by 200000000 in conse-
quence of the motion of our planet, while the other diameter retains
its normal length. Thus, the last little differences find themselves
compensated. And then there still is the hypothesis about forces.
Forces, whatever be their origin, gravity as well as elasticity, would
be reduced in a certain proportion in a world animated by a uniform
translation; or, rather, this would happen for the components perpen-
dicular to the translation; the components parallel would not change.

Resume, then, our example of two electrified bodies; these bodies
repel each other, but at the same time if all is carried along in a
uniform translation, they are equivalent to two parallel currents of
the same sense which attract each other. This electro-dynamic
attraction diminishes, therefore, the electro-static repulsion, and the
total repulsion is more feeble than if the two bodies were at rest.
But since to measure this repulsion we must balance it by another
force, and all these other forces are reduced in the same proportion,
we perceive nothing.

Thus, all is arranged, but are all the doubts dissipated?

What would happen if one could communicate by non-luminous
signals whose velocity of propagation differed from that of light?
If, after having adjusted the watches by the optical procedure, one
wished to verify the adjustment by the aid of these new signals,
then would appear divergences which would render evident the com-
mon translation of the two stations. And are such signals incon-
ceivable, if we admit with Laplace that universal gravitation is
transmitted a million times more rapidly than light?

Thus, the principle of relativity has been valiantly defended in
these latter times, but the very energy of the defense proves how
serious was the attack.

Let us speak now of the principle of Newton, on the equality of
action and reaction.

This is intimately bound up with the preceding, and it seems
indeed that the fall of the one would involve that of the other.
Thus we should not be astonished to find here the same difficulties.

Electrical phenomena, we think, are due to the displacements of
little charged particles, called electrons, immersed in the medium
that we call ether. The movements of these electrons produce per-
turbations in the neighboring ether; these perturbations propagate

PRINCIPLES OF MATHEMATICAL PHYSICS a 13

themselves in every direction with the velocity of light, and in turn
other electrons, originally at rest, are made to vibrate when the
perturbation reaches the parts of the ether which touch them.

The electrons, therefore, act upon one another, but this action is
not direct, it is accomplished through the ether as intermediary.

Under these conditions can there be compensation between action
and reaction, at least for an observer who should take account
only of the movements of matter, that is to say, of the electrons, and
who should be ignorant of those of the ether that he could not see?
Evidently not. Even if the compensation should be exact, it could
not be simultaneous. The perturbation is propagated with a finite
velocity; it, therefore, reaches the second electron only when the
first has long ago entered upon its rest.

This second electron, therefore, will undergo, after a delay, the
action of the first, but certainly it will not react on this, since around
this first electron nothing any longer budges.

The analysis of the facts permits us to be still more precise. Imagine
for example, a Hertzian generator, like those employed in wireless
telegraphy; it sends out energy in every direction; but we can
provide it with a parabolic mirror, as Hertz did with his smallest
generators, so as to send all the energy produced in a single direction.

What happens, then, according to the theory? It is that the
apparatus recoils as if it were a gun and as if the energy it has
projected were a bullet; and that is contrary to the principle of
Newton, since our projectile here has no mass, it is not matter, it
is energy.

It is still the same, moreover, with a beacon light provided with
a reflector, since light is nothing but a perturbation of the electro-
magnetic field. This beacon light should recoil as if the light it
sends out were a projectile. What is the force that this recoil should
produce? It is what one has called the Maxwell-Bartholdi pressure.
It is very minute, and it has been difficult to put it into evidence
even with the most sensitive radiometers; but it suffices that it exists.

If all the energy issuing from our generator falls on a receiver,
this will act as if it had received a mechanical shock, which will
represent in a sense the compensation of the recoil of the generator;
the reaction will be equal to the action, but it will not be simulta-
neous; the receiver will move on but not at the moment when the
generator recoils. If the energy propagates itself indefinitely with-
out encountering a receiver, the compensation will never be made.

Do we say that the space which separates the generator from
the receiver and which the perturbation must pass over in going
from the one to the other is not void, that it is full not only of ether,
but of air; or even in the interplanetary spaces of some fluid subtle
but still ponderable; that this matter undergoes the shock like the

614 APPLIED MATHEMATICS

receiver at the moment when the energy reaches it, and recoils in its
turn when the perturbation quits it? That would save the principle
of Newton, but that is not true.

If energy in its diffusion remained always attached to some ma-
terial substratum, then matter in motion would carry along light
with it, and Fizeau has demonstrated that it does nothing of the
sort, at least for air. This is what Michelson and Morley have since
confirmed.

One may suppose also that the movements of matter, properly
so called, are exactly compensated by those of the ether; but that
would lead us to the same reflections as just now. The principle so
extended would explain everything, since whatever might be the
visible movements, we should always have the power of imagining
hypothetical movements which compensated them.

But if it is able to explain everything, this is because it does
not permit us to foresee anything; it does not enable us to decide
between different possible hypotheses, since it explains everything
beforehand. It therefore becomes useless.

And then the suppositions that it would be necessary to make
on the movements of the ether are not very satisfactory.

If the electric charges double, it would be natural to imagine
that the velocities of the divers atoms of ether double also, and for
the compensation, it would be necessary that the mean velocity of
the ether quadruple.

This is why I have long thought that these consequences of
theory, contrary to the principle of Newton, would end some day
by being abandoned, and yet the recent experiments on the move-
merits of the electrons issuing from radium seem rather to confirm
them.

I arrive at the principle of Lavoisier on the conservation of masses :
in truth this is one not to be touched without unsettling all mechanics.

And now certain persons believe that it seems true to us only
because we consider in mechanics merely moderate velocities, but
that it would cease to be true for bodies animated by velocities com-
parable to that of light. These velocities, it is now believed, have
been realized; the cathode rays or those of radium may be formed
of very minute particles or of electrons which are displaced with
velocities smaller no doubt than that of light, but which might be its
one tenth or one third.

These rays can be deflected, whether by an electric field, or by
a magnetic field, and we are able by comparing these deflections, to
measure at the same time the velocity of the electrons and their mass
(or rather the relation of their mass to their charge). But when
it was seen that these velocities approached that of light, it was
decided that a correction was necessary.

PRINCIPLES OF MATHEMATICAL PHYSICS 615

These molecules, being electrified, could not be displaced with-
out agitating the ether; to put them in motion it is necessary to
overcome a double inertia, that of the molecule itself and that of the
ether. The total or apparent mass that one measures is composed,
therefore, of two parts: the real or mechanical mass of the mole-
cule and the electro-dynamic mass representing the inertia of the
ether.

The calculations of Abraham and the experiments of Kaufmann
have then shown that the mechanical mass, properly so called, is
null, and that the mass of the electrons, or, at least, of the negative
electrons, is of exclusively electro-dynamic origin. This forces us to
change the definition of mass; we cannot any longer distinguish
mechanical mass and electro-dynamic mass, since then the first would
vanish; there is no mass other than electro-dynamic inertia. But
in this case the mass can no longer be constant, it augments with the
velocity, and it even depends on the direction, and a body animated
by a notable velocity will not oppose the same inertia to the forces
which tend to deflect it from its route, as to those which tend to
accelerate or to retard its progress.

There is still a resource; the ultimate elements of bodies are
electrons, some charged negatively, the others charged positively.
The negative electrons have no mass, this is understood; but the
positive electrons, from the little we know of them, seem much
greater. Perhaps they have, besides their electro-dynamic mass,
a true mechanical mass. The veritable mass of a body would, then,
be the sum of the mechanical masses of its positive electrons, the
negative electrons not counting; mass so defined could still be con-
stant.

Alas, this resource also evades us. Recall what we have said
of the principle of relativity and of the efforts made to save it. And
it is not merely a principle which it is a question of saving, such
are the indubitable results of the experiments of Michelson.

Lorentz has been obliged to suppose that all the forces, what-
ever be their origin, were affected with a coefficient in a medium
animated by a uniform translation; this is not sufficient; it is still
necessary, says he, that the masses of all the particles be influenced
by a translation to the same degree as the electro-magnetic masses
of the electrons.

So the mechanical masses will vary in accordance with the same
laws as the electro-dynamic masses; they cannot, therefore, be con-
stant.

Need I point out that the fall of the principle of Lavoisier in-
volves that of the principle of Newton? This latter signifies that
the centre of gravity of an isolated system moves in a straight line;
but if there is no longer a constant mass, there is no longer a centre

616 APPLIED MATHEMATICS

of gravity, we no longer know even what this is. This is why I
said above that the experiments on the cathode rays appeared to
justify the doubts of Lorentz on the subject of the principle of
Newton.

From all these results, if they are confirmed, would arise an
entirely new mechanics, which would be, above all, characterized by
this fact, that no velocity could surpass that of light, any more than
any temperature could fall below the zero absolute, because bodies
would oppose an increasing inertia to the causes, which would tend
to accelerate their motion; and this inertia would become infinite
when one approached the velocity of light.

Nor for an observer carried along himself in a translation he
did not suspect could any apparent velocity surpass that of light;
there would then be a contradiction, if we recall that this observer
would not use the same clocks as a fixed observer, but, indeed, clocks
marking "local time/'

Here we are then facing a question I content myself with stating.
If there is no longer any mass, what becomes of the law of Newton?

Mass has two aspects, it is at the same time a coefficient of iner-
tia and an attracting mass entering as factor into Newtonian attrac-
tion. If the coefficient of inertia is not constant, can the attracting
mass be? That is the question.

At least, the principle of the conservation of energy yet remains
to us, and this seems more solid. Shall I recall to you how it was
in its turn thrown into discredit? This event has made more noise
than the preceding and it is in all the records.

From the first works of Becquerel, and, above all. when the
Curies had discovered radium, one saw that every radio-active body
was an inexhaustible source of radiations. Its activity would seem
to subsist without alteration throughout the months and the years.
This was already a strain on the principles; these radiations were in
fact energy, and from the same morsel of radium this issued and for-
ever issued. But these quantities of energy were too slight to be
measured; at least one believed so and was not much disquieted.

The scene changed when Curie bethought himself to put radium
into a calorimeter; it was seen then that the quantity of heat in-
cessantly created was very notable.

The explanations proposed were numerous; but in so far as no
one of them has prevailed over the others, we cannot be sure there
is a good one among them.

Sir William Ramsay has striven to show that radium is in process
of transformation, that it contains a store of energy enormous but
not inexhaustible.

The transformation of radium, then, would produce a million
times more of heat than all known transformations; radium would

PRINCIPLES OF MATHEMATICAL PHYSICS 617

wear itself out in 1250 years; you see that we are at least certain
to be settled on this point some hundreds of years from now. While
waiting our doubts remain.

In the midst of so many ruins what remains standing? The prin-
ciple of least action has hitherto remained intact, and Larmor appears
to believe that it will long survive the others; in reality, it is still
more vague and more general.

In presence of this general ruin of the principles, what attitude
will mathematical physics take?

And first, before too much perplexity, it is proper to ask if all this
is really true. All these apparent contradictions to the principles are
encountered only among infinitesimals; the microscope is necessary
to see the Brownian movement; electrons are very light; radium is
very rare, and no one has ever seen more than some milligrams of
it at a time.

And, then, it may be asked if, beside the infinitesimal seen, there
be not another infinitesimal unseen counterpoise to the first.

So, there is an interlocutory question, and, as it seems, only
experiment can solve it. We have, therefore, only to hand over the
matter to the experimenters, and, while waiting for them to deter-
mine the question finally, not to preoccupy ourselves with these dis-
quieting problems, but quietly continue our work, as if the princi-
ples were still uncontested. We have much to do without leaving
the domain where they may be applied in all security; we have
enough to employ our activity during this period of doubts.

And as to these doubts, is it indeed true that we can do nothing
to disembarrass science of them? It may be said, it is not alone
experimental physics that has given birth to them; mathematical
physics has well contributed. It is the experimenters who have seen
radium throw out energy, but it is the theorists who have put in
evidence all the difficulties raised by the propagation of light across
a medium in motion; but for these it is probable we should not have
become conscious of them. Well, then, if they have done their best
to put us into this embarrassment, it is proper also that they help us
to get out of it.

They must subject to critical examination all these new views
I have just outlined before you, and abandon the principles only
after having made a loyal effort to save them.

What can they do in this sense? That is what I will try to ex-
plain.

Among the most interesting problems of mathematical physics,
it is proper to give a special place to those relating to the kinetic
theory of gases. Much has already been done in this direction, but
much still remains to be done. This theory is an eternal paradox.
We have reversibility in the premises and irreversibility in the con-

CIS APPLIED MATHEMATICS

elusions; and between the two an abyss' Statistic considerations,
the law of great numbers, do they suffice to fill it? Many points
still remain obscure to which it is necessary to return, and doubtless
many times. In clearing them up, we shall understand better the
sense of the principle of Carnot and its place in the ensemble of
dynamics, and we shall be better armed to interpret properly the
curious experiment of Gouy, of which I spoke above.

Should we not also erideavor to obtain a more satisfactory theory
of the electro-dynamics of bodies in motion? It is there especially,
as I have sufficiently shown above, that difficulties accumulate.
Evidently we must heap up hypotheses, we cannot satisfy all the
principles at once; heretofore, one has succeeded in safeguarding
some only on condition of sacrificing the others; but all hope of
obtaining better results is not yet lost. Let us take, therefore, the
theory of Lorentz, turn it in all senses, modify it little by little, and
perhaps everything will arrange itself.

Thus in place of supposing that bodies in motion undergo a con-
traction in the sense of the motion, and that this contraction is the
same whatever be the nature of these bodies and the forces to which
they are otherwise submitted, could we not make an hypothesis
more simple and more natural?

We might imagine, for example, that it is the ether which is
modified when it is in relative motion in reference to the material
medium which it penetrates, that when it is thus modified, it no
longer transmits perturbations with the same velocity in every direc-
tion. It might transmit more rapidly those which are propagated
parallel to the medium, whether in the same sense or in the opposite
sense, and less rapidly those which are propagated perpendicularly.
The wave surfaces would no longer be spheres, but ellipsoids, and we
could dispense with that extraordinary contraction of all bodies.

I cite that only as an example, since the modifications one might
essay would be evidently susceptible of infinite variation.

It is possible also that the astronomer may some day furnish us data
on this point; he it was in the main who raised the question in
making us acquainted with the phenomenon of the aberration of light.
If we make crudely the theory of aberration, we reach a very curious
result. The apparent positions of the stars differ from their real
positions because of the motion of the earth, and as this motion is
variable, these apparent positions vary. The real position we cannot
know, but we can observe the variations of the apparent position.
The observations of the aberration show us, therefore, not the
movement of the earth, but the variations of this movement; they
cannot, therefore, give us information about the absolute motion
of the earth. At least this is true in first approximation, but it
would be no longer the same if we could appreciate the thousandths

PRINCIPLES OF MATHEMATICAL PHYSICS 619

of a second. Then it would be seen that the amplitude of the oscil-
lation depends not alone on the variation of the motion, variation
which is well known, since it is the motion of our globe on its elliptic
orbit, but on the mean value of this motion; so that the constant of
aberration would not be altogether the same for all the stars, and the
differences would tell us the absolute motion of the earth in space.

This, then, would be, under another form, the ruin of the prin-
ciple of relativity. We are far, it is true, from appreciating the
thousandths of a second, but after all, say some, the total absolute
velocity of the earth may be much greater than its relative velocity
with respect to the sun. If, for example, it were 300 kilometers per
second in place of 30, this would suffice to make the phenomena
observable.

I believe that in reasoning thus we admit a too simple theory
of aberration. Michelson has shown ms, I have told you, that the
physical procedures are powerless to put in evidence absolute mo-
tion; I am persuaded that the same will be true of the astronomic
procedures, however far one pushes precision.

However that may be, the data astronomy will furnish us in
this regard will some day be precious to the physicist. While wait-
ing, I believe the theorists, recalling the experience of Michelson,
may anticipate a negative result, and that they would accomplish
a useful work in constructing a theory of aberration which would
explain this in advance.

But let us come back to the earth. There also we may aid the
experimenters. We can, for example, prepare the ground by study-
ing profoundly the dynamics of electrons; not, be it understood,
in starting from a single hypothesis, but in multiplying hypotheses
as much as possible. It will be, then, for the physicists to utilize
our work in seeking the crucial experiment to decide between these
different hypotheses.

This dynamics of electrons can be approached from many sides,
but among the ways leading thither is one which has been somewhat
neglected, and yet this is one of those which promise us most of sur-
prises. It is the movements of the electrons which produce the line
of the emission spectra; this is proved by the phenomenon of Zee-
mann; in an incandescent body, what vibrates is sensitive to the
magnet, therefore electrified. This is a very important first point,
but no one has gone farther; why are the lines of the spectrum
distributed in accordance with a regular law?

These laws have been studied by the experimenters in their least
details; they are very precise and relatively simple. The first study
of these distributions recalled the harmonics encountered in acous-
tics ; but the difference is great. Not only the numbers of vibrations
are not the successive multiples of one number, but we do not

620 APPLIED MATHEMATICS

even find anything analogous to the roots of those transcendental
equations to which so many problems of mathematical physics con-
duct us: that of the vibrations of an elastic body of any form, that
of the Hertzian oscillations in a generator of any form, the problem
of Fourier for the cooling of a solid body.

The laws are simpler, but they are of wholly other nature, and to
cite only one of these differences, for the harmonics of high order
the number of vibrations tends toward a finite limit, instead of
increasing indefinitely.

That has not yet been accounted for, and I believe that there we
have one of the most important secrets of nature. Lindemann has
made a praiseworthy attempt, but, to my mind, without success;
this attempt should be renewed. Thus we shall penetrate, so to say,
into the inmost recess of matter. And from the particular point of
view which we to-day occupy, when we know why the vibrations
of incandescent bodies differ from ordinary elastic vibrations, why
the electrons do not behave themselves like the matter which is familiar
to us, we shall better comprehend the dynamics of electrons and
it will be perhaps more easy for us to reconcile it with the princi-
ples.

Suppose, now, that all these efforts fail, and after all I do not
believe they will, what must be done? Will it be necessary to seek
to mend the broken principles in giving what we French call a coup
de pouce f That is evidently always possible, and I retract nothing
I have formerly said.

Have you not written, you might say if you wished to seek a
quarrel with me, have you not written that the principles, though of
experimental origin, are now unassailable by experiment because
they have become conventions? And now you have just told us the
most recent conquests of experiment put these principles in danger.
Well, formerly I was right and to-day I am not wrong.

Formerly I was right, and what is now happening is a new proof
of it. Take, for example, the calorimeter experiment of Curie on
radium. Is it possible to reconcile that with the principle of the
conservation of energy?

It has been attempted in many ways; but there is among them
one I should like you to notice.

It has been conjectured that radium was only an intermediary,
that it only stored radiations of unknown nature which flashed
through space in every direction, traversing all bodies, save radium,
without being altered by this passage and without exercising any
action upon them. Radium alone took from them a little of their
energy and afterward gave it out to us in divers forms.

What an advantageous explanation, and how convenient! First,
it is unverifiable and thus irrefutable. Then again it will serve to

PRINCIPLES OF MATHEMATICAL PHYSICS 621

account for any derogation whatever to the principle of Mayer; it
responds in advance not only to the objection of Curie, but to all
the objections that future experimenters might accumulate. This
new and unknown energy would serve for everything. This is just
what I have said, and we are thereby shown that our principle is
unassailable by experiment.

And after all, what have we gained by this coup de pouce ?

The principle is intact, but thenceforth of what use is it?

It permitted us to foresee that in such or such circumstance we
could count on such a total quantity of energy; it limited us; but
now where there is put at our disposition this indefinite provision of
new energy, we are limited by nothing; and as I have written else-
where, if a principle ceases to be fecund, experiment, without con-
tradicting it directly, will be likely to condemn it.

This, therefore, is not what would have to be done, it would be
necessary to rebuild anew.

If we were cornered down to this necessity, we should moreover
console ourselves. It would not be necessary to conclude that science
can weave only a Penelope's web, that it can build only ephemeral
constructions, which it is soon forced to demolish from top to bot-
tom with its own hands.

As I have said, we have already passed through a like crisis. I
have shown you that in the second mathematical physics, that of
the principles, we find traces of the first, that of the central forces;
it will be just the same if we must learn a third.

When an animal exuviates, and breaks its too narrow carapace to
make itself a fresh one, we easily recognize under the new envelope
the essential traits of the organism which have existed.

We cannot foresee in what way we are about to expand; perhaps
it is the kinetic theory of gases which is about to undergo develop-
ment and serve as model to the others. Then, the facts which first
appeared to us as simple, thereafter will be merely results of a very
great number of elementary facts which only the laws of chance
make cooperate for a common end. Physical law will then take an
entirely new aspect; it will no longer be solely a differential equation,
it will take the character of a statistical law.

Perhaps, likewise, we should construct a whole new mechanics,
of which we only succeed in catching a glimpse, where inertia increas-
ing with the velocity, the velocity of light would become an impass-
able limit.

The ordinary mechanics, more simple, would remain a first approx-
imation, since it would be true for velocities not too great, so that we
should still find the old dynamics under the new.

We should not have to regret having believed in the principles,
and even, since velocities too great for the old formulas would always

622 APPLIED MATHEMATICS

be only exceptional, the surest way in practice would-be still to act
as if we continued to believe in them. They are so useful, it would be
necessary to keep a place for them. To determine to exclude them
altogether would be to deprive one's self of a precious weapon. I hasten
to say in conclusion we are not yet there, and as yet nothing proves
that the principles will not come forth from the combat victorious
and intact.

SHORT PAPERS

Three short papers were read in the Section of Applied Mathematics, the first
by Professor Henry T. Eddy, of the University of Minnesota, on " The Electro-
magnetic Theory and the Velocity of Light."

The second paper was presented by Professor Alexander Macfarlane, of Chat-
ham, Ontario, "On the Exponential Notation in Vector-analysis."

The third paper was presented by Professor James McMahon, of Cornell Uni-
versity, " On the Use of N-fold Riemann Spaces in Applied Mathematics."

WORKS OF REFERENCE

(PREPARED THROUGH THE COURTESY OF PROFESSOR GEORGE BRUCE HALSTED,
OF KENYON COLLEGE, AND PROFESSOR LUDWIG BOLTZMANN, OF THE UNIVERSITY
OF VIENNA)

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TODHUNTER AND LEATHEN, Spherical Trigonometry. London, Macmillan, 1901.
WILLSON, F. N., Descriptive Geometry and Mechanical Drawing. New York,

Macmillan, 1904.

WHITEHEAD, A. N., Universal Algebra. Cambridge University Press, 1898.
WITHERS, J. W., Euclid's Parallel Postulate. Chicago, The Open Court Pub.

Co., 1905.

WHITTAKER, E. T., Modern Analysis. Cambridge University Press, 1902.
WOLFFING, ERNST, Mathematische Biicherschatz, Systematische Verzerchniss

der Wichtigsten Deutschen und Aiislandischen Lehrbucher und Monographien

des 19. Jahrhunderts auf dem Gebiete der Mathematischen Wissenschaften.

Leipzig, Teubner, 1903.
Index Du Repertoire Bibliographique des Sciences Math&natiques. 2 ed. 1898.

Paris, Gauthiers-Villars.
Repertoriam der Hoheren Mathematik. i, Thrfil: Analysis, n, Theil: Geometric.

Leipzig, 1902, Teubner.
Encylope'die des Sciences Mathe'matiques pures et applique's. Edition fran^aise,

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SPECIAL WORKS OF REFERENCE
(ACCOMPANYING PARTICULARLY PROFESSOR BOLTZMANN'S ADDRESS)

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Wien, Sitz. Ber. n, 63 p. 397, 1871.
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Uber das Warmegleichgewicht von Gasen, auf welche Aussere Krafte

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p. 503, 1876.
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aus zahlreichen diskreten Theilen bestehenden Systeme. 27 Ein-

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626 SPECIAL BIBLIOGRAPHY

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Phil. Mag. vi, 5, p. 597, 6, p. 720, 1903.
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SCIENCE AND HYPOTHESIS

SCIENCE AND HYPOTHESIS1

BY PROF. JULES HENRI POINCARE, UNIVERSITY OF PARIS

PART I — NUMBER AND MAGNITUDE

On the Nature of Mathematical Reasoning

THE very possibility of mathematical science seems an insoluble
contradiction. If this science is only deductive in appearance, from
whence is derived that perfect rigor which is challenged by none?
If, on the contrary, all the propositions which it enunciates may be
derived in order by the rules of formal logic, how is it that mathe-
matics is not reduced to a gigantic tautology? The syllogism can
teach us nothing essentially new, and if everything must spring from
the principle of identity, then everything should be capable of being
reduced to that principle. Are we then to admit that the enunciations
of all the theorems with which so many volumes are filled, are only
indirect ways of saying that A is A?

No doubt we may refer back to axioms which are at the source of all
these reasonings. If it is felt that they cannot be reduced to the
principle of contradiction, if we decline to see in them any more than
experimental facts which have no part or lot in mathematical neces-
sity, there is still one resource left to us: we may class them among
a priori synthetic views. But this is no solution of the difficulty —
it is merely giving it a name; and even if the nature of the synthetic
views had no longer for us any mystery, the contradiction would not
have disappeared; it would have only been shirked. Syllogistic rea-
soning remains incapable of adding anything to the data that are given
it ; the data are reduced to axioms, and that is all we should find in the
conclusions.

No theorem can be new unless a new axiom intervenes in its dem-
onstration ; reasoning can only give us immediately evident truths
borrowed from direct intuition; it would only be an intermediary

i This is a translation of Prof. Poincare"'s celebrated treatise entitled
La Science et I'Hypothese. It is presented here in the nature of collateral
reading to the lectures on Mathematics and other scientific lectures deliv-
ered at the International Congress of Arts and Science.

630 SCIENCE AND HYPOTHESIS

parasite. Should we not therefore have reason for asking if the syllo-
gistic apparatus serves only to disguise what we have borrowed ?

The contradiction will strike us the more if we open any book on
mathematics; on every page the author announces his intention of
generalizing some proposition already known. Does the mathematical
method proceed from the particular to the general, and, if so, how
can it be called deductive?

Finally, if the science of number were merely analytical, or could
be analytically derived from a few synthetic intuitions, it seems that a
sufficiently powerful mind could with a single glance perceive all its
truths ; nay, one might even hope that some day a language would be
invented simple enough for these truths to be made evident to any
person of ordinary intelligence.

Even if these consequences are challenged, it must be granted that
mathematical reasoning has of itself a kind of creative virtue, and is
therefore to be distinguished from the syllogism. The difference must
be profound. We shall not, for instance, find the key to the mystery
in the frequent use of the rule by which the same uniform operation
applied to two equal numbers will give identical results. All these
modes of reasoning, whether or not reducible to the syllogism, pro-
perly so called, retain the analytical character, and ipso facto, lose
their power.

The argument is an old one. Let us see how Leibnitz tried to show
that two and two make four. I assume the number one to be defined,
and also the operation oH-1 — i.e., the adding of unity to a given num-
ber x. These definitions, whatever they may be, do not enter into the
subsequent reasoning. I next define the numbers 2, 3, 4 by the
equalities : —

(1) 1 + 1 = 2; (2) 2 + 1 = 3; (3) 3 + 1 = 4, and in the same way
I define the operation re + 2 by the relation; (4) x + 2 =(x + 1)+ 1.

Given this, we have : —

2+2=(2+l)+l; (def. 4).
(2+1) +1=3+1 (def. 2).

3+1=4 (def. 3).

whence 2+2=4 Q.E.D.

It cannot be denied that this reasoning is purely analytical. But
if we ask a mathematician, he will reply : " This is not a demonstra-
tion properly so called; it is a verification." We have confined our-
selves to bringing together one or other of two purely conventional
definitions, and we have verified their identity ; nothing new has been
learned. Verification differs from proof precisely because it is analyti-
cal, and because it leads to nothing. It leads to nothing because the
conclusion is nothing but the premisses translated into another lan-
guage. A real proof, on the other hand, is fruitful, because the con-
clusion is in a sense more general than the premisses. The equality

NUMBER AND MAGNITUDE 631

2+2=4 can be verified because it is particular. Each individual enun-
ciation in mathematics may be always verified in the same way. But
if mathematics could be reduced to a series of such verifications it
would not be a science. A chess-player, for instance, does not create
a science by winning a piece. There is no science but the science of the
general. It may even be said that the object of the exact sciences is to
dispense with these direct verifications.

Let us now see the geometer at work, and try to surprise some of
his methods. The task is not without difficulty; it is not enough to
open a book at random and to analyze any proof we may come across.
First of all, geometry must be excluded, or the question becomes
complicated by difficult problems relating to the role of the postulates,
the nature and the origin of the idea of space. For analogous rea-
sons we cannot avail ourselves of the infinitesimal calculus. We must
seek mathematical thought where it has remained pure — i.e., in Arith-
metic. But we still have to choose ; in the higher parts of the theory
of numbers the primitive mathematical ideas have already undergone
so profound an elaboration that it becomes difficult to analyze them.

It is therefore at the beginning of Arithmetic that we must expect
to find the explanation we seek; but it happens that it is precisely in
the proofs of the most elementary theorems that the authors of classic
treatises have displayed the least precision and rigor. We may not
impute this to them as a crime; they have obeyed a necessity. Begin-
ners are not prepared for real mathematical rigor; they would see in
it nothing but empty, tedious subtleties. It would be waste of time to
try to make them more exacting; they have to pass rapidly and without
stopping over the road which was trodden slowly by the founders of
the science.

Why is so long a preparation necessary to habituate oneself to this
perfect rigor, which it would seem should naturally be imposed on
all minds ? This is a logical and psychological problem which is well
worthy of study. But we shall not dwell on it; it is foreign to our
subject. All I wish to insist on is, that we shall fail in our purpose
unless we reconstruct the proofs of the elementary theorems, and give
them, not the rough form in which they are left so as not to weary
the beginner, but the form which will satisfy the skilled geometer.

Definition of Addition

I assume that the operation x-\-l has been defined; it consists in
adding the number 1 to a given number x. Whatever may be said of
this definition, it does not enter into the subsequent reasoning.

We now have to define the operation oH-o, which consists in adding
the number a to any given number x. Suppose that we have defined
the operation x + (a — 1) ; the operation x + a will be defined by

632 SCIENCE AND HYPOTHESIS

the equality: (1) x + a = [x + (a — !)] + !. We shall know what
x + a is when we know what x+(a — 1) is, and as I have assumed
that to start with we know what x + 1 is, we can define suc-
cessively and " by recurrence " the operations x + 2, x + 3, etc. This
definition deserves a moment's attention; it is of a particular nature
which distinguishes it even at this stage from the purely logical defi-
nition; the equality (1), in fact, contains an infinite number of dis-
tinct definitions, each having only one meaning when we know the
meaning of its predecessor.

Properties of Addition

Associative. — I say that a + (& + c) = (a + &)+ c; in fact the theo-
rem is true for c = 1. It may then be written a +(& + 1)= (a + &)
-f- 1 ; which, remembering the difference of notation, is nothing but the
equality (1) by which I have just defined addition. Assume the
theorem true for c — y, I say that it will be true for c = y + 1.
Let (a + &)+y = a+(&+y), it follows that [(a + 6)+y] + l =
[a + (& + y)]+l; or by del (1) — (a + &) + (y + l)=o+(& + y +
l)=a + [&+(y + l)L which shows by a series of purely analytical
deductions that the theorem is true for y + 1. Being true for c = 1, we
see that it is successively true for c = 2, c = 3, etc.

Commutative. — (1)1 say that a + 1 = 1 + a. The theorem is evi-
dently true for a = 1 ; we can verify by purely analytical reasoning that
if it is true for a = y it will be true for a = y + I.1 Now, it is true for
a = 1, and therefore is true for a = 2, a = 3, and so on. This is what
is meant by saying that the proof is demonstrated " by recurrence."

(2) I say that a + b — & + a. The theorem has just been shown to
hold good for & = 1, and it may be verified analytically that if it is true
for b = £, it will be true for b = ft + 1. The proposition is thus estab-
lished by recurrence.

Definition of Multiplication

We shall define multiplication by the equalities: (l)aXl = a. (2)
a X 6 = [a X (& — 1)] -f a. Both of these include an infinite number
of definitions; having defined a X 1, it enables us to define in succession
a X 2, a X 3, and so on.

Properties of Multiplication

Distributive. — I say that (a + &) X c = (a X c) + (6 X c) . We can
verify analytically that the theorem is true for c = 1 ; then if it is true
for c — y, it will be true for c = y + 1. The proposition is then proved
by recurrence.

i For (y+i ) +1= ( i+y ) + 1=1+ (y+ 1 ) .— [TB.J

633

Commutative. — (1) I say that a X 1 — 1 X a. The theorem is
obvious for a = 1. We can verify analytically that if it is true for
a = a, it will be true foi a = a + 1.

(2) I say that a X b ~ b X a. The theorem has just been proved
for b = 1. We can verify analytically that if it be true for b = (3 it
will be true for & = ft + 1.

This monotonous series of reasonings may now be laid aside; but
their very monotony brings vividly to light the process, which is uni-
form, and is met again at every step. The process is proof by recur-
rence. We first show that a theorem is true for n— 1; we then show
that if it is true for n — 1 it is true for n, and we conclude that it is
true for all integers. We have now seen how it may be used for the
proof of the rules of addition and multiplication — that is to say,
for the rules of the algebraical calculus. This calculus is an instru-
ment of transformation which lends itself to many more different
combinations than the simple syllogism; but it is still a purely an-
alytical instrument, and is incapable of teaching us anything new.
If mathematics had no other instrument, it would immediately be ar-
rested in its development; but it has recourse anew to the same
process — i.e., to reasoning by recurrence, and it can continue its
forward march. Then if we look carefully, we find this mode of
reasoning at every step, either under the simple form which we have
just given to it, or under a more or less modified form. It is there-
fore mathematical reasoning par excellence, and we must examine it
closer.

The essential characteristic of reasoning by recurrence is that it
contains, condensed, so to speak, in a single formula, an infinite num-
ber of syllogisms. We shall see this more clearly if we enunci-
ate the syllogisms one after another. They follow one another, if one
may use the expression, in a cascade. The following are the hypo-
thetical syllogisms: — The theorem is true of the number 1. Now, if it
is true of 1, it is true of 2; therefore it is true of 2. Now, if it is
true of 2, it is true of 3; hence it is true of 3, and so on. We see that
the conclusion of each syllogism serves as the minor of its successor.
Further, the majors of all our syllogisms may be reduced to a single
form. If the theorem is true of n — 1, it is true of n.

We see, then, that in reasoning by recurrence we confine ourselves to
the enunciation of the minor of the first syllogism, and the general
formula which contains as particular cases all the majors. This unend-
ing series of syllogisms is thus reduced to a phrase of a few lines.

It is now easy to understand why every particular consequence of
a theorem may, as I have above explained, be verified by purely an-
alytical processes. If, instead of proving that our theorem is true for
all numbers we only wish to show that it is true for
the number 6 for instance, it will be enough to establish the first five

634 SCIENCE AND HYPOTHESIS

syllogisms in our cascade. We shall require 9 if we wish to prove it
for the number 10; for a greater number we shall require more still;
but however great the number may be we shall always reach it, and the
analytical verification will always be possible. But however far we went
we should never reach the general theorem applicable to all numbers,
which alone is the object of science. To reach it we should require
an infinite number of syllogisms, and we should have to cross an abyss
which the patience of the analyst, restricted to the resources of formal
logic, will never succeed in crossing.

I asked at the outset why we cannot conceive of a mind powerful
enough to see at a glance the whole body of mathematical truth. The
answer is now easy. A chess-player can combine for four or five
moves ahead; but, however extraordinary a player he may be, he
cannot prepare for more than a finite number of moves. If he applies
his faculties to Arithmetic, he cannot conceive its general truths by
direct intuition alone ; to prove even the smallest theorem he must use
reasoning by recurrence, for that is the only instrument which enables
us to pass from the finite to the infinite. This instrument is always
useful, for it enables us to leap over as many stages as we wish ; it frees
us from the necessity of long, tedious, and monotonous verifications
which would rapidly become impracticable. Then when we take in
hand the general theorem it becomes indispensable, for otherwise we
should ever be approaching the analytical verification without ever
actually reaching it. In this domain of Arithmetic we may think our-
selves very far from the infinitesimal analysis, but the idea of math-
ematical infinity is already playing a preponderating part, and with-
out it there would be no science at all, because there would be nothing
general.

The views upon which reasoning by recurrence is based may be
exhibited in other forms; we may say, for instance, that in any finite
collection of different integers there is always one which is smaller
than any other. "We may readily pass from one enunciation to an-
other, and thus give ourselves the illusion of having proved that
reasoning by recurrence is legitimate. But we shall always be brought
to a full stop — we shall always come to an indemonstrable axiom,
which will at bottom be but the proposition we had to prove translated
into another language. We cannot therefore escape the conclusion
that the rule of reasoning by recurrence is irreducible to the principle
of contradiction. Nor can the rule come to us from experiment. Ex-
periment may teach us that the rule is true for the first ten or the first
hundred numbers, for instance; it will not bring us to the indefinite
series of numbers, but only to a more or less long, but always limited,
portion of the series.

Now, if that were all that is in question, the principle of contradic-
tion would be sufficient, it would always enable us to develop as many

NUMBEK AND MAGNITUDE 635

syllogisms as we wished. It is only when it is a question of a single
formula to embrace an infinite number of syllogisms that this prin-
ciple breaks down, and there, too, experiment is powerless to aid.
This rule, inaccessible to analytical proof and to experiment, is the
exact type of the a priori synthetic intuition. On the other hand, we
cannot see in it a convention as in the case of the postulates of geom-
etry.

Why then is this view imposed upon us with such an irresistible
weight of evidence? It is because it is only the affirmation of the
power of the mind which knows it can conceive of the indefinite repe-
tition of the same act, when the act is once possible. The mind has a
direct intuition of this power, and experiment can only be for it an
opportunity of using it, and thereby of becoming conscious of it.

But it will be said, if the legitimacy of reasoning by recurrence
cannot be established by experiment alone, is it so with experiment
aided by induction ? We see successively that a theorem is true of the
number 1, of the number 2, of the number 3, and so on — the law is
manifest, we say, and it is so on the same ground that every physical
law is true which is based on a very large but limited number of ob-
servations.

It cannot escape our notice that here is a striking analogy with the
usual processes of induction. But an essential difference exists. In-
duction applied to the physical sciences is always uncertain, because
it is based on the belief in a general order of the universe, an order
which is external to us. Mathematical induction — i.e., proof by re-
currence — is, on the contrary, necessarily imposed on us, because it
is only the affirmation of a property of the mind itself.

Mathematicians, as I have said before, always endeavor to generalize
the propositions they have obtained. To seek no further example, we
have just shown the equality, a+l=l+a, and we then used it to estab-
lish the equality, a+&=&+a, which is obviously more general. Math-
ematics may, therefore, like the other sciences, proceed from the
particular to the general. This is a fact which might otherwise have
appeared incomprehensible to us at the beginning of this study, but
which has no longer anything mysterious about it, since we have
ascertained the analogies between proof by recurrence and ordinary
induction.

No doubt mathematical recurrent reasoning and physical inductive
reasoning are based on different foundations, but they move in par-
allel lines and in the same direction — namely, from the particular to
the general.

Let us examine the case a little more closely. To prove the equality

a+2=2-f-a (1), we need only apply the rale a-f-l^l+a, twice,

and write a+2=a+l+l=l+a+l=l+l+a=2+a (2).

The equality thus deduced by purely analytical means is not, how-

636 SCIENCE AND HYPOTHESIS

ever, a simple particular case. It is something quite different. We
may not therefore even say in the really analytical and deductive part
of mathematical reasoning that we proceed from the general to the
particular in the ordinary sense of the words. The two sides of the
equality (2) are merely more complicated combinations than the two
sides of the equality (1), and analysis only serves to separate the ele-
ments which, enter into these combinations and to study their relations.

Mathematicians therefore proceed " by construction," they " con-
struct " more complicated combinations. When they analyze these
combinations, these aggregates, so to speak, into their primitive ele-
ments, they see the relations of the elements and deduce the relations
of the aggregates themselves. The process is purely analytical, but it
is not a passing from the general to the particular, for the aggregates
obviously cannot be regarded as more particular than their elements.

Great importance has been rightly attached to this process of " con-
struction," and some claim to see in it the necessary and sufficient
condition of the progress of the exact sciences. Necessary, no doubt,
but not sufficient! For a construction to be useful and not mere
waste of mental effort, for it to serve as a stepping-stone to higher
things, it must first of all possess a kind of unity enabling us to see
something more than the juxtaposition of its elements. Or more ac-
curately, there must be some advantage in considering the construc-
tion rather than the elements themselves. What can this advantage
be? Why reason on a polygon, for instance, which is always decom-
posable into triangles, and not on elementary triangles? It is be-
cause there are properties of polygons of any number of sides, and
they can be immediately applied to any particular kind of polygon.
In most cases it is only after long efforts that those properties can be
discovered, by directly studying the relations of elementary triangles.
If the quadrilateral is anything more than the juxtaposition of two
triangles, it is because it is of the polygon type.

A construction only becomes interesting when it can be placed side
by side with other analogous constructions for forming species of the
same genus. To do this we must necessarily go back from the particu-
lar to the general, ascending one or more steps. The analytical
process " by construction " does not compel us to descend, but it
leaves us at the same level. We can only ascend by mathematical
induction, for from it alone can we learn something new. Without
the aid of this induction, which in certain respects differs from, but
is as fruitful as, physical induction, construction would be powerless
to create science.

Let me observe, in conclusion, that this induction is only possible if
the same operation can be repeated indefinitely. That is why the
theory of chess can never become a science, for the different moves of
the same piece are limited and do not resemble each other.

NUMBER AND MAGNITUDE 637

Mathematical Magnitude and Experiment

If we want to know what the mathematicians mean by a continuum,
it is useless to appeal to geometry. The geometer is always seeking,
more or less, to represent to himself the figures he is studying, but
his representations are only instruments to him ; he uses space in his
geometry just as he uses chalk; and further, too much importance
must not bo attached to accidents which are often nothing more than
the whiteness of the chalk.

The pure analyst has not to dread this pitfall. He has disen-
gaged mathematics from all extraneous elements, and he is in a
position to answer our question : — " Tell me exactly what this con-
tinuum is, about which mathematicians reason." Many analysts who
reflect on their art have already done so — M. Tannery, for instance,
in his Introduction a la theorie des Fonctions d'une variable.

Let us start with the integers. Between any two consecutive sets,
intercalate one or more intermediary sets, and then between these sets
others again, and so on indefinitely. We thus get an unlimited num-
ber of term?, and these will be the numbers which we call fractional,
rational, or commensurable. But this is not yet all; between these
terms, which, be it marked, are already infinite in number, other terms
are intercalated, and these are called irrational or incommensurable.

Before going any further, let me make a preliminary remark. The
continuum thus conceived is no longer a collection of individuals ar-
ranged in a certain order, infinite in number, it is true, but external
the one to the other. This is not the ordinary conception in which it
is supposed that between the elements of the continuum exists an inti-
mate connection making of it one whole, in which the point has no
existence previous to the line, but the line does exist previous to the
point. Multiplicity alone subsists, unity has disappeared — " the
continuum is unity in multiplicity," according to the celebrated for-
mula. The analysts have even less reason to define their continuum
as they do, since it is always on this that they reason when they are
particularly proud of their rigor. It is enough to warn the reader that
the real mathematical continuum is quite different from that of the
physicists and from that of the metaphysicians.

It may also be said, perhaps, that mathematicians who are con-
tented with this definition are the dupes of words, that the nature of
each of these sets should be precisely indicated, that it should be ex-
plained how they are to be intercalated, and that it should be shown
how it is possible to do it. This, however, would be wrong; the only
property of the sets which comes into the reasoning is that of pre-
ceding or succeeding these or those other sets ; this alone should there-
fore intervene in the definition. So we need not concern ourselves
with the manner in which the sets are intercalated, and no one will

638 SCIENCE AND HYPOTHESIS

doubt the possibility of the operation if he only remembers that " pos-
sible " in the language of geometers simply means exempt from con-
tradiction. But our definition is not yet complete, and we come back
to it after this rather long digression.

Definition of Incommensurables. — The mathematicians of the Ber-
lin school, and Kronecker in particular, have devoted themselves to
constructing this continuous scale of irrational and fractional num-
bers without using any other materials than the integer. The math-
ematical continuum from this point of view would be a pure creation
of the mind in which experiment would have no part.

The idea of rational number not seeming to present to them any
difficulty, they have confined their attention mainly to defining incom-
mensurable numbers. But before reproducing their definition here, I
must make an observation that will allay the astonishment which this
will not fail to provoke in readers who are but little familiar with the
habits of geometers.

Mathematicians do not study objects, but the relations between
objects; to them it is a matter of indifference if these objects are
replaced by others, provided that the relations do not change. Matter
does not engage their attention, they are interested by form alone.

If we did not remember it, we could hardly understand that Kro-
necker gives the name of incommensurable number to a simple symbol
— that is to say, something very different from the idea we think we
ought to have of a quantity which should be measurable and almost
tangible.

Let us see now what is Kronecker's definition. Commensurable
numbers may be divided into classes in an infinite number of ways,
subject to the condition that any number whatever of the first class is
greater than any number of the second. It may happen that among
the numbers of the first class there is one which is smaller than all
the rest; if, for instance, we arrange in the first class all the numbers
greater than 2, and 2 itself, and in the second class all the numbers
smaller than 2, it is clear that 2 will be the smallest of all the num-
bers of the first class. The number 2 may therefore be chosen as the
symbol of this division.

It may happen, on the contrary, that in the second class there is one
which is greater than all the rest. This is what takes place, for ex-
ample, if the first class comprises all the numbers greater than 2,
and if, in the second, are all the numbers less than 2, and 2 itself.
Here again the number 2 might be chosen as the symbol of this
division.

But it may equally well happen that we can find neither in the
first class a number smaller than all the rest, nor in the second class a
number greater than all the rest. Suppose, for instance, we place in
the first class all the numbers whose squares are greater than 2, and in

NUMBER AND MAGNITUDE 639

the second all the numbers whose squares are smaller than 2. We
know that in neither of them is a number whose square is equal to 2.
Evidently there will be in the first class no number which is smaller
than all the rest, for however near the square of a number may be to 2,
we can always find a commensurable whose square is still nearer to 2.
From Kronecker's point of view, the incommensurable number of 2
is nothing but the symbol of this particular method of division of
commensurable numbers; and to each mode of repartition corresponds
in this way a number, commensurable or not, which serves as a sym-
bol. But to be satisfied with this would be to forget the origin of these
symbols; it remains to explain how we have been led to attribute to
them a kind of concrete existence, and on the other hand, does not
the difficulty begin with fractions? Should we have the notion of
these numbers if we did not previously know a matter which we con-
ceive as infinitely divisible — i.e., as a continuum ?

The Physical Continuum. — We are next led to ask if the idea of
the mathematical continuum is not simply drawn from experiment.
If that be so, the rough data of experiment, which are our sensations,
could be measured. We might, indeed, be tempted to believe that this
is so, for in recent times there has been an attempt to measure them,
and a law has even been formulated, known as Fechner's law, accord-
ing to which sensation is proportional to the logarithm of the stimulus.
But if we examine the experiments by which the endeavor has been
made to establish this law, we shall be led to a diametrically opposite
conclusion. It hap, for instance, been observed that a weight A of 10
grammes and a weight B of 11 grammes produced identical sensa-
tions, that the weight B could no longer be distinguished from a
weight C of 12 grammes, but that the weight A was readily distin-
guished from the weight C. Thus the rough results of the experiments
may be expressed by the following relations: A = B, B = C, A< C,
which may be regarded as the formula of the physical continuum.
But here is an intolerable disagreement with the law of contradiction,
and the necessity of banishing this disagreement has compelled us to
invent the mathematical continuum. We are therefore forced to con-
clude that this notion has been created entirely by the mind, but it is
experiment that has provided the opportunity. We cannot believe that
two quantities which are equal to a third are not equal to one another,
and we are thus led to suppose that A is different from B and B from
C, and that if we have not been aware of this, it is due to the imper-
fections of our senses.

The Creation of the Mathematical Continuum : First Stage. — So
far it would suffice, in order to account for facts, to intercalate between
A and B a small number of terms which would remain discrete. What
happens now if we have recourse to some instrument to make up for
the weakness of our senses? If, for example, we use a microscope?

640 SCIENCE AND HYPOTHESIS

Such terms as A and B, which before were indistinguishable from
one another, appear now to be distinct : but between A and B, which
are distinct, is intercalated another new term D, which we can distin-
guish neither from A nor from B. Although we may use the most
delicate methods, the rough results of our experiments will always
present the characters of the physical continuum with the contradic-
tion which is inherent in it. We only escape from it by incessantly
intercalating new terms between the terms already distinguished, and
this operation must be pursued indefinitely. We might conceive that
it would be possible to stop if we could imagine an instrument power-
ful enough to decompose the physical continuum into discrete ele-
ments, just as the telescope resolves the Milky Way into stars. But
this we cannot imagine; it is always with our senses that we use our
instruments; it is with the eye that we observe the image magnified
by the microscope, and this image must therefore always retain the
characters of visual sensation, and therefore those of the physical
continuum.

Nothing distinguishes a length directly observed from half that
length doubled by the microscope. The whole is homogeneous to the
part ; and there is a fresh contradiction — or rather there would be one
if the number of the terms were supposed to be finite; it is clear
that the part containing less terms than the whole cannot be similar to
the whole. The contradiction ceases as soon as the number of terms is
regarded as infinite. There is nothing, for example, to prevent us
from regarding the aggregate of integers as similar to the aggregate
of even numbers, which is however only a part of it ; in fact, to each
integer corresponds another even number which is its double. But
it is not only to escape this contradiction contained in the empiric
data that the mind is led to create the concept of a continuum formed
of an indefinite number of terms.

Here everything takes place just as in the series of the integers.
We have the faculty of conceiving that a unit may be added to a col-
lection of units. Thanks to experiment, we have had the opportunity
of exercising this faculty and are conscious of it; but from this fact
we feel that our power is unlimited, and that we can count indefinitely,
although we have never had to count more than a finite number of
objects. In the same way, as soon as we have intercalated terms between
two consecutive terms of a series, we feel that this operation may be
continued without limit, and that, so to speak, there is no intrinsic
reason for stopping. As an abbreviation, I may give the name of a
mathematical continuum of the first order to every aggregate of
terms formed after the same law as the scale of commensurable num-
bers. If, then, we intercalate new sets according to the laws of in-
commensurable numbers, we obtain what may be called a continuum
of the second order.

NUMBER AND MAGNITUDE 641

Second Stage. — We have only taken our first step. We have ex-
plained the origin of continuums of the first order; we must now see
why this is not sufficient, and why the incommensurable numbers had
to be invented.

If we try to imagine a line, it must have the characters of the phy-
sical continuum — that is to say, our representation must have a
certain breadth. Two lines will therefore appear to us under the
form of two narrow bands, and if we are content with this rough
image, it is clear that where two lines cross they must have some
common part. But the pure geometer makes one further effort ; with-
out entirely renouncing the aid of his senses, he tries to imagine a
line without breadth and a point without size. This he can do only
by imagining a line as the limit towards which tends a band that is
growing thinner and thinner, and the point as the limit towards
which is tending an area that is growing smaller and smaller. Our
two bands, however narrow they may be, will always have a common
area; the smaller they are the smaller it will be, and its limit is what
the geometer calls a point. This is why it is said that the two lines
which cross must have a common point, and this truth seems intuitive.

But a contradiction would be implied if we conceived of lines as
continuums of the first order — i.e., the lines traced by the geometer
should only give us points, the co-ordinates of which are rational num-
bers. The contradiction would be manifest if we were, for instance, to
assert the existence of lines and circles. It is clear, in fact, that if
the points whose co-ordinates are commensurable were alone regarded
as real, the in-circle of a square and the diagonal of the square would
not intersect, since the co-ordinates of the point of intersection are
incommensurable.

Even then we should have only certain incommensurable numbers,
and not all these numbers.

But let us imagine a line divided into two half-rays (demi-droites) .
Each of these half -rays will appear to our minds as a band of a
certain breadth; these bands will fit close together, because there must
be no interval between them. The common part will appear to us to
be a point which will still remain as we imagine the bands to become
thinner and thinner, so that we admit as an intuitive truth that if a
line be divided into two half-rays the common frontier of these half-
rays is a point. Here we recognize the conception of Kronecker, in
which an incommensurable number was regarded as the common
frontier of two classes of rational numbers. Such is the origin of
the continuum of the second order, which is the mathematical con-
tinuum properly so called.

Summary. — To sum up, the mind has the faculty of creating sym-
bols, and it is thus that it has constructed the mathematical con-
tinuum, which is only a particular system of symbols. The only limit

642 SCIENCE AND HYPOTHESIS

to its power is the necessity of avoiding all contradiction; but the
mind only makes use of it when experiment gives a reason for it.

In the case with which we are concerned, the reason is given by
the idea of the physical continuum, drawn from the rough data of
the senses. But this idea leads to a series of contradictions from each
of which in turn we must be freed. In this way we are forced to
imagine a more and more complicated system of symbols. That
on which we shall dwell is not merely exempt from internal contra-
diction, — it was so already at all the steps we have taken, — but it is
no longer in contradiction with the various propositions which are
called intuitive, and which are derived from more or less elaborate
empirical notions.

Measurable Magnitude. — So far we have not spoken of the measure
of magnitudes; we can tell if any one of them is greater than any
other, but we cannot say that it is two or three times as large.

So far, I have only considered the order in which the terms are
arranged; but that is not sufficient for most applications. We must
learn how to compare the interval which separates any two terms. On
this condition alone will the continuum become measurable, and the
operations of arithmetic be applicable. This can only be done by the
aid of a new and special convention; and this convention is, that in
such a case the interval between the terms A and B is equal to the
interval which separates C and D. For instance, we started with
the integers, and between two consecutive sets we intercalated n in-
termediary sets; by convention we now assume these new sets to be
equidistant. This is one of the ways of defining the addition of two
magnitudes ; for if the interval AB is by definition equal to the inter-
val CD, the interval AD will by definition be the sum of the intervals
AB and AC. This definition is very largely, but not altogether, arbi-
trary. It must satisfy certain conditions — the commutative and as-
sociative laws of addition, for instance; but, provided the definition
we choose satisfies these laws, the choice is indifferent, and we need
not state it precisely.

Remarks. — We are now in a position to discuss several important
questions.

(1) Is the creative power of the mind exhausted by the creation
of the mathematical continuum? The answer is in the negative, and
this is shown in a very striking manner by the work of Du Bois Eey-
mond.

We know that mathematicians distinguish between infinitesimals of
different orders, and that infinitesimals of the second order are in-
finitely small, not only absolutely so, but also in relation to those of
the first order. It is not difficult to imagine infinitesimals of frac-
tional or even of irrational order, and here once more we find the
mathematical continuum which has been dealt with in the preceding

NUMBEK AND MAGNITUDE 643

pages. Further, there are infinitesimals which are infinitely small
with reference to those of the first order, and infinitely large with
respect to the order 1 + e, however small e may be. Here, then, are
new terms intercalated in our series; and if I may be permitted to
revert to the terminology used in the preceding pages, a terminology
which is very convenient, although it has not been consecrated by
usage, I shall say that we have created a kind of continuum of the
third order.

It is an easy matter to go further, but it is idle to do so, for we
would only be imagining symbols without any possible application, and
no one will dream of doing that. This continuum of the third order,
to which we are led by the consideration of the different orders of
infinitesimals, is in itself of but little use and hardly worth quoting.
Geometers look on it as a mere curiosity. The mind only uses its
creative faculty when experiment requires it.

(2) When we are once in possession of the conception of the math-
ematical continuum, are we protected from contradictions analogous to
those which gave it birth ? No, and the following is an instance : —

He is a savant indeed who will not take it as evident that every
curve has a tangent ; and, in fact, if we think of a curve and a straight
line as two narrow bands, we can always arrange them in such a way
that they have a common part without intersecting. Suppose now
that the breadth of the bands diminishes indefinitely: the common
part will still remain, and in the limit, so to speak, the two lines will
have a common point, although they do not intersect — i.e., they will
touch. The geometer who reasons in this way is only doing what we
have done when we proved that two lines which intersect have a com-
mon point, and his intuition might also seem to be quite legitimate.
But this is not the case. We can show that there are curves which
have no tangent, if we define such a curve as an analytical continuum
of the second order. No doubt some artifice analogous to those we
have discussed above would enable us to get rid of this contradiction,
but as the latter is only met with in very exceptional cases, we need
not trouble to do so. Instead of endeavoring to reconcile intuition
and analysis, we are content to sacrifice one of them, and as analysis
must be flawless, intuition must go to the wall.

Tlie Physical Continuum of Several Dimensions. — We have dis-
cussed above the physical continuum as it is derived from the imme-
diate evidence of our senses — , or, if the reader prefers, from the
rough results of Feclmer's experiments; I have shown that these
results are summed up in the contradictory f ormula? : — A= B, B =
C, A > C.

Let us now see how this notion is generalized, and how from it may
be derived the concept of continuums of several dimensions. Consider
any two aggregates of sensations. We can either distinguish between

644 SCIENCE AND HYPOTHESIS

them, or we cannot; just as in Feehner's experiments the weight of
10 grammes could be distinguished from the ; weight of 12 grammes,
but not from the weight of 11 grammes. This is all that is required
to construct the continuum of several dimensions.

Let us call one of these aggregates of sensations an element. It
will be in a measure analogous to the point of the mathematicians,
but will not be, however, the same thing. We cannot say that our
element has no size, for we cannot distinguish it from its immediate
neighbors, and it is thus surrounded by a kind of fog. If the astro-
nomical comparison may be allowed, our " elements " would be like
nebulae, whereas the mathematical points would be like stars.

If this be granted, a system of elements will form a continuum if
we can pass from any one of them to any other by a series of consecu-
tive elements such that each cannot be distinguished from its prede-
cessor. This linear series is to the line of the mathematician what the
isolated element was to the point.

Before going further, I must explain what is meant by a cut. Let us
consider a continuum C, and remove from it certain of its elements,
which for a moment we shall regard as no longer belonging to the
continuum. We shall call the aggregate of elements thus removed a
cut. By means of this cut, the continuum C will be subdivided into
several distinct continuums; the aggregate of elements which remain
will cease to form a single continuum. There will then be on C two
elements, A and B, which we must look upon as belonging to two dis-
tinct continuums; and we see that this must be so, because it will be
impossible to find a linear series of consecutive elements of C (each
of the elements indistinguishable from the preceding, the first being
A and the last B), unless one of the elements of this series is indis-
tinguishable from one of the elements of the cut.

It may happen, on the contrary, that the cut may not be sufficient to
subdivide the continuum C. To classify the physical continuums, we
must first of all ascertain the nature of the cuts which must be made
in order to subdivide them. If a physical continuum, C, may be
subdivided by a cut reducing to a finite number of elements, all dis-
tinguishable the one from the other (and therefore forming neither
one continuum nor several continuums), we shall call C a continuum
of one dimension. If, on the contrary, C can only be subdivided by
cuts which are themselves continuums, we shall say that C is of
several dimensions; if the cuts are continuums of one dimension, then
we shall say that C has two dimensions ; if cuts of two dimensions are
sufficient, we shall say that C is of three dimensions, and so on. Thus
the notion of the physical continuum of several dimensions is defined,
thanks to the very simple fact, that two aggregates of sensations may
be distinguishable or indistinguishable.

The Ufathematicdl Continuum of Several Dimensions. — The con-

NUMBER AND MAGNITUDE 645

ception of the mathematical continuum of n dimensions may be led
up to quite naturally by a process similar to that which we discussed
at the beginning of this chapter. A point of such a continuum is
denned by a system of n distinct magnitudes which we call its co-
ordinates.

The magnitudes need not always be measurable; there is, for in-
stance, one branch of geometry independent of the measure of mag-
nitudes, in which we are only concerned with knowing, for example,
if, on a curve ABC, the point B is between the points A and C, and
in which it is immaterial whether the arc A B is equal to or twice
the arc B C. This branch is called Analysis Situs. It contains
quite a large body of doctrine which has attracted the attention of
the greatest geometers, and from which are derived, one from another,
a whole series of remarkable theorems. What distinguishes these
theorems from those of ordinary geometry is that they are purely
qualitative. They are still true if the figures are copied by an un-
skilful draughtsman, with the result that the proportions are distorted
and the straight lines replaced by lines which are more or less curved.

As soon as measurement is introduced into the continuum we have
just defined, the continuum becomes space, and geometry is born. But
the discussion of this is reserved for Part II.

PAET II — SPACE

Non-Euclidean Geometries

Every conclusion presumes premisses. These premisses are either
self-evident and need no demonstration, or can be established only
if based on other propositions ; and, as we cannot go back in this way
to infinity, every deductive science, and geometry in particular, must
rest upon a certain number of indemonstrable axioms. All treatises
of geometry begin therefore with the enunciation of these axioms.
But there is a distinction to be drawn between them. Some of these,
for example, " Things which are equal to the same thing are equal
to one another," are not propositions in geometry but propositions in
analysis. I look upon them as analytical a priori intuitions, and they
concern me no further. But I must insist on other axioms which are
special to geometry. Of these most treatises explicitly enunciate
three: — (1) Only one line can pass through two points; (2) a
straight line is the shortest distance between two points; (3) through
one point only one parallel can be drawn to a given straight line.
Although we generally dispense with proving the second of these
axioms, it would be possible to deduce it from the other two, and
from those • much more numerous axioms which are implicitly ad-
mitted without enunciation, as I shall explain further on. For a long
time a proof of the third axiom known as Euclid's postulate was
sought in vain. It is impossible to imagine the efforts that have been
spent in pursuit of this chimera. Finally, at the beginning of the
nineteenth century, and almost simultaneously, two scientists, a Eus-
sian and a Bulgarian, Lobatschewsky and Bolyai, showed irrefutably
that this proof is impossible. They have nearly rid us of inventors
of geometries without a postulate, and ever since the Academic des
Sciences receives only about one or two new dmonstrations a year.
But the question was not exhausted, and it was not long before a
great step was taken by the celebrated memoir of Riemann, entitled:
Ueber die Hypothesen welche der Geometric zum Grunde liegen. This
little work has inspired most of the recent treatises to which I shall
later on refer, and among which I may mention those of Beltrami and
Helmholtz.

The Geometry of Lobatschewsky. — If it were possible to deduce

SPACE 647

Euclid's postulate from the several axioms, it is evident that by reject-
ing the postulate and retaining the other axioms we should be led to
contradictory consequences. It would be, therefore, impossible to
found on those premisses a coherent geometry. Now, this is pre-
cisely what Lobatschewsky has done. He assumes at the outset that
several parallels may be drawn through a point to a given straight
line, and he retains all the other axioms of Euclid. From these
hypotheses he deduces a series of theorems between which it is im-
possible to find any contradiction, and he constructs a geometry as
impeccable in its logic as Euclidean geometry. The theorems are very
different, however, from those to which we are accustomed, and at first
will be found a little disconcerting. For instance, the sum of the
angles of a triangle is always less than two right angles, and the differ-
ence between that sum and two right angles is proportional to the
area of the triangle. It is impossible to construct a figure similar to a
given figure but of different dimensions. If the circumference of a
circle be divided into n equal parts, and tangents be drawn at the
points of intersection, the n tangents will form a polygon if the
radius of the circle is small enough, but if the radius is large enough
they will never meet. We need not multiply these examples. Lobat-
schewsky's propositions have no relation to those of Euclid, but they
are none the less logically interconnected.

Riemann's Geometry. — Let us imagine to ourselves a world only
peopled with beings of no thickness, and suppose these " infinitely
flat " animals are all in one and the same plane, from which they
cannot emerge. Let us further admit that this world is sufficiently
distant from other worlds to be withdrawn from their influence, and
while we are making these hypotheses it will not cost us much to en-
dow these beings with reasoning power, and to believe them capable
of making a geometry. In that case they will certainly attribute to
space only two dimensions. But now suppose that these imaginary
animals, while remaining without thickness, have the form of a spher-
ical, and not of a plane figure, and are all on the same sphere, from
which they cannot escape. What kind of a geometry will they con-
struct? In the first place, it is clear that they will attribute to
space only two dimensions. The straight line to them will be the
shortest distance from one point on the sphere to another — that is
to say, an arc of a great circle. In a word, their geometry will be
spherical geometry. What they will call space will be the sphere on
which they are confined, and on which take place all the phenomena
with which they are acquainted. Their space will therefore be
unbounded, since on a sphere one may always walk forward without
ever being brought to a stop, and yet it will be finite; the end will
never be found, but the complete tour can be made. Well, Kiemamrs
geometry is spherical geometry extended to three dimensions. To

648 SCIENCE AND HYPOTHESIS

construct it, the German mathematician had first of all to throw over-
board, not only Euclid's postulate, but also the first axiom that only
one line can pass through two points. On a sphere, through two given
points, we can in general draw only one great circle which, as we have
just seen, would be to our imaginary beings a straight line. But
there was one exception. If the two given points are at the ends of
a diameter, an infinite number of great circles can be drawn through
them. In the same way, in Riemann's geometry — at least in one of
its forms — through two points only one straight line can in general
be drawn, but there are exceptional cases in which through two
points an infinite number of straight lines can be drawn. So there
is a kind of opposition between the geometries of Riemann and Lo-
batschewsky. For instance, the sum of the angles of a triangle is
equal to two right angles in Euclid's geometry, less than two right
angles in that of Lobatschewsky, and greater than two right angles
in that of Riemann. The number of parallel lines that can be drawn
through a given point to a given line is one in Euclid's geometry, none
in Riemann's, and an infinite number in the geometry of Lobatschew-
sky. Let us add that Riemann's space is finite, although unbounded
in the sense which we have above attached to these words.

Surfaces with Constant Curvature. — One objection, however, re-
mains possible. There is no contradiction between the theorems
of Lobatschewsky and Riemann; but however numerous are the other
consequences that these geometers have deduced from their hypothe-
ses, they had to arrest their course before they exhausted them all,
for the number would be infinite; and who can say that if they had
carried their deductions further they would not have eventually
reached some contradiction? This difficulty does not exist for Rie-
mann's geometry, provided it is limited to two dimensions. As we
have seen, the two-dimensional geometry of Riemann, in fact, does
not differ from spherical geometry, which is only a branch of ordinary
geometry, and is therefore outside all contradiction. Beltrami, by
showing that Lobatschewsky's two-dimensional geometry was only a
branch of ordinary geometry, has equally refuted the objection as
far as it is concerned. This is the course of his argument: Let us
consider any figure whatever on a surface. Imagine this figure to be
traced on a flexible and inextensible canvas applied to the surface, in
such a way that when the canvas is displaced and deformed the differ-
ent lines of the figure change their form without changing their
length. As a rule, this flexible and inextensible figure cannot be
displaced without leaving the surface. But there are certain surfaces
for which such a movement would be possible. They are surfaces of
constant curvature. If we resume the comparison that we made
just now, and imagine beings without thickness living on one of these
surfaces, they will regard as possible the motion of a figure all the

SPACE 649

lines of which remain of a constant length. Such a movement would
appear absurd, on the other hand, to animals without thickness living
on a surface of variable curvature. These surfaces of constant curva-
ture are of two kinds. The curvature of some is positive, and they
may be deformed so as to be applied to a sphere. The geometry of
these surfaces is therefore reduced to spherical geometry — namely,
Riemann's. The curvature of others is negative. Beltrami has shown
that the geometry of these surfaces is identical with that of Lobat-
schewsky. Thus the two-dimensional geometries of Riemann and
Lobatschewsky are connected with Euclidean geometry.

Interpretation of Non-Euclidean Geometries. — Thus vanishes the
objection so far as two-dimensional geometries arc concerned. It
would be easy to extend Beltrami's reasoning to three dimensional
geometries, and minds which do not recoil before space of four dimen-
sions will see no difficulty in it; but such minds are few in number.
I prefer, then, to proceed otherwise. Let us consider a certain plane,
which I shall call the fundamental plane, and let us construct a kind
of dictionary by making a double series of terms written in two col-
umns, and corresponding each to each, just as in ordinary diction-
aries the Avords in two languages which have the same signification
correspond to one another : —

Space The portion of space situated above the

fundamental plane.

Plane Sphere cutting orthogonally the funda-
mental plane.

Line Circle cutting orthogonally the funda-
mental plane.

Sphere Sphere.

Circle Circle.

Angle Angle.

Distance between

two points Logarithm of the anharmonic ratio of these

two points and of the intersection of the
fundamental plane with the circle pass-
ing through these two points and cutting
it orthogonally.
Etc. Etc.

Let us now take Lobatschewsky's theorems and translate them by
the aid of this dictionary, as we would translate a German text with
the aid of a German-French dictionary. We shall then obtain the the-
orems of ordinary geometry. For instance, Lobatschewsky's theo-
rem : " The sum of the angles of a triangle is less than two right
angles," may be translated thus : " If a curvilinear triangle has for
its sides arcs of circles which if produced would cut orthogonally the
fundamental plane, the sum of the angles of this curvilinear triangle

650 SCIENCE AND HYPOTHESIS

will be less than two right angles." Thus, however far the conse-
quences of Lobatschewsky's hypotheses are carried, they will never
lead to a contradiction; in fact, if two of Lobatschewsky's theorems
were contradictory, the translations of these two theorems made by
the aid of our dictionary would be contradictory also. But these
translations are theorems of ordinary geometry, and no one doubts that
ordinary geometry is exempt from contradiction. Whence is the
certainty derived, and how far is it justified? That is a question
upon which I cannot enter here, but it is a very interesting question,
and I think not insoluble. Nothing, therefore, is left of the objection
I formulated above. But this is not all. Lobatschewsky's geometry
being susceptible of a concrete interpretation, ceases to be a useless
logical exercise, and may be applied. I have no time here to deal
with these applications, nor with what Herr Klein and myself have
done by using them in the integration of linear equations. Further,
this interpretation is not unique, and several dictionaries may be
constructed analogous to that above, which will enable us by a simple
translation to convert Lobatschewsky's theorems into the theorems of
ordinary geometry.

Implicit Axioms. — Are the axioms implicitly enunciated in our
text-books the only foundation of geometry? We may be assured of
the contrary when we see that, when they are abandoned one after
another, there are still left standing some propositions whicTi are com-
mon to the geometries of Euclid, Lobatschewsky, and Kiemamu
These propositions must be based on premisses that geometers admit
without enunciation. It is interesting to try and extract them from
the classical proofs.

John Stuart Mill asserted 1 that every definition contains an axiom,
because by defining we implicitly affirm the existence of the object
defined. That is going rather too far. It is but rarely in mathematics
that a definition is given without following it up by the proof of
the existence of the object defined, and when this is not done it is
generally because the reader can easily supply it; and it must not be
forgotten that the word " existence " has not the same meaning when
it refers to a mathematical entity as when it refers to a material
object.

A mathematical entity exists provided there is no contradiction
implied in its definition, eithe ' in itself, or with the propositions pre-
viously admitted. But if the observation of John Stuart Mill cannot
be applied to all definitions, it is none the less true for some of them.
A plane is sometimes defined in the following manner : — The plane is
a surface such that the line which joins any two points upon it lies
wholly on that surface. Now, there is obviously a new axiom con-
cealed in this definition. It is true we might change it, and that
i Logic, c. viii., cf. Definitions, § 5-6. — TB.

SPACE 651

would be preferable, but then we should have to enunciate the axiom
explicitly. Other definitions may give rise to no less important reflec-
tions, such as, for example, that of the equality of two figures. Two
figures are equal when they can be superposed. To superpose them,
one of them must be displaced until it coincides with the other. But
how must it be displaced? If we asked that question, no doubt we
should be told that it ought to be done without deforming it, and as
an invariable solid is displaced. The vicious circle would then be
evident. As a matter of fact, this definition defines nothing. It has
no meaning to a being living in a world in which there are only fluids.
If it seems clear to us, it is because we are accustomed to the pro-
perties of natural solids which do not much differ from those of the
ideal solids, all of whose dimensions are invariable. However, im-
perfect as it may be, this definition implies an axiom. The possi-
bility of the motion of an invariable figure is not a self-evident truth.
At least it is only so in the application to Euclid's postulate, and not
as an analytical a priori intuition would be. Moreover, when we study
the definitions and the proofs of geometry, we see that we are com-
pelled to admit without proof not only the possibility of this motion,
but also some of its properties. This first arises in the definition of
the straight line. Many defective definitions have been given, but
the true one is that which is understood in all the proofs in which the
straight line intervenes. " It may happen that the motion of an in-
variable figure may be such that all the points of a line belonging to
the figure are motionless, while all the points situate outside that line
are in motion. Such a line would be called a straight line." "We
have deliberately in this enunciation separated the definition from the
axiom which it implies. Many proofs such as those of the cases of
the equality of triangles, of the possibility of drawing a perpen-
dicular from a point to a straight line, assume propositions the
enunciations of which are dispensed with, for they necessarily imply
that it is possible to move a figure in space in a certain way.

The Fourth Geometry. — Among these explicit axioms there is one
which seems to me to deserve some attention, because when we aban-
don it we can construct a fourth geometry as coherent as those of
Euclid, Lobatschewsky, and Eiemann. To prove that we can always
draw a perpendicular at a point A to a straight line A B, we consider
a straight line A C movable about the point A, and initially identical
with the fixed straight line A B. We then can make it turn about the
point A until it lies in A B produced. Thus we assume two propo-
sitions — first, that such a rotation is possible, and then that it may
continue until the two linos lie the one in the other produced. If
the first point is conceded and the second rejected, we are led to a
series of theorems even stranger than those of Lobatschewsky and
Riemann, but equally free from contradiction. I shall give only one

652 SCIENCE AND HYPOTHESIS

of these theorems, and I shall not choose the least remarkable of
them. A real straight line may be perpendicular to itself.

Lie's Theorem. — The number of axioms implicity introduced into
classical proofs is greater than necessary, and it would be interesting
to reduce them to a minimum. It may be asked, in the first place, if
this reduction is possible — if the number of necessary axioms and
that of imaginable geometries is not infinite? A theorem due to
Sophus Lie is of weighty importance in this discussion. It may be
enunciated in the following manner : — Suppose the following prem-
isses are admitted: (1) space has n dimensions; (2) the movement of
an invariable figure is possible; (3) p conditions are necessary to
determine the position of this figure in space.

The number of geometries compatible with these premisses will be
limited. I may even add that if n is given, a superior limit can be
assigned to p. If, therefore, the possibility of the movement is granted,
we can only invent a finite and even a rather restricted number of
three-dimensional geometries.

Riemann's Geometries. — However, this result seems contradicted
by Eiemann, for that scientist constructs an infinite number of geo-
metries, and that to which his name is usually attached is only a par-
ticular case of them. All depends, he says, on the manner in which
the length of a curve is defined. Now, there is an infinite number of
ways of defining this length, and each of them may be the starting-
point of a new geometry. That is perfectly true, but most of these
definitions are incompatible with the movement of a variable figure
such as we assume to be possible in Lie's theorem. These geometries
of Eiemann, so interesting on various grounds, can never be, there-
fore, purely analytical, and would not lend themselves to proofs analo-
gous to those of Euclid.

On the Nature of Axioms. — Most mathematicians regard Lobat-
schewsky's geometry as a mere logical curiosity. Some of them have,
however, gone further. If several geometries are possible, they say,
is it certain that our geometry is the one that is true ? Experiment no
doubt teaches us that the sum of the angles of a triangle is equal to
two right angles, but this is because the triangles we deal with are
too small. According to Lobatschewsky, the difference is proportional
to the area of the triangle, and will not this become sensible when we
operate on much larger triangles, and when our measurements become
more accurate? Euclid's geometry would thus be a provisory geo-
metry. Now, to discuss this view we must first of all ask ourselves,
what is the nature of geometrical axioms? Are they aynthetic
a priori intuitions, as Kant affirmed? They would then be imposed
upon us with such a force that we could not conceive of the contrary
proposition, nor could we build upon it a theoretical edifice. There
would be no non-Euclidean geometry. To convince ourselves of this,

SPACE 653

let us take a true synthetic a priori intuition — the following, for
instance, which played an important part in the first chapter : — If a
theorem is true for the number 1, and if it has been proved that it is
true of Ti+1, provided it is true of n, it will be true for all positive
integers. Let us next try to get rid of this, and while rejecting this
proposition let us construct a false arithmetic analogous to non-
Euclidean geometry. We shall not be able to do it. We shall be
even tempted at the outset to look upon these intuitions as analytical.
Besides, to take up again our fiction of animals without thickness, we
can scarcely admit that these beings, if their minds are like ours,
would adopt the Euclidean geometry, which would be contradicted by
all their experience. Ought we, then, to conclude that the axioms of
geometry are experimental truths? But we do not make experiments
on ideal lines or ideal circles; we can only make them on material
objects. On what, therefore, would experiments serving as a founda-
tion for geometry be based? The answer is easy. We have seen above
that we constantly reason as if the geometrical figures behaved like
solids. What geometry would borrow from experiment would be
therefore the properties of these bodies. The properties of light and
its propagation in a straight line have also given rise to some of the
propositions of geometry, and in particular to those of projective
geometry, so that from that point of view one would be tempted to
say that metrical geometry is the study of solids, and projective
geometry that of light. But a difficulty remains, and is unsurmount-
able. If geometry were an experimental science, it would not be
an exact science. It would be subjected to continual revision. Nay,
it would from that day forth be proved to be erroneous, for we
know that no rigorously invariable solid exists. Tine geometrical
axioms are therefore neither synthetic a priori intuitions nor experi-
mental facts. They are conventions. Our choice among all possible
conventions is guided by experimental facts; but it remains free, and
is only limited by the necessity of avoiding every contradiction, and
thus it is that postulates may remain rigorously true even when the
experimental laws which have determined their adoption are only ap-
proximate. In other words, the axioms of geometry (I do not speak of
those of arithmetic) are only definitions in disguise. What, then, are
we to think of the question: Is Euclidean geometry true? It has
no meaning. We might as well ask if the metric system is true,
and if the old weights and measures are false ; if Cartesian co-ordinates
are true and polar co-ordinates false. One geometry cannot be more
true than another; it can only be more convenient. Now Euclidean
geometry is, and will remain, the most convenient: 1st, because it is
the simplest, and it is not so only because of our mental habits or
because of the kind of direct intuition that we have of Euclidean
space; it is the simplest in itself, just as a polynomial of the first de-

654 SCIENCE AND HYPOTHESIS

gree is simpler than a polynomial of the second degree; 2nd, because
it sufficiently agrees with the properties of natural solids, those bodies
which we can compare and measure by means of our senses.

Space and Geometry

Let us begin with a little paradox. Beings whose minds were made
as ours, and with senses like ours, but without any preliminary edu-
cation, might receive from a suitably-chosen external world impres-
sions which would lead them to construct a geometry other than that
of Euclid, and to localize the phenomena of this external world in a
non-Euclidean space, or even in space of four dimensions. As for us,
whose education has been made by our actual world, if we were sud-
denly transported into this new world, we should have no difficulty in
referring phenomena to our Euclidean space. Perhaps somebody may
appear on the scene some day who will devote his life to it, and be
able to represent to himself the fourth dimension.

Geometrical Space and Representative Space. — It is often said
that the images we form of external objects are localized in space,
and even that they can only be formed on this condition. It is also
said that this space, which thus serves as a kind of framework ready
prepared for our sensations and representations, is identical with
the space of the geometers, having all the properties of that space.
To all clear-headed men who think in this way, the preceding state-
ment might well appear extraordinary ; but it is as well to see if they
are not the victims of some illusion which closer analysis may be able
to dissipate. In the first place, what are the properties of space pro-
perly so called? I mean of that space which is the object of geo-
metry, and which I shall call geometrical space. The following are
some of the more essential : —

1st, it is continuous; 2nd, it is infinite; 3rd, it is of three dimen-
sions; 4th, it is homogeneous — that is to say, all its points are
identical one with another; 5th, it is isotropic. Compare this now
with the framework of our representations and sensations, which I
may call representative space.

Visual Space. — First of all let us consider a purely visual impres-
sion, due to an image formed on the back of the retina. A cursory
analysis shows us this image as continuous, but as possessing only
two dimensions, which already distinguishes purely visual from what
may be called geometrical space. On the Bother hand, the image is
enclosed within a limited framework; and there is a no less important
difference: this pure visual space is not homogeneous. All the points
on the retina, apart from the images which may be formed, do not
play the same role. The yellow spot can in no way be regarded as
identical with a point on the edge of the retina. Not only does the

SPACE 655

same object produce on it much brighter impressions, but in the
whole of the limited framework the point which occupies the centre
will not appear identical with a point near one of the edges. Closer
analysis no doubt would show us that this continuity of visual space
and its two dimensions are but an illusion. It would make visual
space even more different than before from geometrical space, but we
may treat this remark as incidental.

However, sight enables us to appreciate distance, and therefore to
perceive a third dimension. But every one knows that this perception
of the third dimension reduces to a sense of the effort of accommoda-
tion which must be made, and to a sense of the convergence of the
two eyes, that must take place in order to perceive an object distinctly.
These are muscular sensations quite different from the visual sensa-
tions which have given us the concept of the two first dimensions.
The third dimension will therefore not appear to us as playing the
same role as the two others. What may be called complete visual space
is not therefore an isotropic space. It has, it is true, exactly three
dimensions; which means that the elements of our visual sensations
(those at least which concur in forming the concept of extension)
will be completely defined if we know three of them ; or, in mathemat-
ical language, they will be functions of three independent variables.
But let us look at the matter a little closer. The third dimension is
revealed to us in two different ways: by the effort of accommodation,
and by the convergence of the eyes. No doubt these two indications
are always in harmony; there is between them a constant relation; or,
in mathematical language, the two variables which measure these two
muscular sensations do not appear to us as independent. Or, again,
to avoid an appeal to mathematical ideas which are already rather too
refined, we may go back to the language of the preceding chapter and
enunciate the same fact as follows : — If two sensations of conver-
gence A and B are indistinguishable, the two sensations of accom-
modation A' and B' which accompany them respectively will also be
indistinguishable. But that is, so to speak, an experimental fact. Noth-
ing prevents us a priori from assuming the contrary, and if the con-
trary takes place, if these two muscular sensations both vary inde-
pendently, we must take into account one more independent variable,
and complete visual space will appear to us as a physical continuum
of four dimensions. And so in this there is also a fact of external
experiment. Nothing prevents us from assuming that a being with
a mind like ours, with the same sense-organs as ourselves, may be
placed in a world in which light would only reach him after being
passed through refracting media of complicated form. The two indi-
cations which enable us to appreciate distances would cease to be con-
nected by a constant relation. A being educating his senses in such a

656 SCIENCE AND HYPOTHESIS

world would no doubt attribute four dimensions to complete visual
space.

Tactile and Motor Space. — " Tactile space " is more complicated
still than visual space, and differs even more widely from geometrical
space. It is useless to repeat for the sense of touch my remarks
on the sense of sight. But outside the data of sight and touch there
are other sensations which contribute as much and more than they do
to the genesis of the concept of space. They are those which every-
body knows, which accompany all our movements, and which we usu-
ally call muscular sensations. The corresponding framework con-
stitutes what may be called motor space. Each muscle gives rise to a
special sensation which may be increased or diminished so that the
aggregate of our muscular sensations will depend upon as many
variables as we have muscles. From this point of view motor space
would have as many dimensions as we have muscles. I know that it is
said that if the muscular sensations contribute to form the concept
of space, it is because we have the sense of the direction of each move-
ment, and that this is an integral part of the sensation. If this were
so, and if a muscular sense could not be aroused unless it were accom-
panied by this geometrical sense of direction, geometrical space would
certainly be a form imposed upon our sensitiveness. But I do not see
this at all when I analyze my sensations. What I do see is that the
sensations which correspond to movements in the same direction are
connected in my mind by a simple association of ideas. It is to this
association that what we call the sense of direction is reduced. We
cannot therefore discover this sense in a single sensation. This asso-
ciation is extremely complex, for the contraction of the same muscle
may correspond, according to the position of the limbs, to very differ-
ent movements of direction. Moreover, it is evidently acquired; it is
like all associations of ideas, the result of a habit. This habit itself
is the result of a very large number of experiments, and no doubt if
the education of our senses had taken place in a different medium,
where we would have been subjected to different impressions, then
contrary habits would have been acquired, and our muscular sensations
would have been associated according to other laws.

Characteristics of Representative Space. — Thus representative
space in its triple form — visual, tactile, and motor — differs essen-
tially from geometrical space. It is neither homogeneous nor iso-
tropic ; we cannot even say that it is of three dimensions. It is often
said that we " project " into geometrical space the objects of our
external perception ; that we " localize " them. Now, has that any
meaning, and if so what is that meaning? Does it mean that we
represent to ourselves external objects in geometrical space ? Our rep-
resentations are only the reproduction of our sensations; they cannot
therefore be arranged in the same framework — that is to say, in

SPACE 657

representative space. II is also just as impossible for us to repre-
sent to ourselves external objects in geometrical space, as it is impos-
sible for a painter to paint on a flat surface objects with their three
dimensions. Bepresentative space is only an image of geometrical
space, an image deformed by a kind of perspective, and we can only
represent to ourselves objects by making them obey the laws of this
perspective. Thus we do not represent to ourselves external bodies in
geometrical space, but we reason about these bodies as if they were sit-
uated in geometrical space. When it is said, on the other hand, that
we " localize " such an object in such a point of space, what does it
mean ? It simply means that we represent to ourselves the movements
that must take place to reach that object. And it does not mean that
to represent to ourselves these movements they must be projected into
space, and that the concept of space must therefore pre-exist. When I
say that we represent to ourselves these movements, I only mean that
we represent to ourselves the muscular sensations which accompany
them, and which have no geometrical character, and which therefore
in no way imply the pre-existence of the concept of space.

Changes of State and Changes of Position. — But, it may be said,
if the concept of geometrical space is not imposed upon our minds, and
if, on the other hand, none of our sensations can furnish us with that
concept, how then did it ever come into existence? This is what we
have now to examine, and it will take some time; but I can sum up
in a few words the attempt at explanation which I am going to develop.
None of our sensations, if isolated, could have brought us to the con-
cept of space; we are brought to it solely by studying the laws by which
those sensations succeed one another. We see at first that our im-
pressions are subject to change ; but among the changes that we ascer-
tain, we are very soon led to make a distinction. Sometimes we say
that the objects, the causes of these impressions, have changed their
state, sometimes that they have changed their position, that they have
only been displaced. Whether an object changes its state or only its
position, this is always translated for us in the same manner, by a
modification in an aggregate of impressions. How then have we been
enabled to distinguish them ? If there were only change of position,
we could restore the primitive aggregate of inpressions by making
movements which would confront us with the movable object in the
same relative situation. We thus correct the modification which was
produced, and we re-establish the initial state by an inverse modi-
fication. If, for example, it were a question of the sight, and if
an object be displaced before our eyes, we can " follow it with
the eye/' and retain its image on the same point of the retina by
appropriate movements of the eyeball. These movements we are
conscious of because they are voluntary, and because they are accom-
panied by muscular sensations. But that does not mean that we

658 SCIENCE AND HYPOTHESIS

represent them to ourselves in geometrical space. So what charac-
terizes change of position, what distinguishes it from change of state,
is that it can always be corrected by this means. It may therefore
happen that we pass from the aggregate of impressions A to the aggre-
gate B in two different ways. First, involuntarily and without ex-
periencing muscular sensations — which happens when it is the
object that is displaced; secondly, voluntarily, and with muscular
sensation — which happens when the object is motionless, but when
we displace ourselves in such a way that the object has relative motion
with respect to us. If this be so, the translation of the aggregate A
to the aggregate B is only a change of position. It follows that sight
and touch could not have given us the idea of space without the help
of the " muscular sense." Not only could this concept not be derived
from a single sensation, or even from a series of sensations; but a
motionless being could never have acquired it, because, not being
able to correct by his movements the effects of the change of position
of external objects, he would have had no reason to distinguish them
from changes of state. Nor would he have been able to acquire it if
his movements had not been voluntary, or if they were unaccompanied
by any sensations whatever.

Conditions of Compensation. — How is such a compensation possible
in such a way that two changes, otherwise mutually independent, may
be reciprocally corrected? A mind already familiar with geometry
would reason as follows : — If there is to be compensation, the differ-
ent parts of the external object on the one hand, and the different
organs of our senses on the other, must be in the same relative position
after the double change. And for that to be the case, the different
parts of the external body on the one hand, and the different organs
of our senses on the other, must have the same relative position to each
other after the double change; and so with the different parts of our
body with respect to each other. In other words, the external object
in the first change must be displaced as an invariable solid would be
displaced, and it must also be so with the whole of our body in the
second change, which is to correct the first. Under these conditions
compensation may be produced. But we who as yet know nothing of
geometry, whose ideas of space are not yet formed, we cannot reason
in this way — we cannot predict a priori if compensation is possible.
But experiment shows us that it sometimes does take place, and we
start from this experimental fact in order to distinguish changes of
state from changes of position.

Solid Bodies and Geometry. — Among surrounding objects there
are some which frequently experience displacements that may be thus
corrected by a correlative movement of our own body — namely, solid
bodies. The other objects, whose form is variable, only in exceptional
circumstances undergo similar displacement (change of position with-
out change of form). When the displacement of a body takes place

SPACE 659

with deformation, we can no longer by appropriate movements place
the organs of our body in the same relative situation with respect to
this body; we can no longer, therefore, reconstruct the primitive ag-
gregate of impressions.

It is only later, and after a series of new experiments, that we
learn how to decompose a body of variable form into smaller elements
such that each is displaced approximately according to the same laws
as solid bodies. We thus distinguish " deformations " from other
changes of state. In these deformations each element undergoes a
simple change of position which may be corrected; but the modifica-
tion of the aggregate is more profound, and can no longer be corrected
by a correlative movement. Such a concept is very complex even at
this stage, and has been relatively slow in its appearance. It would
not have been conceived at all had not the observation of solid bodies
shown us beforehand how to distinguish changes of position.

//, then, there were no solid bodies in nature there would be no
geontetry.

Another remark deserves a moment's attention. Suppose a solid
body to occupy successively the positions a and /? ; in the first position
it will give us an aggregate of impressions A, and in the second posi-
tion the aggregate of impressions B. Now let there be a second solid
body, of qualities entirely different from the first — of different color,
for instance. Assume it to pass from the position a, where it gives
us the aggregate of impressions A' to the position /?, where it gives
the aggregate of impressions B'. In general, the aggregate A will
have nothing in common with the aggregate A', nor will the aggregate
B have anything in common with the aggregate B'. The transition
from the aggregate A to the aggregate B, and that of the aggregate A'
to the aggregate B', are therefore two changes which in themselves have
in general nothing in common. Yet we consider both these changes
as displacements; and, further, we consider them the same displace-
ment. How can this be? It is simply because they may be both cor-
rected by the same correlative movement of our body. " Correlative
movement," therefore, constitutes the sole connection between two
phenomena which otherwise we should never have dreamed of con-
necting.

On the other hand, our body, thanks to the number of its articula-
tion? and muscles, may have a multitude of different movements, but
all are not capable of " correcting " a modification of external objects;
those alone are capable of it in which our whole body, or at least all
those in which the organs of our senses enter into play are displaced
en bloc — - i.e., without any variation of their relative positions, as in
the case of a solid body.
To sum up :
1. In the first place, we distinguish two categories of phenomena : —

660 SCIENCE AND HYPOTHESIS

The first involuntary, unaccompanied by muscular sensations, and
attributed to external objects — they are external changes ; the second,
of opposite character and attributed to the movements of our own
body, are internal changes.

2. We notice that certain changes of each in these categories may
be corrected by a correlative change of the other category.

3. We distinguish among external changes those that have a corre-
lative in the other category — which we call displacements; and in
the same way we distinguish among the internal changes those which
have a correlative in the first category.

Thus by means of this reciprocity is defined a particular class of
phenomena called displacements. The laws of these phenomena are
the object of geometry.

Law of Homogeneity. — The first of these laws is the law of homo-
geneity. Suppose that by an external change we pass from the aggre-
gate of impressions A to the aggregate B, and that then this change
a is corrected by a correlative voluntary movement ft, so that we are
brought back to the aggregate A. Suppose now that another external
change a brings us again from the aggregate A to the aggregate B.
Experiment then shows us that this change a', like the change a, may
be corrected by a voluntary correlative movement ft', and that this
Momevent ft' corresponds to the same muscular sensations as the
movement ft which corrected a.

This fact is usually enunciated as follows : — Space is homogeneous
and isotropic. We may also say that a movement which is once pro-
duced may be repeated a second and a third time, and so on, without
any variation of its properties. In the first chapter, in which we
discussed the nature of mathematical reasoning, we saw the import-
ance that should be attached to the possibility of repeating the same
operation indefinitely. The virtue of mathematical reasoning is due
to this repetition; by means of the law of homogeneity geometrical
facts are apprehended. To be complete, to the law of homogeneity
must be added a multitude of other laws, into the details of which I
do not propose to enter, but which mathematicians sum up by saying
that these displacements form a " group."

The Non-Euclidean World. — If geometrical space were a frame-
work imposed on each of our representations considered individually,
it would be impossible to represent to ourselves an image without this
framework, and we should be quite unable to change our geometry.
But this is not the case; geometry is only the summary of the laws
by which these images succeed each other. There is nothing, there-
fore, to prevent us from imagining a series of representations, similar
in every way to our ordinary representations, but succeeding one
another according to laws which differ from those to which we are
accustomed. We may thus conceive that beings whose education has

SPACE 661

taken place in a medium in which those laws would be so different,
might have a very different geometry from ours.

Suppose, for example, a world enclosed in a large sphere and
subject to the following laws: — The temperature is not uniform; it
is greatest at the centre, and gradually decreases as we move towards
the circumference of the sphere, where it is absolute zero. The law
of this temperature is as follows : — If E be the radius of the sphere,
and r the distance of the point considered from the centre, the abso-
lute temperature will be proportional to R2 — r2. Further, I shall sup-
pose that in this world all bodies have the same co-efficient of dilata-
tion, so that the linear dilatation of any body is proportional to its ab-
solute temperature. Finally, I shall assume that a body transported
from one point to another of different temperature is instantaneously
in thermal equilibrium with its new environment. There is nothing
in these hypotheses either contradictory or unimaginable. A moving
object will become smaller and smaller as it approaches the circum-
ference of the sphere. Let us observe, in the first place, that although
from the point of view of our ordinary geometry this world is finite,
to its inhabitants it will appear infinite. As they approach the sur-
face of the sphere they become colder, and at the same time smaller
and smaller. The steps they take are therefore also smaller and
smaller, so that they can never reach the boundary of the sphere. If
to us geometry is only the study of the laws according to which in-
variable solids move, to these imaginary beings it will be the study
of the laws of motion of solids deformed by the differences of tem-
perature alluded to.

No doubt, in our world, natural solids also experience variations of
form and volume due to differences of temperature. But in laying
the foundations of geometry we neglect these variations; for besides
being but small they are irregular, and consequently appear to us to
be accidental. In our hypothetical world this will no longer be the
case, the variations will obey very simple and regular laws. On the
other hand, the different solid parts of which the bodies of these
inhabitants are composed will undergo the same variations of form
and volume.

Let me make another hypothesis: suppose that light passes through
media of different refractive indices, such that the index of refrac-
tion is inversely proportional to ft* — r2. Under these conditions it is
clear that the rays of light will no longer be rectilinear but circular.
To justify what lias been said, we have to prove that certain changes
in the position of external objects may be corrected by correlative
movements of the beings which inhabit this imaginary world; and in
such a way as to restore the primitive aggregate of the impressions
experienced by these sentient beings. Suppose, for example, that an
object is displaced and deformed, not like an invariable solid, but like

662 SCIENCE AND HYPOTHESIS

a solid subjected to unequal dilatations in exact conformity with
the law of temperature assumed above. To use an abbreviation, we
shall call such a movement a non-Euclidean displacement.

If a sentient being be in the neighborhood of such a displacement
of the object, his impressions will be modified; but by moving in a
suitable manner, he may reconstruct them. For this purpose, all that
is required is that the aggregate of the sentient being and the object,
considered as forming a single body, shall experience one of those
special displacements which I have just called non-Euclidean. This
is possible if we suppose that the limbs of these beings dilate accord-
ing to the same laws as the other bodies of the world they inhabit.

Although from the point of view of our ordinary geometry there is
a deformation of the bodies in this displacement, and although their
different parts are no longer in the same relative position, neverthe-
less we shall see that the impressions of the sentient being remain the
same as before; in fact, though the mutual distances of the different
parts have varied, yet the parts which at first were in contact are
still in contact. It follows that tactile impressions will be unchanged.
On the other hand, from the hypothesis as to refraction and the curva-
ture of the rays of light, visual impressions will also be unchanged.
These imaginary beings will therefore be led to classify the phenomena
they observe, and to distinguish among them the " changes of posi-
tion," which may be corrected by a voluntary correlative movement,
just as we do.

If they construct a geometry, it will not be like ours, which is the
study of the movements of our invariable solids; it will be the study
of the changes of position which they will have thus distinguished, and
will be " non-Euclidean displacements," and this will be non-Euclidean
geometry. So that beings like ourselves, educated in such a world,
will not have the same geometry as ours.

The World of Four Dimensions. — Just as we have pictured to our-
selves a non-Euclidean world, so we may picture a world of four di-
mensions.

The sense of light, even with one eye, together with the muscular
sensations relative to the movements of the eyeball, will suffice to
enable us to conceive of space of three dimensions. The images of
external objects are painted on the retina, which is a plane of two
dimensions; these are perspectives. But as eye and objects are mov-
able, we see in succession different perspectives of the same body
taken from different points of view. We find at the same time that
the transition from one perspective to another is often accompanied
by muscular sensations. If the transition from the perspective A
to the perspective B, and that of the perspective A' to the perspec-
tive B' are accompanied by the same muscular sensations, we con-
nect them as we do other operations of the same nature. Then when

SPACE 663

we study the laws according to which these operations are combined
we see that they form a group, which has the same structure as that
of the movements of invariable solids. Now, we have seen that it IB
from the properties of this group that we derive the idea of geomet-
rical space and that of three dimensions. We thus understand how
these perspectives gave rise to the conception of three dimensions,
although each perspective is of only two dimensions, — because they
succeed each other according to certain laws. Well, in the same way
that we draw the perspective of a three-dimensional figure on a plane,
so we can draw that of a four-dimensional figure on a canvas of three
(or two) dimensions. To a geometer this is but child's play. We can
even draw several perspectives of the same figure from several different
points of view. We can easily represent to ourselves these perspective,
since they are of only three dimensions. Imagine that the different
perspectives of one and the same object occur in succession, and
that the transition from one to the other is accompanied by muscular
sensations. It is understood that we shall consider two of these tran-
sitions as two operations of the same nature when they are associated
with the same muscular sensations. There is nothing, then, to prevent
us from imagining that these operations are combined according to
any law we choose — for instance, by forming a group with the same
structure as that of the movements of an invariable four-dimensional
solid. In this there is nothing that we cannot represent to ourselves,
and, moreover, these sensations are those which a being would experi-
ence who has a retina of two dimensions, and who may be displaced
in space of four dimensions. In this sense we may say that we can
represent to ourselves the fourth dimension.

Conclusions. — It is seen that experiment plays a considerable role
in the genesis of geometry ; but it would be a mistake to conclude from
that that geometry is, even in part, an experimental science. If it were
experimental, it would only be approximate and provisory. And
what a rough approximation it would be! Geometry would be only
the study of the movements of solid bodies; but, in reality, it is not
concerned with natural solids: its object is certain ideal solids, abso-
lutely invariable, which are but a greatly simplified and very remote
image of them. The concept of these ideal bodies is entirely mental,
and experiment is but the opportunity which enables us to reach the
idea. The object of geometry is the study of a particular " group " ;
but the general concept of group pre-exists in our minds, at least
potentially. It is imposed on us not as a form of our sensitiveness, but
as a form of our understanding ; only, from among all possible groups,
we must choose one that will be the standard, so to speak, to which
we shall refer natural phenomena.

Experiment guides us in this choice, which it does not impose on
us. It tells us not what is the truest, but what is the most convenient

664 SCIENCE AND HYPOTHESIS

geometry. It will be noticed that my description of these fantastic
worlds has required no language other than that of ordinary geo-
metry. Then, were we transported to those worlds, there would be
no need to change that language. Beings educated there would no
doubt find it more convenient to create a geometry different from ours,
and better adapted to their impressions ; but as for us, in the presence
of the same impressions, it is certain that we should not find it more
convenient to make a change.

Experiment and Geometry

1. I have on several occasions in the preceding pages tried to show
how the principles of geometry are not experimental facts, and that in
particular Euclid's postulate cannot be proved by experiment. How-
ever convincing the reasons already given may appear to me, I feel I
must dwell upon them, because there is a profoundly false conception
deeply rooted in many minds.

2. Think of a material circle, measure its radius and circumference,
and see if the ratio of the two lengths is equal to IT. What have we
done? We have made an experiment on the properties of the matter
with which this roundness has been realized, and of which the measure
we used is made.

3. Geometry and Astronomy. — The same question may also be
asked in another way. If Lobatschewsky's geometry is true, the paral-
lax of a very distant star will be finite. If Kiemann's is true, it will
be negative. These are the results which seem within the reach of
experiment, and it is hoped that astronomical observations may enable
us to decide between the two geometries. But what we call a straight
line in astronomy is simply the path of a ray of light. If, therefore,x
we were to discover negative parallaxes, or to prove that all parallaxes
are higher than a certain limit, we should have a choice between two
conclusions: we could give up Euclidean geometry, or modify the
laws of optics, and suppose that light is not rigorously propagated in
a straight line. It is needless to add that every one would look upon
this solution as the more advantageous. Euclidean geometry, there-
fore, has nothing to fear from fresh experiments.

4. Can we maintain that certain phenomena which are possible
in Euclidean space would be impossible in non-Euclidean space, so
that experiment in establishing these phenomena would directly con-
tradict the non-Euclidean hypothesis? I think that such a question
cannot be seriously asked. To me it is exactly equivalent to the fol-
lowing, the absurdity of which is obvious : — There are lengths which
can be expressed in metres and centimetres, but cannot be measured in
toises, feet, and inches; so that experiment, by ascertaining the exist-
ence of these lengths, would directly contradict this hypothesis, that

SPACE 665

there are toises divided into six feet. Let us look at the question a
little more closely. I assume that the straight line in Euclidean space
possesses any two properties, which I shall call A and B ; that in non-
Euclidean space it still possesses the property A, but no longer pos-
sesses the property B; and, finally, I assume that in both Euclidean
and non-Euclidean space the straight line is the only line that pos-
sesses the property A. If this were so, experiment would be able to
decide between the hypotheses of Euclid and Lobatschewsky. It would
be found that some concrete object, upon which we can experiment —
for example, a pencil of rays of light — possesses the property A.
We should conclude that it is rectilinear, and we should then
endeavor to find out if it does, or does not, possess the property B.
But it is not so. There exists no property which can, like this pro-
perty A, be an absolute criterion enabling us to recognize the straight
line, and to distinguish it from every other line. Shall we say, for
instance, " This property will be the following : the straight line is a
line such that a figure of which this line is a part can move without
the mutual distances of its points varying, and in such a way that all
the points in this straight line remain fixed " ? Now, this is a pro-
perty which in either Euclidean or non-Euclidean space belongs to
the straight line, and belongs to it alone. But how can we ascer-
tain by experiment if it belongs to any particular concrete object?
Distances must be measured, and how shall we know that any concrete
magnitude which I have measured with my material instrument really
represents the abstract distance? We have only removed the diffi-
culty a little farther off. In reality, the property that I have just
enunciated is not a property of the straight line alone; it is a pro-
perty of the straight line and of distance. For it to serve as an ab-
solute criterion, we must be able to show, not only that it does not
also belong to any other line than the straight line and to distance, but
also that it does not belong to any other line than the straight line, and
to any other magnitude than distance. Now, that is not true, and if
we are not convinced by these considerations, I challenge any one to
give me a concrete experiment which can be interpreted in the Eucli-
dean system, and which cannot be inteipreted in the system of Lobat-
schewsky. As I am well aware that this challenge will never be ac-
cepted, I may conclude that no experiment will ever be in contradic-
tion with Euclid's postulate: but, on the other hand, no experiment
will ever be in contradiction with Lobatschewsky's postulate.

5. But it is not sufficient that the Euclidean (or non-Euclidean)
geometry can ever be directly contradicted by experiment. Nor could
it happen that it can cnly agree with experiment by a violation of
the principle of sufficient reason, and of that of the relativity of space.
Let me explain myself. Consider any material system whatever. We
have to consider on the one hand the " state " of the various bodies of

666 SCIENCE AND HYPOTHESIS

this system — for example, their temperature, their electric potential,
etc.; and on the other hand their position in space. And among the
data which enable us to define this position we distinguish the mutual
distances of these bodies that define their relative positions, and the
conditions which define the absolute position of the system
and its absolute orientation in space. The law of the pheno-
mena which will be produced in this system will depend on the
state of these bodies, and on their mutual distances; but because
of the relativity and the inertia of space, they will not depend on the
absolute position and orientation of the system. In other words, the
state of the bodies and their mutual distances at any moment will
solely depend on the state of the same bodies and on their mutual dis-
tances at the initial moment, but will in no way depend on the abso-
lute initial position of the system and on its absolute initial orientation.
This is what we shall call, for the sake of abbreviation, the law of rela-
tivity.

So far I have spoken as a Euclidean geometer. But I have said
that an experiment, whatever it may be, requires an interpretation
on the Euclidean hypothesis; it equally requires one on the non-
Euclidean hypothesis. Well, we have made a series of experiments.
We have interpreted them on the Euclidean hypothesis, and we have
recognized that these experiments thus interpreted do not violate this
" law of relativity." We now interpret them on the non-Euclidean
hypothesis. This is always possible, only the non-Euclidean distances
of our different bodies in this new interpretation will not generally
be the same as the Euclidean distances in the primitive interpretation.
Will our experiment interpreted in this new manner be still in agree-
ment with our " law of relativity," and if this agreement had not
taken place, would we not still have the right to say that experiment
has proved the falsity of non-Euclidean geometry? It is easy to see
that this is an idle fear. In fact, to apply the law of relativity in all
its rigor, it must be applied to the entire universe; for if we were to
consider only a part of the universe, and if the absolute position of
this part were to vary, the distances of the other bodies of the universe
would equally vary; their influence on the part of the universe con-
sidered might therefore increase or diminish, and this might modify
the laws of the phenomena which take place in it. But if our system
is the entire universe, experiment is powerless to give us any opinion
on its position and its absolute orientation in space. All that our
instruments, however perfect they may be, can let us know will be the
state of the different parts of the universe, and their mutual distances.
Hence, our law of relativity may be enunciated as follows : — The
readings that we can make with our instruments at any given moment
will depend only on the readings that we were able to make on the
same instruments at the initial moment. Now such an enunciation is

SPACE 667

independent of all interpretation by experiments. If the law is true
in the Euclidean interpretation, it will be also true in the non-Eucli-
dean interpretation. Allow me to make a short digression on this
point. I have spoken above of the data which define the position of
the different bodies of the system. I might also have spoken of those
which define their velocities. I should then have to distinguish the
velocity with which the mutual distances of the different bodies are
changing, and on the other hand the velocities of translation and
rotation of the system; that is to say, the velocities with which its
absolute position and orientation are changing. For the mind to be
fully satisfied, the law of relativity would have to be enunciated as
follows : — The state of bodies and their mutual distances at any
given moment, as well as the velocities with which those distances are
changing at that moment, will depend only on the state of those bodies,
on their mutual distances at the initial moment, and on the velocities
with which those distances were changing at the initial moment. But
they will not depend on the absolute initial position of the system
nor on its absolute orientation, nor on the velocities with which that
absolute position and orientation were changing at the initial moment.
Unfortunately, the law thus enunciated does not agree with experi-
ments — at least, as they are ordinarily interpreted. Suppose a man
were translated to a planet, the sky of which was constantly covered
with a thick curtain of clouds, so that he could never see the other
stars. On that planet he would live as if it were isolated in space.
But he would notice that it revolves, either by measuring its ellipticity
(which is ordinarily done by mean? of astronomical observations, but
which could be done bv purely geodesic means), or by repeating the
experiment of Foucault's pendulum. The absolute rotation of this
planet might be clearly shown in this way. Now, here is a fact which
shocks the philosopher, but which the physicist is compelled to accept.
We know from this fact Newton concluded the existence of absolute
space. I myself cannot accept this way of looking at it. I shall ex-
plain why in Part III., but for the moment it is not my intention to
discuss this difficulty. I must therefore resign myself, in the enun-
ciation of the law of relativity, to including velocities of every kind
among the data which define the state of the bodies. However that
may be, the difficulty is the same for both Euclid's geometry and for
Lobatschewsky's. I need not therefore trouble about it further, and
I have only mentioned it incidentally. To sum up, whichever way we
look at it, it is impossible to discover in geometric empiricism a ra-
tional meaning.

6. Experiments only teach us the relations of bodies to one another.
They do not and cannot give us the relations of bodies and space, nor
the mutual relations of the different parts of space. " Yes ! " you
reply, " a single experiment is not enough, because it only gives us

668 SCIENCE AND HYPOTHESIS

one equation with several unknowns; but when I have made enough
experiments I shall have enough equations to calculate all my un-
knowns." If I know the height of the main-mast, that is not sufficient
to enable me to calculate the age of the captain. When you have
measured every fragment of wood in a ship you will have many equa-
tions, but you will be no nearer knowing the captain's age. All your
measurements bearing on your fragments of wood can tell you only
what concerns those fragments ; and similarly, your experiments, how-
ever numerous they may be, referring only to the relations of bodies
with one another, will tell you nothing about the mutual relations of
the different parts of space.

7. Will you say that if the experiments have reference to the bodies,
they at least have reference to the geometrical properties of the bodies ?
First, what do you understand by the geometrical properties of bodies ?
I assume that it is a question of the relations of the bodies to space.
These properties therefore are not reached by experiments which only
have reference to the relations of bodies to one another, and that is
enough to show that it is not of those properties that there can be a
question. Let us therefore begin by making ourselves clear as to the
sense of the phrase: geometrical properties of bodies. Wlien I say
that a body is composed of several parts, I presume that I am thus
enunciating a geometrical property, and that will be true even if I
agree to give the improper name of points to the very small parts I
am considering. When I say that this or that part of a certain body
is in contact with this or that part of another bod)", I am enunciating
a proposition which concerns the mutual relations of the two bodies,
and not their relations with space. I assume that you will agree
with me that these are not geometrical properties. I am sure that at
least you will grant that these properties are independent of all know-
ledge of metrical geometry. Admitting this, I suppose that we have
a solid body formed of eight thin iron rods, oa, ob, oc, od, oe, of,
og, oh, connected at one of their extremities, o. And let us take a
second solid body — for example, a piece of wood, on which are
marked three little spots of ink which I shall call a ft y. I now
suppose that we find that we can bring into contact a (3 y with ago;
by that I mean a with a, and at the same time ft with g, and y
with o. Then we can successively bring into contact a/3y with bgo,
ego, dgo, ego, fgo, then with oho, bho, cho, dho, eho, fho; and then
ay successively with ab, be, cd, de, ef, fa. Now these are observations
that can be made without having any idea beforehand as to the form
or the metrical properties of space. They have no reference whatever
to the " geometrical properties of bodies." These observations will not
be possible if the bodies on which we experiment move in a group
having the same structure as the Lobatschewskian group (I mean
according to the same laws as solid bodies in Lobatschewsky's geo-

SPACE 669

metry). They therefore suffice to prove that these bodies move ac-
cording to the Euclidean group; or at least that they do not move
according to the Lobatschewskian group. That they may be com-
patible with the Euclidean group is easily seen; for we might make
them so if the body a(3y were an invariable solid of our ordinary geo-
metry in the shape of a right-angled triangle and if the points
abcdefgh were the vertices of the polyhedron formed of two regular
hexagonal pyramids of our ordinary geometry having abcdef
as their common base, and having the one g and the other h as their
vertices. Suppose now, instead of the previous observations, we note
that we can as before apply a/3y successively to ago, bgo, ego, dgo, ego,
igo, oho, bho, clw, dho, eho, fho, and then that we can apply a/3 (and
no longer ay) successively to ab, be, cd, de, cf, and fa. These are
observations that could be made if non-Euclidean geometry were true,
if the bodies a(3y, oabcdefgh were invariable solids, if the former
were a right-angled triangle, and the latter a double regular hexagonal
pyramid of suitable dimensions. These new verifications are there-
fore impossible if the bodies move according to the Euclidean group ;
but they become possible if we suppose the bodies to move according
to the Lobatschewskian group. They would therefore suffice to show,
if we carried them out, that the bodies in question do not move ac-
cording to the Euclidean group. And so, without making any hypo-
thesis on the form and the nature of space, on the relations of the
bodies and space, and without attributing to bodies any geometrical
property, I have made observations which have enabled me to show
in one case that the bodies experimented upon move according to
a group, the structure of which is Euclidean, and in the other case that
they move in a group, the structure of which is Lobatschewskian. It
cannot be said that all the first observations would constitute an exper-
iment proving that space is Euclidean, and the second an experiment
proving space is non-Euclidean: in fact, it might be imagined (note
that I use the word imagined) that there are bodies moving in such a
manner as to render possible the second series of observations: and
the proof is that the first mechanic who came our way could con-
struct it if he would only take the trouble. But you must not con-
clude, however, that space is non-Euclidean. In the same way, just
as ordinary solid bodies would continue to exist when the mechanic
had constnicted the strange bodies I have just mentioned, he would
have to conclude that space is both Euclidean and non-Euclidean.
Suppose, for instance, that we have a large sphere of radius E, and
that its temperature decreases from the centre to the surface of the
sphere according to the law of which I spoke when I was describing
the non-Euclidean world. We might have bodies whose dilatation is
negligible, and which would behave as ordinary invariable solids;
and, on the other hand, we might have very dilatable bodies, which

670

would behave as non-Euclidean solids. We might have two double
pyramids oabcdefgh and o a V c <f e' f g h', and two triangles a ft y
and a ft' y. The first double pyramid would be rectilinear, and the
second curvilinear. The triangle afty would consist of undilatable
matter, and the other of very dilatable matter. We might therefore
make our first observations with the double pyramid o' a h' and the
triangle a ft' y.

And then the experiment would seem to show — first, that Eucli-
dean geometry is true, and then that it is false. Hence, experiments
have reference not to space but to bodies.

8. To round the matter off, I ought to speak of a very delicate
question, which will require considerable development; but I shall
confine myself to summing up what I have written in the Revue de
Metaphysique et de Morale and in the Monist. When we say that
space has three dimensions, what do we mean? We have seen the im-
portance of these " internal changes " which are revealed to us by
our muscular sensations. They may serve to characterize the different
attitudes of our body. Let us take arbitrarily as our origin one of
these attitudes, A. When we pass from this initial attitude to another
attitude B we experience a series of muscular sensations, and this series
S of muscular sensations will define B. Observe, however, that we
shall often look upon two series S and S' as defining the same attitude
B (since the initial and final attitudes A and B remaining the same,
the intermediary attitudes of the corresponding sensations may differ).
How then can we recognize the equivalence of these two series? Be-
cause they may serve to compensate for the same external change, or
more generally, because when it is a question of compen-
sation for an external change, one of the series may be replaced
by the other. Among these series we have distinguished those which
can alone compensate for an external change, and which we have
called " displacements." As we cannot distinguish two displacements
which are very close together, the aggregate of these displacements
presents the characteristics of a physical continuum. Experience
teaches us that they are the characteristics of a physical continuum of
six dimensions; but we do not know as yet how many dimensions
space itself possesses, so we must first of all answer another question.
What is a point in space ? Every one thinks he knows, but that is an
illusion. What we see when we try to represent to ourselves a point
in space is a black spot on white paper, a spot of chalk on a blackboard,
always an object. The question should therefore be understood as
follows : — What do I mean when I say the object B is at the point
which a moment before was occupied by the object A? Again, what
criterion will enable me to recognize it ? I mean that although I have
not moved (my muscular sense tells me this), my finger, which just
now touched the object A, is now touching the object B. I might have

SPACE 671

used other criteria — for instance, another finger or the sense of sight
— but the first criterion is sufficient. I know that if it answers in the
affirmative all other criteria will give the same answer. I know
it from experiment. I cannot know it a priori. For the same reason
I say that touch cannot be exercised at a distance ; that is another way
of enunciating the same experimental fact. If I say, on the contrary,
that sight is exercised at a distance, it means that the criterion fur-
nished by sight may give an affirmative answer while the others reply
in the negative.

To sum up. For each attitude of my body my finger determines a
point, and it is that and that only which defines a point
in space. To each attitude corresponds in this way a point. But
it often happens that the same point corresponds to several different
attitudes (in this case we say that our finger has not moved, but the
rest of our body has). We distinguish, therefore, among changes of
attitude those in which the finger does not move. How are we led to
this? It is because we often remark that in these changes the object
which is in touch with the finger remains in contact with it. Let us
arrange then in the same class all the attitudes which are deduced one
from the other by one of the changes that we have thus distinguished.
To all these attitudes of the same class will correspond the same point
in space. Then to each class will correspond a point, and to each
point a class. Yet it may be said that what we get from this experi-
ment is not the point, but the class of changes, or, better still, the
corresponding class of muscular sensations. Thus, when we say that
space has three dimensions, we merely mean that the aggregate of
these classes appears to us with the characteristics of a physical con-
tinuum of three dimensions. Then if, instead of defining the points
in space with the aid of the first finger, I use, for example, another
finger, would the results be the same? That is by no means a priori
evident. But, as we have seen, experiment has shown us that all our
criteria are in agreement, and this enables us to answer in the affirma-
tive. If we recur to what we have called displacements, the aggre-
gate of which forms, as we have seen, a group, we shall be brought
to distinguish those in which a finger does not move, and by what has
preceded, those are the displacements which characterize a point in
space, and their aggregate will form a sub-group of our group. To
each sub-group of this kind, then, will correspond a point in space.
We might be tempted to conclude that experiment has taught us the
number of dimensions of space; but in reality our experiments have
referred not to space, but to our body and its relations with neighbor-
ing objects. What is more, our experiments are exceeding crude. In
our mind the latent idea of a certain number of groups pre-existed;
these are the groups with which Lie's theory is concerned. Which

672 SCIENCE AND HYPOTHESIS

shall we choose to form a kind of standard by which to compare nat-
ural phenomena? And when this group is chosen, which of the sub-
groups shall we take to characterize a point in space ? Experiment has
guided us by showing us what choice adapts itself best to the proper-
ties of our body ; but there its role ends.

PAET III — FOKCE

The Classical Mechanics.

THE English teach mechanics as an experimental science; on the
Continent it is taught always more or less as a deductive and a priori
science. The English are right, no doubt. How is it that the other
method has been persisted in for so long; how is it that Continental
scientists who have tried to escape from the practice of their predeces-
sors have in most cases been unsuccessful ? On the other hand, if the
principles of mechanics are only of experimental origin, are they not
merely approximate and provisory? May we not be some day com-
pelled by new experiments to modify or even to abandon them ? These
are the questions which naturally arise, and the difficulty of solution
is largely due to the fact that treatises on mechanics do not clearly
distinguish between what is experiment, what is mathematical rea-
soning, what is convention, and what is hypothesis. This is not all.

1. There is no absolute space, and we only conceive of relative
motion; and yet in most cases mechanical facts are enunciated as if
there is an absolute space to which they can be referred.

2. There is no absolute time. When we say that two periods are
equal, the statement has no meaning, and can only acquire a meaning
by a convention.

3. Not only have we no direct intuition of the equality of two
periods, but we have not even direct intuition of the simultaneity of
two events occurring in two different places. I have explained this in
an article entitled " Mesure du Temps."1

4. Finally, is not our Euclidean geometry in itself only a kind of
convention of language? Mechanical facts might be enunciated with
reference to a non-Euclidean space which would be less convenient but
quite as legitimate as our ordinary space; the enunciation would be-
come more complicated, but it still would be possible.

Thus, absolute space, absolute time, and even geometry are not con-
ditions which are imposed on mechanics. All these things no more
existed before mechanics than the French language can be logicallj
said to have existed before the truths which are expressed in French.

i Revue de Metaphysique et de Morale, t. vi., pp. 1-13, January, 1898.

674 SCIENCE AND HYPOTHESIS

We might endeavor to enunciate the fundamental law of mechanics
in a language independent of all these conventions; and no doubt we
should in this way get a clearer idea of those laws in themselves. This
is what M. Andrade has tried to do, to some extent at any rate, in his
Lemons de Mecanique physique. Of course the enunciation of these
laws would become much more complicated, because all these con-
ventions have been adopted for the very purpose of abbreviating and
simplifying the enunciation. As far as we are concerned, I shall
ignore all these difficulties; not because I disregard them, far from it;
but because they have received sufficient attention in the first two
parts of the book. Provisionally, then, we shall admit absolute time
and Euclidean geometry.

The Principle of Inertia. — A body under the action of no force can
only move uniformly in a straight line. Is this a truth imposed on
the mind a priori ? If this be so, how is it that the Greeks ignored it ?
How could they have believed that motion ceases with the cause of
motion? or, again, that every body, if there is nothing to prevent it,
will move in a circle, the noblest of all forms of motion ?

If it be said that the velocity of a body cannot change, if there is
no reason for it to change, may we not just as legitimately maintain
that the position of a body cannot change, or that the curvature of its
path cannot change, without the agency of an external cause? Is,
then, the principle of inertia, which is not an a priori truth, an exper-
imental fact? Have there ever been experiments on bodies acted on
by no forces ? and, if so, how did we know that no forces were acting ?
The usual instance is that of a ball rolling for a very long time on a
marble table ; but why do we say it is under the action of no force ?
Is it because it is too remote from all other bodies to experience any
sensible action? It is not further from the earth than if it were
thrown freely into the air; and we all know that in that case it
would be subject to the attraction of the earth. Teachers of mechan-
ics usually pass rapidly over the example of the ball, but they add that
the principle of inertia is verified indirectly by its consequences. This
is very badly expressed; they evidently mean that various conse-
quences may be verified by a more general principle, of which the
principle of inertia is only a particular case. I shall propose for
this general principle the following enunciation: — The acceleration
of a body depends only on its position and that of neighboring bodies,
and on their velocities. Mathematicians would say that the move-
ments of all the material molecules of the universe depend on
differential equations of the second order. To make it clear that this
is really a generalization of the law of inertia we may again have
recourse to our imagination. The law of inertia, as I have said above,
is not imposed on us a priori; other laws would be just as compatible
with the principle of sufficient reason. If a body is not acted upon

FOECE 675

by force, instead of supposing that its velocity is unchanged we may
suppose that its position or its acceleration is unchanged.

Let us for a moment suppose that one of these two laws is a law
of nature, and substitute it for a law of inertia: what will be the
natural generalization? A moment's reflection will show us. In
the first case, we may suppose that the velocity of a body depends only
on its position and that of neighboring bodies; in the. second case,
that the variation of the acceleration of a body depends only on the
position of the body and of neighboring bodies, on their velocities
and accelerations; or, in mathematical terms, the differential equations
of the motion would be of the first order in the first case and of the
third order in the second.

Let us now modify our supposition a little. Suppose a world ana-
logous to our solar system, but one in which by a singular chance the
orbits of all the planets have neither eccentricity nor inclination;
and further, suppose that the masses of the planets are too small
for their mutual perturbations to be sensible. Astronomers living
in one of these planets would not hesitate to conclude that the orbit
of a star can only be circular and parallel to a certain plane ; the posi-
tion of a star at a given moment would then be sufficient to determine
its velocity and path. The law of inertia which they would adopt
would be the former of the two hypothetical laws I have mentioned.

Now, imagine this system to be some day crossed by a body of vast
mass and immense velocity coming from distant constellations. All
the orbits would be profoundly disturbed. Our astronomers would not
be greatly astonished. They would guess that this new star is in
itself quite capable of doing all the mischief; but, they would say, as
soon as it has passed by. order will again be established. No doubt
the distances of the planets from the sun will not be the same as
before the cataclysm, but the orbits will become circular again as soon
as the disturbing cause has disappeared. It would be only when \he
perturbing body is remote, and when the orbits, instead of being
circular are found to be elliptical, that the astronomers would find
out their mistake, and discover the necessity of reconstructing their
mechanics.

I have dwelt on these hypotheses, for it seems to me that we can
clearly understand our generalized law of inertia only by opposing it
to a contrary hypothesis.

Has this generalized law of inertia been verified by experiment, and
can it be so verified ? When Newton wrote the Principia, he certainly
regarded this truth as experimentally acquired and demonstrated.
It was so in his eyes, not only from the anthropomorphic conception
to which I shall later refer, but also because of the work of Galileo. It
was so proved by the laws of Kepler. According to those laws, in fact,
the path of a planet is entirely determined by its initial position and

676 SCIENCE AND HYPOTHESIS

initial velocity; this, indeed, is what our generalized law of inertia
requires.

For this principle to be only true in appearance — lest we should
fear that some day it must be replaced by one of the analogous prin-
ciples which I opposed to it just now — we must have been led astray
by some amazing chance such as that which had led into error our
imaginary astronomers. Such an hypothesis is so unlikely that it need
not delay us. No one will believe that there can be such chances; no
doubt the probability that two eccentricities are both exactly zero is
not smaller than the probability that one is 0.1 and the other 0.2.
The probability of a simple event is not smaller than that of a
complex one. If, however, the former does occur, we shall not attri-
bute its occurrence to chance; we shall not be inclined to believe that
nature has done it deliberately to deceive us. The hypothesis of an
error of this kind being discarded, we may admit that so far as astro-
nomy is concerned our law has been verified by experiment.

But Astronomy is not the whole of Physics. May we not fear that
some day a new. experiment will falsify the law in some domain of
physics? An experimental law is always subject to revision; we may
always expect to see it replaced by some other and more exact law. But
no one seriously thinks that the law of which we speak will ever be
abandoned or amended. Why? Precisely because it will never be
submitted to a decisive test.

In the first place, for this test to be complete, all the bodies of the
universe must return with their initial velocities to their initial posi-
tions after a certain time. We ought then to find that they would re-
sume their original paths. But this test is impossible; it can be
only partially applied, and even when it is applied there will still be
some bodies which will not return to their original positions. Thus
there will be a ready explanation of any breaking down of the law.

Yet this is not all. In Astronomy we see, the bodies whose motion
we are studying, and in most cases we grant that they are not subject
to the action of other invisible bodies. Under these conditions, our
law must certainly be either verified or not. But it is not so in
Physics. If physical phenomena are due to motion, it is to the motion
of molecules which we cannot see. If, then, the acceleration of bodies
we cannot see depends on something else than the positions or veloci-
ties of other visible bodies or of invisible molecules, the existence of
which we have been led previously to admit, there is nothing to
prevent us from supposing that this something else is the position or
velocity of other molecules of which we have not so far suspected
the existence. The law will be safeguarded. Let me express the same
thought in another form in mathematical language. Suppose we are
observing n molecules, and find that their 3n co-ordinates satisfy a
system of 3n differential equations of the fourth order (and not of

FOKCE 677

the second, as required by the law of inertia) . We know that by intro-
ducing 3n variable auxiliaries, a system of 3n equations of the fourth
order may be reduced to a system of 6n equations of the second order.
If, then, we suppose that the 3n auxiliary variables represent the co-
ordinates of n invisible molecules, the result is again conformable to
the law of inertia. To sum up, this law, verified experimentally in
some particular cases, may be extended fearlessly to the most general
cases; for we know that in these general cases it can neither be con-
firmed nor contradicted by experiment.

The Law of Acceleration. — The acceleration of a body is equal to
the force which acts on it divided by its mass.

Can this law be verified by experiment? If so, we have to measure
the three magnitudes mentioned in the enunciation : acceleration, force,
and mass. I admit that acceleration may be measured, because I pass
over the difficulty arising from the measurement of time. But how
are we to measure force and mass? We do not even know what they
are. What is mass ? Newton replies : " The product of the volume
and the density." " It were better to say," answer Thomson and Tait,
" that density is the quotient of the mass by the volume." What is
force ? " It is," replies Lagrange, " that which moves or tends to
move a body." " It is," according to Kirchoff, " the product of the
mass and the acceleration." Then why not say that mass is the quo-
tient of the force by the acceleration? These difficulties are insur-
mountable.

When we say force is the cause of motion, we are talking metaphy-
sics ; and this definition, if we had to be content with it, would be
absolutely fruitless, would lead to absolutely nothing. For a defini-
tion to be of any use it must tell us how to measure force; and that
is quite sufiicient, for it is by no means necessary to tell what force
is in itself, nor whether it is the cause or the effect of motion. We
must therefore first define what is meant by the equality of two
forces. When are two forces equal? We are told that it is when they
give the same acceleration to the same mass, or when acting in oppo-
site directions they are in equilibrium. This definition is a sham. A
force applied to a body cannot be uncoupled and applied to another
body as an engine is uncoupled from one train and coupled to an-
other. It is therefore impossible to say what acceleration such a force,
applied to such a body, would give to another body if it were applied
to it. It is impossible to tell how two forces which are not acting in
exactly opposite directions would behave if they were acting in oppo-
site directions. It is this definition which we try to materialize, as
it were, when we measure a force with a dynamometer or with a bal-
ance. Two forces, F and F', which I suppose, for simplicity, to be
acting vertically upwards, are respectively applied to two bodies, C
and f '. T attach a body weighing P first to C and then to C'; if

678 SCIENCE AND HYPOTHESIS

there is equilibrium in both cases I conclude that the two forces F and
F' are equal, for they are both equal to the weight of the body P. But
am I certain that the body P has kept its weight when I transferred it
from the first body to the second? Far from it. I am certain of the
contrary. I know that the magnitude of the weight varies from one
point to another, and that it is greater, for instance, at the pole than
at the equator. No doubt, the difference is very small, and we neglect
it in practice; but a definition must have mathematical rigor; this
rigor does not exist. What I say of weight would apply equally to
the force of the spring of a dynamometer, which would vary accord-
ing to temperature and many other circumstances. Nor is this all.
We cannot say that the weight of the body P is applied to the body
C and keeps in equilibrium the force F. What is applied to the body
C is the action of the body P on the body C. On the other hand, the
body P is acted on by its weight, and by the reaction E of the body C
on P the forces F and A are equal, because they are in equilibrium;
the forces A and R are equal by virtue of the principle of action
and reaction; and finally, the force R and the weight P are equal be-
cause they are in equilibrium. From these three equalities we deduce
the equality of the weight P and the force F.

Thus we are compelled to bring into our definition of the equality
of two forces the principle of the equality of action and reaction;
hence this principle can no longer "be regarded as an experimental law
but only as a definition.

To recognize the equality of two forces we are then in possession of
two rules: the equality of two forces in equilibrium and the equality
of action and reaction. But, as we have seen, these are not sufficient,
and we are compelled to have recourse to a third rule, and to admit
that certain forces — the weight of a body, for instance — are con-
stant in magnitude and direction. But this third rule is an experi-
mental law. It is only approximately true : it is a bad definition. We
are therefore reduced to Kirchoff's definition: force is the product of
the mass and the acceleration. This law of Newton in its turn
ceases to be regarded as an experimental law, it is now only a defini-
tion. But as a definition it is insufficient, for we do not know what
mass is. It enables us, no doubt, to calculate the ratio of two forces
applied at different times to the same body, but it tells us nothing
about the ratio of two forces applied to two different bodies. To fill
up the gap we must have recourse to Newton's third law, the equality
of action and reaction, still regarded not as an experimental law but
as a definition. Two bodies, A and B, act on each other ; the accelera-
tion of A, multiplied by the mass of A, is equal to the action of B
on A ; in the same way the acceleration of B, multiplied by the mass
of B, is equal to the -reaction of A on B. As, by definition, the action
and the reaction are equal, the masses of A and B are respectively in

FOECE 679

the inverse ratio of their masses. Thus is the ratio of the two masses
defined, and it is for experiment to verify that the ratio is constant.

This would do very well if the two bodies were alone and could be
abstracted from the action of the rest of the world; but this is by no
means the case. The acceleration of A is not solely due to the action
of B, but to that of a multitude of other bodies, C, D, . . . To apply
the preceding rule we must decompose the acceleration of A into many
components, and find out which of these components is due to the
action of B. The decomposition would still be possible if we suppose
that the action of C on A is simply added to that of B on A, and that
the presence of the body C does not in any way modify the action of
B on A, or that the presence of B does not modify the action of C
on A ; that is, if we admit that any two bodies attract each other, that
their mutual action is along their join, and is only dependent on their
distance apart ; if, in a word, we admit the hypothesis of central forces.

We know that to determine the masses of the heavenly bodies we
adopt quite a different principle. The law of gravitation teaches us
that the attraction of two bodies is proportional to their masses; if r
is their distance apart, ra and in their masses, Jc, a constant, then their
attraction will be Jcmm'/r2. What we are measuring is therefore not
mass, the ratio of the force to the acceleration, but the attracting
mass; not the inertia of the body, but its attracting power. It is an
indirect process, the use of which is not indispensable theoretically.
We might have said that the attraction is inversely proportional to
the square of the distance, without being proportional to the product
of the masses, that it is equal to f/r2 and not to Tcmm. If it were so,
we should nevertheless, by observing the relative motion of the celestial
bodies, be able to calculate the masses of these bodies.

But have we any right to admit the hypothesis of central forces ? Is
this hypothesis rigorously accurate ? Is it certain that it will never be
falsified by experiment? Who will venture to make such an asser-
tion? And if we must abandon this hypothesis, the building which
has been so laboriously erected must fall to the ground.

We have no longer any right to speak of the component of the ac-
celeration of A which is due to the action of B. We have no means of
distinguishing it from that which is due to the action of C or of any
other body. The rule becomes inapplicable in the measurement of
masses. What then is left of the principle of the equality of action
and reaction? If we reject the hypothesis of central forces this prin-
ciple must go too; the geometrical resultant of all the forces applied
to the different bodies of a system abstracted from all external action
will be zero. In other words, the motion of the centre of gravity of
this system will be uniform and in a straight line. Here would seem
to be a means of defining mass. The position of the centre of gravity
evidently depends on the values given to the masses; we must select

680 SCIENCE AND HYPOTHESIS

these values so that the motion of the centre of gravity is uniform and
rectilinear. This will always be possible if Newton's third law holds
good, and it will be in general possible only in one way. But no sys-
tem exists which is abstracted from all external action; every part
of the universe is subject, more or less, to the action of the other parts.
The law of the motion of the centre of gravity is only rigorously true
when applied to the whole universe.

But then, to obtain the values of the masses we must find the motion
of the centre of gravity of the universe. The absurdity of this con-
clusion is obvious ; the motion of the centre of gravity of the universe
will be forever to us unknown. Nothing, therefore, is left, and our
efforts are fruitless. There is no escape from the following definition,
which is only a confession of failure: Masses are co-efficients, which
it is found convenient to introduce into calculations.

We could reconstruct our mechanics by giving to our masses differ-
ent values. The new mechanics would be in contradiction neither with
experiment nor with the general principles of dynamics (the principle
of inertia, proportionality of masses and accelerations, equality of
action and reaction, uniform motion of the centre of gravity in a
straight line, and areas). But the equations of this mechanics would
not be so simple. Let us clearly understand this. It would be only
the first terms which would be less simple — i.e., those we already
know through experiment ; perhaps the small masses could be slightly
altered without the complete equations gaining or losing in sim-
plicity.

Hertz has inquired if the principles of mechanics are rigorously
true. " In the opinion of many physicists it seems inconceivable that
experiment will ever alter the impregnable principles of mechanics;
and yet, what is due to experiment may always be rectified by experi-
ment." From what we have just seen these fears would appear to be
groundless. The principles of dynamics appeared to us first as ex-
perimental truths, but we have been compelled to use them as defini-
tions. It is by definition that force is equal to the product of the mass
and the acceleration; this is a principle which is henceforth beyond
the reach of any future experiment. Thus it is by definition that
action and reaction are equal and opposite. But then it will be said,
these unverifiable principles are absolutely devoid of any significance.
They cannot be disproved by experiment, but we can learn from them
nothing of any use to us ; what then is the use of studying dynamics ?
This somewhat rapid condemnation would be rather unfair. There is
not in Nature any system perfectly isolated, perfectly abstracted from
all external action; but there are systems which are nearly isolated.
If we observe such a system, we can study not only the relative motion
of its different parts with respect to each other, but the motion of
its centre of gravity with respect to the other parts of the universe.

FORCE 681

We then find that the motion of its centre of gravity is nearly uni-
form and rectilinear in conformity with Newton's Third Law. This
is an experimental fact, which cannot be invalidated by a more accu-
rate experiment. What, in fact, would a more accurate experiment
teach us? It would teach us that the law is only approximately true,
and we know that already. Thus is explained how experiment may
serve as a basis for the principles of mechanics, and yet will never
invalidate them.

Anthropomorphic Mechanics. — It will be said that Kirchoff has
only followed the general tendency of mathematicians towards nom-
inalism; from this his skill as a physicist has not saved him. He
wanted a definition of a force, and he took the first that came handy;
but we do not require a definition of force ; the idea of force is primi-
tive, irreducible, indefinable; we all know what it is; of it we have
direct intuition. This direct intuition arises from the idea of effort
which is familiar to us from childhood. But in the first place, even
if this direct intuition made known to us the real nature of force in
itself, it would prove to be an insufficient basis for mechanics; it
would, moreover, be quite useless. The important thing is not to
know what force is, but how to measure it. Everything which does
not teach UP how to measure it is as useless to the mechanician as,
for instance, the subjective idea of heat and cold to the student of
heat. This subjective idea cannot be translated into numbers, and is
therefore useless: a scientist whose skin is an absolutely bad con-
ductor of heat, and who, therefore, has never felt the sensation of
heat or cold, would read a thermometer in just the same way as any
one else, and would have enough material to construct the whole of
the theory of heat.

Now this immediate notion of effort is of no use to us in the
measurement of force. It is clear, for example, that I shall experi-
ence more fatigue in lifting a weight of 100 Ib. than a man who is
accustomed to lifting heavy burdens. But there is more than this.
This notion of effort does not teach us the nature of force; it is
definitely reduced to a recollection of muscular sensations, and no
one will maintain that the sun experiences a muscular sensation when
it attracts the earth. All that we can expect to find from it is a sym-
bol, less precise and less convenient than the arrows (to denote
direction) used by geometers, and quite as remote from reality.

Anthropomorphism plays a considerable historic role in the genesis
of mechanics; perhaps it may yet furnish us with a symbol which
some minds may find convenient; but it can be the foundation of
nothing of a really scientific or philosophical character.

The Thread School. — M. Andrade, in his Legons de Mecanique
physique, has modernized anthropomorphic mechanics. To the school

682 SCIENCE AND HYPOTHESIS

of mechanics with which Kirchoff is identified, he opposes a school
which is quaintly called the " Thread School."

This school tries to reduce everything to the consideration of
certain material systems of negligible mass, regarded in a state of
tension and capable of transmitting considerable effort to distant
bodies — systems of which the ideal type is the fine string, wire, or
thread. A thread which transmits any force is slightly lengthened in
the direction of that force ; the direction of' the thread tells us the
direction of the force, and the magnitude of the force is measured by
the lengthening of the thread.

We may imagine such an experiment as the following : — A body A
is attached to a thread; at the other extremity of the thread acts a
force which is made to vary until the length of the thread is increased
by a, and the acceleration of the body A is recorded. A is then de-
tached, and a body B is attached to the same thread, and the same
or another force is made to act until the increment of length again is a,
and the acceleration of B is noted. The experiment is then renewed
with both A and B until the increment of length is (3. The four ac-
celerations observed should be proportional. Here we have an ex-
perimental verification of the law of acceleration enunciated above.
Again, we may consider a body under the action of several threads in
equal tension, and by experiment we determine the direction of those
threads when the body is in equilibrium. This is an experimental
verification of the law of the composition of forces. But, as a matter
of fact, what have we done ? We have defined the force acting on the
string by the deformation of the thread, Avhich is reasonable enough;
we have then assumed that if a body is attached to this thread, the
effort which is transmitted to it by the thread is equal to the action
exercised by the body on the thread ; in fact, we have used the principle
of action and reaction by considering it, not as an experimental
truth, but as the very definition of force. This definition is quite as
conventional as that of Kirchoff, but it is much less general.

All the forces are not transmitted by the thread (and to compare
them they would all have to be transmitted by identical threads). If
we even admitted that the earth is attached to the sun by an invisible
thread, at any rate it will be agreed that we have no means of mea-
suring the increment of the thread. Nine times out of ten, in con-
sequence, our definition will be in default; no sense of any kind can
be attached to it, and we must fall back on that of Kirchoff. Why
then go on in this roundabout way? You admit a certain definition
of force which has a meaning only in certain particular cases. In
those cases you verify by experiment that it leads to the law of accel-
eration. On the strength of these experiments you then take the law
of acceleration as a definition of force in all the other cases.

Would it not be simpler to consider the law of acceleration as a

FOKCE 683

definition in all cases, and to regard the experiments in question, not
as verifications of that law, but as verifications of the principle of
action and reaction, or as proving the deformations of an elastic body
depend only on the forces acting on that body? Without taking
into account the fact that the conditions in which your definition
could be accepted can only be very imperfectly fulfilled, that a thread
is never without mass, that it is never isolated from all other forces
than the reacticn of the bodies attached to its extremities.

The ideas expounded by M. Andrade are none the less very interest-
ing. If they do not satisfy our logical requirements, they give us a
better view of the historical genesis of the fundamental ideas of
mechanics. The reflections they suggest show us how the human
mind passed from a naive anthropomorphism to the present concep-
tion of science.

We see that we end with an experiment which is very particular,
and as a matter of fact very crude, and we start with a perfectly
general law, perfectly precise, the truth of which we regard as abso-
lute. We have, so to speak, freely conferred this certainty on it by
looking upon it as a convention.

Are the laws of acceleration and of the composition of forces only
arbitrary conventions ? Conventions, yes ; arbitrary, no — they would
be so if we lost sight of the experiments which led the founders of
the science to adopt them, and which, imperfect as they were, were
sufficient to justify their adoption. It is well from time to time to
let our attention dwell on the experimental origin of these conventions.

Relative and Absolute Motion

TJic Principle of Relative Motion. — Sometimes endeavors have
been made to connect the law of acceleration with a more general
principle. The movement of any system whatever ought to obey the
same laws, whether it is referred to fixed axes or to the movable axes
which are implied in uniform motion in a straight line. This is
the principle of relative motion; it is imposed upon us for two rea-
sons: the commonest experiment confirms it; the consideration of the
contrary hypothesis is singularly repugnant to the mind.

Let us admit it then, and consider a body under the action of a
force. The relative motion of this body with respect to an observer
moving with a uniform velocity equal to the initial velocity of the
body, should be identical with what would be its absolute motion if
it started from rest. We conclude that its acceleration must not de-
pend upon its absolute velocity, and from that we attempt to deduce
the complete law of acceleration.

For a long time there have been traces of this proof in the regula-
tion? for the degree of B.Sc. It is clear that the attempt has

684 SCIENCE AND HYPOTHESIS

failed. The obstacle which prevented us from proving the law of
acceleration is that we have no definition of force. This obstacle
subsists in its entirety, since the principle invoked has not furnished
us with the missing definition. The principle of relative motion is
none the less very interesting, and deserves to be considered for its
own sake. Let us try to enunciate it in an accurate manner. We have
said above that the accelerations of the different bodies which form
part of an isolated system only depend on their velocities and their
relative positions, and not on their velocities and their absolute posi-
tions, provided that the movable axes to which the relative motion is
referred move uniformly in a straight line ; or, if it is preferred, their
accelerations depend only on the differences of their velocities and the
differences of their co-ordinates, and not on the absolute values of
these velocities and co-ordinates. If this principle is true for relative
accelerations, or rather for differences of acceleration, by combining
it with the law of reaction we shall deduce that it is true for
absolute accelerations. It remains to be seen how we can prove
that differences of acceleration depend only on differences of
velocities and co-ordinates; or, to speak in mathematical language,
that these differences of co-ordinates satisfy differential equations of
the second order. Can this proof be deduced from experiment or
from a priori conditions? Remembering what we have said before,
the reader will give his own answer. Thus enunciated, in fact, the
principle of relative motion curiously resembles what I called above
the generalized principle of inertia; it is not quite the same thing,
since it is a question of differences of co-ordinates, and not of the co-
ordinates themselves. The new principle teaches us something more
than the old, but the same discussion applies to it, and would lead
to the same conclusions. We need not recur to it.

Newton's Argument. — Here we find a very important and even
slightly disturbing question. I have said that the principle of relative
motion was not for us simply a result of experiment ; and that a priori
every contrary hypothesis would be repugnant to the mind. But, then,
why is the principle only true if the motion of the movable axes is
uniform and in a straight line? It seems that it should be imposed
upon us with the same force if the motion is accelerated, or at any rate
if it reduces to a uniform rotation. In these two cases, in fact, the
principle is not true. I need not dwell on the case in which the mo-
tion of the axes is in a straight line and not uniform. The paradox
does not bear a moment's examination. If I am in a railway carriage,
and if the train, striking against any obstacle whatever, is suddenly
stopped, I shall be projected on to the opposite side, although I have
not been directly acted upon by any force. There is nothing mysteri-
ous in that, and if I have not been subject to the action of any exter-
nal force, the train has experienced an external impact. There can be

FORCE 685

nothing paradoxical in the relative motion of two bodies being dis-
turbed when the motion of one or the other is modified by an external
cause. NOT need I dwell on the case of relative motion referring to
axes which rotate uniformly. If the sky were forever covered with
clouds, and if we had no means of observing the stars, we might,
nevertheless, conclude that the earth turns round. We should be
warned of this fact by the flattening at the poles, or by the experiment
of Foucault's pendulum. And yet, would there in this case be any
meaning in saying that the earth turns round? If there is no abso-
lute space, can a thing turn without turning with respect to some-
thing; and, on the other hand, how can we admit Newton's conclu-
sion and believe in absolute space? But it is not sufficient to state
that all possible solutions are equally unpleasant to us. We must
analyze in each case the reason of our dislike, in order to make our
choice with the knowledge of the cause. The long discussion which
follows must, therefore, be excused.

Let us resume our imaginary story. Thick clouds hide the stars
from men who cannot observe them, and even are ignorant of their
existence. How will those men know that the earth turns round ? No
doubt, for a longer period than did our ancestors, they will regard the
soil on which they stand as fixed and immovable. They will wait a
much longer time than we did for the coming of a Copernicus; but
this Copernicus will come at last. How will he come? In the first
place, the mechanical school of this world would not run their heads
against an absolute contradiction. In the theory of relative motion we
observe, besides real forces, two imaginary forces, which we call ordi-
nary centrifugal force and compounded centrifugal force. Our imag-
inary scientists can thus explain everything by looking upon these two
forces as real, and they would not see in this a contradiction of the
generalized principle of inertia, for these forces would depend, the one
on the relative positions of the different parts of the system, such as
real attraction?, and the other on their relative velocities, as in the
case of real frictions. Many difficulties, however, would before long
awaken their attention. If they succeeded in realizing an isolated
system, the centre of gravity of this system would not have an ap-
proximately rectilinear path. They could invoke, to explain this fact,
the centrifugal forces which they would regard as real, and which,
no doubt, they would attribute to the mutual actions of the bodies —
only they would not see these forces vanish at great distances — that is
to say, in proportion as the isolation is better realized. Far from it.
Centrifugal force increases indefinitely with distance. Already this
difficulty would seem to them sufficiently serious, but it would not
detain them for long. They would soon imagine some very subtle
medium analogous to our ether, in which all bodies would be bathed,
and which would exercise on them a repulsive action. But that is not

686

all. Space is symmetrical — yet the laws of motion would present
no symmetry. They should be able to distinguish between right and
left. They would see, for instance, that cyclones always turn in the
same direction, while for reasons of symmetry they should turn in-
differently in any direction. If our scientists were able by dint of
much hard work to make their universe perfectly symmetrical, this
symmetry would not subsist, although there is no apparent reason
why it should be disturbed in one direction more than in another.
They would extract this from the situation no doubt — they would
invent something which would not be more extraordinary than the
glass spheres of Ptolemy, and would thus go on accumulating compli-
cations until the long-expected Copernicus would sweep them all
away with a single blow, saying it is much more simple to admit that
the earth turns round. Just as our Copernicus said to us : " It is
more convenient to suppose that the earth turns round, because the
laws of astronomy are thus expressed in a more simple language," so
he would say to them : " It is more convenient to suppose that the
earth turns round, because the laws of mechanics are thus expressed in
much more simple language." That does not prevent absolute space —
that is to say, the point to which we must refer the earth to know
if it really does turn round — from having no objective existence.
And hence this affirmation : " the earth turns round," has no mean-
ing, since it cannot be verified by experiment; since such an experi-
ment not only cannot be realized or even dreamed of by the most
daring Jules Verne, but cannot even be conceived of without contra-
diction ; or, in other words, these two propositions, " the earth turns
round," and " it is more convenient to suppose that the earth turns
round," have one and the same meaning. There is nothing more in
one than in the other. Perhaps they will not be content with this,
and may find it surprising that among all the hypotheses, or rather
all the conventions, that can be made on this subject there is one
which is more convenient than the rest. But if we have admitted it
without difficulty when it is a question of the laws of astronomy, why
should we object when it is a question of the laws of mechanics? We
have seen that the co-ordinates of bodies are determined by differ-
ential equations of the second order, and that so are the differences
of these co-ordinates. This is what we have called the generalized
principle of inertia, and the principle of relative motion. If the dis-
tances of these bodies were determined in the same way by equations
of the second order, it seems that the mind should be entirely satisfied.
How far does the mind receive this satisfaction, and why is it not
content with it ? To explain this we had better take a simple example.
I assume a system analogous to our solar system, but in which fixed
stars foreign to this system cannot be perceived, so that astronomers
can only observe the mutual distances of planets and the sun, and not

FORCE 687

the absolute longitudes of the planets. If we deduce directly from
Newton's law the differential equations which define the variation of
these distances, these equations will not be of the second order. I mean
that if, outside Newton's law, we knew the initial values of these dis-
tances and of their derivatives with respect to time — that would
not be sufficient to determine the values of these same distances at an
ulterior moment. A datum would be still lacking, and this datum
might be, for example, what astronomers call the area-constant. But
here we may look at it from two different points of view. We may
consider two kinds of constants. In the eyes of the physicist the
world reduces to a series of phenomena depending, on the one hand,
solely on initial phenomena, and, on the other hand, on the laws con-
necting consequence and antecedent. If observation then teaches us
that a certain quantity is a constant, we shall have a choice of two
ways of looking at it. So let us admit that there is a law which re-
quires that this quantity shall not vary, but that by chance it has
been found to have had in the beginning of time this value rather
than that, a value that it has kept ever since. This quantity might
then be called an accidental constant. Or again, let us admit on the
contrary that there is a law of nature which imposes on this quantity
this value and not that. We shall then have what may be called an
essential constant. For example, in virtue of the laws of Newton the
duration of the revolution of the earth must be constant. But if it is
366 and something sidereal days, and not 300 or 400, it is because of
some initial chance or other. It is an accidental constant. If, on the
other hand, the exponent of the distance which figures in the expres-
sion of the attractive force is eqiial to —2 and not to —3, it is not by
chance, but because it is required by Newton's law. It is an essential
constant. I do not know if this manner of giving to chance its share
is legitimate in itself, and if there is not some artificiality about this
distinction; but it is certain at least that in proportion as Nature
has secrets, she will be strictly arbitrary and always uncertain in their
application. As far as the area-constant is concerned, we are accus-
tomed to look upon it as accidental. Is it certain that our imaginary
astronomers would do the same? If they were able to compare two
different solar systems, they would get the idea that this constant
may assume several different values. But I supposed at the outset, as
I was entitled to do, that their system would appear isolated, and that
they would see no star which was foreign to their system. Under these
conditions they could only detect a single constant, which would have
an absolutely invariable, unique value. They would be led no doubt
to look upon it as an essential constant.

One word in passing to forestall an objection. The inhabitants
of this imaginary world could neither observe nor define the area-
constant as we do, because absolute longitudes escape their notice;

688 SCIENCE AND HYPOTHESIS

but that would not prevent them from being rapidly led to remark
a certain constant which would be naturally introduced into their
equations, and which would be nothing but what we call the area-
constant. But then what would happen? If the area-constant is
regarded as essential, as dependent upon a law of nature, then in
order to calculate the distances of the planets at any given moment it
would be sufficient to know the initial values of these distances and
those of their first derivatives. From this new point of view, dis-
tances will be determined by differential equations of the second order.
Woiild this completely satisfy the minds of these astronomers? I
think not. In the first place, they would very soon see that in differ-
entiating their equations so as to raise them to a higher order, these
equations would become much more simple, and they would be espe-
cially struck by the difficulty which arises from symmetry. They
would have to admit different laws, according as the aggregate of the
planets presented the figure of a certain polyhedron or rather of a
regular polyhedron, and these consequences can only be escaped by
regarding the area-constant as accidental. I have taken this particular
example, because I have imagined astronomers who would not be in
the least concerned with terrestrial mechanics and whose vision would
be bounded by the solar system. But our conclusions apply in all
cases. Our universe is more extended than theirs, since we have fixed
stars ; but it, too, is very limited, so we might reason on the whole of
our universe just as these astronomers do on their solar system.
We thus see that we should be definitively led to conclude
that the equations which define distances are of an order
higher than the second. Why should this alarm us — why do we
find it perfectly natural that the sequence of phenomena depends on
initial values of the first derivatives of these distances, while we
hesitate to admit that they may depend on the initial values of the
second derivatives? It can only be because of mental habits created
in us by the constant study of the generalized principle of inertia
and of its consequences. The values of the distances at any given
moment depend upon their initial values, on that of their first deriva-
tives, and something else. What is that something else? If we do
not want it to be merely one of the second derivatives, we have only
the choice of hypotheses. Suppose, as is usually done, that this some-
thing else is the absolute orientation of the universe in space, or the
rapidity with which this orientation varies ; this may be, it certainly is,
the most convenient solution for the geometer. But it is not the most
satisfactory for the philosopher, because this orientation does not exist.
We may assume that this something else is the position or the velocity
of some invisible body, and this is what is done by certain persons,
who have even called the body Alpha, although we are destined to never
know anything about this body except its name. This is an artifice en-

FOECE 689

tirely analogous to that of which I spoke at the end of the paragraph
containing my reflections on the principle of inertia. But as a matter
of fact the difficulty is artificial. Provided that the future indica-
tions of our instruments can only depend on the indications which
they have given us, or that they might have formerly given us, such is
all we want, and with these conditions we may rest satisfied.

Energy and Thermo-Dynamics

Energetics. — The difficulties raised by the classical mechanics have
led certain minds to prefer a new system which they call Energetics.
Energetics took its rise in consequence of the discovery of the prin-
ciple of the conservation of energy. Helmholtz gave it its definite
form. We begin by defining two quantities which play a fundamental
part in this theory. They are kinetic energy, or vis viva, and potential
energy. Every change that the bodies of nature can undergo is regu-
lated by two experimental laws. First, the sum of the kinetic and
potential energies is constant. This is the principle of the conserva-
tion of energy. Second, if a system of bodies is at A at the time t0, and
at B at the time ti} it always passes from the first position to the
second by such a path that the mean value of the difference between
the two kinds of energy in the interval of time which separates the
two epochs, t0 and £t is a minimum. This is Hamilton's principle, and
is one of the forms of the principle of least action. The energetic
theory has the following advantages over the classical. First, it is
less incomplete — that is to say, the principles of the conservation of
energy and of Hamilton teach us more than the fundamental princi-
ples of the classical theory, and exclude certain motions which do not
occur in nature and which would be compatible with the classical
theory. Second, it frees us from the hypothesis of atoms, which it
was almost impossible to avoid with the classical theory. But in its
turn it raises fresh difficulties. The definitions of the two kinds of
energy would raise difficulties almost as great as those of force and
mass in the first system. However, we can get out of these difficulties
more easily, at any rate in the simplest cases. Assume an isolated
system formed of a certain number of material points. Assume that
these points are acted upon by forces depending only on their relative
position and their distances apart, and independent of their velocities.
In virtue of the principle of the conservation of energy there must
be a function of forces. In this simple case the enunciation of the
principle of the conservation of energy is of extreme simplicity. A
certain quantity, which may be determined by experiment, must re-
main constant. This quantity is the sum of two terms. The first
depends only on the position of the material points, and is inde-
pendent of their velocities; the second is proportional to the squares

690 SCIENCE AND HYPOTHESIS

of these velocities. This decomposition can only take place in one
way. The first of these terms, which I shall call U, will be potential
energy ; the second, which I shall call T, will be kinetic energy. It is
true that if T+U is constant, so is any function of T+U, <f> (T+U).
But this function <j> (T+U) will not be the sum of two terms, the
one independent of the velocities, and the other proportional to the
square of the velocities. Among the functions which remain constant
there is only one which enjoys this property. It is T+U (or a linear
function of T+U), it matters not which, since this linear function
may always be reduced to T+U by a change of unit and of origin.
This, then, is what we call energy. The first term we shall call poten-
tial energy, and the second kinetic energy. The definition of the two
kinds of energy may therefore be carried through without any am-
biguity.

So it is with the definition of mass. Kinetic energy, or vis viva,
is expressed very simply by the aid of the masses, and of the relative
velocities of all the material points with reference to one of them.
These relative velocities may be observed, and when we have the ex-
pression of the kinetic energy as a function of these relative veloci-
ties, the co-efficients of this expression will give us the masses. So in
this simple case the fundamental ideas can be defined without diffi-
culty. But the difficulties reappear in the more complicated cases if
the forces, instead of depending solely on the distances, depend also
on the velocities. For example, Weber supposes the mutual action of
two electric molecules to depend not only on their distance but on their
velocity and on their acceleration. If material points attracted each
other according to an analogous law, U would depend on the velocity,
and it might contain a term proportional to the square of the velocity.
How can we detect among such terms those that arise from T or U?
and how, therefore, can we distinguish the two parts of the energy?
But there is more than this. How can we define energy itself? "We
have no reason to take as our definition T + U rather than any
other function of T+U, when the property which characterized T+U
has disappeared — namely, that of being the sum of two terms of a
particular form. But that is not all. We must take account, not only
of mechanical energy properly so called, but of the other forms of
energy — heat, chemical energy, electrical energy, etc. The principle
of the conservation of energy must be written T+U+Q— a con-
stant, where T is the sensible kinetic energy, U the potential energy
of position, depending only on the position of the bodies, Q the internal
molecular energy under the thermal, chemical, or electrical form.
This would be all right if the three terms were absolutely distinct;
if T were proportional to the square of the velocities, U independent
of these velocities and of the state of the bodies, Q independent of the
velocities and of the positions of the bodies, and depending only on

FORCE 691

their internal state. The expression for the energy could be decom-
posed in one way only into three terms of this form. But this is not
the case. Let us consider electrified bodies. The electro-static en-
ergy due to their mutual action will evidently depend on their charge
— i. e., on their state; but it will equally depend on their position.
If these bodies are in motion, they will act electro-dynamically on one
another, and the electro-dynamic energy will depend not only on their
state and their position, but on their velocities. We have therefore
no means of making the selection of the terms which should form part
of T, and U, and Q, and of separating the three parts of the energy.
If T+U+Q is constant, the same is true of any function whatever, <£
(T+U+Q).

If T+U+Q were of the particular form that I have suggested above,
no ambiguity would ensue. Among the functions $ (T+U+Q) which
remain constant, there is only one that would be of this particular
form, namely the one which I would agree to call energy. But I
have said this is not rigorously the case. Among the functions that
remain constant there is not one which can rigorously be placed in
this particular form. How then can we choose from among them
that which should be called energy? We have no longer any guide
in our choice.

Of the principle of the conservation of energy there is nothing left
then but an enunciation : — There is something which remains con-
stant. In this form it, in its turn, is outside the bounds of experi-
ment and reduced to a kind of tautology. It is clear that if the world
is governed by laws there will be quantities which remain constant.
Like Newton's laws, and for an analogous reason, the principle of the
conservation of energy being based on experiment, can no longer be
invalidated by it.

This discussion shows that, in passing from the classical system to
the energetic, an advance has been made, but it shows, at the same
time, that we have not advanced far enough.

Another objection seems to be still more serious. The principle of
least action is applicable to reversible phenomena, but it is by no
means satisfactory as far as irreversible phenomena are concerned.
Helmholtz attempted to extend it to this class of phenomena, but he
did not and could not succeed. So far as this is concerned all has yet
to be done. The very enunciation of the principle of least action is
objectionable. To move from one point to another, a material mole-
cule, acted upon' by no force, but compelled to move on a surface, will
take as its path the geodesic line — i.e., the shortest path. This mole-
cule seems to know the point to which we want to take it, to foresee
the time that it will take it to reach it by such a path, and then to
know how to choose the most convenient path. The enunciation of
the principle presents it to us, so to speak, as a living and free entity.

692 SCIENCE AND HYPOTHESIS

It is clear that it would be better to replace it by a less objectionable
enunciation, one in which, as philosophers would say, final effects do
not seem to be substituted for acting causes.

Thermo-dynamics. — The role of the two fundamental principles of
thermo-dynamics becomes daily more important in all branches of
natural philosophy. Abandoning the ambitious theories of forty years
ago, encumbered as they were with molecular hypotheses, we now try
to rest on thermo-dynamics alone the entire edifice of mathematical
physics. Will the two principles of Mayer and of Clausius assure to it
foundations solid enough to last for some time? We all feel it, but
whence does our confidence arise? An eminent physicist said to me
one day, apropos of the law of errors : — every one stoutly believes it,
because mathematicians imagine that it is an effect of observation, and
observers imagine that it is a mathematical theorem. And this was
for a long time the case with the principle of the conservation of
energy. It is no longer the same now. There is no one who does not
know that it is an experimental fact. But then who gives us the
right of attributing to the principle itself more generality and more
precision than to the experiments which have served to demonstrate it ?
This is asking, if it is legitimate to generalize, as we do every day,
empiric data, and I shall not be so foolhardy as to discuss this ques-
tion, after so many philosophers have vainly tried to solve it. One
thing alone is certain. If this permission were refused to us, science
could not exist; or at least would be reduced to a kind of inventory, to
the ascertaining of isolated facts. It would no longer be to us of
any value, since it could not satisfy our need of order and harmony,
and because it would be at the same time incapable of prediction. As
the circumstances which have preceded any fact whatever will never
again, in all probability, be simultaneously reproduced, we already
require a first generalization to predict whether the fact will be
renewed as soon as the least of these circumstances is changed. But
every proposition may be generalized in an infinite number of ways.
Among all possible generalizations we must choose, and we cannot but
choose the simplest. We are therefore led to adopt the same course
as if a simple law were, other things being equal, more probable than
a complex law. A century ago it was frankly confessed and pro-
claimed abroad that Nature loves simplicity; but Nature has proved
the contrary since then on more than one occasion. We no longer con-
fess this tendency, and we only keep of it what is indispensable, so
that science may not become impossible. In formulating a general,
simple, and formal law, based on a comparatively small number of
not altogether consistent experiments, we have only obeyed a neces-
sity from which the human mind cannot free itself. But there is
something more, and that is why I dwell on this topic. No one doubts
that Mayer's principle is not called upon to survive all the particular

FORCE 693

laws from which it was deduced, in the same way that Newton's law
has survived the laws of Kepler from which it was derived, and
which are no longer anything but approximations, if we take pertur-
bations into account. Now why does this principle thus occupy a kind
of privileged position among physical laws? There are many reasons
for that. At the outset we think that we cannot reject it, or even
doubt, its absolute rigor, without admitting the possibility of perpetual
motion; we certainly feel distrust at such a prospect, and we believe
ourselves less rash in affirming it than in denying it. That perhaps
is not quite accurate. The impossibility of perpetual motion only
implies the conservation of energy for reversible phenomena. The
imposing simplicity of Mayer's principle equally contributes to
strengthen our faith. In a law immediately deduced from experi-
ments, such as Mariotte's law, this simplicity would rather appear to
us a reason for distrust ; but here this is no longer the case. We take
elements which at the first glance are unconnected; these arrange
themselves in an unexpected order, and form a harmonious whole.
We cannot believe that this unexpected harmony is a mere result of
chance. Our conquest appears to be valuable to us in proportion to the
efforts it has cost, and we feel the more certain of having snatched its
true secret from Nature in proportion as Nature has appeared more
jealous of our attempts to discover it. But these are only small rea-
sons. Before we raise Mayer's law to the dignity of an absolute prin-
ciple, a deeper discussion is necessary. But if we embark on this dis-
cussion we see that this absolute principle is not even easy to enunciate.
In every particular case we clearly see what energy is, and we can give
it at least a provisory definition ; but it is impossible to find a general
definition of it. If we wish to enunciate the principle in all its gen-
eiality and apply it to the universe, we see it vanish, so to speak, and
nothing is left but this — there is something which remains constant.
But has this a meaning? In the determinist hypothesis the state of
the universe is determined by an extremely large number n of par-
ameters, which I shall call a*j, a'2, x3 . . . xn. As soon as we know
at a given moment the values of these n parameters, we also know
their derivatives with respect to time, and we can therefore calculate
the rallies of these same parameters at an anterior or ulterior moment.
Tn other words, these n parameters specify n differential equations of
the first order. These equations have n — 1 integrals, and therefore
there are n — 1 functions of .r,, .r2, #:i, . . . xn, which remain con-
stant. If we say then, there is something which remains constant, we
are only enunciating a tautology. We would be even embarrassed to
decide which among all our integrals is that which should retain the
name of energy. Besides, it is not in this sense that Mayer's principle
is understood when it is applied to a limited system. We admit, then,
that p of our n parameters vary independently so that we have only

694 SCIENCE AND HYPOTHESIS

n — p relations, generally linear, between our n parameters and their
derivatives. Suppose, for the sake of simplicity, that the sum of the
work done by the external forces is zero, as well as that of all the
quantities of heat given off from the interior: what will then be the
meaning of our principle ? There is a combination of these n — p rela-
tions, of which the first member is an exact differential; and then this
differential vanishing in virtue of our n— p relations, its integral is a
constant, and it is this integral which we call energy. But how can it
be that there are several parameters whose variations are independent ?
That can only take place in the case of external forces (although we
have supposed, for the sake of simplicity, that the algebraical sum
of all the work done by these forces has vanished). If, in fact, the
system were completely isolated from all external action, the values of
our n parameters at a given moment would suffice to determine the
state of the system at any ulterior moment whatever, provided that we
still clung to the determinist hypothesis. We should therefore fall
back on the same difficulty as before. If the future state of the system
is not entirely determined by its present state, it is because it further
depends on the state of bodies external to the system. But then, is it
likely that there exist among the parameters x which define the state
of the system of equations independent of this state of the external
bodies? and if in certain cases we think we can find them, is it not
only because of our ignorance, and because the influence of these
bodies is too weak for our experiment to be able to detect it? If the
system is not regarded as completely isolated, it is probable that the
rigorously exact expression of its internal energy will depend upon
the state of the external bodies. Again, I have supposed above that
the sum of all the external work is zero, and if we wish to be free
from this rather artificial restriction the enunciation becomes still
more difficult. To formulate Mayer's principle by giving it an abso-
lute meaning, we must extend it to the whole universe, and then we
find ourselves face to face with the very difficulty we have endeavored
to avoid. To sum up, and to use ordinary language, the law of the
conservation of energy can have only one significance, because there is
in it a property common to all possible properties; but in the deter-
minist hypothesis there is only one possible, and then the law has no
meaning. In the indeterminist hypothesis, on the other hand, it
would have a meaning even if we wished to regard it in an absolute
sense. It would appear as a limitation imposed on freedom.

But this word warns me that I am wandering from the subject, and
that I am leaving the domain of mathematics and physics. I check
myself, therefore, and I wish to retain only one impression of the
whole of this discussion, and that is, tnat Mayer's law is a form subtle
enough for us to be able to put into it almost anything we like. I do
not mean by that that it corresponds to no objective reality, nor that it

FOECE 695

is reduced to mere tautology; since, in each particular case, and pro-
vided we do not wish to extend it to the absolute, it has a perfectly
clear meaning. This subtlety is a reason for believing that it will last
long; and as, on the other hand, it will only disappear to be blended
in a higher harmony, we may work with confidence and utilize it,
certain beforehand that our work will not be lost.

Almost everything that I have just said applies to the principle of
Clausius. What distinguishes it is, that it is expressed by an in-
equality. It will be said perhaps that it is the same with all physical
laws, since their precision is always limited by errors of observation.
But they at least claim to be first approximations, and we hope to
replace them little by little by more exact laws. If, on the other hand,
the principle of Clausius reduces to an inequality, this is not caused
by the imperfection of our means of observation, but by the very nature
of the question.

General Conclusions on Part III. — The principles of mechanics are
therefore presented to us under two different aspects. On the one
hand, there are truths founded on experiment, and verified approxi-
mately as far as almost isolated systems are concerned; on the other
hand, there are postulates applicable to the whole of the universe and
regarded as rigorously true. If these postulates possess a generality
and a certainty which falsify the experimental truths from which they
were deduced, it is because they reduce in final analysis to a simple
convention that we have a right to make, because we are certain before-
hand that no experiment can contradict it. This convention, however,
is not absolutely arbitrary; it is not the child of our caprice. We
admit it because certain experiments have shown us that it will be
convenient, and thus is explained how experiment has built up the
principles of mechanics, and why, moreover, it cannot reverse them.
Take a comparison with geometry. The fundamental propositions of
geometry, for instance, Euclid's postulate, are only conventions, and it
is quite as unreasonable to ask if they are true or false as to ask if the
metric system is true or false. Only, these conventions are convenient,
and there are certain experiments which prove it to us. At the first
glance, the analogy is complete, the role of experiment seems the same.
We shall therefore be tempted to say, either mechanics must be looked
upon as experimental science and then it should be the same with
geometry; or, on the contrary, geometry is a deductive science, and
then we can say the same of mechanics. Such a conclusion would
be illegitimate. The experiments which have led us to adopt as more
convenient the fundamental conventions of geometry refer to bodies
which have nothing in common with those that are studied by geo-
metry. They refer to the properties of solid bodies and to the propa-
gation of light in a straight line. These are mechanical, optical expe-
riments. In no way can they be regarded' as geometrical experiments.

696 SCIENCE AND HYPOTHESIS

And even the probable reason why our geometry seems convenient to
us is, that our bodies, our hands, and our limbs enjoy the properties
of solid bodies. Our fundamental experiments are pre-eminently phy-
siological experiments which refer, not to the space which is the object
that geometry must study, but to our body — that is to say, to the
instrument which we use for that study. On the other hand, the fun-
damental conventions of mechanics and the experiments which prove
to us that they are convenient, certainly refer to the same objects or
to analogous objects. Conventional and general principles are the
natural and direct generalizations of experimental and particular prin-
ciples. Let it not be said that I am thus tracing artificial frontiers
between the sciences; that I am separating by a barrier geometry
properly so called from the study of solid bodies. I might just as well
raise a barrier between experimental mechanics and the conventional
mechanics of general principles. Who does not see, in fact, that by
separating these two sciences we mutilate both, and that what will
remain of the conventional mechanics when it is isolated will be but
very little, and can in no way be compared with that grand body of
doctrine which is called geometry.

We now understand why the teaching of mechanics should remain
experimental. Thus only can we be made to understand the genesis
of the science, and that is indispensable for a complete knowledge of
the science itself. Besides, if we study mechanics, it is in order to
apply it ; and we can only apply it if it remains objective. Now, as we
have seen, when principles gain in generality and certainty they lose
in objectivity. It is therefore especially with the objective side of
principles that we must be early familiarized, and this can only be
by passing from the particular to the general, instead of from the
general to the particular.

Principles are conventions and definitions in disguise. They are,
however, deduced from experimental laws, and these laws have, so to
speak, been erected into principles to which our mind attributes an
absolute value. Some philosophers have generalized far too much.
They have thought that the principles were the whole of science, and
therefore that the whole of science was conventional. This paradoxi-
cal doctrine, which is called Nominalism, cannot stand examination.
How can a law become a principle? It expressed a relation between
two real terms, A and B; but it was not rigorously true, it was only
approximate. We introduce arbitrarily an intermediate term, C, more
or less imaginary, and C is by definition that which has with A exactly
the relation expressed by the law. So our law is decomposed into an
absolute and rigorous principle which expresses the relation of A to C,
and an approximate experimental and revisable law which expresses
the relation of C to B. But it is clear that however far this decom-
position may be carried, laws will always remain. We shall now enter
into the domain of laws properly so called.

PART IV. — NATURE
Hypotheses in Physics

The Role of Experiment and Generalization. — Experiment is the
sole source of truth. It alone can teach us something new; it alone
can give us certainty. These are two points that cannot be ques-
tioned. But then, if experiment is everything, what place is left for
mathematical physics? What can experimental physics do with such
an auxiliary — an auxiliary, moreover, which seems useless, and even
may be dangerous?

However, mathematical physics exists. It has rendered undeniable
service, and that is a fact which has to be explained. It is not suffi-
cient merely to observe; we must use our observations, and for that
purpose we must generalize. This is what has always been done, only
as the recollection of past errors has made man more and more circum-
spect, he has observed more and more and generalized less and less.
Every age has scoffed at its predecessor, accusing it of having gener-
alizet too boldly and too naively. Descartes used to commiserate the
lonians. Descartes in his turn makes us smile, and no doubt some
day our children will laugh at us. Is there no way of getting at once
to the gist of the matter, and thereby escaping the raillery which we
foresee? Cannot we be content with experiment alone? No, that is
impossible; that would be a complete misunderstanding of the true
character of science. The man of science must work with method.
Science is built up of facts, as a house is built of stones ; but an accu-
mulation of facts is no more a science than a heap of stones is a
house. Most important of all, the man of science must exhibit fore-
sight. Carlyle has written somewhere something after this fashion.
" Nothing but facts are of importance. John Lackland passed by
here. Here is something that is admirable. Here is a reality for
which I would give all the theories in the world." 1 Carlyle was a
compatriot of Bacon, and, like him, he wished to proclaim his worship
of the God of Things as they are.

But Bacon would not have said that. That is the language of the
historian. The physicist would most likely have said : " John Lack-

i V. Past and Present, end of Chapter I., Book II. — [TR.]

698

land passed by here. It is all the same to me, for he will not pass
this way again."

We all know that there are good and bad experiments. The latter
accumulate in vain. Whether there are a hundred or a thousand,
one single piece of work by a real master — by a Pasteur, for example
— will be sufficient to sweep them into oblivion. Bacon would have
thoroughly understood that, for he invented the phrase experimentum
cruets; but Carlyle would not have understood it. A fact is a fact.
A student has read such and such a number on his thermometer. He
has taken no precautions. It does not matter; he has read it, and if
it is only the fact which counts, this is a reality that is as much
entitled to be called a reality as the peregrinations of King John
Lackland. What, then, is a good experiment? It is that which
teaches us something more than an isolated fact. It is that which
enables us to predict, and to generalize. Without generalization, pre-
diction is impossible. The circumstances under which one has op-
erated will never again be reproduced simultaneously. The fact ob-
served will never be repeated. All that can be affirmed is that under
analogous circumstances an analogous fact will be produced. To pre-
dict it, we must therefore invoke the aid of analogy — that is to say,
even at this stage, we must generalize. However timid we may be,
there must be interpolation. Experiment only gives us a certain num-
ber of isolated points. They must be connected by a continuous line,
and this is a true generalization. But more is done. The curve thus
traced will pass between and near the points observed; it will not
pass through the points themselves. Thus we are not restricted to
generalizing our experiment, we correct it; and the physicist who
would abstain from these corrections, and really content himself with
experiment pure and simple, would be compelled to enunciate very
extraordinary laws indeed. Detached facts cannot therefore satisfy us,
and that is why our science must be ordered, or, better still, general-
ized.

It is often said that experiments should be made without precon-
ceived ideas. That is impossible. Not only would it make every
experiment fruitless, but even if we wished to do so, it could not be
done. Every man has his own conception of the world, and this he
cannot so easily lay aside. We must, for example, use language, and
our language is necessarily steeped in preconceived ideas. Only they
are unconscious preconceived ideas, which are a thousand times the
most dangerous of all. Shall we say, that if we cause others to inter-
vene of which we are fully conscious, that we shall only aggravate the
evil? I do not think so. I am inclined to think that they will serve
as ample counterpoises — I was almost going to say antidotes. They
will generally disagree, they will enter into conflict one with another,
and ipso facto, they will force us to look at things under different

NATURE 699

aspects. This is enough to free us. He is no longer a slave who can
choose his master.

Thus, by generalization, every fact observed enables us to predict
a large number of others; only we ought not to forget that the first
alone is certain, and that all the others are merely probable. How-
ever solidly founded a prediction may appear to us, we are never
absolutely sure that experiment will not prove it to be baseless if we
set to work to verify it. But the probability of its accuracy is often
so great that practically we may be content with it. It is far better
to predict without certainty, than never to have predicted at all. We
should never, therefore, disdain to verify when the opportunity pre-
sents itself. But every experiment is long and difficult, and the la-
borers are few, and the number of facts which we require to predict
is enormous; and besides this mass, the number of direct verifications
that we can make will never be more than a negligible quantity. Of
this little that we can directly attain we must choose the best. Every
experiment must enable us to make a maximum number of predic-
tions having the highest possible degree of probability. The problem
is, so to speak, to increase the output of the scientific machine. I may
be permitted to compare science to a library which must go on in-
creasing indefinitely ; the librarian has limited funds for his purchases,
and he must, therefore, strain every nerve not to waste them. Exper-
imental physics has to make the purchases, and experimental physics
alone can enrich the library. As for mathematical physics, her duty
is to draw up the catalogue. If the catalogue is well done the library
is none the richer for it; but the reader will be enabled to utilize its
riches ; and also by showing the librarian the gaps in his collection, it
will help him to make a judicious use of his funds, which is all the
more important, inasmuch as those funds are entirely inadequate.
That is the role of mathematical physics. It must direct generaliza-
tion, so as to increase what I called just now the output of science.
By what means it does this, and how it may do it without danger, is
what we have now to examine.

The Unity of Nature. — Let us first of all observe that every gen-
eralization supposes in a certain measure a belief in the unity and
simplicity of Nature. As far as the unity is concerned, there can be
no difficulty. If the different parts of the universe were not as the
organs of the same body, they would not re-act one upon the other;
they would mutually ignore each other, and we in particular should
only know one part. We need not, therefore, ask if Nature is one,
but how she is one.

As for the second point, that is not so clear. It is not certain that
Nature is simple. Can we without danger act as if she were?

There was a time when the simplicity of Mariotte's law was an argu-
ment in favor of its accuracy : when Fresnel himself, after having

700 SCIENCE AND HYPOTHESIS

said in a conversation with Laplace that Nature cares naught for
analytical difficulties, was compelled to explain his words so as not to
give offence to current opinion. Nowadays, ideas have changed con-
siderably; but those who do not believe that natural laws must be
simple, are still often obliged to act as if they did believe it. They
cannot entirely dispense with this necessity without making all gen-
eralization, and therefore all science, impossible. It is clear that
any fact can be generalized in an infinite number of ways, and it is a
question of choice. The choice can only be guided by considerations of
simplicity. Let us take the most ordinary case, that of interpolation.
We draw a continuous line as regularly as possible between the points
given by observation. Why do we avoid angular points and inflections
that are too sharp? Why do we not make our curve describe the
most capricious zigzags? It is because we know beforehand, or think
we know, that the law we have to express cannot be so complicated as
all that. The mass of Jupiter may be deduced either from the move-
ments of his satellites, or from the perturbations of the major planets,
or from those of the minor planets. If we take the mean of the deter-
minations obtained by these three methods, we find three numbers
very close together, but not quite identical. This result might be in-
terpreted by supposing that the gravitation constant is not the same
in the three cases; the observations would be certainly much better
represented. Why do we reject this interpretation? Not because it
is absurd, but because it is uselessly complicated. We shall only
accept it when we are forced to, and it is not imposed upon us yet.
To sum up, in most cases every law is held to be simple until the
contrary is proved.

This custom is imposed upon physicists by the reasons that I have
indicated, but how can it be justified in the presence of discoveries
which daily show us fresh details, richer and more complex? How
can we even reconcile it with the unity of nature? For if all things
are interdependent, the relations in which so many different objects
intervene can no longer be simple.

If we study the history of science we see produced two phenomena
which are, so to speak, each the inverse of the other. Sometimes it is
simplicity which is hidden under what is apparently complex; some-
times, on the contrary, it is simplicity which is apparent, and which
conceals extremely complex realities. What is there more complicated
than the disturbed motions of the planets, and what more simple than
Newton's law ? There, as Fresnel said, Nature playing with analytical
difficulties, only uses simple means, and creates by their combination
I know not what tangled skein. Here it is the hidden simplicity which
must be disentangled. Examples to the contrary abound. In the
kinetic theory of gases, molecules of tremendous velocity are dis-
cussed, whose paths, deformed by incessant impacts, have the most

NATUKE 701

capricious shapes, and plough their way through space in every direc-
tion. The result observable is Mariotte's simple law. Each individual
fact was complicated. The law of great numbers has re-established
simplicity in the mean. Here the simplicity is only apparent, and the
coarseness of our senses alone prevents us from seeing the complexity.

Many phenomena obey a law of proportionality. But why? Be-
cause in these phenomena there is something which is very small. The
simple law observed is only the translation of the general analytical
rule by which the infinitely small increment of a function is pro-
portional to the increment of the variable. As in reality our incre-
ments are not infinitely small, but only very small, the law of pro-
portionality is only approximate, and simplicity is only apparent.
What I have just said applies to the law of the superposition of small
movements, which is so fruitful in its applications and which is the
foundation of optics.

And Newton's law itself ? Its simplicity, so long undetected, is per-
haps only apparent. Who knows if it be not due to some complicated
mechanism, to the impact of some subtle matter animated by irregu-
lar movements, and if it has not become simple merely through the
play of averages and large numbers ? In any case, it is difficult not to
suppose that the true law contains complementary terms which may
become sensible at small distances. If in astronomy they are negligi-
ble, and if the law thus regains its simplicity, it is solely on account
of the enormous distances of the celestial bodies. No doubt, if our
means of investigation became more and more penetrating, we should
discover the simple beneath the complex, and then the com-
plex from the simple, and then again the simple beneath the
complex, and so on, without ever being able to predict what the last
term will be. We must stop somewhere, and for science to be possible
we must stop where we have found simplicity. That is the only
ground on which we can erect the edifice of our generalizations. But,
this simplicity being only apparent, will the ground be solid enough?
That is what we have now to discover.

For this purpose let us see what part is played in our generalizations
by the belief in simplicity. We have verified a simple law in a con-
siderable number of particular cases. We refuse to admit that this
coincidence, so often repeated, is a result of mere chance, and we
conclude that the law must be true in the general case.

Kepler remarks that the positions of a planet observed by Tycho
are all on the same ellipse. Not for one moment does he think that,
by a singular freak of chance, Tycho had never looked at the heavens
except at the very moment when the path of the planet happened to
cut that ellipse. What does it matter then if the simplicity be real
or if it hide a complex truth? Whether it be due to the influence of
great numbers which reduces individual differences to a level, or to the

702 SCIENCE AND HYPOTHESIS

greatness or the smallness of certain quantities which allow of certain
terms to be neglected — in no case is it due to chance. This simplicity,
real or apparent, has always a cause. We shall therefore always be
able to reason in the same fashion, and if a simple law has been ob-
served in several particular cases, we may legitimately suppose that it
still will be true in analogous cases. To refuse to admit this would
be to attribute an inadmissible role to chance. However, there is a
difference. If the simplicity were real and profound it would bear
the test of the increasing precision of our methods of measurement.
If, then, we believe Nature to be profoundly simple, we must conclude
that it is an approximate and not a rigorous simplicity. This is what
was formerly done, but it is what we have no longer the right to do.
The simplicity of Kepler's laws, for instance, is only apparent; but
that does not prevent them from being applied to almost all systems
analogous to the solar system, though that prevents them from being
rigorously exact.

Role of Hypothesis. — Every generalization is a hypothesis. Hy-
pothesis therefore plays a necessary role, which no one has ever con-
tested. Only, it should always be as soon as possible submitted to
verification. It goes without saying that, if it cannot stand this test,
it must be abandoned without any hesitation. This is, indeed, what
is generally done ; but sometimes with a certain impatience. Ah well !
this impatience is not justified. The physicist who has just given
up one of his hypotheses should, on the contrary, rejoice, for he found
an unexpected opportunity of discovery. His hypothesis, I imagine,
had not been lightly adopted. It took into account all the known
factors which seem capable of intervention in the phenomenon. If
it is not verified it is because there is something unexpected and
extraordinary about it, because we are on the point of finding some-
thing unknown and new. Has the hypothesis thus rejected been
sterile? Far from it. It may be even said that it has rendered more
service than a true hypothesis. Not only has it been the occasion of
a decisive experiment, but if this experiment had been made by chance,
without the hypothesis, no conclusion could have been drawn ; nothing
extraordinary would have been seen ; and only one fact the more would
have been catalogued, without deducing from it the remotest conse-
quence.

Now, under what conditions is the use of hypothesis without dan-
ger ? The proposal to submit all to experiment is not sufficient. Some
hypotheses are dangerous, — first and foremost those which are tacit
and unconscious. And since we make them without knowing them, we
cannot get rid of them. Here again, there is a service that mathemat-
ical physics may render us. By the precision which is its character-
istic, we are compelled to formulate all the hypotheses that we would
unhesitatingly make without its aid. Let us also notice that it is

NATURE 703

important not to multiply hypotheses indefinitely. If we construct a
theory based upon multiple hypotheses, and if experiment condemns it,
which of the premisses must be changed? It is impossible to tell.
Conversely, if the experiment succeeds, must we suppose that it has
verified all these hypotheses at once? Can several unknowns be de-
termined from a single equation?

We must also take care to distinguish between the different kinds of
hypotheses. First of all, there are those which are quite natural and
necessary. It is difficult not to suppose that the influence of very dis-
tant bodies is quite negligible, that small movements obey a linear
law, and that effect is a continuous function of its cause. I will
say as much for the conditions imposed by symmetry. All these
hypotheses affirm, so to speak, the common basis of all the theories of
mathematical physics. They are the last that should be abandoned.
There is a second category of hypotheses which I shall qualify as indif-
ferent. In most questions the analyst assumes, at the beginning of
his calculations, either that matter is continuous, or the reverse, that
it is formed of atoms. In either case, his results would have been the
same. On the atomic supposition he has a little more difficulty in
obtaining them — that is all. If, then, experiment confirms his con-
clusions, will he suppose that he has proved, for example, the real
existence of atoms?

In optical theories two vectors are introduced, one of which we
consider as a velocity and the other as a vortex. This again is an
indifferent hypothesis, since we should have arrived at the same con-
clusions by assuming the former to be a vortex and the latter to be
a velocity. The success of the experiment cannot prove, therefore, that
the first vector is really a velocity. It only proves one thing — namely,
that it is a vector ; and that is the only hypothesis that has really been
introduced into the premisses. To give it the concrete appearance
that the fallibility of our minds demands, it was necessary to consider
it either as a velocity or as a vortex. In the same way, it was neces-
sary to represent it by an a; or a y. but the result will not prove that
we were right or wrong in regarding it as a velocity; nor will it
prove we are right or wrong in calling it x and not y.

These indifferent hypotheses are never dangerous provided their
characters are not misunderstood. They may be useful, either as arti-
fices for calculation, or to assist our understanding by concrete image-s,
to fix the ideas, as we say. They need not therefore be rejected. The
hypotheses of the third category are real generalizations. They must
be confirmed or invalidated by experiment. Whether verified or con-
demned, they will always be fruitful; but, for the reasons I rjave
given, they will only be so if they are not too numerous.

Origin of Mathematical Physics. — Let us go further and study
more closely the conditions which have assisted the development of

704 SCIENCE AND HYPOTHESIS

mathematical physics. We recognize at the outset that the efforts of
men of science have always tended to resolve the complex phenomenon
given directly by experiment into a very large number of elementary
phenomena, and that in three different ways.

First, with respect to time. Instead of embracing in its entirety
the progressive development of a phenomenon, we simply try to
connect each moment with the one immediately preceding. We admit
that the present state of the world only depends on the immediate
past, without being directly influenced, so to speak, by the recollection
of a more distant past. Thanks to this postulate, instead of studying
directly the whole succession of phenomena, we may confine ourselves
to writing down its differential equation; for the laws of Kepler we
substitute the law of Newton.

Next, we try to decompose the phenomena in space. What experi-
ment gives us is a confused aggregate of facts spread over a scene of
considerable extent. We must try to deduce the elementary phenom-
enon, which will still be localized in a very small region of space.

A few examples perhaps will make my meaning clearer. If we
wished to study in all its complexity the distribution of temperature in
a cooling solid, we could never do so. This is simply because, if
we only reflect that a point in the solid can directly impart some of
its heat to a neighboring point, it will immediately impart that heat
only to the nearest points, and it is but gradually that the flow of
heat will reach other portions of the solid. The elementary pheno-
menon is the interchange of heat between two contiguous points. It
is strictly localized and relatively simple if, as is natural, we admit
that it is not influenced by the temperature of the molecules whose
distance apart is small.

I bend a rod: it takes a very complicated form, the direct investi-
gation of which would be impossible. But I can attack the problem,
however, if I notice that its flexure is only the resultant of the
deformations of the very small elements of the rod, and that the
deformation of each of these elements only depends on the forces
which are directly applied to it, and not in the least on those which
may be acting on the other elements.

In all these examples, which may be increased without difficulty, it
is admitted that there is no action at a distance or at great distances.
That is an hypothesis. It is not always true, as the law of gravitation
proves. It must therefore be verified. If it is confirmed, even ap-
proximately, it is valuable, for it helps us to use mathematical physics,
at any rate by successive approximations. If it does not stand the
test, we must seek something else that is analogous, for there are
other means of arriving at the elementary phenomenon. If several
bodies act simultaneously, it may happen that their actions are inde-
pendent, and may be added one to the other, either as vectors or as

NATURE 705

scalar quantities. The elementary phenomenon is then the action of
an isolated body. Or suppose, again, it is a question of small move-
ments, or more generally of small variations which obey the well-
known law of mutual or relative independence. The movement ob-
served will then be decomposed into simple movements — for example,
sound into its harmonics, and white light into its monochromatic
components. When we have discovered in which direction to seek for
the elementary phenomenon, by what means may we reach it ? First, it
will often happen that in order to predict it, or rather in order to
predict what is useful to us, it will not be necessary to know its mech-
anism. The law of great numbers will suffice. Take for example the
propagation of heat. Each molecule radiates towards its neighbor —
we need not inquire according to what law; and if we make any
supposition in this respect, it will be an indifferent hypothesis, and
therefore useless and unverifiable. In fact, by the action of averages
and thanks to the symmetry of the medium, all differences are levelled,
and, whatever the hypothesis may be, the result is always the same.

The same feature is presented in the theory of elasticity, and in that
of capillarity. The neighboring molecules attract and repel each other,
we need not inquire by what law. It is enough for us that this at-
traction is sensible at small distances only, and that the molecules
are very numerous, that the medium is symmetrical, and we have
only to let the law of great numbers come into play.

Here again the simplicity of the elementary phenomenon is hidden
beneath the complexity of the observable resultant phenomenon ; but in
its turn this simplicity was only apparent and disguised a very com-
plex mechanism. Evidently the best means of reaching the elementary
phenomenon would be experiment. It would be necessary by experi-
mental artifices to dissociate the complex system which nature offers
for our investigations and carefully to study the elements as disso-
ciated as possible; for example, natural white light would be decom-
posed into monochromatic lights by the aid of the prism, and into
polarized lights by the aid of the polarizer. Unfortunately, that is
neither always possible nor always sufficient, and sometimes the mind
must run ahead of experiment. I shall only give one example which
has always struck me rather forcibly. If I decompose white light, I
shall be able to isolate a portion of the spectrum, but however small it
may be, it will always be a certain width. In the same way the nat-
ural lights which are called monochromatic give us a very fine a ray,
but a ray which is not, however, infinitely fine. It might be supposed
that in the experimental study of the properties of these natural lights*
by operating with finer and finer rays, and passing on at last to the
limit, so to speak, we should eventually obtain the properties of a rigor-
ously monochromatic light. That would not be accurate. I assume
that two rays emanate from the same source, that they are first polar-

706 SCIENCE AND HYPOTHESIS

ized in planes at right angles, that they are then brought back again
to the same plane of polarization, and that we try to obtain inter-
ference. If the light were rigorously monochromatic, there would be
interference; but with our nearly monochromatic lights, there will be
no interference, and that, however narrow the ray may be. For it to be
otherwise, the ray would have to be several million times finer than
the finest known rays.

Here then we should be led astray by proceeding to the limit. The
mind has to run ahead of the experiment, and if it has done so with
success, it is because it has allowed itself to be guided by the instinct of
simplicity. The knowledge of the elementary fact enables us to state
the problem in the form of an equation. It only remains to deduce
from it by combination the observable and verifiable complex fact.
That is what we call integration, and it is the province of the mathe-
matician. It might be asked, why in physical science generalization
so readily takes the mathematical form. The reason is now easy to
see. It is not only because we have to express numerical laws;
it is because the observable phenomenon is due to the superposition of
a large number of elementary phenomena which are all similar to each
other; and in this way differential equations are quite naturally intro-
duced. It is not enough that each elementary phenomenon should
obey simple laws: all those that we have to combine must obey the
same law; then only is the intervention of mathematics of any use.
Mathematics teaches us, in fact, to combine like with like. Its object
is to define the result of a combination without having to reconstruct
that combination element by element. If we have to repeat the same
operation several times, mathematics enables us to avoid this repetition
by telling the result beforehand by a kind of induction. This I have
explained before in the chapter on mathematical reasoning. But for
that purpose all these operations must be similar ; in the contrary case
we must evidently make up our minds to working them out in full
one after the other, and mathematics will be useless. It is therefore,
thanks to the approximate homogeneity of the matter studied by phy-
sicists, that mathematical physics came into existence. In the nat-
ural sciences the following conditions are no longer to be found: —
homogeneity, relative independence of remote parts, simplicity of the
elementary fact; and that is why the student of natural science is
compelled to have recourse to other modes of generalization.

The Theories of Modern Physics

Significance of Physical Theories. — The ephemeral nature of scien-
tific theories takes by surprise the man of the world. Their brief period
of prosperity ended, he sees them abandoned one after another; he
sees ruins piled upon ruins; he predicts that the theories in fashion

NATURE 707

to-day will in a short time succumb in their turn, and he concludes
that they are absolutely in vain. This is what he calls the bank-
ruptcy of science.

His skepticism is superficial; he does not take into account the
object of scientific theories and the part they play, or he would under-
stand that the ruins may be still good for something. No theory
seemed established on firmer ground than Fresnel's, which attributed
light to the movements of the ether. Then if Maxwell's theory is
to-day preferred, does that mean that Fresnel's work was in vain?
No; for Fresnel's object was not to know whether there really is an
ether, if it is or is not formed of atoms, if these atoms really move in
this way or that; his object was to predict optical phenomena.

This Fresnel's theory enables us to do to-day as well as it did before
Maxwell's time. The differential equations are always true, they
may be always integrated by the same methods, and the results of this
integration still preserve their value. It cannot be said that this is
reducing physical theories to simple practical recipes; these equations
express relations, and if the equations remain true, it is because the
relations preserve their reality. They teach us now, as they did then,
that there is such and such a relation between this thing and that;
only, the something which we then called motion, we now call electric
current. But these are merely names of the images we substituted for
the real objects which Nature will hide forever from our eyes. The
true relations between these real objects are the only reality we can
attain, and the sole condition is that the same relations shall exist be-
tween these objects as between the images we are forced to put in
their place. If the relations are known to us, what does it matter if
we think it convenient to replace one image by another?

That a given periodic phenomenon (an electric oscillation, for in-
stance) is really due to the vibration of a given atom, which, behaving
like a pendulum, is really displaced in this manner or that, all this is
neither certain nor essential. But that there is between the electric
oscillation, the movement of the pendulum, and all periodic phenom-
ena an intimate relationship which corresponds to a profound reality;
that this relationship, this similarity, or rather this parallelism, is
continued in the details; that it is a consequence of more general
principles such as that of the conservation of energy, and that of
least action; this we may affirm; this is the truth which will ever
remain the same in whatever garb we may see fit to clothe it.

Many theories of dispersion have been proposed. The first were
imperfect, and contained but little truth. Then came that of Helm-
holtz, and this in its turn was modified in different ways; its author
himself conceived another theory, founded on Maxwell's principles.
But the remarkable thing is, that all the scientists who followed Helm-
holtz obtain the same equations, although their starting-points were

708 SCIENCE AND HYPOTHESIS

to all appearance widely separated. I venture to say that these theo-
ries are all simultaneously true ; not merely because they express a true
relation — that between absorption and abnormal dispersion. In the
premisses of these theories the part that is true is the part common
to all: it is the affirmation of this or that relation between certain
things, which some call by one name and some by another.

The kinetic theory of gases has given rise to many objections, to
which it would be difficult to find an answer were it claimed that the
theory is absolutely true. But all these objections do not alter the
fact that it has been useful, particularly in revealing to us one true
relation which would otherwise have remained profoundly hidden —
the relation between gaseous and osmotic pressures. In this sense,
then, it may be said to be true.

When a physicist finds a contradiction between two theories which
are equally dear to him, he sometimes says : " Let us not be troubled,
but let us hold fast to the two ends of the chain, lest we lose the
intermediate links." This argument of the embarrassed theologian
would be ridiculous if we were to attribute to physical theories the
interpretation given them by the man of the world. In case of con-
tradiction one of them at least should be considered false. But this is
no longer the case if we only seek in them what should be sought. It
is quite possible that they both express true relations, and that the
contradictions only exist in the images we have formed to ourselves of
reality. To those who feel that we are going too far in our limitations
of the domain accessible to the scientist, I reply: These questions
which we forbid you to investigate, and which you so regret, are not
only insoluble, they are illusory and devoid of meaning.

Such a philosopher claims that all physics can be explained by the
mutual impact of atoms. If he simply means that the same relations
obtain between physical phenomena as between the mutual im-
pact of a large number of billiard balls — well and good! this is
verifiable, and perhaps is true. But he means something more, and
we think we understand him, because we think we know what an
impact is. Why? Simply because we have often watched a game of
billiards. Are we to understand that God experiences the same sensa-
tions in the contemplation of His work that we do in watching a
game of billiards? If it is not our intention to give his assertion
this fantastic meaning, and if we do not wish to give it the more re-
stricted meaning I have already mentioned, which is the sound mean-
ing, then it has no meaning at all. Hypotheses of this kind have there-
fore only a metaphorical sense. The scientist should no more banish
them than a poet banishes metaphor ; but he ought to know what they
are worth. They may be useful to give satisfaction to the mind, and
they will do no harm as long as they are only indifferent hypotheses.

These considerations explain to us why certain theories, that were

NATURE 709

thought to be abandoned and definitively condemned by experiment,
are suddenly revived from their ashes and begin a new life. It is be-
cause they expressed true relations, and had not ceased to do so when
for some reason or other we felt it necessary to enunciate the same
relations in another language. Their life had been latent, as it were.

Barely fifteen years ago, was there anything more ridiculous, more
quaintly old-fashioned, than the fluids of Coulomb? And yet, here
they are re-appearing under the name of electrons. In what do
these permanently electrified molecules differ from the electric mole-
cules of Coulomb? It is true that in the electrons the electricity is
supported by a little, a very little matter; in other words, they have
mass. Yet Coulomb did not deny mass to his fluids, or if he did, it
was with reluctance. It would be rash to affirm that the belief in
electrons will not also undergo an eclipse, but it was none the less
curious to note this unexpected renaissance.

But the most striking example is Carnot's principle. Carnot estab-
lished it, starting from false hypotheses. When it was found that
heat was indestructible, and may be converted into work, his ideas
were completely abandoned; later, Clausius returned to them, and
to him is due their definitive triumph. In its primitive form, Car-
not's theory expressed in addition to true relations, other inexact rela-
tions, the debris of old ideas; but the presence of the latter did not
alter the reality of the others. Clausius had only to separate them,
just as one lops off dead branches.

The result was the second fundamental law of thermodynamics.
The relations were always the same, although they did not hold, at
least to all appearance, between the same objects. This was sufficient
for the principle to retain its value. Xor have the reasonings of Car-
not perished on this arcount ; they were applied to an imperfect con-
ception of matter, but their form — i.e., the essential part of them,
remained correct. What I have just said throws some light at the
same time on the role of general principles, such as those of the prin-
ciple of least action or of the conservation of energy. These principles
are of very •jreat value. They were obtained in the search for
what there was in common in the enunciation of numerous
physical laws: they thus represent the quintessence of innumer-
able observations. However, from their very generality results a
consequence to which I have called attention in Chapter VIII.
— namely, that they arc no longer capable of verification. As we
cannot give a general definition of energy, the principle of the con-
servation of energy simply signifies that there is a something which
remains constant. Whatever fresh notions of the world may be given
us by future experiments, we are certain beforehand that there is
something which remains constant, and which may be called energy.
Does this mean that the principle has no meaning and vanishes into

710 SCIENCE AND HYPOTHESIS

a tautology ? Not at all. It means that the different things to which
we give the name of energy are connected by a true relationship; it
affirms between them a real relation. But then, if this principle haa
a meaning, it may be false ; it may be that we have no right to extend
indefinitely its applications, and yet it is certain beforehand to be
verified in the strict sense of the word. How, then, shall we know
when it has been extended as far as is legitimate? Simply when it
ceases to be useful to us — t.£.,when we can no longer use it to pre-
dict correctly new phenomena. We shall be certain in such a case that
the relation affirmed is no longer real, for otherwise it would be fruit-
ful ; experiment without directly contradicting a new extension of the
principle will nevertheless have condemned it.

Physics and Mechanism. — Most theorists have a constant predilec-
tion for. explanations borrowed from physics, mechanics, or dynamics.
Some would be satisfied if they could account for all phenomena by
the motion of molecules attracting one another according to certain
laws. Others are more exact; they would suppress attractions acting
at a distance; their molecules would follow rectilinear paths, from
which they would only be deviated by impacts. Others again, such
as Hertz, suppress the forces as well, but suppose their molecules
subjected to geometrical connections analogous, for instance, to those
of articulated systems; thus, they wish to reduce dynamics to a kind
of kinematics. In a word, they all wish to bend nature into a certain
form, and unless they can do this they cannot be satisfied. Is Nature
flexible enough for this?

We shall examine this question hereafter under the head of Max-
well's Theory. Every time that the principles of least action and
energy are satisfied, we shall see that not only is there always a
mechanical explanation possible, but that there is an unlimited num-
ber of such explanations. By means of a well-known theorem due to
Konigs, it may be shown that we can explain everything in an un-
limited number of ways, by connections after the manner of Hertz,
or, again, by central forces. No doubt it may be just as easily demon-
strated that everything may be explained by simple impacts. For
this, let us bear in mind that it is not enough to be content with the
ordinary matter of which we are aware by means of our senses, and
the movements of which we observe directly. We may conceive of
ordinary matter as either composed of atoms, whose internal move-
ments escape us, our senses being able to estimate only the displacement
of the whole; or we may imagine one of those subtle fluids, which
under the name of ether or other names, have from all time played
so important a role in physical theories. Often we go further, and
regard the ether as the only primitive, or even as the only true matter.
The more moderate consider ordinary matter to be condensed ether,
and there is nothing startling in this conception; but others only re-

NATURE 711

duce its importance still further, and see in matter nothing more than
the geometrical locus of singularities in the ether. Lord Kelvin, for
instance, holds what we call matter to be only the locus of those points
at which the ether is animated by vortex motions. Riemann believes it
to be the locus of those points at which ether is constantly destroyed;
to Wiechert or Larmor, it is the locus of the points at which the ether
has undergone a kind of torsion of a very particular kind. Taking
any one of these points of view, I ask by what right do we apply to
the ether the mechanical properties observed in ordinary matter,
which is but false matter ? The ancient fluids, caloric, electricity, etc.,
were abandoned when it was seen that heat is not indestructible. But
they were also laid aside for another reason. In materializing them,
their individuality was, so to speak, emphasized — gaps were opened
between them ; and these gaps had to be filled in when the sentiment
of the unity of Nature became stronger, and when the intimate rela-
tions which connect all the parts were perceived. In multiplying the
fluids, not only did the ancient physicists create unnecessary entities,
but they destroyed real ties. It is not enough for a theory not to
affirm false relations; it must not conceal true relations.

Does our ether actually exist? We know the origin of our belief
in the ether. If light takes several years to reach us from a distant
star, it is no longer on the star, nor is it on the earth. It must be
somewhere, and supported, so to speak, by some material agency.

The same idea may be expressed in a more mathematical and more
abstract form. What we note are the changes undergone by the mate-
rial molecules. We see, for instance, that the photographic plate ex-
periences the consequences of a phenomenon of which the incan-
descent mass of a star was the scene several years before. Now, in
ordinary mechanics, the state of the system under consideration de-
pends only on its state at the moment immediately preceding; the
system therefore satisfies certain differential equations. On the other
hand, if we did not believe in the ether, the state of the material uni-
verse would depend not only on the state immediately preceding, but
also on much older states ; the system would satisfy equations of finite
differences. The ether was invented to escape this breaking down
of the laws of general mechanics.

Still, this would only compel us to fill the interplanetary space with
ether, but not to make it penetrate into the midst of the material
media. Fizeau's experiment goes further. By the interference of
rays which have passed through the air or water in motion, it seems
to show us two different media penetrating each other, and yet being
displaced with respect to each other. The ether is all but in our
grasp. Experiments can be conceived in which we come closer still
to it. Assume that Newton's principle of the equality of action and
re-action is not true if applied to matter alone, and that this can be

712 SCIENCE AND HYPOTHESIS

proved. The geometrical sum of all the forces applied to all the
molecules would no longer be zero. If we did not wish to change the
whole of the science of mechanics, we should have to introduce the
ether, in order that the action which matter apparently undergoes
should be counterbalanced by the re-action of matter on something.

Or again, suppose we discover that optical and electrical phenomena
are influenced by the motion of the earth. It would follow that those
phenomena might reveal to us not only the relative motion of material
bodies, but also what would seem to be their absolute motion. Again,
it would be necessary to have an ether in order that these so-called
absolute movements should not be their displacements with respect
to empty space, but with respect to something concrete.

Will this ever be accomplished? I do not think so, and I shall
explain why ; and yet, it is not absurd, for others have entertained this
view. For instance, if the theory of Lorentz, of which I shall speak
in more detail in Chapter XIII., were true, Newton's principle would
not apply to matter alone, and the difference would not be very far
from being within reach of experiment. On the other hand, many
experiments have been made on the influence of the motion of the
earth. The results have always been negative. But if these experi-
ments have been undertaken, it is because we have not been certain
beforehand; and indeed, according to current theories, the compensa-
tion would be only approximate, and we might expect to find accurate
methods giving positive results. I think that such a hope is illusory;
it was none the less interesting to show that a success of this kind
would, in a certain sense, open to us a new world.

And now allow me to make a digression; I must explain why I
do not believe, in spite of Lorentz, that more exact observations will
ever make evident anything else but the relative displacements of ma-
terial bodies. Experiments have been made that should have dis-
closed the terms of the first order; the results were nugatory.
Could that have been by chance ? No one has admitted this ; a general
explanation was sought, and Lorentz found it. He showed that the
terms of the first order should cancel each other, but not the terms
of the second order. Then more exact experiments were made, which
were also negative; neither could this be the result of chance. An ex-
planation was necessary, and was forthcoming; they always are; hypo-
theses are what we lack the least. But this is not enough. Who is
there who does not think that this leaves to chance far too important a
role? Would it not also be a chance that this singular concurrence
should cause a certain circumstance to destroy the terms of the first
order, and that a totally different but very opportune circumstance
should cause those of the second order to vanish? No; the same
explanation must be found for the two cases, and everything tends to
show that this explanation would serve equally well for the terms of

NATURE 713

the higher order, and that the mutual destruction of these terms will
be rigorous and absolute.

The Present State of Physics. — Two opposite tendencies may be
distinguished in the history of the development of physics. On the
one hand, new relations are continually being discovered between
objects which seemed destined to remain forever unconnected ; scattered
facts cease to be strangers to each other and tend to be marshalled
into an imposing synthesis. The march of science is towards unity
and simplicity.

On the other hand, new phenomena are continually being revealed;
it will be long before they can be assigned their place — sometimes it
may happen that to find them a place a corner of the edifice must be
demolished. In the same way, we are continually perceiving details
ever more varied in the phenomena we know, where our crude senses
used to be unable to detect any lack of unity. What we thought to
be simple becomes complex, and the march of science seems to be
towards diversity and complication.

Here, then, are two opposing tendencies, each of which seems to
triumph in turn. Which will win? If the first wins, science is pos-
sible; but nothing proves this a priori, and it may be that after un-
successful efforts to bend Nature to our ideal of unity in spite of her-
self, we shall be submerged by the ever- rising flood of our new riches
and compelled to renounce all idea of classification — to abandon our
ideal, and to reduce science to the mere recording of innumerable
recipes.

In fact, we can give this question no answer. All that we can do
is to observe the science of to-day, and compare it with that of yes-
terday. Xo doubt after this examination we shall be in a position to
offer a few conjectures.

Half-a-century ago hopes ran high indeed. The unity of force had
just been revealed to us by the discovery of the conservation of energy
and of its transformation. This discovery also showed that the phe-
nomena of heat could be explained by molecular movements. Although
the nature of these movements was not exactly known, no one
doubted but that they would be ascertained before long. As for light,
the work seemed entirely completed. So far as electricity was con-
cerned, there was not so great an advance. Electricity had just an-
nexed magnetism. This was a considerable and a definitive step
towards unity. But how was electricity in its turn to be brought into
the general unity, and how was it to be included in the general uni-
versal mechanism? No one had the slightest idea. As to the possi-
bility of the inclusion, all were agreed ; they had faith. Finally, as
far as the molecular properties of material bodies are concerned, the
inclusion seemed easier, but the details were very hazy. In a word,
hopes were vast and strong, but vague.

714 SCIENCE AND HYPOTHESIS

To-day, what do we see ? In the first place, a step in advance — im-
mense progress. The relations between light and electricity are now
known; the three domains of light, electricity, and magnetism, for-
merly separated, are now one ; and this annexation seems definitive.

Nevertheless the conquest has caused us some sacrifices. Optical
phenomena become particular cases in electric phenomena; as long as
the former remained isolated, it was easy to explain them by move-
ments which were thought to be known in all their details. That
was easy enough; but any explanation to be accepted must now cover
the whole domain of electricity. This cannot be done without diffi-
culty.

The most satisfactory theory is that of Lorentz ; it is unquestionably
the theory that best explains the known facts, the one that throws into
relief the greatest number of known relations, the one in which we
find most traces of definitive construction. That it still possesses a
serious fault I have shown above. It is in contradiction with New-
ton's law that action and re-action are equal and opposite — or rather,
this principle according to Lorentz cannot be applicable to matter
alone; if it be true, it must take into account the action of the ether
on matter, and the re-action of the matter on the ether. Now, in the
new order, it is very likely that things do not happen in this way.

However this may be, it is due to Lorentz that the results of
Fizeau on the optics of moving bodies, the laws of normal and abnor-
mal dispersion and of absorption are connected with each other and
with the other properties of the ether, by bonds which no doubt will
not be readily severed. Look at the ease with which the new Zeeman
phenomenon found its place, and even aided the classification of
Faraday's magnetic rotation, which had defied all Maxwell's efforts.
This facility proves that Lorentz's theory is not a mere artificial com-
bination which must eventually find its solvent. It will probably have
to be modified, but not destroyed.

The only object of Lorentz was to include in a single whole all
the optics and electro-dynamics of moving bodies; he did not claim
to give a mechanical explanation. Larmor goes further; keeping the
essential part of Lorentz's theory, he grafts upon it, so to speak, Mac-
Cullagh's ideas on the direction of the movement of the
ether. MacCullagh held that the velocity of the ether is the
same in magnitude and direction as the magnetic force. Ingenious
as is this attempt, the fault in Lorentz's theory remains, and is even
aggravated. According to Lorentz, we do not know what the move-
ments of the ether are; and because we do not know this, we may
suppose them to be movements compensating those of matter, and re-
affirming that action and re-action are equal and opposite. According
to Larmor we know the movements of the ether, and we can prove
that the compensation does not take place.

NATURE 715

If Larmor has failed, as in my opinion he has, does it necessarily
follow that a mechanical explanation is impossible? Far from it. I
eaid above that as long as a phenomenon obeys the two principles of
energy and least action, so long it allows of an unlimited number of
mechanical explanations. And so with the phenomena of optics and
electricity.

But this is not enough. For a mechanical explanation to be good
it must be simple; to choose it from among all the explanations that
are possible there must be other reasons than the necessity of making
a choice. Well, we have no theory as yet which will satisfy this con-
dition and consequently be of any use. Are we then to complain?
That would be to forget the end we seek, which is not the mechanism ;
the true and only aim is unity.

We ought therefore to set some limits to our ambition. Let us not
seek to formulate a mechanical explanation; let us be content to show
that we can always find one if we wish. In this we have succeeded.
The principle of the conservation of energy has always been con-
firmed, and now it has a fellow in the principle of least action, stated
in the form appropriate to physics. This has also been verified, at
least as far as concerns the reversible phenomena which obey La-
grange's equations — in other words, which obey the most general
laws of physics. The irreversible phenomena are much more difficult
to bring into line; but they, too, are being co-ordinated and tend to
come into the unity. The light which illuminates them comes from
Carnot's principle. For a long time thermo-dynamics was confined to
the study of the dilatations of bodies and of their change of state.
For some time past it has been . growing bolder, and has considerably
extended its domain. We owe to it the theories of the voltaic cell
and of their thermo-electric phenomena; there is not a corner in phy-
sics which it has not explored, and it has even attacked chemistry
itself. The same laws hold good; everywhere, disguised in some form
or other, we find Carnot's principle ; everywhere also appears that emi-
nently abstract concept of entropy which is as universal as the con-
cept of energy, and like it, seems to conceal a reality. It seemed that
radiant heat must escape, but recently that, too, has been brought
under the same laws.

In this way fresh analogies are revealed which may be often pur-
sued in detail; electric resistance resembles the viscosity of fluids; hys-
teresis would rather be like the friction of solids.. In all cases friction
appears to be the type most imitated by the most diverse irreversible
phenomena, and this relationship is real and profound.

A strictly mechanical explanation of these phenomena has also been
sought, but, owing to their nature, it is hardly likely that it will be
found. To find it, it has been necessary to suppose that the irreversi-
bility is but apparent, that the elementary phenomena are reversible

716 SCIENCE AND HYPOTHESIS

and obey the known laws of dynamics. But the elements are extremely
numerous, and become blended more and more, so that to our crude
sight all appears to tend towards uniformity — i.e., all seems to
progress in the same direction, and that without hope of return. The
apparent irreversibility is therefore but an effect of the law of great
numbers. Only a being of infinitely subtle senses, such as Maxwell's
demon, could unravel this tangled skein and turn back the course of
the universe.

This conception, which is connected with the kinetic theory of
gases, has cost great effort and has not, on the whole, been fruitful;
it may become so. This is not the place to examine if it leads to con-
tradictions, and if it is in conformity with the true nature of things.

Let us notice, however, the original ideas of M. Gouy on the Brown-
ian movement. According to this scientist, this singular movement
does not obey Carnot's principle. The particles which it sets moving
would be smaller than the meshes of that tightly drawn net; they
would thus be ready to separate them, and thereby to set back the
course of the universe. One can almost see Maxwell's demon at work. 1

To resume, phenomena long known are gradually being better clas-
sified, but new phenomena come to claim their place, and most of
them, like the Zeeman effect, find it at once. Then we have the
cathode rays, the X-rays, uranium and radium rays; in fact, a whole
world of which none had suspected the existence. How many unex-
pected guests to find a place for! No one can yet predict the place
they will occupy, but I do not believe they will destroy the general
unity; I think that they will rather complete it. On the one hand,
indeed, the new radiations seem to be connected with the phenomena
of luminosity; not only do they excite fluorescence, but they some-
times come into existence under the same conditions as that property ;
neither are they unrelated to the cause which produces the electric
spark under the action of ultra-violet light. Finally, and most im-
portant of all, it is believed that in all these phenomena there exist
ions, animated, it is true, with velocities far greater than those of
electrolytes. All this is very vague, but it will all become clearer.

Phosphorescence and the action of light on the spark were regions
rather isolated, and consequently somewhat neglected by investigators.
It is to be hoped that a new path will now be made which will facili-
tate their communications with the rest of science. Not only do we
discover new phenomena, but those we think we know are revealed in
unlooked-for aspects. In the free ether the laws preserve their majes-
tic simplicity, but matter properly so called seems more and more

i Clerk-Maxwell imagined some supernatural agency at work, sorting mole-
cules in a gas of uniform temperature into (a) those possessing kinetic en-
ergy above the average, (6) those possessing kinetic energy below the average.

-[TB.]

NATURE 717

complex; all we can say of it is but approximate, and our formulae
are constantly requiring new terms.

But the ranks are unbroken, the relations that we have discovered
between objects we thought simple still hold good between the same
objects when their complexity is recognized, and that alone is the
important thing. Our equations become, it is true, more and more
complicated, so as to embrace more closely the complexity of nature;
but nothing is changed in the relations which enable these equations
to be derived from each other. In a word, the form of these equa-
tions persists. Take for instance the laws of reflection. Fresnel estab-
lished them by a simple and attractive theory which experiment
seemed to confirm. Subsequently, more accurate researches have
shown that this verification was but approximate; traces of elliptic
polarization were detected everywhere. But it is owing to the first ap-
proximation that the cause of these anomalies was found in the exist-
ence of a transition layer, and all the essentials of Fresnel's theory
have remained. We cannot help reflecting that all these relations
would never have been noted if there had been doubt in the first
place as to the complexity of the objects they connect. Long ago it
was said: If Tycho had had instruments ten times as precise, we
would never have had a Kepler, or a Newton, or Astronomy. It is
a misfortune for a science to be born too late, when the means of
observation have become too perfect. That is what is happening at
this moment with respect to physical chemistry; the founders are
hampered in their general grasp by third and fourth decimal places;
happily they are men of robust faith. As we get to know the proper-
ties of matter better we see that continuity reigns. From the work
of Andrews and Van der Waals, we see how the transition from the
liquid to the gaseous state is made, and that it is not abrupt. Sim-
ilarly, there is no gap between the liquid and solid states, and in the
proceedings of a recent Congress we see memoirs on the rigidity of
liquids side by side with papers on the flow of solids.

With this tendency there is no doubt a loss of simplicity. Such and
such an effect was represented by straight lines ; it is now necessary to
connect these lines by more or less complicated curves. On the other
hand, unity is gained. Separate categories quieted but did not sat-
isfy the mind.

Finally, a new domain, that of chemistry, has been invaded by
the method of physics, and we see the birth of physical chemistry. It
is still quite young, but already it has enabled us to connect such
phenomena as electrolysis, osmosis, and the movements of ions.

From this cursory exposition what can we conclude? Taking all
things into account, we have approached the realization of unity.
This has not been done as quickly as was hoped fifty years ago, and

718 SCIENCE AND HYPOTHESIS

the path predicted has not always been followed; but, on the whole,
much ground has been gained.

The Calculus of Probabilities

No doubt the reader will be astonished to find reflections on the
calculus of probabilities in such a volume as this. What has that
calculus to do with physical science? The questions I shall raise —
without, however, giving them a solution — are naturally raised by
the philosopher who is examining the problems of physics. So far is
this the case, that in the two preceding chapters I have several times
used the words " probability " and " chance." " Predicted facts," as
I said above, " can only be probable." However solidly founded a
prediction may appear to be, we are never absolutely certain that ex-
periment will not prove it false; but the probability is often so great
that practically it may be accepted. And a little farther on I added :
" See what a part the belief in simplicity plays in our generalizations.
We have verified a simple law in a large number of particular cases,
and we refuse to admit that this so-often repeated coincidence is a
mere effect of chance." Thus, in a multitude of circumstances the
physicist is often in the same position as the gambler who reckons
up his chances. Every time that he reasons by induction, he more or
less consciously requires the calculus of probabilities, and that is why
I am obliged to open this chapter parenthetically, and to interrupt
our discussion of method in the physical sciences in order to examine
a little closer what this calculus is worth, and what dependence we
may place upon it. The very name of the calculus of probabilities is
a paradox. Probability as opposed to certainty is what one does not
know, and how can we calculate the unknown? Yet many eminent
scientists have devoted themselves to this calculus, and it cannot be
denied that science has drawn therefrom no small advantage. How
can we explain this apparent contradiction? Has probability been
defined? Can it even be defined? And if it cannot, how can we
venture to reason upon it? The definition, it will be said, is very
simple. The probability of an event is the ratio of the number of
cases favorable to the event to the total number of possible cases. A
simple example will show how incomplete this definition is : — I
throw two dice. What is the probability that one of the two at least
turns up a 6 ? Each can turn up in six different ways ; the number of
possible cases is 6X6=36. The number of favorable cases is 11; the
probability i|. That is the correct solution. But why cannot we
just as well proceed as follows ? — The points which turn up on the
two dice form *-^- =21 different combinations. Among these com-
binations, six are favorable; the probability is ^. Now why is the
first method of calculating the number of possible cases more legiti-

NATURE 719

mate than the second? In any case it is not the definition that tells
us. We are therefore bound to complete the definition by saying,
" ... to the total number of possible cases, provided the cases are
equally probable." So we are compelled to define the probable by
the probable. How can we know that two possible cases are equally
probable ? Will it be by a convention ? If we insert at the beginning
of every problem an explicit convention, well and good! We then
have nothing to do but to apply the rules of arithmetic and algebra,
and we complete our calculation, when our result cannot be called in
question. But if we wish to make the slightest application of this
result, we must prove that our convention is legitimate, and we shall
find ourselves in the presence of the very difficulty we thought we had
avoided. It may be said that common-sense is enough to show us the
convention that should be adopted. Alas ! M. Bertrand has amused
himself by discussing the following simple problem : — " What is the
probability that a chord of a circle may be greater than the side of
the inscribed equilateral triangle ? " The illustrious geometer suc-
cessively adopted two conventions which seemed to be equally impera-
tive in the eyes of common-sense, and with one convention he finds
1-2, and with the other 1-3. The conclusion which seems to follow
from this is that the calculus of probabilities is a useless science, that
the obscure instinct which we call common-sense, and to which we
appeal for the legitimization of our conventions, must be distrusted.
But to this conclusion we can no longer subscribe. We cannot do
without that obscure instinct. Without it, science would be impos-
sible, and without it we could neither discover nor apply a law.
Have we any right, for instance, to enunciate Newton's law? No
doubt numerous observations are in agreement with it, but is not that
a simple fact of chance? and how do we know, besides, that this law
which has been true for so many generations will not be untrue in the
next? To this objection the only answer you can give is: It is very
improbable. But grant the law. By means of it I can calculate the
position of Jupiter in a year from now. Yet have I any right to say
this? Who can tell if a gigantic mass of enormous velocity is not
going to pass near the solar system and produce unforeseen perturba-
tions? Here again the only answer is: It is very improbable. From
this point of view all the sciences would only be unconscious applica-
tions of the calculus of probabilities. And if this calculus be con-
demned, then the whole of the sciences must also be condemned.
I shall not dwell at length on scientific problems in which the inter-
vention of the calculus of probabilities is more evident. In the fore-
front of these is the problem of interpolation, in which, knowing a
certain number of values of a function, we try to discover the inter-
mediary values. I may also mention the celebrated theory of errors
of observation, to which I shall return later; the kinetic theory of

720 SCIENCE AND HYPOTHESIS

gases, a well-known hypothesis wherein each gaseous molecule is sup-
posed to describe an extremely complicated path, but in which,
through the effect of great numbers, the mean phenomena which are
all we observe obey the simple laws of Mariotte and Gay-Lussac.
All these theories are based upon the laws of great numbers, and the
calculus of probabilities would evidently involve them in its ruin.
It is true that they have only a particular interest, and that, save
as far as interpolation is concerned, they are sacrifices to which we
might readily be resigned. But I have said above, it would not be
these partial sacrifices that would be in question; it would be the
legitimacy of the whole of science that would be challenged. I quite
see that it might be said : We do not know, and yet we must act. As
for action, we have not time to devote ourselves to an inquiry that
will suffice to dispel our ignorance. Besides, such an inquiry would
demand unlimited time. We must therefore make up our minds
without knowing. This must be often done whatever may happen,
and we must follow the rules although we may have but little confi-
dence in them. What I know is, not that such a thing is true, but
that the best course for me is to act as if it were true. The calculus
of probabilities, and therefore science itself, would be no longer of
any practical value.

Unfortunately the difficulty does not thus disappear. A gambler
wants to try a coup, and he asks my advice. If I give it him, I use
the calculus of probabilities; but I shall not guarantee success. That
is what I shall call subjective probability. In this case we might be
content with the explanation of which I have just given a sketch.
But assume that an observer is present at the play, that he knows of
the coup, and that play goes on for a long time, and that he makes
a summary of his notes. He will find that events have taken place
in conformity with the laws of the calculus of probabilities. That
is what I shall call objective probability, and it is this phenomenon
which has to be explained. There are numerous Insurance Societies
which apply the rules of calculus of probabilities, and they distribute
to their shareholders dividends, the objective reality of which cannot
be contested. In order to explain them, we must do more than invoke
our ignorance and the necessity of action. Thus, absolute scepti-
cism is not admissible. We may distrust, but we cannot condemn
en bloc. Discussion is necessary.

I. Classification of the Problems of Probability. — In order to
classify the problems which are presented to us with reference to
probabilities, we must look at them from different points of view, and
first of all, from that of generality. I said above that probability is
the ratio of the number of favorable to the number of possible cases.
What for want of a better term I call generality will increase with the
number of possible cases. This number may be finite, as, for instance,

NATUEE 721

if we take a throw of the dice in which the number of possible cases
is 36. That is the first degree of generality. But if we ask, for in-
stance, what is the probability that a point within a circle is within
the inscribed square, there are as many possible cases as there are
points in the circle — that is to say, an infinite number. This is
the second degree of generality. Generality can be pushed further
still. We may ask the probability that a function will satisfy a given
condition. There are then as many possible cases as one can imagine
different functions. This is the third degree of generality, which we
reach, for instance, when we try to find the most probable law after a
finite number of observations. Yet we may place ourselves at a quite
different point of view. If we were not ignorant there would be no
probability, there could only be certainty. But our ignorance cannot
be absolute, for then there would be no longer any probability at all.
Thus the problems of probability may be classed according to the
greater or less depth of this ignorance. In mathematics we may set
ourselves problems in probability. What is the probability that the
fifth decimal of a logarithm taken at random from a table is a 9?
There is no hesitation in answering that this probability is l-10th.
Here we possess all the data of the problem. We can calculate our
logarithm without having recourse to the table, but we need not give
ourselves the trouble. This is the first degree of ignorance. In the
physical sciences our ignorance is already greater. The state of a
system at a given moment depends on two things — its initial state,
and the law according to which that state varies. If we know both
this law and this initial state, we have a simple mathematical problem
to solve, and we fall back upon our first degree of ignorance. Then it
often happens that we know the law and do not know the initial
state. It may be asked, for instance, what is the present distribution
of the minor planets? We know that from all time they have obeyed
the laws of Kepler, but we do not know what was their initial dis-
tribution. In the kinetic theory of gases we assume that the gaseous
molecules follow rectilinear paths and obey the laws of impact and
elastic bodies; yet as we know nothing of their initial velocities, we
know nothing of their present velocities. The calculus of probabili-
ties alone enables us to predict the mean phenomena which will result
from a combination of these velocities. This is the second degree of
ignorance. Finally it is possible, that not only the initial conditions
but the laws themselves are unknown. We then reach the third
degree of ignorance, and in general we can no longer affirm anything
at all as to the probability of a phenomenon. It often happens that
instead of trying to discover an event by means of a more or less im-
perfect knowledge of the law, the events may be known, and we want
to find the law; or that, instead of deducing effects from causes, we
wish to deduce the causes from the effects. Now, these problems are

722 SCIENCE AND HYPOTHESIS

classified as probability of causes, and are the most interesting of all
from their scientific applications. I play at ecarte with a gentleman
whom I know to be perfectly honest. What is the chance that he
turns up the king? It is 1-8. This is a problem of the probability
of effects. I play with a gentleman whom I do not know. He has
dealt ten times, and he has turned the king up six times. What is the
chance that he is a sharper ? This is a problem in the probability of
causes. It may be said that it is the essential problem of the experi-
mental method. I have observed n values of x and the corresponding
values of y. I have found that the ratio of the latter to the former
is practically constant. There is the event; what is the cause? Is it
probable that there is a general law according to which y would be
proportional to x, and that small divergences are due to errors of
observation? This is the type of question that we are ever asking,
and which we unconsciously solve whenever we are engaged in scientific
work. I am now going to pass in review these different categories of
problems by discussing in succession what I have called subjective and
objective probability.

II. Probability in Mathematics. — The impossibility of squaring
the circle was shown in 1885, but before that date all geometers con-
sidered this impossibility as so " probable " that the Academic des
Sciences rejected without examination the, alas! too numerous mem-
oirs on this subject that a few unhappy madmen sent in every year.
Was the Academic wrong? Evidently not, and it knew perfectly well
that by acting in this manner it did not run the least risk of stifling
a discovery of moment. The Academic could not have proved that it
was right, but it knew quite well that its instinct did not deceive it.
If you had asked the Academicians, they would have answered : " We
have compared the probability that an unknown scientist should have
found out what has been vainly sought for so long, with the proba-
bility that there is one madman the more on the earth, and the latter
has appeared to us the greater." These are very good reasons, but
there is nothing mathematical about them; they are purely psycho-
logical. If you had pressed them further, they would have added:
" Why do you expect a particular value of a transcendental function
to be an algebraical number; if •*• be the root of an algebraical equa-
tion, why do you expect this root to be a period of the function sin 2x,
and why is it not the same with the other roots of the same equation ?"
To sum up, they would have invoked the principle of sufficient reason
in its vaguest form. Yet what information could they draw from it ?
At most a rule of conduct for the employment of their time, which
would be more usefully spent at their ordinary work than in reading
a lucubration that inspired in them a legitimate distrust. But what
I called above objective probability has nothing in common with this
first problem. It is otherwise with the second. Let us consider the

NATURE 723

first 10,000 Igarithms that we find in a table. Among these 10,000
logarithms I take one at random. What is the probability that its
third decimal is an even number? You will say without any hesita-
tion that the probability is 1-2, and in fact if you pick out in a table
the third decimals in these 10,000 numbers you will find nearly as
many even digits as odd. Or, if you prefer it, let us write 10,000
numbers corresponding to our 10,000 logarithms, writing down for
each of these numbers + 1 if the third decimal of the corresponding
logarithm is even, and — 1 if odd ; and then let us take the mean of
these 10,000 numbers. I do not hesitate to say that the mean of these
10,000 units is probably zero, and if I were to calculate it practically,
I would verify that it is extremely small. But this verification is
needless. I might have rigorously proved that this mean is smaller
than 0.003. To prove this result I should have had to make a rather
long calculation for which there is no room here, and for which I may
refer the reader to an article that I published in the Revue generate des
Sciences, April 15th, 1899. The only point to which I wish to draw
attention is the following. In this calculation I had occasion to rest
my case on only two facts — namely, that the first and second deriva-
tives of the logarithm remain, in the interval considered, between
certain limits. Hence our first conclusion is that the property is not
only true of the logarithm but of any continuous function what-
ever, since the derivatives of every continuous function are limited.
If I was certain beforehand of the result, it is because I have often
observed analogous facts for other continuous functions; and next,
it is because I went through in my mind in a more or less uncon-
scious and imperfect manner the reasoning which led me to the
preceding inequalities, just as a skilled calculator before finishing his
multiplication takes into account what it ought to come to approx-
imately. And besides, since what I call my intuition was only an
incomplete summary of a piece of true reasoning, it is clear that
observation has confirmed my predictions, and that the objective and
subjective probabilities are in agreement. As a third example I shall
choose the following : — The number u is taken at random and n
is a given very large integer. What is the mean value of sin nuf
This problem has no meaning by itself. To give it one, a convention
is required — namely, we agree that the probability for the number
u to lie between a and a + da is <f> (a) da; that it is therefore propor-
tional to the infinitely small interval da, and is equal to this multi-
plied by a function </> (a), only depending on a. As for this function
I choose it arbitrarily, but I must assume it to be continuous. The
value of sin nu remaining the same when u increases by 2ir, I may
without loss of generality assume that u lies between 0 and 2ir, and I
shall thus be led to suppose that <f> (a) is a periodic function whose
period is 2 TT . The mean value that we seek is readily expressed by

724 SCIENCE AND HYPOTHESIS

a s.'mple integral, aod il is easy to show that this integral is smaller

than 2 ^Mg UK being the maximum value of the nth derivative ot

*K
<f> (u). We see then that if the K& derivative is finite, our mean

value will tend towards zero when n increases indefinitely, and

that more rapidly than ~^ . The mean value of sin nu when n is

n

very large is therefore zero. To define this value I required a con-
vention, but the result remains the same whatever that convention
may be. I have imposed upon myself but slight restrictions when I
assumed that the function </> (a) is continuous and periodic, and these
hypotheses are so natural that we may ask ourselves how they can be
escaped. Examination of the three preceding examples, so different
in all respects, has already given us a glimpse on the one hand of
the role of what philosophers call the principle of sufficient reason,
and on the other hand of the importance of the fact that certain pro-
perties are common to all continuous functions. The study of prob-
ability in the physical sciences will lead us to the same result.

III. Probability in the Physical Sciences. — We now come to the
problems which are connected with what I have called the second
degree of ignorance — namely, those in which we know the law but
do not know the initial state of the system. I could multiply exam-
ples, but I shall take only one. What is the probable present distri-
bution of the minor planets on the zodiac ? We know they obey the
laws of Kepler. We may even, without changing the nature of the
problem, suppose that their orbits are circular and situated in the
same plane, a plane which we are given. On the other hand, we
know absolutely nothing about their initial distribution. However,
we do not hesitate to affirm, that this distribution is now nearly uni-
form. Why? Let b be the longitude of a minor planet in the
initial epoch — that is to say, the epoch zero. Let a be its mean
motion. Its longitude at the present time — i.e., at the time t will
be at + b. To say that the present distribution is uniform is to say
that the mean value of the sines and cosines of multiples of at+b
is zero. Why do we assert this? Let us represent our minor planet
by a point in a plane — namely, the point whose co-ordinates are
a and b. All these representative points will be contained in a cer-
tain region of the plane, but as they are very numerous this region
will appear dotted with points. We know nothing else about the dis-
tribution of the points. Now what do we do when we apply the
calculus of probabilities to such a question as this? What is the
probability that one or more representative points may be found in a
certain portion of the plane? In our ignorance we are compelled to
make an arbitrary hypothesis. To explain the nature of this hypo-
thesis I may be allowed to use, instead of a mathematical formula, a
crude but concrete image. Let us suppose that over the surface of
our plane has been spread imaginary matter, the density of which is

NATURE 725

variable, but varies continuously. We shall then agree to say that
the probable number of representative points to be found on a certain
portion of the plane is proportional to the quantity of this imaginary
matter which is found there. If there are, then, two regions of the
plane of the same extent, the probabilities that a representative point
of one of our minor planets is in one or other of these regions will be
as the mean densities of the imaginary matter in one or other of the re-
gions. Here then are two distributions, one real, in which the represent-
ative points are very numerous, very close together, but discrete like
molecules of matter in atomic hypothesis ; the other remote from reality
in which our representative points are replaced by imaginary continuous
matter. We know that the latter cannot be real, but we are forced
to adopt it through our ignorance. If, again, we had some idea of the
real distribution of the representative points, we could arrange it so
that in a region of some extent the density of this imaginary continu-
ous matter may be nearly proportional to the number of representa-
tive points, or if it is preferred, to the number of atoms which are
contained in that region. Even that is impossible, and our ignorance
is so great that we are forced to choose arbitrarily the function which
defines the density of our imaginary matter. We shall be compelled
to adopt a hypothesis from which we can hardly get away; we shall
suppose that this function is continuous. That is sufficient, as we
shall see, to enable us to reach our conclusion.

What is at the instant t the probable distribution of the minor
planets — or rather, what is the mean value of the sine of the longi-
tude at the moment i — i.e., of sin (at+b) ? We made at the outset
an arbitrary convention, but if we adopt it, this probable value is en-
tirely denned. Let us decompose the plane into elements of surface.
Consider the value of sin (ai+b) at the centre of each of these ele-
ments. Multiply this value by the surface of the element and by the
corresponding density of the imaginary matter. Let us then take the
sum for all the elements of the plane. This sum, by definition, will
be the probable mean value we seek, which will thus be expressed by a
double integral. It may be thought at first that this mean value
depends on the choice of the function </> which defines the density of
the imaginary matter, and as this function <f> is arbitrary, we can,
according to the arbitrary choice which we make, obtain a certain
mean value. But this is not the case. A simple calculation shows us
that our double integral decreases very rapidly as t increases. Thus,
I cannot tell what hypothesis to make as to the probability of this or
that initial distribution, but when once the hypothesis is made the
result will be the same, and this gets me out of my difficulty. What-
ever the function <£ may be, the mean value tends towards zero as t
increases, and as the minor planets have certainly accomplished a
very large number of revolutions, I may assert that this mean value is

726 SCIENCE AND HYPOTHESIS

very small. I may give to <£ any value I choose, with one restriction :
this function must be continuous; and, in fact, from the point of
view of subjective probability, the choice of a discontinuous function
would have been unreasonable. What reason could I have, for in-
stance, for supposing that the initial longitude might be exactly o°,
but that it could not lie between o° and 1° ?

The difficulty reappears if we look at it from the point of view of
objective probability; if we pass from our imaginary distribution in
which the supposititious matter was assumed to be continuous, to the
real distribution in which our representative points are formed as
discrete atoms. The mean value of sin (at + b) will be represented
quite simply by

J2 "n (at+b),

n being the number of minor planets. Instead of a double integral
referring to a continuous function, we shall have a sum of discrete
terms. However, no one will seriously doubt that this mean value is
practically very small. Our representative points being very close
together, our discrete sum will in general differ very little from an
integral. An integral is the limit towards which a sum of
terms tends when the number of these terms is indefinitely in-
creased. If the terms are very numerous, the sum will differ
very little from its limit — that is to say, from the integral,
and what I said of the latter will still be true of the sum
itself. But there are exceptions. If, for instance, for all the minor
planets b = ?- — at, the longitude of all the planets at the time t
would be £ , and the mean value in question would be evidently unity.
For this to be the case at the time o, the minor planets must have all
been lying on a kind of spiral of peculiar form, with its spires very
close together. All will admit that such an initial distribution is
extremely improbable (and even if it were realized, the distribution
would not be uniform at the present time — for example, on the 1st
January 1900; but it would become so a few years later). Why,
then, do we think this initial distribution improbable ? This must be
explained, for if we are wrong in rejecting as improbable this absurd
hypothesis, our inquiry breaks down, and we can no longer affirm any-
thing on the subject of the probability of this or that present distri-
bution. Once more we shall invoke the principle of sufficient reason,
to which we must always recur. We might admit that at the begin-
ning the planets were distributed almost in a straight line. We
might admit that they were irregularly distributed. But it seems
to us that there is no sufficient reason for the unknown cause that
gave them birth to have acted along a curve so regular and yet so
complicated, which would appear to have been expressly chosen so
that the distribution at the present day would not be uniform.

NATUEE 727

IV. Rouge et Noir. — The questions raised by games of chance,
such as roulette, are, fundamentally, quite analogous to those we
have just treated. For example, a wheel is divided into thirty-seven
equal compartments, alternately red and black. A ball is spun round
the wheel, and after having moved round a number of times, it stops
in front of one of these sub-divisions. The probability that the divi-
sion is red is obviously 1-2. The needle describes an angle 6, in-
cluding several complete revolutions. I do not know what is the
probability that the ball is spun with such a force that this angle
should lie between 6 and 0 +d 6, but I can make a convention. I
can suppose that this probability is <$>(6)dO. As for the function
<f>(0), I can choose it in an entirely arbitrary manner. I have noth-
ing to guide me in my choice, but I am naturally induced to suppose
the function to be continuous. Let e be a length (measured on the
circumference of the circle of radius unity) of each red and black
compartment. We have to calculate the integral of <f>(6)d&, ex-
tending it on the one hand to all the red, and on the other hand to all
the black compartments, and to compare the results. Consider an
interval 2 e comprising two consecutive red and black compartments.
Let M and m be the maximum and minimum values of the function
<f> (0) in this interval. The integral extended to the red com-
partments will be smaller than ^ Me; extended to the black it will
be greater than ^ me- The difference will therefore be smaller than
^ (M — m)e. But if the function </> is supposed continuous, and
if on the other hand the interval e is very small with respect to the
total angle described by the needle, the difference M — m will be
very small. The difference of the two integrals will be therefore very
small, and the probability will be very nearly 1-2. We see that with-
out knowing anything of the function <f> we must act as if the prob-
ability were 1-2. And on the other hand it explains why, from the
objective point of view, if I watch a certain number of coups, ob-
servation will give me almost as many black coups as red. All the
players know this objective law; but it leads them into a remarkable
error, which has often been exposed, but into which they are always
falling. When the red has won, for example, six times running, they
bet on black, thinking that they are playing an absolutely safe game,
because they say it is a very rare thing for the red to win seven times
running. In reality their probability of winning is still 1-2. Ob-
servation shows, it is true, that the series of seven consecutive reds is
very rare, but series of six reds followed by a black are also very
rare. They have noticed the rarity of the series of seven reds; if
they have not remarked the rarity of six reds and a black, it is only
because such series strike the attention less.

V. The Probability of Causes. — We now come to the problems
of the probability of causes, the most important from the point of view

728 SCIENCE AND HYPOTHESIS

of scientific applications. Two stars, for instance, are very close
together on the celestial sphere. Is this apparent contiguity a mere
effect of chance? Are these stars, although almost on the same
visual ray, situated at very different distances from the earth, and
therefore very far indeed from one another ? or does the apparent cor-
respond to a real contiguity? This is a problem on the probability
of causes.

First of all, I recall that at the outset of all problems of probability
of effects that have occupied our attention up to now, we have had
to use a convention which was more or less justified; and if in most
cases the result was to a certain extent independent of this convention,
it was only the condition of certain hypotheses which enabled us
a priori to reject discontinuous functions, for example, or certain
absurd conventions. We shall again find something analogous to this
when we deal with the probability of causes. An effect may be pro-
duced by the cause a or by the cause b. The effect has just been ob-
served. We ask the probability that it is due to the cause a. This
is an <z posteriori probability of cause. But I could not calculate it,
if a convention more or less justified did not tell me in advance
what is the a priori probability for the cause a to come into play —
I mean the probability of this event to some one who had not ob-
served the effect. To make my meaning clearer, I go back to the
game of ecarte mentioned before. My adversary deals for the first
time and turns up a king. What is the probability that he is a
sharper? The formulae ordinarily taught give 8-9. a result which is
obviously rather surprising. If we look at it closer, we see that the
conclusion is arrived at as if, before sitting down at the table, I
had considered that there was one chance in two that my adversary
was not honest. An absurd hypothesis, because in that case I should
certainly not have played with him; and this explains the absurdity
of the conclusion. The function on the a priori probability was un-
justified, and that is why the conclusion of the a posteriori prob-
ability led me into an inadmissible result. The importance of this
preliminary convention is obvious. I shall even add that if none
were made, the problem of the a posteriori probability would have
no meaning. It must be always made either explicitly or tacitly.

Let us pass on to an example of a more scientific character. 1
require to detennine an experimental law; this law, when discovered,
can be represented by a curve. I make a certain number of isolated
observations, each of which may be represented by a point. When I
have obtained these different points, I draw a curve between them as
carefully as possible, giving my curve a regular form, avoiding sharp
angles, accentuated inflections, and any sudden variation of the ra-
dius of curvature. This curve will represent to me the probable law,
and not only will it give me the values of the functions intermediary

NATURE 729

to those which have been observed, but it also gives me the observed
values more accurately than direct observation does; that is why I
make the curve pass near the points and not through the points
themselves.

Here, then, is a problem in the probability of causes. The effects
are the measurements I have recorded; they depend on the com-
bination of two causes — the true law of the phenomenon and errors
of observation. Knowing the effects, we have to find the probability
that the phenomenon shall obey this law or that, and that the observa-
tions have been accompanied by this or that error. The most probable
law, therefore, corresponds to the curve we have traced, and the most
probable error is represented by the distance of the corresponding point
from that curve. But the problem has no meaning if before the observa-
tions I had an a priori idea of the probability of this law or that,
or of the chances of error to which I am exposed. If my instruments
are good (and I knew whether this is so or not before beginning the
observations), I shall not draw the curve far from the points which
represent the rough measurements. If they are inferior, I may draw
it a little farther from the points, so that I may get a less sinuous
•curve; much will be sacrificed to regularity.

Why, then, do I draw a curve without sinuosities? Because I
consider a priori a law represented by a continuous function (or
function the derivatives of which to a high order are small), as
more probable than a law not satisfying those conditions. But for
this conviction the problem would have no meaning; interpolation
would be impossible ; no law could be deduced from a finite number
of observations; science would cease to exist.

Fifty years ago physicists considered, other things being equal, a
simple law as more probable than a complicated law.- This prin-
ciple was even invoked in favor of Mariotte's law as against that of
Regnault. But this belief is now repudiated; and yet, how many
times are we compelled to act as though we still held it! However
that may be, what remains of this tendency is the belief in continuity,
and as we have just seen, if the belief in continuity were to disap-
pear, experimental science would become impossible.

VI. The Theory of Errors. — We are thus brought to consider the
theory of errors which is directly connected with the problem of the
probability of causes. Here again we find effect* — to wit, a certain
number of irreconcilable observations, and we try to find the causes
which are, on the one hand, the true value of the quantity to be
measured, and, on the other, the error made in each isolated observa-
tion. We must calculate the probable a posteriori value of each error,
and therefore the probable value of the quantity to be measured.
But, as I have just explained, we cannot undertake this calculation
unless we admit a priori — i.e., before any observations are made —

730 SCIENCE AND HYPOTHESIS

that there is a law of the probability of errors. Is there a law of
errors? The law to which all calculators assent is Gauss's law, that
is represented by a certain transcendental curve known as the " bell."

But it is first of all necessary to recall the classic distinction be-
tween systematic and accidental errors. If the metre with which we
measure a length is too long, the number we get will be too small,
and it will be no use to measure several times — that is a systematic
error. If we measure with an accurate metre, we may make a mis-
take, and find the length sometimes too large and sometimes too
small, and when we take the mean of a large number of measurements,
the error will tend to grow small. These are accidental errors.

It is clear that systematic errors do not satisfy Gauss's
law, but do accidental errors satisfy it? Numerous proofs have
been attempted, almost all of them crude paralogisms. But start-
ing from the following hypotheses we may prove Gauss's law:
the error is the result of a very large number of partial and inde-
pendent errors; each partial error is very small and obeys any law
of probability whatever, provided the probability of a positive error
is the same as that of an equal negative error. It is clear that these
conditions will be often, but not always, fulfilled, and we may reserve
the name of accidental for errors which satisfy them.

We see that the method of least squares is not legitimate in every
case; in general, physicists are more distrustful of it than astrono-
mers. This is no doubt because the latter, apart from the systematic
errors to which they and the physicists are subject alike, have to con-
tend with an extremely important source of error which is entirely
accidental — I mean atmospheric undulations. So it is very curi-
ous to hear a discussion between a physicist and an astronomer about
a method of observation. The physicist, persuaded that one good
measurement is worth more than many bad ones, is pre-eminently
concerned with the elimination by means of every precaution of the
final systematic errors ; the astronomer retorts : " But you can only
observe a small number of stars, and accidental errors will not dis-
appear."

What conclusion must we draw ? Must we continue to use the
method of least squares ? We must distinguish. We have eliminated
all the systematic errors of which we have any suspicion ; we are quite
certain that there are others still, but we cannot detect them ; and yet
we must make up our minds and adopt a definitive value which will
be regarded as the probable value; and for that purpose it is clear
that the best thing we can do is to apply Gauss's law. We have only
applied a practical rule referring to subjective probability. And there
is no more to be said.

Yet we want to go farther and say that not only the probable
value is so much, but that the probable error in the re-

NATUEE 731

suit is so much. This is absolutely invalid: it would be true
only if we were sure that all the systematic errors were eliminated,
and of that we know absolutely nothing. We have two series of ob-
servations; by applying the law of least squares we find that the
probable error in the first series is twice as small as in the second.
The second series may, however, be more accurate than the first,
because the first is perhaps affected by a large systematic error. All
that we can say is, that the first series is probably better than the
second because its accidental error is smaller, and that we have no
reason for affirming that the systematic error is greater for one of
the series than for the other, our ignorance on this point being ab-
solute.

VII. Conclusion. — In the preceding lines I have set several pro-
blems, and have given no solution. I do not regret this, for perhaps
they will invite the reader to reflect on these delicate questions.

However that may be, there are certain points which seem to be
well established. To undertake the calculation of any probability, and
even for that calculation to have any meaning at all, we must admit,
as a point of departure, an hypothesis or convention which has always
something arbitrary about it. In the choice of this convention we
can be guided only by the principle of sufficient reason. Unfortu-
nately, this principle is very vague and very elastic, and in the cur-
sory examination we have just made we have seen it assume different
forms. The form under which we meet it most often is the belief in
continuity, a belief which it would be difficult to justify by apodeictic
reasoning, but without which all science would be impossible. Finally,
the problems to which the calculus of probabilities may be applied
with profit are those in which the result is independent of the hypo-
thesis made at the outset, provided only that this hypothesis satisfies
the condition of continuity.

Optics and Electricity l

Fresnel's Theory. — The best example that can be chosen is the
theory of light and its relations to the theory of electricity. It is
owing to Fresnel that the science of optics is more advanced than
any other branch of physics. The theory called the theory of undu-
lations forms a complete whole, which is satisfying to the mind; but
we must not ask from it what it cannot give us. The object of math-
ematical theories is not to reveal to us the real nature of things; that
would be an unreasonable claim. Their only object is to co-ordinate
the physical laws with which physical experiment makes us acquainted,

i This section is mainly taken from the prefaces of two of my books —
Thdorie Hathtmatique de la lumicre (Paris: Naud, 1889). and Electricity et
Optique (Paris: Naud, 1901).

732 SCIENCE AND HYPOTHESIS

enunciation of which, without the aid of mathematics, we should be
unable to effect. Whether the ether exists or not matters little —
let us leave that to the metaphysicians; what is essential for us is,
that everything happens as if it existed, and that this hypothesis
is found to be suitable for the explanation of phenomena. After all,
have we any other reason for believing in the existence of material
objects? That, too, is only a convenient hypothesis; only, it will
never cease to be so, while some day, no doubt, the ether will be
thrown aside as useless.

But at the present moment the laws of optics, and the equations
which translate them into the language of analysis, hold good — at
least as a first approximation. It will therefore be always useful to
study a theory which brings these equations into connection.

The undulatory theory is based on a molecular hypothesis; this is
an advantage to those who think they can discover the cause under
the law. But others find in it a reason for distrust; and this distrust
seems to me as unfounded as the illusions of the former. These
hypotheses play but a secondary role. They may be sacrificed, and
the sole reason why this is not generally done is, that it would involve
a certain loss of lucidity in the explanation. In fact, if we look at it
a little closer we shall see that we borrow from molecular hypotheses
but two things — the principle of the conservation of energy, and
the linear form of the equations, which is the general law of small
movements as of all small variations. This explains why most of
the conclusions of Fresnel remain unchanged when we adopt the
electro-magnetic theory of light.

Maxwell's Theory. — We all know that it was Maxwell who con-
nected by a slender tie two branches of physics — optics and electricity
— until then unsuspected of having anything in common. Thus
blended in a larger aggregate, in a higher harmony, Fresnel's theory
of optics did not perish. Parts of it are yet alive, and their mutual
relations are still the same. Only, the language which we use to
express them has changed; and, on the other hand, Maxwell has re-
vealed to us other relations, hitherto unsuspected, between the differ-
ent branches of optics and the domain of electricity.

The first time a French reader opens Maxwell's book, his admira-
tion is tempered with a feeling of uneasiness, and often of distrust.

It is only after prolonged study, and at the cost of much effort, that
this feeling disappears. Some minds of high calibre never lose this
feeling. Why is it so difficult for the ideas of this English scientist
to become acclimatized among us? No doubt the education received
by most enlightened Frenchmen predisposes them to appreciate pre-
cision and logic more than any other qualities. In this respect the
old theories of mathematical physics gave us complete satisfaction.
All our masters, from Laplace to Cauchy, proceeded along the same

NATURE 733

lines. Starting with clearly enunciated hypotheses, they deduced
from them all their consequences with mathematical rigor, and then
compared them with experiment. It seemed to be their aim to give to
each of the branches of physics the same precision as to celestial
mechanics.

A mind accustomed to admire such models is not easily satisfied
with a theory. Not only will it not tolerate the least appearance of
contradiction, but it will expect the different parts to be logically
connected with one another, and will require the number of hypo-
theses to be reduced to a minimum.

This is not all; there will be other demands which appear to me
to be less reasonable. Behind the matter of which our senses are
aware, and which is made known to us by experiment, such a thinker
will expect to see another kind of matter — the only true matter in
his opinion — which will no longer have anything but purely geo-
metrical qualities, and the atoms of which will be mathematical points
subject to the laws of dynamics alone. And yet he will try to repre-
sent to himself, by an unconscious contradiction, these invisible and
colorless atoms, and therefore to bring them as close as possible to
ordinary matter.

Then only will he be thoroughly satisfied, and he will then imagine
that he has penetrated the secret of the universe. Even if the satis-
faction is fallacious, it is none the less difficult to give it up. Thus, on
opening the pages of Maxwell, a Frenchman expects to find a theo-
retical whole, as logical and as precise as the physical optics that is
founded on the hypothesis of the ether. He is thus preparing for
himself a disappointment which I should like the reader to avoid;
so I will warn him at once of what he will find and what he will not
find in Maxwell.

Maxwell does not give a mechanical explanation of electricity and
magnetism; he confines himself to showing that such an explanation
is possible. He shows that the phenomena of optics are only a par-
ticular case of electro-magnetic phenomena. From the whole theory
of electricity a theory of light can be immediately deduced. Unfor-
tunately the converse is not true; it is not always easy to find a
complete explanation of electrical phenomena. In particular it is
not easy if we take as our starting-point Fresnel's theory; to do so,
no doubt, would be impossible; but none the less we must ask our-
selves if we are compelled to surrender admirable results which we
thought we had definitively acquired. That seems a step backwards,
and many sound intellects will not willingly allow of this.

Should the reader consent to set some bounds to his hopes, he will
still come across other difficulties. The English scientist does not
try to erect a unique, definitive, and well-arranged building; he seems
to raise rather a large number of provisional and independent con-

734 SCIENCE AND HYPOTHESIS

structions, between which communication is difficult and sometimes
impossible. Take, for instance, the chapter in which electrostatic at-
tractions are explained by the pressures and tensions of the dielectric
medium. This chapter might be suppressed without the rest of the
book being thereby less clear or less complete, and yet it contains a
theory which is self-sufficient, and which can be understood without
reading a word of what precedes or follows. But it is not only inde-
pendent of the rest of the book; it is difficult to reconcile it with
the fundamental ideas of the volume. Maxwell does not even attempt
to reconcile it ; he merely says : " I have not been able to makie the
next step — namely, to account by mechanical considerations for
these stresses in the dielectric."

This example will be sufficient to show what I mean; I could quote
many others. Thus, who would suspect on reading the pages devoted
to magnetic rotatory polarization that there is an identity between
optical and magnetic phenomena?

We must not flatter ourselves that we have avoided every contra-
diction, but we ought to make up our minds. Two contradictory
theories, provided that they are kept from overlapping, and that
we do not look to find in them the explanation of things, may, in
fact, be very useful instruments of research ; and perhaps the reading
of Maxwell would be less suggestive if he had not opened up to us
so many new and divergent ways. But the fundamental idea is
masked, as it were. So far is this the case, that in most works that
are popularized, this idea is the only point which is left completely
untouched. To show the importance of this, I think I ought to
explain in what this fundamental idea consists; but for that purpose
a short digression is necessary.

The Mechanical Explanation of Physical Phenomena. — In every
physical phenomenon there is a certain number of parameters which
are reached directly by experiment, and which can be measured. I
shall call them the parameters q. Observation next teaches us the
laws of the variations of these parameters, and these laws can be
generally stated in the form of differential equations which connect
together the parameters q and time. What can be done to give a
mechanical interpretation to such a phenomenon? We may endeavor
to explain it, either by the movements of ordinary matter, or by those
of one or more hypothetical fluids. These fluids will be considered as
formed of a very large number of isolated molecules m. When may
we say that we have a complete mechanical explanation of the phe-
nomenon? It will be, on the one hand, when we know the differen-
tial equations which are satisfied by the co-ordinates of these hypo-
thetical molecules m, equations which must, in addition, conform to
the laws of dynamics; and, on the other hand, when we know the
relations which define the co-ordinates of the molecules m as func-

NATURE 735

tions of the parameters q, attainable by experiment. These equations,
as I have said, should conform to the principles of dynamics, and, in
particular, to the principle of the conservation of energy, and to that
of least action.

The first of these two principles teaches us that the total energy is
constant, and may be divided into two parts :

(1) Kinetic energy, or vis viva, which depends on the masses of
the hypothetical molecules m, and on their velocities. This I shall
call T. (2) The potential energy which depends only on the co-
ordinates of these molecules, and this I shall call U. It is the sum
of the energies T and U that is constant.

Now what are we taught by the principle of least action? It
teaches us that to pass from the initial position occupied at the in-
stant t0 to the final position occupied at the instant t1} the system
must describe such a path that in the interval of time between the
instant t0 and t1} the mean value of the action — i.e., the difference
between the two energies T and U, must be as small as possible. The
first of these two principles is, moreover, a consequence of the second.
If we know the functions T and U, this second principle is sufficient
to determine the equations of motion.

Among the paths which enable us to pass from one position to
another, there is clearly one for which the mean value of the action
is smaller than for all the others. In addition, there is only such
path; and it follows from this, that the principle of least action is
sufficient to determine the path followed, and therefore the equations
of motion. We thus obtain what are called the equations of La-
grange. In these equations the independent variables are the co-ordi-
nates of the hypothetical molecules m; but I now assume that we take
for variables the parameters q, which are directly accessible to experi-
ment.

The two parts of the energy should then be expressed as a function
of the parameters q and their derivatives; it is clear that it is under
this form that they will appear to the experimenter. The latter will
naturally endeavor to define kinetic and potential energy by the aid
of quantities he can directly observe.1 If this be granted, the system
will always proceed from one position to another by such a path that
the mean value of the action is a minimum. It matters little that T
and U are now expressed by the aid of the parameters q and their
derivatives ; it matters little that it is also by the aid of these parame-
ters that we define the initial and final positions; the principle of
least action will always remain true.

Now here again, of the whole of the paths which lead from one

1 We may add that U will depend only on the q parameters, that T will
depend on them and their derivatives with respect to time, and will be
a homogeneous polynomial of the second degree with respect to these deriva-
tives.

736 SCIENCE AND HYPOTHESIS

position to another, there is one and only one for which the mean
action is a minimum. The principle of least action is therefore suffi-
cient for the determination of the differential equations which define
the variations of the parameters q. The equations thus obtained are
another form of Lagrange's equations.

To form these equations we need not know the relations which
connect the parameters g with the co-ordinates of the hypothetical
molecules, nor the masses of the molecules, nor the expression of U
as a function of the co-ordinates of these molecules. All we need
know is the expression of TJ as a function of the parameters q, and
that of T as a function of the parameters q and their derivatives —
i.e., the expressions of the kinetic and potential energy in terms of
experimental data.

One of two things must now happen. Either for a convenient choice
of T and U the Lagrangian equations, constructed as we have indi-
cated, will be identical with the differential equations deduced from
experiment, or there will be no functions T and U for which this
identity takes place. In the latter case it is clear that no mechanical
explanation is possible. The necessary condition for a mechanical
explanation to be possible is therefore this: that we may choose the
functions T and U so as to satisfy the principle of least action, and
of the conservation of energy. Besides, this condition is sufficient.
Suppose, in fact, that we have found a function U of the parameters
q, which represents one of the parts of energy, and that the part of
the energy which we represent by T is a function of the parameters q
and their derivatives; that it is a polynomial of the second degree
with respect to its derivatives, and finally that the Lagrangian equa-
tions formed by the aid of these two functions T and U are in con-
formity with the data of the experiment. How can we deduce from
this a mechanical explanation? U must be regarded as the potential
energy of a system of which T is the kinetic energy. There is no
difficulty as far as U is concerned, but can T be regarded as the vis
viva of a material system?

It is easily shown that this is always possible, and in an unlimited
number of ways. I will be content with referring the reader to the
pages of the preface of my Electricite et Optique for further details.
Thus, if the principle of least action cannot be satisfied, no mechanical
explanation is possible; if it can be satisfied, there is not only one
explanation, but an unlimited number, whence it follows that since
there is one there must be an unlimited number.

One more remark. Among the quantities that may be reached by
experiment directly we shall consider some as the co-ordinates of our
hypothetical molecules, some will be our parameters q, and the rest
will be regarded as dependent not only on the co-ordinates but on
the velocities — or what comet? to the same thing, we look on them

NATUKE 737

as derivatives of the parameters q, or as combinations of these para-
meters and their derivatives.

Here then a question occurs : among all these quantities measured
experimentally which shall we choose to represent the parameters q?
and which shall we prefer to regard as the derivatives of these para-
meters? This choice remains arbitary to a large extent, but a me-
chanical explanation will be possible if it is done so as to satisfy the
principle of least action.

Next, Maxwell apks: Can this choice and that of the two energies
T and U be made so that electric phenomena will satisfy this prin-
ciple? Experiment shows us that the energy of an electro-magnetic
field decomposes into electro-static and electro-dynamic energy. Max-
well recognized that if we regard the former as the potential energy
U, and the latter as the kinetic energy T, and that if on the other
hand we take the electro-static charges of the conductors as the para-
meters q, and the intensity of the currents as derivatives of other
parameters q — under these conditions, Maxwell has recognized that
electric phenomena satisfies the principle of least action. He was
then certain of a mechanical explanation. If he had expounded this
theory at the beginning of his first volume, instead of relegating it to
a corner of the second, it would not have escaped the attention of
most readers. If therefore a phenomenon allows of a complete me-
chanical explanation, it allows of an unlimited number of others,
which will equally take into account all the particulars revealed by
experiment. And this is confirmed by the history of every branch of
physics. In Optics, for instance, Fresnel believed vibration to be
perpendicular to the plane of polarization; Neumann holds that it is
parallel to that plane. For a long time an experimentum crucis was
sought for, which would enable us to decide between these two theo-
ries, but in vain. In the same way, without going out of the domain
of electricity, we find that the theory of two fluids and the single fluid
theory equally account in a satisfactory manner for all the laws of
electro-statics. All these facts are easily explained, thanks to the
properties of the Lagrange equations.

Jt is easy now to understand Maxwell's fundamental idea. To
demonstrate the possibility of a mechanical explanation of electricity
we need not trouble to find the explanation itself; we need only know
the expression of the two functions T and U, which are the two
parts of energy, and to form with these two functions Lagrange'ft
equations, and then to compare these equations with the experimental
laws.

How shall we choose from all the possible explanations one in
which the help of experiment will be wanting? The day will per-
haps come when physicists will no longer concern themselves with
questions which are inaccessible to positive methods, and will leave

738 SCIENCE AND HYPOTHESIS

them to the metaphysicians. That day has not yet come; man does
not so easily resign himself to remaining for ever ignorant of the
causes of things. Our choice cannot be therefore any longer guided
by considerations in which personal appreciation plays too large a
part. There are, however, solutions which all will reject because of
their fantastic nature, and others which all will prefer because of their
simplicity. As far as magnetism and electricity are concerned, Max-
well abstained from making any choice. It is not that he has a
systematic contempt for all that positive methods cannot reach, as
may be seen from the time he has devoted to the kinetic theory of
gases. I may add that if in his magnum opu& he develops no complete
explanation, he has attempted one in an article in the Philosophical
Magazine. The strangeness and the complexity of the hypotheses he
found himself compelled to make, led him afterwards to withdraw it.
The same spirit is found throughout his whole work. He throws
into relief the essential — i.e., what is common to all theories ; every-
thing that suits only a particular theory is passed over almost in
silence. The reader therefore finds himself in the presence of form
nearly devoid of matter, which at first he is tempted to take as a
fugitive and unassailable phantom. But the efforts he is thus com-
pelled to make force him to think, and eventually he sees that there
is often something rather artificial in the theoretical " aggregates "
which he once admired.

Electro-Dynamics

The history of electro-dynamics is very instructive from our point
of view. The title of Ampere's immortal work is, Theorie des phe-
nomenes electro-dynamiques, uniquement fondee sur experience. He
therefore imagined that he had made no hypotheses; but as we shall
not be long in recognizing, he was mistaken; only, of these hypothe-
ses he was quite unaware. On the other hand, his successors see
them clearly enough, because their attention is attracted by the weak
points in Ampere's solution. They made fresh hypotheses, but this
time deliberately. How many times they had to change them before
they reached the classic system, which is perhaps even now not quite
definitive, we shall see.

I. Ampere's Theory. — In Ampere's experimental study of the
mutual action of currents, he has operated, and he could operate only,
with closed currents. This was not because he denied the existence
or possibility of open currents. If two conductors are positively and
negatively charged and brought into communication by a wire, a cur-
rent is set up which passes from one to the other until the two po-
tentials are equal. According to the ideas of Ampere's time, this was
considered to be an open current; the current was known to pass

NATURE 739

from the first conductor to the second, but they did not know it
returned from the second to the first. All currents of this kind were
therefore considered by Ampere to be open currents — for instance,
the currents of discharge of a condenser; he was unable to experi-
ment on them, their duration being too short. Another kind of open
current may be imagined. Suppose we have two conductors A and
B connected by a wire AMB. Small conducting masses in motion
are first of all placed in contact with the conductor B, receive an elec-
tric charge, and leaving B are set in motion along a path BNA, carry-
ing their charge with them. On coming into contact with A they
lose their charge, which then returns to B along the wire AMB. Now
here we have, in a sense, a closed circuit, since the electricity describes
the closed circuit BNAMB ; but the two parts of the current are quite
different. In the wire AMB the electricity is displaced through a
fixed conductor like a voltaic current, overcoming an ohmic resistance
and developing heat; we say that it is displaced by conduction. In
the part BJSTA the electricity is carried by a moving conductor, and
is said to be displaced by convection. If therefore the convection cur-
rent is considered to be perfectly analogous to the conduction current,
the circuit BNAMB is closed; if on the contrary the convection cur-
rent is not a " true current," and, for instance, does not act on the
magnet, there is only the conduction current AMB, which is open.
For example, if we connect by a wire the poles of a Holtz machine,
the charged rotating disc transfers the electricity by convection from
one pole to the other, and it returns to the first pole by conduction
through the wire. But currents of this kind are very difficult to pro-
duce with appreciable intensity; in fact, with the means at Ampere's
disposal we may almost say it was impossible.

To sum up, Ampere could conceive of the existence of two kinds of
open currents, but he could experiment on neither, because they were
not strong enough, or because their duration was too short. Experi-
ment therefore could only show him the action of a closed current
on a closed current — or more accurately, the action of a closed cur-
rent on a portion of current, because a current can be made to de-
scribe a closed circuit, of which part may be in motion and the other
part fixed. The displacements of the moving part may be studied
under the action of another closed current. On the other hand, Am-
pere had no means of studying the action of an open current either
on a closed or on another open current.

1. The Case of Closed Currents. — In the case of the mutual action
of two closed currents, experiment revealed to Ampere remarkably
simple laws. The following will be useful to us in the sequel : —

(1) // the intensity of the currents is kept constant, and if the
two circuits, after having undergone any displacements and deforma-
tions whatever, return finally to their initial positions, the total work

740 SCIENCE AND HYPOTHESIS

done by the electro-dynamical actions is zero. In other words, there
is an electro-dynamical potential of the two circuits proportional to
the product of their intensities, and depending on the form and rela-
tive positions of the circuits; the work done by the electro-dynamical
actions is equal to the change of this potential.

(2) The action of a closed solenoid is zero.

(3) The action of a circuit C on another voltaic circuit C' depends
only on the " magnetic field " developed by the circuit C. At each
point in space we can, in fact, define in magnitude and direction a
certain force called "magnetic force," which enjoys the following
properties : —

(a) The force exercised by C on a magnetic pole is applied to that
pole, and is equal to the magnetic force multiplied by the magnetic
.mass of the pole.

(&) A very short magnetic needle tends to take the direction of the
magnetic force, and the couple to which it tends to reduce is propor-
tional to the product of the magnetic force, the magnetic moment
of the needle, and the sine of the dip of the needle.

(c) If the circuit C' is displaced, the amount of the work done by
the electro-dynamic action of C on C' will be equal to the increment of
" flow of magnetic force " which passes through the circuit.

2. Action of a Closed Current on a Portion of Current. — Ampere
being unable to produce the open current properly so called, had only
one way of studying the action of a closed current on a portion of
current. This was by operating on a circuit C composed of two parts,
one movable and the other fixed. The movable part was, for instance,
a movable wire aft, the ends a and ft of which could slide along a
fixed wire. In one of the positions of the movable wire the end a
rested on the point A, and the end ft on the point B of the fixed
wire. The current ran from a to ft — i.e., from A to B along the
movable wire, and then from B to A along the fixed wire. This cur-
rent was therefore closed.

In the second position, the movable wire having slipped, the points
a and ft were respectively at A' and B' on the fixed wire. The current
ran from a to ft — i.e., from A' to B' on the movable wire, and re-
turned from B' to B, and then from B to A, and then from A to A' —
all on the fixed wire. This current was also closed. If a similar
circuit be exposed to the action of a closed current C, the movable
part will be displaced just as if it were acted on by a force. Ampere
admits that the force, apparently acting on the movable part A B,
representing the action of C on the portion aft of the current, re-
mains the same whether an open current runs through aft, stopping at
a and ft, or whether a closed current runs first to ft, and then returns
to a through the fixed portion of the circuit. This hypothesis seemed
natural enough, and Ampere innocently assumed it; nevertheless the

NATUEE 741

hypothesis is not a necessity, for we shall presently see that Helmholtz
rejected it. However that may be, it enabled Ampere, although he
had never produced an open current, to lay down the laws of the
action of a closed current on an open current, or even on an element
of current. They are simple :

(1) The force acting on an element of current is applied to that
element; it is normal to the element and to the magnetic force, and
proportional to that component of the magnetic force which is nor-
mal to the element.

(2) The action of a closed solenoid on an element of current is
zero. But the electro-dynamic potential has disappeared — i.e., when
a closed and an open current of constant intensities return to their
initial positions, the total work done is not zero.

3. Continuous Rotations. — The most remarkable electro- dynamical
experiments are those in which continuous rotations are produced,
and which are called unipolar induction experiments. A magnet may
turn about its axis; a current passes first through a fixed wire and
then enters the magnet by the pole N, for instance, passes through
half the magnet, and emerges by a sliding contact and re-enters the
fixed wire. The magnet then begins to rotate continuously. This is
Faraday's experiment. How is it possible? If it were a question of
two circuits of invariable form, C fixed and C' movable about an axis,
the latter would never take up a position of continuous rotation; in
fact, there is an electro-dynamical potential; there must therefore be
a position of equilibrium when the potential is a maximum. Con-
tinuous rotations are therefore possible only when the circuit C' is
composed of two parts — one fixed, and the other movable about an
axis, as in the case of Faraday's experiment. Here again it is con-
venient to draw a distinction. The passage from the fixed to the
movable part, or vice versa, may take place either by simple contact,
the same point of the movable part remaining constantly in contact
with the same point of the fixed part, or by sliding contact, the
same point of the movable part coming successively into contact with
the different points of the fixed part.

It is only in the second case that there can be continuous rotation.
This is what then happens : — the system tends to take up a position
of equilibrium ; but, when at the point of reaching that position, the
sliding contact puts the moving part in contact with a fresh point in
the fixed part; it changes the connections and therefore the conditions
of equilibrium, so that as the position of equilibrium is ever eluding,
PO to speak, the system which is trying to reach it, rotation may take
place indefinitely.

Ampere admits that the action of the circuit on the movable part
of C' is the same as if the fixed part of C' did not exist, and therefore
as if the current passing through the movable part were an open

742 SCIENCE AND HYPOTHESIS

current. He concluded that the action of a closed on an open cur-
rent, or vice versa, that of an open current on a fixed current, may
give rise to continuous rotation. But this conclusion depends on
the hypothesis which I have enunciated, and to which, as I said
above, Helmholtz declined to subscribe.

4. Mutual Action of Two Open Currents. — As far as the mutual
action of two open currents, and in particular that of two elements of
current, is concerned, all experiment breaks down. Ampere falls
back on hypothesis. He assumes: (1) that the mutual action of two
elements reduces to a force acting along their join; (2) that the
action, of two closed currents is the resultant of the mutual actions of
their different elements, which are the same as if these elements
were isolated.

The remarkable thing is that here again Ampere makes two hypo-
theses without being aware of it. However that may be, these two
hypotheses, together with the experiments on closed currents, suffice
to determine completely the law of mutual action of two elements.
But then, most of the simple laws we have met in the case of closed
currents are no longer true. In the first place, there is no electro-
dynamical potential; nor was there any, as we have seen, in the case
of a closed current acting on an open current. Next, there is, pro-
perly speaking, no magnetic force; and we have above defined this
force in three different ways: (1) By the action on a magnetic pole;
(2) by the director couple which orientates the magnetic needle; (3)
by the action on an element of current.

In the case with which we are immediately concerned, not only
are these three definitions not in harmony, but each has lost its
meaning : —

( 1 ) A magnetic pole is no longer acted on by a unique force applied
to that pole. We have seen, in fact, the action of an element of cur-
rent on a pole is not applied to the pole but to the element; it may,
moreover, be replaced by a force applied to the pole and by a couple.

(2) The couple which acts on the magnetic needle is no longer a
simple director couple, for its moment with respect to the axis of the
needle is not zero. It decomposes into a director couple, properly so
called, and a supplementary couple which tends to produce the con-
tinuous rotation of which we have spoken above.

(3) Finally, the force acting on an element of a current is not
normal to that element. In other words, the unity of the magnetic
force has disappeared.

Let us see in what this unity consists. Two systems which exercise
the same action on a magnetic pole will also exercise the same action
on an indefinitely small magnetic needle, or on an element of cur-
rent placed at the point in space at which the pole is. Well, this is
true if the two systems only contain closed currents, and according

NATUKE 743

to Ampere it would not be true if the systems contained open cur-
rents. It is sufficient to remark, for instance, that if a magnetic pole
is placed at A and an element at B, the direction of the element being
in AB produced, this element, which will exercise no action on the pole,
will exercise an action either on a magnetic needle placed at A, or on
an element of current at A.

5. Induction. — We know that the discovery of electro-dynamical
induction followed not long after the immortal work of Ampere. As
long as it is only a question of closed currents there is no difficulty,
and Helmholtz has even remarked that the principle of the conserva-
tion of energy is sufficient for us to deduce the laws of induction from
the electro-dynamical laws of Ampere. But on the condition, as
Bertrand has shown, — that we make a certain number of hypothe-
ses.

The same principle again enables this deduction to be made in the
case of open currents, although the result cannot be tested by experi-
ment, since such currents cannot be produced.

If we wish to compare this method of analysis with Ampere's
theorem on open currents, we get results which are calculated to
surprise us. In the first place, induction cannot be deduced from
the variation of the magnetic field by the well-known formula of
scientists and practical men; in fact, as I have said, properly speaking,
there is no magnetic field. But further, if a circuit C is subjected to
the induction of a variable voltaic system S, and if this system S be
displaced and deformed in any way whatever, so that the intensity of
the currents of this system varies according to any law whatever,
then so long as after these variations the system eventually returns to
its initial position, it seems natural to suppose that the mean electro-
motive force induced in the current C is zero. This is true if the
circuit C is closed, and if the system S only contains closed currents.
It is no longer true if we accept the theory of Ampere, since there
would be open currents. So that not only will induction no longer
be the variation of the flow of magnetic force in any of the usual
senses of the word, but it cannot be represented by the variation of
that force whatever it may be.

II. Ilelmlioliz's Theory. — I have dwelt upon the consequences of
Ampere's theory and on his method of explaining the action of open
currents. It is difficult to disregard the paradoxical and artificial
character of the propositions to which we are thus led. We feel bound
to think " it cannot be so." We may imagine then that Helmholtz
has been led to look for something else. He rejects the fundamental
hypothesis of Ampere — namely, that the mutual action of two ele-
ments of current reduces to a force along their join. He admits that
an element of current is not acted upon by a single force but by a
force and a couple, and this is what gave rise to the celebrated polemic

744 SCIENCE AND HYPOTHESIS

between Bertrand and Helmholtz. Helmholtz replaces Ampere's hypo-
thesis by the following : — Two elements of current always admit of
an electro-dynamic potential, depending solely upon their position
and orientation; and the work of the forces that they exercise one on
the other is equal to the variation of this potential. Thus Helmholtz
can no more do without hypothesis than Ampere, but at least he does
not do so without explicitly announcing it. In the case of closed
currents, which alone are accessible to experiment, the two theories
agree; in all other cases they differ. In the first place, contrary to
what Ampere supposed, the force which seems to act on the movable
portion of a closed current is not the same as that acting on the
movable portion if it were isolated and if it constituted an open cur-
rent. Let us return to the circuit C', of which we spoke above, and
which was formed of a movable wire sliding on a fixed wire. In the
only experiment that can be made the movable portion a /3 is not
isolated, but is part of a closed circuit. When it passes from AB to
A'B', the total electro-dynamic potential varies for two reasons. First,
it has a slight increment because the potential of A'B' with respect to
the circuit C is not the same as that of AB ; secondly, it has a second
increment because it must be increased by the potentials of the ele-
ments AA' and B'B with respect to C. It is this double increment
which represents the work of the force acting upon the portion AB.
If, on the contrary, aft be isolated, the potential would only have the
first increment, and this first increment alone would measure the work
of the force acting on AB. In the second place, there could be no
continuous rotation without sliding contact, and in fact, that, as we
have seen in the case of closed currents, is an immediate consequence
of the existence of an electro-dynamic potential. In Faraday's experi-
ment, if the magnet is fixed, and if the part of the current external
to the magnet runs along a movable wire, that movable wire may
undergo continuous rotation. But it does not mean that, if the con-
tacts of the wire with the magnet were suppressed, and an open cur-
rent were to run along the wire, the wire would still have a movement
of continuous rotation. I have just said, in fact, that an isolated
element is not acted on in the same way as a movable element making
part of a closed circuit. But there is another difference. The action
of a solenoid on a closed current is zero according to experiment and
according to the two theories. Its action on an open current would be
zero according to Ampere, and it would not be zero according to
Helmholtz. From this follows an important consequence. We have
given above three definitions of the magnetic force. The third has
no meaning here, since an element of current is no longer acted upon
by a single force. Nor has the first any meaning. What, in fact, is
a magnetic pole ? It is the extremity of an indefinite linear magnet.
This magnet may be replaced by an indefinite solenoid. For

NATUEE 745

the definition of magnetic force to have any meaning, the action exer-
cised by an open current on an indefinite solenoid would only depend
on the position of the extremity of that solenoid — i.e., that the action
of a closed solenoid is zero. Now we have just seen that this is not
the case. On the other hand, there is nothing to prevent us from
adopting the second definition which is founded on the measurement
of the director couple which tends to orientate the magnetic needle;
but, if it is adopted, neither the effects of induction nor electro-
dynamic effects will depend solely on the distribution of the lines of
force in this magnetic field.

III. Difficulties raised by these Theories. — Helmholtz's theory is
an advance on that of Ampere; it is necessary, however, that every
difficulty should be removed. In both, the name " magnetic field "
has no meaning, or, if we give it one by a more or less artificial con-
vention, the ordinary laws so familiar to electricians no longer apply;
and it is thus that the electro-motive force induced in a wire is no
longer measured by the number of lines of force met by that wire.
And our objections do not proceed only from the fact that it is diffi-
cult to give up deeply-rooted habits of language and thought. There
is something more. If we do not believe in actions at a distance, elec-
tro-dynamic phenomena must be explained by a modification of the
medium. And this medium is precisely what we call " magnetic field,"
and then the electro-magnetic effects should only depend on that field.
All these difficulties arise from the hypothesis of open currents.

IV. Maxwell's Theory. — Such were the difficulties raised by the
current theories, when Maxwell with a stroke of the pen caused them
to vanish. To his mind, in fact, all currents are closed currents.
Maxwell admits that if in a dielectric, the electric field happens to
vary, this dielectric becomes the seat of a particular phenomenon
acting on the galvanometer like a current and called the current of
displacement. If, then, two conductors bearing positive and negative
charges are placed in connection by means of a wire, during the dis-
charge there is an open current of conduction in that wire; but there
are produced at the same time in the surrounding dielectric cur-
rents of displacement which close this current of conduction. We
know that Maxwell's theory leads to the explanation of optical phe-
nomena which would be due to extremely rapid electrical oscillations.
At that period such a conception was only a daring hypothesis which
could be supported by no experiment; but after twenty years Max-
well's ideas received the confirmation of experiment. Hertz succeeded
in producing systems of electric oscillations which reproduce all the
properties of light, and only differ by the length of their wave —
that is to say, as violet differs from red. In some measure he made
a synthesis of light. It might be said that Hertz has not directly
proved Maxwell's fundamental idea of the action of the current of

746 SCIENCE AND HYPOTHESIS

displacement on the galvanometer. That is true in a sense. What
he has shown directly is that electro-magnetic induction is not in-
stantaneously propagated, as was supposed, but its speed is the speed
of light. Yet, to suppose there is no current of displacement, and
that induction is with the speed of light; or, rather, to suppose that
the currents of displacement produce inductive effects, and that the
induction takes place instantaneously — comes to the same thing.
This cannot be seen at the first glance, but it is proved by an analysis
of which I must not even think of giving even a summary here.

V. Rowland's Experiment. — But, as I have said above, there are
two kinds of open conduction currents. There are first the currents
of discharge of a condenser, or of any conductor whatever. There
are also cases in which the electric charges describe a closed contour,
being displaced by conduction in one part of the circuit and by con-
vection in the other part. The question might be regarded as solved
for open currents of the first kind; they were closed by currents of
displacement. For open currents of the second kind the solution
appeared still more simple.

It seemed that if the current were closed it could only be by the
current of convection itself. For that purpose it was sufficient to
admit that a " convection current " — i.e., a charged conductor in
motion — could act on the galvanometer. But experimental confirma-
tion was lacking. It appeared difficult, in fact, to obtain a sufficient
intensity even by increasing as much as possible the charge and the
velocity of the conductors. Eowland, an extremely skilful experi-
mentalist, was the first to triumph, or to seem to triumph, over these
difficulties. A disc received a strong electrostatic charge and a very
high speed of rotation. An astatic magnetic system placed beside the
disc underwent deviations. The experiment was made twice by Eow-
land, once in Berlin and once at Baltimore. It was afterwards re-
peated by Himstedt. These physicists even believed that they could
announce that they had succeeded in making quantitative measure-
ments. For twenty years Eowland's law was admitted without ob-
jection by all physicists, and, indeed, everything seemed to confirm it.
The spark certainly does produce a magnetic effect, and does it not
seem extremely likely that the spark discharged is due to particles
taken from one of the electrodes and transferred to the other electrode
with their charge? Is not the very spectrum of the spark, in which
we recognize the lines of the metal of the electrode, a proof of it?
The spark would then be a real current of induction.

On the other hand, it is also admitted that in an electrolyte the
electricity is carried by the ions in motion. The current in an electro-
lyte would therefore also be a current of convection; but it acts on
the magnetic needle. And in the same way for cathodic rays ; Crooks
attributed these rays to very subtle matter charged with negative elec-

NATUKE 747

tricity and moving with very high velocity. He looked upon them, in
other words, as currents of convection. Now, these cathodic rays are
deviated by the magnet. In virtue of the principle of action and re-
action, they should in their turn deviate the magnetic needle. It is
true that Hertz believed he had proved that the cathodic rays do not
carry negative electricity, and that they do not act on the magnetic
needle; but Hertz was wrong. First of all, Perrin succeeded in col-
lecting the electricity carried by these rays — electricity of which
Hertz denied the existence; the German scientist appears to have
been deceived by the effects due to the action of the X-rays, which
were not yet discovered. Afterwards, and quite recently, the action
of the cathodic rays on the magnetic needle has been brought to light.
Thus all these phenomena looked upon as currents of convection,
electric sparks, electrolytic currents, cathodic rays, act in the same
manner on the galvanometer and in conformity to Eowland's law.

VI. Lorentz's Theory. — We need not go much further. Accord-
ing to Lorentz's theory, currents of conduction would themselves be
true convection currents. Electricity would remain indissolubly con-
nected with certain material particles called electrons. The circula-
tion of these electrons through bodies would produce voltaic currents,
and what would distinguish conductors from insulators would be that
the one could be traversed by these electrons, while the others would
check the movement of the electrons. Lorentz's theory is very attrac-
tive. It gives a very simple explanation of certain phenomena, which
the earlier theories — even Maxwell's in its primitive form — could
only deal with in an unsatisfactory manner; for example, the aberra-
tion of light, the partial impulse of luminous waves, magnetic polari-
zation, and Zeeman's experiment.

A few objections still remained. The phenomena of an electric sys-
tem seemed to depend on the absolute velocity of translation of the
centre of gravity of this system, which is contrary to the idea that we
have of the relativity of space. Supported by M. Cremieu, M. Lipp-
man has presented this objection in a very striking form. Imagine
two charged conductors with the same velocity of translation. They
are relatively at rest. However, each of them being equivalent to a
current of convection, they ought to attract one another, and by
measuring this attraction we could measure their absolute velocity.
" N"o !" replied the partisans of Lorentz. " What we could measure in
that way is not their absolute velocity, but their relative velocity with
respect to the ether, so that the principle of relativity is safe." What-
ever there may be in these objections, the edifice of electro- dynamics
seemed, at any rate in its broad line?, definitively constructed. Every-
thing was presented under the most satisfactory aspect. The theories
of Ampere and Helmholtz, which were made for the open currents
that no longer existed, seem to have no more than purely historic in-

748

terest, and the inextricable complications to which these theories led
have been almost forgotten. This quiescence has been recently dis-
turbed by the experiments of M. Cr6mieu, which have contradicted,
or at least have seemed to contradict, the results formerly obtained
by Eowland. Numerous investigators have endeavored to solve the
question, and fresh experiments have been undertaken. What result
will they give ? I shall take care not to risk a prophecy which might
be falsified between the day this treatise is ready for the press and the
day on which it is placed before the public.

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