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+% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %
+%                                                                         %
+% The Project Gutenberg EBook of Science and hypothesis, by Henri Poincaré%
+%                                                                         %
+% This eBook is for the use of anyone anywhere at no cost and with        %
+% almost no restrictions whatsoever.  You may copy it, give it away or    %
+% re-use it under the terms of the Project Gutenberg License included     %
+% with this eBook or online at www.gutenberg.net                          %
+%                                                                         %
+%                                                                         %
+% Title: Science and hypothesis                                           %
+%                                                                         %
+% Author: Henri Poincaré                                                  %
+%                                                                         %
+% Release Date: August 21, 2011 [EBook #37157]                            %
+% Most recently updated: June 11, 2021                         %
+%                                                                         %
+% Language: English                                                       %
+%                                                                         %
+% Character set encoding: UTF-8                                           %
+%                                                                         %
+% *** START OF THIS PROJECT GUTENBERG EBOOK SCIENCE AND HYPOTHESIS ***    %
+%                                                                         %
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+%%   OCR text for this ebook was obtained on July 30, 2011, from    %%
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+%%                                                                  %%
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+%%%%%%%%%%%%%%%%%%%%%%%% START OF DOCUMENT %%%%%%%%%%%%%%%%%%%%%%%%%%
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+%% PG BOILERPLATE %%
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+\begin{center}
+\begin{minipage}{\textwidth}
+\small
+\begin{PGtext}
+The Project Gutenberg EBook of Science and hypothesis, by Henri Poincaré
+
+This eBook is for the use of anyone anywhere at no cost and with
+almost no restrictions whatsoever.  You may copy it, give it away or
+re-use it under the terms of the Project Gutenberg License included
+with this eBook or online at www.gutenberg.net
+
+
+Title: Science and hypothesis
+
+Author: Henri Poincaré
+
+Release Date: August 21, 2011 [EBook #37157]
+Most recently updated: June 11, 2021
+
+Language: English
+
+Character set encoding: UTF-8     
+
+*** START OF THIS PROJECT GUTENBERG EBOOK SCIENCE AND HYPOTHESIS ***
+\end{PGtext}
+\end{minipage}
+\end{center}
+\newpage
+%% Credits and transcriber's note %%
+\begin{center}
+\begin{minipage}{\textwidth}
+\begin{PGtext}
+Produced by Andrew D. Hwang
+\end{PGtext}
+\end{minipage}
+\vfill
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+
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+\end{minipage}
+%%%%%%%%%%%%%%%%%%%%%%%%%%% FRONT MATTER %%%%%%%%%%%%%%%%%%%%%%%%%%
+\PageSep{i}
+\FrontMatter
+\ifthenelse{\boolean{ForPrinting}}{%
+\null\vfill
+{\Large SCIENCE AND HYPOTHESIS}
+\vfill
+\cleardoublepage
+}{}% Omit half-title in screen version
+\PageSep{ii}
+%[Blank page]
+\PageSep{iii}
+\begin{center}
+\huge SCIENCE \\
+AND HYPOTHESIS
+\vfill
+
+\normalsize
+{\footnotesize BY} \\
+H. POINCARÉ, \\
+{\scriptsize MEMBER OF THE INSTITUTE OF FRANCE.}
+\vfill
+
+{\small\textsc{With a Preface by}} \\
+\textsc{J. LARMOR, D.Sc., Sec. R.S.}, \\
+{\scriptsize\textit{Lucasian Professor of Mathematics in the University of Cambridge.}}
+\vfill\vfill
+
+\textswab{London and Newcastle-on-Tyne:} \\
+THE WALTER SCOTT PUBLISHING CO., LTD \\
+{\footnotesize NEW YORK: 3 EAST 14TH STREET.} \\
+\oldstylenums{1905}.
+\end{center}
+\newpage
+\PageSep{iv}
+%[Blank page]
+\PageSep{v}
+\TableofContents
+\iffalse
+CONTENTS.
+
+                                       PAGE
+Translator's Note........................ix
+
+Introduction.............................xi
+
+Author's Preface........................xxi
+
+
+PART I.
+
+NUMBER AND MAGNITUDE.
+
+CHAPTER I.
+On the Nature of Mathematical Reasoning...1
+
+CHAPTER II.
+Mathematical Magnitude and Experiment....17
+
+
+PART II.
+
+SPACE.
+
+CHAPTER III.
+Non-Euclidean Geometries.................35
+\PageSep{vi}
+
+CHAPTER IV.
+Space and Geometry.......................51
+
+CHAPTER V.
+Experiment and Geometry..................72
+
+
+PART III.
+
+FORCE.
+
+CHAPTER VI.
+The Classical Mechanics..................89
+
+CHAPTER VII.
+Relative and Absolute Motion............111
+
+CHAPTER VIII.
+Energy and Thermo-dynamics..............123
+
+
+PART IV.
+
+NATURE.
+
+CHAPTER IX.
+Hypotheses in Physics...................140
+\PageSep{vii}
+
+CHAPTER X.
+The Theories of Modern Physics..........160
+
+CHAPTER XI.
+The Calculus of Probabilities...........183
+
+CHAPTER XII.
+Optics and Electricity..................211
+
+CHAPTER XIII.
+Electro-dynamics........................225
+\fi
+\PageSep{viii}
+%[Blank page]
+\PageSep{ix}
+
+
+\OtherChapter{Translator's Note.}
+
+\First{The} translator wishes to express his indebtedness
+to Professor Larmor, for kindly consenting
+to introduce the author of \Title{Science and Hypothesis}
+to English readers; to Dr.~F.~S. Macaulay and
+Mr.~C.~S. Jackson,~M.A., who have read the whole
+of the proofs and have greatly helped by suggestions;
+also to Professor G.~H. Bryan,~F.R.S., who
+has read the proofs of \ChapRef{VIII}., and whose
+criticisms have been most valuable.
+
+%[** TN: William John Greenstreet (1861--1930), not identified by name]
+\Signature{W.\;J.\;G.}{\textit{February} 1905.}
+\PageSep{x}
+%[Blank page]
+\PageSep{xi}
+
+
+\OtherChapter{Introduction.}
+
+\First{It} is to be hoped that, as a consequence of the
+present active scrutiny of our educational aims
+and methods, and of the resulting encouragement
+of the study of modern languages, we shall not
+remain, as a nation, so much isolated from
+ideas and tendencies in continental thought and
+literature as we have been in the past. As things
+are, however, the translation of this book is
+doubtless required; at any rate, it brings vividly
+before us an instructive point of view. Though
+some of M.~Poincaré's chapters have been collected
+from well-known treatises written several years
+ago, and indeed are sometimes in detail not quite
+up to date, besides occasionally suggesting the
+suspicion that his views may possibly have been
+modified in the interval, yet their publication in
+a compact form has excited a warm welcome in
+this country.
+
+It must be confessed that the English language
+\PageSep{xii}
+hardly lends itself as a perfect medium for the
+rendering of the delicate shades of suggestion
+and allusion characteristic of M.~Poincaré's play
+around his subject; notwithstanding the excellence
+of the translation, loss in this respect is
+inevitable.
+
+There has been of late a growing trend of
+opinion, prompted in part by general philosophical
+views, in the direction that the theoretical constructions
+of physical science are largely factitious,
+that instead of presenting a valid image of the
+relations of things on which further progress can
+be based, they are still little better than a mirage.
+The best method of abating this scepticism is to
+become acquainted with the real scope and modes
+of application of conceptions which, in the popular
+language of superficial exposition---and even in
+the unguarded and playful paradox of their
+authors, intended only for the instructed eye---often
+look bizarre enough. But much advantage
+will accrue if men of science become their own
+epistemologists, and show to the world by critical
+exposition in non-technical terms of the results
+and methods of their constructive work, that more
+than mere instinct is involved in it: the community
+has indeed a right to expect as much as
+this.
+\PageSep{xiii}
+
+It would be hard to find any one better
+qualified for this kind of exposition, either
+from the profundity of his own mathematical
+achievements, or from the extent and freshness
+of his interest in the theories of physical science,
+than the author of this book. If an appreciation
+might be ventured on as regards the later chapters,
+they are, perhaps, intended to present the stern
+logical analyst quizzing the cultivator of physical
+ideas as to what he is driving at, and whither he
+expects to go, rather than any responsible attempt
+towards a settled confession of faith. Thus, when
+M.~Poincaré allows himself for a moment to
+indulge in a process of evaporation of the
+Principle of Energy, he is content to sum up:
+``Eh bien, quelles que soient les notions nouvelles
+que les expériences futures nous donneront sur le
+monde, nous sommes sûrs d'avance qu'il y aura
+quelque chose qui demeurera constant et que nous
+pourrons appeler \Foreign{énergie}'' (\Pageref{166}), and to leave
+the matter there for his readers to think it out.
+Though hardly necessary in the original French, it
+may not now be superfluous to point out that
+independent reflection and criticism on the part
+of the reader are tacitly implied here as elsewhere.
+
+An interesting passage is the one devoted to
+\PageSep{xiv}
+Maxwell's theory of the functions of the æther,
+and the comparison of the close-knit theories of
+the classical French mathematical physicists with
+the somewhat loosely-connected \Foreign{corpus} of ideas by
+which Maxwell, the interpreter and successor of
+Faraday, has (posthumously) recast the whole
+face of physical science. How many times has
+that theory been re-written since Maxwell's day?
+and yet how little has it been altered in essence,
+except by further developments in the problem of
+moving bodies, from the form in which he left it!
+If, as M.~Poincaré remarks, the French instinct
+for precision and lucid demonstration sometimes
+finds itself ill at ease with physical theories of
+the British school, he as readily admits (\Pagerefs{223}{224}),
+and indeed fully appreciates, the advantages
+on the other side. Our own mental philosophers
+have been shocked at the point of view indicated
+by the proposition hazarded by Laplace, that a
+sufficiently developed intelligence, if it were made
+acquainted with the positions and motions of the
+atoms at any instant, could predict all future
+history: no amount of demur suffices sometimes
+to persuade them that this is not a conception
+universally entertained in physical science. It
+was not so even in Laplace's own day. From
+the point of view of the study of the evolution
+\PageSep{xv}
+of the sciences, there are few episodes more
+instructive than the collision between Laplace
+and Young with regard to the theory of capillarity.
+The precise and intricate mathematical
+analysis of Laplace, starting from fixed preconceptions
+regarding atomic forces which were
+to remain intact throughout the logical development
+of the argument, came into contrast with the
+tentative, mobile intuitions of Young; yet the
+latter was able to grasp, by sheer direct mental
+force, the fruitful though partial analogies of this
+recondite class of phenomena with more familiar
+operations of nature, and to form a direct picture
+of the way things interacted, such as could only
+have been illustrated, quite possibly damaged or
+obliterated, by premature effort to translate it
+into elaborate analytical formulas. The \Foreign{aperçus}
+of Young were apparently devoid of all cogency
+to Laplace; while Young expressed, doubtless in
+too extreme a way, his sense of the inanity of the
+array of mathematical logic of his rival. The
+subsequent history involved the Nemesis that the
+fabric of Laplace was taken down and reconstructed
+in the next generation by Poisson; while
+the modern cultivator of the subject turns, at any
+rate in England, to neither of those expositions
+for illumination, but rather finds in the partial
+\PageSep{xvi}
+and succinct indications of Young the best starting-point
+for further effort.
+
+It seems, however, hard to accept entirely
+the distinction suggested (\Pageref{213}) between the
+methods of cultivating theoretical physics in
+the two countries. To mention only two
+transcendent names which stand at the very
+front of two of the greatest developments of
+physical science of the last century, Carnot and
+Fresnel, their procedure was certainly not on the
+lines thus described. Possibly it is not devoid of
+significance that each of them attained his first
+effective recognition from the British school.
+
+It may, in fact, be maintained that the part
+played by mechanical and such-like theories---analogies
+if you will---is an essential one. The
+reader of this book will appreciate that the human
+mind has need of many instruments of comparison
+and discovery besides the unrelenting logic of the
+infinitesimal calculus. The dynamical basis which
+underlies the objects of our most frequent experience
+has now been systematised into a great
+calculus of exact thought, and traces of new real
+relationships may come out more vividly when
+considered in terms of our familiar acquaintance
+with dynamical systems than when formulated
+under the paler shadow of more analytical abstractions.
+\PageSep{xvii}
+It is even possible for a constructive
+physicist to conduct his mental operations entirely
+by dynamical images, though Helmholtz, as well
+as our author, seems to class a predilection in this
+direction as a British trait. A time arrives when,
+as in other subjects, ideas have crystallised out
+into distinctness; their exact verification and
+development then becomes a problem in mathematical
+physics. But whether the mechanical
+analogies still survive, or new terms are now
+introduced devoid of all naïve mechanical bias,
+it matters essentially little. The precise determination
+of the relations of things in the
+rational scheme of nature in which we find
+ourselves is the fundamental task, and for its
+fulfilment in any direction advantage has to be
+taken of our knowledge, even when only partial,
+of new aspects and types of relationship which
+may have become familiar perhaps in quite
+different fields. Nor can it be forgotten that the
+most fruitful and fundamental conceptions of
+abstract pure mathematics itself have often been
+suggested from these mechanical ideas of flux
+and force, where the play of intuition is our
+most powerful guide. The study of the historical
+evolution of physical theories is essential to the
+complete understanding of their import. It is in
+\PageSep{xviii}
+the mental workshop of a Fresnel, a Kelvin, or
+a Helmholtz, that profound ideas of the deep
+things of Nature are struck out and assume
+form; when pondered over and paraphrased by
+philosophers we see them react on the conduct
+of life: it is the business of criticism to polish
+them gradually to the common measure of human
+understanding. Oppressed though we are with
+the necessity of being specialists, if we are
+to know anything thoroughly in these days of
+accumulated details, we may at any rate profitably
+study the historical evolution of knowledge
+over a field wider than our own.
+
+The aspect of the subject which has here been
+dwelt on is that scientific progress, considered
+historically, is not a strictly logical process, and
+does not proceed by syllogisms. New ideas
+emerge dimly into intuition, come into consciousness
+from nobody knows where, and become
+the material on which the mind operates, forging
+them gradually into consistent doctrine, which
+can be welded on to existing domains of knowledge.
+But this process is never complete: a
+crude connection can always be pointed to by a
+logician as an indication of the imperfection of
+human constructions.
+
+If intuition plays a part which is so important,
+\PageSep{xix}
+it is surely necessary that we should possess a firm
+grasp of its limitations. In M.~Poincaré's earlier
+chapters the reader can gain very pleasantly a
+vivid idea of the various and highly complicated
+ways of docketing our perceptions of the relations
+of external things, all equally valid, that were
+open to the human race to develop. Strange to
+say, they never tried any of them; and, satisfied
+with the very remarkable practical fitness of the
+scheme of geometry and dynamics that came
+naturally to hand, did not consciously trouble
+themselves about the possible existence of others
+until recently. Still more recently has it been
+found that the good Bishop Berkeley's logical
+jibes against the Newtonian ideas of fluxions and
+limiting ratios cannot be adequately appeased in
+the rigorous mathematical conscience, until our
+apparent continuities are resolved mentally into
+discrete aggregates which we only partially
+apprehend. The irresistible impulse to atomize
+everything thus proves to be not merely a disease
+of the physicist; a deeper origin, in the nature
+of knowledge itself, is suggested.
+
+Everywhere want of absolute, exact adaptation
+can be detected, if pains are taken, between the
+various constructions that result from our mental
+activity and the impressions which give rise to
+\PageSep{xx}
+them. The bluntness of our unaided sensual
+perceptions, which are the source in part of the
+intuitions of the race, is well brought out in this
+connection by M.~Poincaré. Is there real contradiction?
+Harmony usually proves to be recovered
+by shifting our attitude to the phenomena.
+All experience leads us to interpret the totality of
+things as a consistent cosmos---undergoing evolution,
+the naturalists will say---in the large-scale
+workings of which we are interested spectators
+and explorers, while of the inner relations and
+ramifications we only apprehend dim glimpses.
+When our formulation of experience is imperfect
+or even paradoxical, we learn to attribute the
+fault to our point of view, and to expect that
+future adaptation will put it right. But Truth
+resides in a deep well, and we shall never get
+to the bottom. Only, while deriving enjoyment
+and insight from M.~Poincaré's Socratic exposition
+of the limitations of the human outlook on
+the universe, let us beware of counting limitation
+as imperfection, and drifting into an inadequate
+conception of the wonderful fabric of human
+knowledge.
+
+\Signature{J. LARMOR.}{}
+\PageSep{xxi}
+
+
+\OtherChapter{Author's Preface.}
+
+\First{To} the superficial observer scientific truth is unassailable,
+the logic of science is infallible; and if
+scientific men sometimes make mistakes, it is
+because they have not understood the rules of
+the game. Mathematical truths are derived from
+a few self-evident propositions, by a chain of
+flawless reasonings; they are imposed not only on
+us, but on Nature itself. By them the Creator is
+fettered, as it were, and His choice is limited to
+a relatively small number of solutions. A few
+experiments, therefore, will be sufficient to enable
+us to determine what choice He has made. From
+each experiment a number of consequences will
+follow by a series of mathematical deductions,
+and in this way each of them will reveal to us a
+corner of the universe. This, to the minds of most
+people, and to students who are getting their first
+ideas of physics, is the origin of certainty in
+science. This is what they take to be the rôle of
+\PageSep{xxii}
+experiment and mathematics. And thus, too, it
+was understood a hundred years ago by many
+men of science who dreamed of constructing the
+world with the aid of the smallest possible amount
+of material borrowed from experiment.
+
+But upon more mature reflection the position
+held by hypothesis was seen; it was recognised that
+it is as necessary to the experimenter as it is to the
+mathematician. And then the doubt arose if all
+these constructions are built on solid foundations.
+The conclusion was drawn that a breath would
+bring them to the ground. This sceptical attitude
+does not escape the charge of superficiality. To
+doubt everything or to believe everything are two
+equally convenient solutions; both dispense with
+the necessity of reflection.
+
+Instead of a summary condemnation we should
+examine with the utmost care the rôle of hypothesis;
+we shall then recognise not only that it is
+necessary, but that in most cases it is legitimate.
+We shall also see that there are several kinds of
+hypotheses; that some are verifiable, and when
+once confirmed by experiment become truths of
+great fertility; that others may be useful to us in
+fixing our ideas; and finally, that others are
+hypotheses only in appearance, and reduce to
+definitions or to conventions in disguise. The
+\PageSep{xxiii}
+latter are to be met with especially in mathematics
+and in the sciences to which it is applied. From
+them, indeed, the sciences derive their rigour;
+such conventions are the result of the unrestricted
+activity of the mind, which in this domain recognises
+no obstacle. For here the mind may affirm
+because it lays down its own laws; but let us
+clearly understand that while these laws are
+imposed on \emph{our} science, which otherwise could
+not exist, they are not imposed on Nature. Are
+they then arbitrary? No; for if they were, they
+would not be fertile. Experience leaves us our
+freedom of choice, but it guides us by helping us to
+discern the most convenient path to follow. Our
+laws are therefore like those of an absolute
+monarch, who is wise and consults his council of
+state. Some people have been struck by this
+characteristic of free convention which may be
+recognised in certain fundamental principles of
+the sciences. Some have set no limits to their
+generalisations, and at the same time they have
+forgotten that there is a difference between liberty
+and the purely arbitrary. So that they are compelled
+to end in what is called \emph{nominalism}; they
+have asked if the \Foreign{savant} is not the dupe of his
+own definitions, and if the world he thinks he has
+discovered is not simply the creation of his own
+\PageSep{xxiv}
+caprice.\footnote
+  {Cf.\ M.~le~Roy: ``Science et Philosophie,'' \Title{Revue de Métaphysique
+  et de Morale}, 1901.}
+Under these conditions science would
+retain its certainty, but would not attain its object,
+and would become powerless. Now, we daily see
+what science is doing for us. This could not be
+unless it taught us something about reality; the
+aim of science is not things themselves, as the
+dogmatists in their simplicity imagine, but the
+relations between things; outside those relations
+there is no reality knowable.
+
+Such is the conclusion to which we are led; but
+to reach that conclusion we must pass in review
+the series of sciences from arithmetic and
+geometry to mechanics and experimental physics.
+What is the nature of mathematical reasoning?
+Is it really deductive, as is commonly supposed?
+Careful analysis shows us that it is nothing of the
+kind; that it participates to some extent in the
+nature of inductive reasoning, and for that reason
+it is fruitful. But none the less does it retain its
+character of absolute rigour; and this is what
+must first be shown.
+
+When we know more of this instrument which
+is placed in the hands of the investigator by
+mathematics, we have then to analyse another
+fundamental idea, that of mathematical magnitude.
+\PageSep{xxv}
+Do we find it in nature, or have we ourselves
+introduced it? And if the latter be the
+case, are we not running a risk of coming to
+incorrect conclusions all round? Comparing the
+rough data of our senses with that extremely complex
+and subtle conception which mathematicians
+call magnitude, we are compelled to recognise a
+divergence. The framework into which we wish
+to make everything fit is one of our own construction;
+but we did not construct it at random, we
+constructed it by measurement so to speak; and
+that is why we can fit the facts into it without
+altering their essential qualities.
+
+Space is another framework which we impose
+on the world. Whence are the first principles of
+geometry derived? Are they imposed on us by
+logic? Lobatschewsky, by inventing non-Euclidean
+geometries, has shown that this is not the case.
+Is space revealed to us by our senses? No; for
+the space revealed to us by our senses is absolutely
+different from the space of geometry. Is geometry
+derived from experience? Careful discussion will
+give the answer---no! We therefore conclude that
+the principles of geometry are only conventions;
+but these conventions are not arbitrary, and if
+transported into another world (which I shall
+call the non-Euclidean world, and which I shall
+\PageSep{xxvi}
+endeavour to describe), we shall find ourselves
+compelled to adopt more of them.
+
+In mechanics we shall be led to analogous conclusions,
+and we shall see that the principles of
+this science, although more directly based on
+experience, still share the conventional character
+of the geometrical postulates. So far, nominalism
+triumphs; but we now come to the physical
+sciences, properly so called, and here the scene
+changes. We meet with hypotheses of another
+kind, and we fully grasp how fruitful they are.
+No doubt at the outset theories seem unsound,
+and the history of science shows us how ephemeral
+they are; but they do not entirely perish, and of
+each of them some traces still remain. It is these
+traces which we must try to discover, because in
+them and in them alone is the true reality.
+
+The method of the physical sciences is based
+upon the induction which leads us to expect the
+recurrence of a phenomenon when the circumstances
+which give rise to it are repeated. If all
+the circumstances could be simultaneously reproduced,
+this principle could be fearlessly applied;
+but this never happens; some of the circumstances
+will always be missing. Are we absolutely certain
+that they are unimportant? Evidently not! It
+may be probable, but it cannot be rigorously
+\PageSep{xxvii}
+certain. Hence the importance of the rôle that is
+played in the physical sciences by the law of
+probability. The calculus of probabilities is therefore
+not merely a recreation, or a guide to the
+baccarat player; and we must thoroughly examine
+the principles on which it is based. In this connection
+I have but very incomplete results to lay
+before the reader, for the vague instinct which
+enables us to determine probability almost defies
+analysis. After a study of the conditions under
+which the work of the physicist is carried on, I
+have thought it best to show him at work. For
+this purpose I have taken instances from the
+history of optics and of electricity. We shall thus
+see how the ideas of Fresnel and Maxwell took
+their rise, and what unconscious hypotheses were
+made by Ampère and the other founders of
+electro-dynamics.
+\PageSep{xxviii}
+%[Blank page]
+\PageSep{1}
+\MainMatter
+%[** TN: Commented text is printed by the \Part macro]
+% SCIENCE AND HYPOTHESIS.
+
+
+\Part{I.}{Number and Magnitude.}
+
+\Chapter[Nature of Mathematical Reasoning.]{I.}{On the Nature of Mathematical Reasoning.}
+
+\Section{I.}
+
+\First{The} very possibility of mathematical science seems
+an insoluble contradiction. If this science is only
+deductive in appearance, from whence is derived
+that perfect rigour 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 mathematics 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
+\PageSep{2}
+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 necessity, there is still one
+resource left to us: we may class them among
+\Foreign{à~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 reasoning 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 demonstration; reasoning can
+only give us immediately evident truths borrowed
+from direct intuition; it would only be an intermediary
+parasite. Should we not therefore have
+reason for asking if the syllogistic 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 generalising some
+proposition already known. Does the mathematical
+method proceed from the particular to the general,
+and, if so, how can it be called deductive?
+\PageSep{3}
+
+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, properly so called,
+retain the analytical character, and \Foreign{ipso facto}, lose
+their power.
+
+\Section{II.}
+
+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~$x + 1$---\ie, the adding of unity
+to a given number~$x$. These definitions, whatever
+they may be, do not enter into the subsequent
+reasoning. I next define the numbers $2$,~$3$,~$4$ by
+the equalities\Chg{:---}{}
+%[** TN: Numbered eqns displayed in the French, but not in the English transl.]
+\[
+\Tag{(1)} 1 + 1 = 2;\qquad
+\Tag{(2)} 2 + 1 = 3;\qquad
+\Tag{(3)} 3 + 1 = 4\Chg{,}{;}
+\]
+and in
+\PageSep{4}
+the same way I define the operation~$x + 2$ by the
+relation\Chg{;}{}
+\[
+\Tag{(4)}
+x + 2 = (x + 1) + 1.
+\]
+
+Given this, we have\Chg{:---}{}
+\begin{alignat*}{2}
+      2 + 2 &= (2 + 1) + 1\Chg{;}{,}\ &&\text{(def.~4)\Chg{.}{;}} \\
+(2 + 1) + 1 &= 3 + 1\Add{,}  &&\text{(def.~2)\Chg{.}{;}} \\
+      3 + 1 &= 4\Add{,}      &&\text{(def.~3)\Chg{.}{;}} \\
+\text{whence } 2 + 2 &= 4\Add{,}&&\quad\QED
+\end{alignat*}
+
+It cannot be denied that this reasoning is purely
+analytical. But if we ask a mathematician, he will
+reply: ``This is not a demonstration properly so
+called; it is a verification.'' We have confined
+ourselves to bringing together one or other of two
+purely conventional definitions, and we have verified
+their identity; nothing new has been learned.
+\emph{Verification} differs from proof precisely because it
+is analytical, and because it leads to nothing. It
+leads to nothing because the conclusion is nothing
+but the premisses translated into another language.
+A real proof, on the other hand, is fruitful, because
+the conclusion is in a sense more general than the
+premisses. The equality $2 + 2 = 4$ can be verified
+because it is particular. Each individual enunciation
+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.
+\PageSep{5}
+
+\Section{III.}
+
+Let us now see the geometer at work, and try
+%[** TN: "surprise" is correct: "...cherchons à surprendre ses procédés."]
+to surprise some of his methods. The task is
+not without difficulty; it is not enough to open a
+book at random and to analyse any proof we may
+come across. First of all, geometry must be excluded,
+or the question becomes complicated by
+difficult problems relating to the rôle of the
+postulates, the nature and the origin of the idea
+of space. For analogous reasons we cannot
+avail ourselves of the infinitesimal calculus. We
+must seek mathematical thought where it has
+remained pure---\ie, in Arithmetic. 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 analyse
+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 rigour. We may not impute this to
+them as a crime; they have obeyed a necessity.
+Beginners are not prepared for real mathematical
+rigour; 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
+\PageSep{6}
+which was trodden slowly by the founders of the
+science.
+
+Why is so long a preparation necessary to
+habituate oneself to this perfect rigour, 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.
+
+\Subsection{Definition of Addition.}
+
+I assume that the operation~$x + 1$ 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~$x + a$, which
+consists in adding the number~$a$ to any given
+number~$x$. Suppose that we have defined the
+operation
+%[** TN: Displayed eqns as in the French; all are inline in the English transl.]
+\[
+x + (a - 1);
+\]
+the operation~$x + a$ will be
+defined by the equality\Chg{:}{}
+\[
+\Tag{(1)}
+x + a = \bigl[x + (a - 1)\bigr] + 1.
+\]
+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
+successively and ``by recurrence'' the operations
+$x + 2$, $x + 3$,~etc. This definition deserves a moment's
+\PageSep{7}
+attention; it is of a particular nature which
+distinguishes it even at this stage from the purely
+logical definition; the equality~(1), in fact, contains
+an infinite number of distinct definitions, each
+having only one meaning when we know the
+meaning of its predecessor.
+
+\Subsection{Properties of Addition.}
+
+\Par{Associative.}---I say that
+%[** TN: Displayed eqns as in the French; all are inline in the English transl.]
+\[
+a + (b + c) = (a + b) + c;
+\]
+in
+fact, the theorem is true for $c = 1$. It may then be
+written
+\[
+a + (b + 1) = (a + b) + 1;
+\]
+which, remembering
+the difference of notation, is nothing but the equality~(1)
+by which I have just defined addition. Assume
+the theorem true for $c = \gamma$, I say that it will be true for
+$c = \gamma + 1$. Let
+\[
+(a + b) + \gamma = a + (b + \gamma)\Chg{,}{;}
+\]
+it follows that
+\[
+\bigl[(a + b) + \gamma\bigr] + 1 = \bigl[a + (b + \gamma)\bigr] + 1;
+\]
+or by def.~(1)\Chg{---}{,}
+\[
+(a + b) + (\gamma + 1)
+  = a + (b + \gamma + 1)
+  = a + \bigl[b + (\gamma + 1)\bigr]\Chg{,}{;}
+\]
+which shows by a series of purely analytical deductions
+that the theorem is true for $\gamma + 1$. Being
+true for $c = 1$, we see that it is successively true for
+$c = 2$, $c = 3$,~etc.
+
+\Par{Commutative.}---(1)~I say that
+%[** TN: Displayed eqns as in the French; all are inline in the English transl.]
+\[
+a + 1 = 1 + a.
+\]
+The
+theorem is evidently true for $a = 1$; we can \emph{verify}
+by purely analytical reasoning that if it is true for
+$a = \gamma$ it will be true for $a = \gamma + 1$.\footnote
+  {For $(\gamma + 1) + 1 = (1 + \gamma) + 1 = 1 + (\gamma + 1)$.\Transl}
+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
+\[
+a + b = b + a.
+\]
+The theorem has just
+\PageSep{8}
+been shown to hold good for $b = 1$, and it may be
+verified analytically that if it is true for $b = \beta$, it
+will be true for $b = \beta + 1$. The proposition is thus
+established by recurrence.
+
+\Subsection{Definition of Multiplication.}
+
+We shall define multiplication by the equalities\Chg{:}{}
+\begin{gather*}
+\Tag{(1)}
+a × 1 = a\Chg{.}{;} \\
+\Tag{(2)}
+a × b = \bigl[a × (b - 1)\bigr] + a.
+\end{gather*}
+Both of
+these include an infinite number of definitions;
+having defined~$a × 1$, it enables us to define in
+succession $a × 2$, $a × 3$, and so on.
+
+\Subsection{Properties of Multiplication.}
+
+\Par{Distributive.}---I say that
+%[** TN: Displayed eqns as in the French; all are inline in the English transl.]
+\[
+(a + b) × c = (a × c) + (b × c).
+\]
+We can verify analytically that the theorem
+is true for $c = 1$; then if it is true for $c = \gamma$, it will be
+true for $c = \gamma + 1$. The proposition is then proved
+by recurrence.
+
+\Par{Commutative.}---(1) I say that
+\[
+a × 1 = 1 × a.
+\]
+The
+theorem is obvious for $a = 1$. We can verify
+analytically that if it is true for $a = \alpha$, it will be
+true for $a = \alpha + 1$.
+
+(2)~I say that
+\[
+a × b = b × a.
+\]
+The theorem has
+just been proved for $b = 1$. We can verify analytically
+that if it be true for $b = \beta$ it will be true for
+$b = \beta + 1$.
+
+\Section{IV.}
+
+This monotonous series of reasonings may now
+be laid aside; but their very monotony brings
+vividly to light the process, which is uniform,
+\PageSep{9}
+and is met again at every step. The process is
+proof by recurrence. 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 instrument of transformation which lends
+itself to many more different combinations than
+the simple syllogism; but it is still a purely analytical
+instrument, and is incapable of teaching us
+anything new. If mathematics had no other instrument,
+it would immediately be arrested in its
+development; but it has recourse anew to the
+same process---\ie, 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 therefore mathematical reasoning
+\Foreign{par excellence}, and we must examine it closer.
+
+\Section{V.}
+
+The essential characteristic of reasoning by recurrence
+is that it contains, condensed, so to
+speak, in a single formula, an infinite number of
+syllogisms. We shall see this more clearly if we
+enunciate the syllogisms one after another. They
+follow one another, if one may use the expression,
+in a cascade. The following are the hypothetical
+\PageSep{10}
+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 unending 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 analytical
+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 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
+\PageSep{11}
+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 ourselves very far from the infinitesimal
+analysis, but the idea of mathematical infinity is
+already playing a preponderating part, and without
+it there would be no science at all, because there
+would be nothing general.
+\PageSep{12}
+
+\Section{VI.}
+
+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 another, 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. Experiment 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 contradiction would be sufficient, it
+would always enable us to develop as many
+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 principle 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 \Foreign{à~priori}
+\PageSep{13}
+synthetic intuition. On the other hand, we
+cannot see in it a convention as in the case of the
+postulates of geometry.
+
+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 repetition 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 observations.
+
+It cannot escape our notice that here is a
+striking analogy with the usual processes of
+induction. But an essential difference exists.
+Induction 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---\ie,
+proof by recurrence---is, on the contrary,
+necessarily imposed on us, because it is only the
+affirmation of a property of the mind itself.
+\PageSep{14}
+
+\Section{VII.}
+
+Mathematicians, as I have said before, always
+endeavour to generalise the propositions they have
+obtained. To seek no further example, we have
+just shown the equality\Chg{,}{}
+%[** TN: Displayed eqns as in the French; all are inline in the English transl.]
+\[
+a + 1 = 1 + a,
+\]
+and we then
+used it to establish the equality\Chg{,}{}
+\[
+a + b = b + a,
+\]
+which
+is obviously more general. Mathematics 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 parallel 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
+\[
+\Tag{(1)}
+a + 2 = 2 + a,
+\]
+we need
+only apply the rule
+\[
+a + 1 = 1 + a\Chg{,}{}
+\]
+twice, and write
+\[
+\Tag{(2)}
+a + 2 = a + 1 + 1 = 1 + a + 1 = 1 + 1 + a = 2 + a.
+\]
+
+The equality thus deduced by purely analytical
+means is not, however, 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
+\PageSep{15}
+the equality~(2) are merely more complicated
+combinations than the two sides of the equality~(1),
+and analysis only serves to separate the elements
+which enter into these combinations and to
+study their relations.
+
+Mathematicians therefore proceed ``by construction,''
+they ``construct'' more complicated combinations.
+When they analyse these combinations,
+these aggregates, so to speak, into their primitive
+elements, 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 ``construction,'' 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 accurately,
+there must be some advantage in considering the
+construction rather than the elements themselves.
+What can this advantage be? Why reason on a
+polygon, for instance, which is always decomposable
+into triangles, and not on elementary
+triangles? It is because there are properties of
+\PageSep{16}
+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 particular 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.
+\PageSep{17}
+
+
+\Chapter[Mathematical Magnitude.]{II.}{Mathematical Magnitude and Experiment.}
+
+\First{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 be 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 disengaged mathematics from all extraneous
+elements, and he is in a position to answer
+our question:---``Tell me exactly what this continuum
+is, about which mathematicians reason.''
+Many analysts who reflect on their art have
+already done so---M.~Tannery, for instance, in
+his \Title{Introduction à la théorie 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
+\PageSep{18}
+again, and so on indefinitely. We thus get an
+unlimited number of terms, 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 arranged 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
+intimate 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 formula. 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 rigour. 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 contented with this definition are the
+dupes of words, that the nature of each of these
+sets should be precisely indicated, that it should
+\PageSep{19}
+be explained 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
+preceding or succeeding these or those other sets;
+this alone should therefore intervene in the definition.
+So we need not concern ourselves with the
+manner in which the sets are intercalated, and
+no one will doubt the possibility of the operation
+if he only remembers that ``possible'' in the
+language of geometers simply means exempt from
+contradiction. But our definition is not yet complete,
+and we come back to it after this rather long
+digression.
+
+\Par{Definition of Incommensurables.}---The mathematicians
+of the Berlin school, and Kronecker
+in particular, have devoted themselves to constructing
+this continuous scale of irrational and
+fractional numbers without using any other
+materials than the integer. The mathematical
+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 incommensurable
+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.
+\PageSep{20}
+
+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 Kronecker 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 numbers 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 example,
+if the first class comprises all the numbers greater
+than~$2$, and if, in the second, are all the numbers
+\PageSep{21}
+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 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~$\sqrt{2}$ 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 symbol. 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 conceive as infinitely
+divisible---\ie, as a continuum?
+
+\Par{The Physical Continuum.}---We are next led to ask
+\PageSep{22}
+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, according to which sensation is proportional
+to the logarithm of the stimulus. But if we
+examine the experiments by which the endeavour
+has been made to establish this law, we shall be
+led to a diametrically opposite conclusion. It has,
+for instance, been observed that a weight~$A$ of $10$~grammes
+and a weight~$B$ of $11$~grammes produced
+identical sensations, that the weight~$B$ could no
+longer be distinguished from a weight~$C$ of $12$~grammes,
+but that the weight~$A$ was readily
+distinguished from the weight~$C$. Thus the rough
+results of the experiments may be expressed by
+the following relations\Chg{:}{}
+%[** TN: Not displayed in the English translation]
+\[
+A = B,\qquad B = C,\qquad A < C,
+\]
+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 conclude 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~$B$,
+\PageSep{23}
+and $B$~from~$C$, and that if we have not been
+aware of this, it is due to the imperfections of our
+senses.
+
+\Par{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?
+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 distinguish 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 contradiction 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
+powerful enough to decompose the physical continuum
+into discrete elements, 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
+\PageSep{24}
+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 collection 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
+\PageSep{25}
+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 numbers.
+If, then, we intercalate new sets according to the
+laws of incommensurable numbers, we obtain
+what may be called a continuum of the second
+order.
+
+\Par{Second Stage.}---We have only taken our first
+step. We have explained 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 physical 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; without 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
+\PageSep{26}
+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---\ie,
+the lines traced by the geometer should only
+give us points, the co-ordinates of which are
+rational numbers. 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
+(\Foreign{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 recognise the conception of Kronecker,
+\PageSep{27}
+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 continuum
+properly so called.
+
+\Par{Summary.}---To sum up, the mind has the faculty
+of creating symbols, and it is thus that it has constructed
+the mathematical continuum, which is
+only a particular system of symbols. The only
+limit 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 contradiction,---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.
+
+\Par{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
+\PageSep{28}
+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 $n$~intermediary
+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 interval~$CD$, the interval~$AD$ will by
+definition be the sum of the intervals $AB$~and~$AC$.
+This definition is very largely, but not altogether,
+arbitrary. It must satisfy certain conditions---the
+commutative and associative 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.
+
+\Par{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
+Reymond.
+\PageSep{29}
+
+We know that mathematicians distinguish
+between infinitesimals of different orders, and that
+infinitesimals of the second order are infinitely
+small, not only absolutely so, but also in relation
+to those of the first order. It is not difficult to
+imagine infinitesimals of fractional or even of
+irrational order, and here once more we find the
+mathematical continuum which has been dealt
+with in the preceding 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~$1 + \epsilon$, however
+small~$\epsilon$ 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 mathematical continuum, are
+\PageSep{30}
+we protected from contradictions analogous to
+those which gave it birth? No, and the following
+is an instance:---
+
+He is a \Foreign{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---\ie, 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 common 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 endeavouring 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.
+
+\Par{The Physical Continuum of several Dimensions.}---We
+have discussed above the physical continuum
+as it is derived from the immediate evidence of our
+\PageSep{31}
+senses---or, if the reader prefers, from the rough
+results of Fechner's experiments; I have shown
+that these results are summed up in the contradictory
+formulæ\Chg{:---}{}
+\[
+A = B,\qquad B = C,\qquad A < C.
+\]
+
+Let us now see how this notion is generalised,
+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 them, or we cannot; just as in
+Fechner'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 \emph{element}. It will be in a measure analogous to
+the \emph{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 neighbours, and it is thus
+surrounded by a kind of fog. If the astronomical
+comparison may be allowed, our ``elements''
+would be like nebulæ, 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 consecutive
+elements such that each cannot be distinguished
+from its predecessor. This \emph{linear} series is to the
+\emph{line} of the mathematician what the isolated \emph{element}
+was to the point.
+\PageSep{32}
+
+Before going further, I must explain what is
+meant by a \emph{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 \emph{cut}. By means of this
+cut, the continuum~$C$ will be \emph{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 distinct 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~$B$), \emph{unless
+one of the elements of this series is indistinguishable
+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 distinguishable
+the one from the other (and therefore
+forming neither one continuum nor several continuums),
+we shall call~$C$ a continuum \emph{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
+\PageSep{33}
+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.
+
+\Par{The Mathematical Continuum of Several Dimensions.}---The
+conception 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 defined by a system of $n$~distinct
+magnitudes which we call its co-ordinates.
+
+The magnitudes need not always be measurable;
+there is, for instance, one branch of geometry
+independent of the measure of magnitudes, 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~$AB$ is equal to or
+twice the arc~$BC$. This branch is called \emph{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 unskilful draughtsman, with the result that
+\PageSep{34}
+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.
+\PageSep{35}
+
+
+\Part{II.}{Space.}
+
+\Chapter{III.}{Non-Euclidean Geometries.}
+
+\First{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 \Foreign{à~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
+\PageSep{36}
+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 admitted 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,
+%[** TN: Correct ("Hongrois") in the French edition]
+two scientists, a Russian and a \Reword{Bulgarian}{Hungarian}, 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 Académic des Sciences receives only
+about one or two new demonstrations 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: \Title{Ueber
+die Hypothesen welche der Geometrie 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.
+
+\Par{The Geometry of Lobatschewsky.}---If it were
+possible to deduce Euclid's postulate from the
+several axioms, it is evident that by rejecting
+the postulate and retaining the other axioms we
+\PageSep{37}
+should be led to contradictory consequences. It
+would be, therefore, impossible to found on those
+premisses a coherent geometry. Now, this is
+precisely 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 impossible 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 difference 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. Lobatschewsky's propositions
+have no relation to those of Euclid,
+but they are none the less logically interconnected.
+
+\Par{Riemann's Geometry.}---Let us imagine to ourselves
+a world only peopled with beings of no
+thickness, and suppose these ``infinitely flat''
+\PageSep{38}
+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 endow 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
+spherical, 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 construct? 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 \emph{unbounded}, since on a sphere one may
+always walk forward without ever being brought
+to a stop, and yet it will be \emph{finite}; the end will
+never be found, but the complete tour can be
+made. Well, Riemann's geometry is spherical
+geometry extended to three dimensions. To construct
+it, the German mathematician had first of
+all to throw overboard, not only Euclid's postulate
+\PageSep{39}
+but also the first axiom that \emph{only one line can pass
+through two points}. On a sphere, through two
+given points, we can \emph{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 Lobatschewsky. 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 Lobatschewsky.
+Let us add that Riemann's space is
+finite, although unbounded in the sense which we
+have above attached to these words.
+
+\Par{Surfaces with Constant Curvature.}---One objection,
+however, remains possible. There is no contradiction
+between the theorems of Lobatschewsky and
+Riemann; but however numerous are the other
+consequences that these geometers have deduced
+\PageSep{40}
+from their hypotheses, 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
+Riemann'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 different 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
+lines of which remain of a constant length. Such
+\PageSep{41}
+a movement would appear absurd, on the other
+hand, to animals without thickness living on a
+surface of variable curvature. These surfaces of
+constant curvature are of two kinds. The
+curvature of some is \emph{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 \emph{negative}. Beltrami has shown
+that the geometry of these surfaces is identical
+with that of Lobatschewsky. Thus the two-dimensional
+geometries of Riemann and Lobatschewsky
+are connected with Euclidean geometry.
+
+\Par{Interpretation of Non-Euclidean Geometries.}---Thus
+vanishes the objection so far as two-dimensional
+geometries are concerned. It would be easy to
+extend Beltrami's reasoning to three-dimensional
+geometries, and minds which do not recoil before
+space of four dimensions 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
+columns, and corresponding each to each, just as
+in ordinary dictionaries the words in two languages
+which have the same signification correspond to
+one another:---
+\Dict{Space}{\raggedright The portion of space situated
+above the fundamental
+plane.}
+\PageSep{42}
+\Dict{Plane}{\raggedright Sphere cutting orthogonally
+the fundamental plane.}
+\Dict{Line}{\raggedright Circle cutting orthogonally
+the fundamental plane.}
+\Dict{Sphere}{Sphere.}
+\Dict{Circle}{Circle.}
+\Dict{Angle}{Angle.}
+\Dict{Distance between
+two points}{Logarithm of the anharmonic
+ratio of these two points
+and of the intersection
+of the fundamental plane
+with the circle passing
+through these two points
+and cutting it orthogonally.}
+\Dict{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. \emph{We shall then
+obtain the theorems of ordinary geometry.} For
+instance, Lobatschewsky's theorem: ``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 will be less than two right angles.'' Thus,
+however far the consequences of Lobatschewsky's
+hypotheses are carried, they will never lead to a
+\PageSep{43}
+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.
+
+\Par{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 which are common to the geometries
+of Euclid, Lobatschewsky, and Riemann. These
+propositions must be based on premisses that
+\PageSep{44}
+geometers admit without enunciation. It is interesting
+to try and extract them from the classical
+proofs.
+
+John Stuart Mill asserted\footnote
+  {\Title{Logic}, c.~viii., cf.\ Definitions, §5--6.\Transl}
+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, either in
+itself, or with the propositions previously 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 concealed in this definition.
+It is true we might change it, and that would be
+preferable, but then we should have to enunciate
+the axiom explicitly. Other definitions may give
+rise to no less important reflections, such as, for
+example, that of the equality of two figures. Two
+\PageSep{45}
+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
+properties of natural solids which do not much
+differ from those of the ideal solids, all of whose
+dimensions are invariable. However, imperfect as
+it may be, this definition implies an axiom. The
+possibility 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 \Foreign{à~priori} intuition would be. Moreover,
+when we study the definitions and the proofs
+of geometry, we see that we are compelled 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 invariable 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
+\PageSep{46}
+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 perpendicular
+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.
+
+\Par{The Fourth Geometry.}---Among these explicit
+axioms there is one which seems to me to deserve
+some attention, because when we abandon it we
+can construct a fourth geometry as coherent as
+those of Euclid, Lobatschewsky, and Riemann.
+To prove that we can always draw a perpendicular
+at a point~$A$ to a straight line~$AB$, we consider a
+straight line~$AC$ movable about the point~$A$, and
+initially identical with the fixed straight line~$AB$.
+We then can make it turn about the point~$A$ until
+it lies in~$AB$ produced. Thus we assume two
+propositions---first, that such a rotation is possible,
+and then that it may continue until the two lines
+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 of these
+theorems, and I shall not choose the least remarkable
+of them. \emph{A real straight line may be perpendicular
+to itself.}
+
+\Par{Lie's Theorem.}---The number of axioms implicitly
+\PageSep{47}
+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 premisses 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.
+
+\emph{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.
+
+\Par{Riemann's Geometries.}---However, this result
+seems contradicted by Riemann, for that scientist
+constructs an infinite number of geometries, and
+that to which his name is usually attached is only
+a particular 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.
+\PageSep{48}
+These geometries of Riemann, so interesting on
+various grounds, can never be, therefore, purely
+analytical, and would not lend themselves to
+proofs analogous to those of Euclid.
+
+\Par{On the Nature of Axioms.}---Most mathematicians
+regard Lobatschewsky'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
+geometry. Now, to discuss this view we must
+first of all ask ourselves, what is the nature of
+geometrical axioms? Are they synthetic \Foreign{à~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, let us take a true synthetic
+\Foreign{à~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~$n + 1$, provided it is
+\PageSep{49}
+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 foundation 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 unsurmountable.
+If geometry were an experimental science, it would
+not be an exact science. It would be subjected to
+\PageSep{5O}
+continual revision. Nay, it would from that day
+forth be proved to be erroneous, for we know that
+no rigorously invariable solid exists. \emph{The geometrical
+axioms are therefore neither synthetic \Foreign{à~priori}
+intuitions nor experimental facts.} They are conventions.
+Our choice among all possible conventions
+is \emph{guided} by experimental facts; but it remains
+\emph{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 approximate. In other words,
+\emph{the axioms of geometry} (I do not speak of those of
+arithmetic) \emph{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 degree 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.
+\PageSep{51}
+
+
+\Chapter{IV.}{Space and Geometry.}
+
+\First{Let} us begin with a little paradox. Beings whose
+minds were made as ours, and with senses like
+ours, but without any preliminary education,
+might receive from a suitably-chosen external
+world impressions which would lead them to
+construct a geometry other than that of Euclid,
+and to localise 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
+suddenly 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.
+
+\Par{Geometrical Space and Representative Space.}---It is
+often said that the images we form of external
+objects are localised 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
+\PageSep{52}
+geometers, having all the properties of that space.
+To all clear-headed men who think in this way,
+the preceding statement 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 properly so called?
+I mean of that space which is the object of
+geometry, 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 dimensions; 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 \emph{representative
+space}.
+
+\Par{Visual Space.}---First of all let us consider a
+purely visual impression, 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 other hand, the image is enclosed within
+a limited framework; and there is a no less
+important difference: \emph{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 rôle. The yellow spot can in no
+way be regarded as identical with a point on the
+\PageSep{53}
+edge of the retina. Not only does the same object
+produce on it much brighter impressions, but in
+the whole of the \emph{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
+accommodation 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 sensations which have
+given us the concept of the two first dimensions.
+The third dimension will therefore not appear to us
+as playing the same rôle as the two others. What
+may be called \emph{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 mathematical language, they will be functions
+of three independent variables. But let us
+look at the matter a little closer. The third
+\PageSep{54}
+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 convergence $A$~and~$B$
+are indistinguishable, the two sensations of
+accommodation $A'$~and~$B'$ which accompany them
+respectively will also be indistinguishable. But
+that is, so to speak, an experimental fact. Nothing
+prevents us \Foreign{à~priori} from assuming the contrary,
+and if the contrary takes place, if these two
+muscular sensations both vary independently, 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 \emph{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 indications which enable us to
+appreciate distances would cease to be connected
+by a constant relation. A being educating his
+\PageSep{55}
+senses in such a world would no doubt attribute
+four dimensions to complete visual space.
+
+\Par{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 everybody knows, which accompany all our
+movements, and which we usually call muscular
+sensations. The corresponding framework constitutes
+what may be called \emph{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 \emph{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 \emph{direction} of each movement, 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 accompanied 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 analyse my sensations. What I do see is that
+the sensations which correspond to movements in
+\PageSep{56}
+the same direction are connected in my mind by a
+simple \emph{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 association is extremely complex,
+for the contraction of the same muscle may correspond,
+according to the position of the limbs,
+to very different movements of direction. Moreover,
+it is evidently acquired; it is like all
+associations of ideas, the result of a \emph{habit}. This
+habit itself is the result of a very large number of
+\emph{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.
+
+\Par{Characteristics of Representative Space.}---Thus representative
+space in its triple form---visual,
+tactile, and motor---differs essentially from geometrical
+space. It is neither homogeneous nor
+isotropic; 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 ``localise'' them. Now, has
+that any meaning, and if so what is that meaning?
+Does it mean that we \emph{represent} to ourselves external
+objects in geometrical space? Our representations
+are only the reproduction of our sensations;
+they cannot therefore be arranged in the
+same framework---that is to say, in representative
+\PageSep{57}
+space. It is also just as impossible for us to represent
+to ourselves external objects in geometrical
+space, as it is impossible for a painter to paint on
+a flat surface objects with their three dimensions.
+Representative 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 \emph{represent} to ourselves
+external bodies in geometrical space, but we
+\emph{reason} about these bodies as if they were situated
+in geometrical space. When it is said, on the
+other hand, that we ``localise'' such an object in
+such a point of space, what does it mean? \emph{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.
+
+\Par{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
+\PageSep{58}
+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. \emph{None of our sensations, if isolated,
+could have brought us to the concept 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 impressions are subject to change; but among
+the changes that we ascertain, 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, \emph{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 impressions by making movements which
+would confront us with the movable object in
+the same \emph{relative} situation. We thus \emph{correct} the
+modification which was produced, and we re-establish
+the initial state by an inverse modification.
+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 accompanied by
+\PageSep{59}
+muscular sensations. But that does not mean
+that we represent them to ourselves in geometrical
+space. So what characterises change of position,
+what distinguishes it from change of state, is that
+it can always be \emph{corrected} by this means. It may
+therefore happen that we pass from the aggregate
+of impressions~$A$ to the aggregate~$B$ in two different
+ways. First, involuntarily and without experiencing
+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
+\emph{a series of sensations}; but a \emph{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.
+
+\Par{Conditions of Compensation.}---How is such a
+compensation possible in such a way that two
+\PageSep{60}
+changes, otherwise mutually independent, may be
+reciprocally corrected? A mind \emph{already familiar
+with geometry} would reason as follows:---If there
+is to be compensation, the different parts of the
+external object on the one hand, and the different
+organs of our senses on the other, must be in the
+same \emph{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
+\Foreign{à~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.
+
+\Par{Solid Bodies and Geometry.}---Among surrounding
+objects there are some which frequently experience
+displacements that may be thus corrected by a
+\emph{correlative} movement of our own body---namely,
+\emph{solid bodies}. The other objects, whose form is variable,
+\PageSep{61}
+only in exceptional circumstances undergo
+similar displacement (change of position without
+change of form). When the displacement of a
+body takes place with deformation, we can no
+longer by appropriate movements place the organs
+of our body in the same \emph{relative} situation with
+respect to this body; we can no longer, therefore,
+reconstruct the primitive aggregate 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
+modification 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.
+
+\emph{If, then, there were no solid bodies in nature there
+would be no geometry.}
+
+Another remark deserves a moment's attention.
+Suppose a solid body to occupy successively the
+positions $\alpha$~and~$\beta$; in the first position it will give
+us an aggregate of impressions~$A$, and in the second
+position the aggregate of impressions~$B$. Now let
+\PageSep{62}
+there be a second solid body, of qualities entirely
+different from the first---of different colour, for
+instance. Assume it to pass from the position~$\alpha$,
+where it gives us the aggregate of impressions~$A'$ to
+the position~$\beta$, 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 \emph{in themselves} have in general
+nothing in common. Yet we consider both
+these changes as displacements; and, further, we
+consider them the \emph{same} displacement. How can
+this be? It is simply because they may be both
+corrected by the \emph{same} correlative movement of our
+body. ``Correlative movement,'' therefore, constitutes
+the \emph{sole connection} between two phenomena
+which otherwise we should never have dreamed of
+connecting.
+
+On the other hand, our body, thanks to the
+number of its articulations 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 \Foreign{en bloc}---\ie, without any variation of
+their relative positions, as in the case of a solid
+body.
+\PageSep{63}
+
+To sum up:\Add{---}
+
+1. In the first place, we distinguish two categories
+of phenomena:---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 correlative 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.
+\emph{The laws of these phenomena are the object of
+geometry.}
+
+\Par{Law of Homogeneity.}---The first of these laws
+is the law of homogeneity. Suppose that by an
+external change we pass from the aggregate of
+impressions~$A$ to the aggregate~$B$, and that then
+this change~$\alpha$ is corrected by a correlative
+voluntary movement~$\beta$, so that we are brought
+back to the aggregate~$A$. Suppose now that
+another external change~$\alpha'$ brings us again from
+the aggregate~$A$ to the aggregate~$B$. Experiment
+then shows us that this change~$\alpha'$, like the change~$\alpha$,
+may be corrected by a voluntary correlative
+movement~$\beta'$, and that this movement~$\beta'$ corresponds
+\PageSep{64}
+to the same muscular sensations as the
+movement~$\beta$ which corrected~$\alpha$.
+
+This fact is usually enunciated as follows:---\emph{Space
+is homogeneous and isotropic.} We may also say that a
+movement which is once produced 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 importance 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.''
+
+\Par{The Non-Euclidean World.}---If geometrical space
+were a framework imposed on \emph{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, therefore, 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
+\PageSep{65}
+may thus conceive that beings whose education
+has 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
+$R$~be the radius of the sphere, and $r$~the distance
+of the point considered from the centre, the absolute
+temperature will be proportional to~$R^{2} - r^{2}$.
+Further, I shall suppose that in this world all bodies
+have the same co-efficient of dilatation, so that the
+linear dilatation of any body is proportional to its
+absolute 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 circumference
+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
+surface 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
+\PageSep{66}
+sphere. If to us geometry is only the study of the
+laws according to which invariable solids move, to
+these imaginary beings it will be the study of the
+laws of motion of solids \emph{deformed by the differences
+of temperature} 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 refraction is inversely
+proportional to~$R^{2} - r^{2}$. Under these conditions it
+is clear that the rays of light will no longer be
+rectilinear but circular. To justify what has 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 a
+\PageSep{67}
+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 neighbourhood 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 according 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,
+nevertheless 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 curvature 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 position,'' which may be corrected
+\PageSep{68}
+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 \emph{this will be non-Euclidean geometry}.
+So that beings like ourselves, educated in
+such a world, will not have the same geometry as
+ours.
+
+\Par{The World of Four Dimensions.}---Just as we have
+pictured to ourselves a non-Euclidean world, so we
+may picture a world of four dimensions.
+
+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 \emph{perspectives}.
+But as eye and objects are movable, 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 perspective~$B'$ are accompanied
+by the same muscular sensations, we
+connect them as we do other operations of the
+same nature. Then when we study the laws
+\PageSep{69}
+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 is
+from the properties of this group that we derive
+the idea of geometrical 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 \emph{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
+perspectives, since they are of only three dimensions.
+Imagine that the different perspectives of
+one and the same object to 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 transitions
+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.
+\PageSep{70}
+In this there is nothing that we cannot represent
+to ourselves, and, moreover, these sensations are
+those which a being would experience 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.
+
+\Par{Conclusions.}---It is seen that experiment plays a
+considerable rôle 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 approximative
+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, absolutely
+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 \emph{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
+\PageSep{71}
+truest, but what is the most convenient geometry.
+It will be noticed that my description of these
+fantastic worlds has required no language other
+than that of ordinary geometry. 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.
+\PageSep{72}
+
+
+\Chapter{V.}{Experiment and Geometry.}
+
+\ParSkip1. 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.
+However 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~$\pi$. What have we done? We
+have made an experiment on the properties of the
+matter with which this \emph{roundness} has been realised,
+and of which the measure we used is made.
+
+\Par[3.\ ]{Geometry and Astronomy.}---The same question
+may also be asked in another way. If Lobatschewsky's
+geometry is true, the parallax of a very
+distant star will be finite. If Riemann'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
+%[** TN: "...les trois géométries" in the French edition]
+us to decide between the \Reword{two}{three} geometries. But
+\PageSep{73}
+what we call a straight line in astronomy is simply
+the path of a ray of light. If, therefore, 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, therefore, 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 contradict the non-Euclidean hypothesis?
+I think that such a question cannot be seriously
+asked. To me it is exactly equivalent to the following,
+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
+existence of these lengths, would directly contradict
+this hypothesis, that 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 possesses the property~$B$; and, finally, I
+\PageSep{74}
+assume that in both Euclidean and non-Euclidean
+space the straight line is the only line that possesses
+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 endeavour
+to find out if it does, or does not, possess the property~$B$.
+But \emph{it is not so}. There exists no
+property which can, like this property~$A$, be an
+absolute criterion enabling us to recognise 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 property which in either Euclidean or
+non-Euclidean space belongs to the straight line,
+and belongs to it alone. But how can we ascertain
+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 difficulty a little farther
+off. In reality, the property that I have just
+enunciated is not a property of the straight line
+\PageSep{75}
+alone; it is a property of the straight line and of
+distance. For it to serve as an absolute 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 Euclidean system, and which cannot be interpreted
+in the system of Lobatschewsky. As I
+am well aware that this challenge will never be
+accepted, I may conclude that no experiment will
+ever be in contradiction 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 only 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 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,
+\PageSep{76}
+and the conditions which define the absolute position
+of the system and its absolute orientation in
+space. The law of the phenomena 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
+distances at the initial moment, but will in no
+way depend on the absolute initial position of
+the system and of its absolute initial orientation.
+This is what we shall call, for the sake of
+abbreviation, \emph{the law of relativity}.
+
+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 recognised
+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
+\PageSep{77}
+be still in agreement 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 rigour, 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 considered 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 independent of all
+interpretation by experiments. If the law is true
+in the Euclidean interpretation, it will be also true
+in the non-Euclidean interpretation. Allow me
+to make a short digression on this point. I have
+spoken above of the data which define the position
+\PageSep{78}
+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 experiments---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 means of astronomical observations, but which
+\PageSep{79}
+could be done by 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 that
+from this fact Newton concluded the existence of
+absolute space. I myself cannot accept this way
+of looking at it. I shall explain why in Part~III.,
+but for the moment it is not my intention to
+discuss this difficulty. I must therefore resign
+myself, in the enunciation 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 rational
+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 one equation with
+several unknowns; but when I have made enough
+experiments I shall have enough equations to
+calculate all my unknowns.'' If I know the height
+of the main-mast, that is not sufficient to enable
+\PageSep{80}
+me to calculate the age of the captain. When
+you have measured every fragment of wood in a
+ship you will have many equations, 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, however 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. When
+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 body, I am
+\PageSep{81}
+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 knowledge 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 $\alpha\ \beta\ \gamma$. I now suppose
+that we find that we can bring into contact $\Chg{\alpha\ \beta\ \gamma}{\alpha\beta\gamma}$
+with~$ago$; by that I mean $\alpha$~with~$a$, and at the
+same time $\beta$~with~$g$, and $\gamma$~with~$o$. Then we can
+successively bring into contact $\alpha\beta\gamma$ with $bgo$, $cgo$,
+$dgo$, $ego$, $fgo$, then with $aho$, $bho$, $cho$, $dho$, $eho$, $fho$;
+and then $\alpha\gamma$ successively with $ab$, $bc$, $cd$, $de$, $ef$, $fa$.
+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
+geometry). They therefore suffice to prove that
+these bodies move according to the Euclidean
+group; or at least that they do not move according
+\PageSep{82}
+to the Lobatschewskian group. That they may
+be compatible with the Euclidean group is easily
+seen; for we might make them so if the body~$\alpha\beta\gamma$
+were an invariable solid of our ordinary
+geometry in the shape of a right-angled triangle,
+and if the points $abcdefgh$ were the vertices of
+a 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~$\alpha\beta\gamma$ successively to~$ago$,
+$bgo$, $cgo$, $dgo$, $ego$, $fgo$, $aho$, $bho$, $cho$, $dho$, $eho$, $fho$,
+and then that we can apply~$\alpha\beta$ (and no longer~$\alpha\gamma$)
+successively to~$ab$, $bc$, $cd$, $de$, $ef$, and~$fa$. These are
+observations that could be made if non-Euclidean
+geometry were true. If the bodies~$\alpha\beta\gamma$, $oabcdefgh$
+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 therefore 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
+according to the Euclidean group. And so, without
+making any hypothesis 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
+\PageSep{83}
+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 experiment proving that space is
+Euclidean, and the second an experiment proving
+that space is non-Euclidean; in fact, it might be
+imagined (note that I use the word \emph{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 construct it if he would only take
+the trouble. But you must not conclude, however,
+that space is non-Euclidean. In the same way,
+just as ordinary solid bodies would continue
+to exist when the mechanic had constructed 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~$R$, 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 \Typo{negligeable}{negligible}, and which would behave
+as ordinary invariable solids; and, on the other
+hand, we might have very dilatable bodies, which
+would behave as non-Euclidean solids. We
+might have two double pyramids~$oabcdefgh$ and
+\PageSep{84}
+$o'a'b'c'd'e'f'g'h'$, and two triangles $\alpha\beta\gamma$~and~$\alpha'\beta'\gamma'$.
+The first double pyramid would be rectilinear, and
+the second curvilinear. The triangle~$\alpha\beta\gamma$ 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~$\alpha'\beta'\gamma'$.
+
+And then the experiment would seem to show---first,
+that Euclidean geometry is true, and then
+that it is false. Hence, \emph{experiments have reference
+not to space but to bodies}.
+
+\Subsection{Supplement.}
+
+\ParSkip8. 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 \Title{Revue
+de Métaphysique et de Morale} and in the \Title{Monist}.
+When we say that space has three dimensions,
+what do we mean? We have seen the importance
+of these ``internal changes'' which are revealed to
+us by our muscular sensations. They may serve
+to characterise 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
+\PageSep{85}
+initial and final attitudes $A$~and~$B$ remaining the
+same, the intermediary attitudes of the corresponding
+sensations may differ). How then can
+we recognise the equivalence of these two series?
+Because they may serve to compensate for the same
+external change, or more generally, because, when
+it is a question of compensation 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 recognise it? I mean that \emph{although I have
+not moved} (my muscular sense tells me this), my
+finger, which just now touched the object~$A$, is
+\PageSep{86}
+now touching the object~$B$. I might have 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 \Foreign{à~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 furnished 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
+\PageSep{87}
+each point a class. Yet it may be said that what
+we get from this experiment 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 continuum
+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 \Foreign{à~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
+affirmative. If we recur to what we have called
+displacements, the aggregate 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 characterise 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 neighbouring
+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.
+\PageSep{88}
+Which shall we choose to form a kind of
+standard by which to compare natural phenomena?
+And when this group is chosen, which
+of the sub-groups shall we take to characterise a
+point in space? Experiment has guided us by
+showing us what choice adapts itself best to the
+properties of our body; but there its rôle ends.
+\PageSep{89}
+
+
+\Part{III.}{Force.}
+
+\Chapter{VI.}{The Classical Mechanics.}
+
+\First{The} English teach mechanics as an experimental
+science; on the Continent it is taught always more
+or less as a deductive and \Foreign{à~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 predecessors 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
+compelled 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 reasoning, what is
+convention, and what is hypothesis. This is not
+all.
+\PageSep{90}
+
+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.''\footnote
+  {\Title{Revue de Métaphysique et de Morale}, t.~vi., pp.~1--13, January,
+  1898.}
+
+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 become
+more complicated, but it still would be possible.
+
+Thus, absolute space, absolute time, and even
+geometry are not conditions which are imposed on
+mechanics. All these things no more existed
+before mechanics than the French language can
+be logically said to have existed before the truths
+which are expressed in French. We might
+endeavour to enunciate the fundamental law of
+mechanics in a language independent of all these
+\PageSep{91}
+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 \Title{Leçons de Mécanique
+physique}. Of course the enunciation of these laws
+would become much more complicated, because all
+these conventions 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.
+
+\Par{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
+\Foreign{à~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 \Foreign{à~priori} truth, an
+experimental fact? Have there ever been experiments
+\PageSep{92}
+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 mechanics 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 consequences 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
+neighbouring bodies, and on their velocities.
+Mathematicians would say that the movements
+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
+generalisation 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
+\Foreign{à~priori}; other laws would be just as compatible
+with the principle of sufficient reason. If a body
+is not acted upon by a force, instead of supposing
+\PageSep{93}
+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
+the law of inertia: what will be the natural
+generalisation? 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 neighbouring bodies; in the second case,
+that the variation of the acceleration of a body
+depends only on the position of the body and of
+neighbouring 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 analogous 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, I 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
+position 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.
+\PageSep{94}
+
+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 the 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 generalised
+law of inertia only by opposing it to a contrary
+hypothesis.
+
+Has this generalised law of inertia been verified
+by experiment, and can it be so verified?
+When Newton wrote the \Title{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
+\PageSep{95}
+planet is entirely determined by its initial position
+and initial velocity; this, indeed, is what our
+generalised 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 principles 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 attribute 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
+astronomy 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?
+\PageSep{96}
+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 positions after
+a certain time. We ought then to find that they
+would resume 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 \emph{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 velocities 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
+\PageSep{97}
+that their $3n$~co-ordinates satisfy a system of $3n$~differential
+equations of the fourth order (and
+not of the second, as required by the law of
+inertia). We know that by introducing $3n$~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 confirmed nor contradicted by
+experiment.
+
+\Par{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
+\PageSep{98}
+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 quotient of the force by the acceleration?
+These difficulties are insurmountable.
+
+When we say force is the cause of motion, we
+are talking metaphysics; and this definition, if we
+had to be content with it, would be absolutely
+fruitless, would lead to absolutely nothing. For a
+definition to be of any use it must tell us how to
+measure force; and that is quite sufficient, 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 opposite 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 another. 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 opposite directions.
+It is this definition which we try to materialise, as
+it were, when we measure a force with a dynamometer
+or with a balance. Two forces, $F$~and~$F'$,
+\PageSep{99}
+which I suppose, for simplicity, to be acting
+vertically upwards, are respectively applied to two
+bodies, $C$~and~$C'$. I attach a body weighing~$P$
+first to~$C$ and then to~$C'$; if 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 rigour; this
+rigour does not exist. What I say of weight
+would apply equally to the force of the spring of
+a dynamometer, which would vary according 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~$R$
+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 because they
+are in equilibrium. From these three equalities
+\PageSep{100}
+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; \emph{hence this
+principle can no longer be regarded as an experimental
+law but only as a definition}.
+
+To recognise 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 constant in
+magnitude and direction. But this third rule is
+an experimental law. It is only approximately
+true: \emph{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 definition. 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 acceleration
+\PageSep{101}
+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~$B$ arc respectively in 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, $C$,~$D$,~\ldots. 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 \emph{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
+\PageSep{102}
+attraction of two bodies is proportional to their
+masses; if $r$~is their distance apart, $m$~and~$m'$ their
+masses, $k$~a constant, then their attraction will be~$kmm'/r^{2}$.
+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
+%[** TN: "mais sans que l'on eût f = kmm'"]
+masses, that it is equal to~$f/r^{2}$ \Reword{and not to~$kmm'$}{but without having $f = kmm'$}.
+If it were so, we should nevertheless, by observing
+the \emph{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 assertion? 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 acceleration 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 principle
+\PageSep{103}
+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, \emph{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 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 system
+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. \emph{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 conclusion
+is obvious; the motion of the centre of gravity
+of the universe will be for ever 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:
+\emph{Masses are co-efficients which it is found convenient to
+introduce into calculations.}
+
+We could reconstruct our mechanics by giving
+to our masses different values. The new mechanics
+would be in contradiction neither with
+\PageSep{104}
+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 \emph{would not be so simple}.
+Let us clearly understand this. It would be only
+the first terms which would be less simple---\ie,
+those we already know through experiment;
+perhaps the small masses could be slightly altered
+without the \emph{complete} equations gaining or losing
+in simplicity.
+
+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 experiment.'' From
+what we have just seen these fears would appear
+to be groundless. The principles of dynamics
+appeared to us first as experimental truths, but
+we have been compelled to use them as definitions.
+It is \emph{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
+\PageSep{105}
+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 \emph{perfectly} isolated, perfectly abstracted from
+all external action; but there are systems which
+are \emph{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. We then find that
+the motion of its centre of gravity is \emph{nearly} uniform
+and rectilinear in conformity with Newton's Third
+Law. This is an experimental fact, which cannot
+be invalidated by a more accurate 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.
+\emph{Thus is explained how experiment may serve as a basis
+for the principles of mechanics, and yet will never
+invalidate them.}
+
+\Par{Anthropomorphic Mechanics.}---It will be said that
+Kirchoff has only followed the general tendency of
+mathematicians towards nominalism; 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 primitive, 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
+\PageSep{106}
+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 us 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 conductor 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 experience more fatigue in
+lifting a weight of $100$~lb.\ 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 definitively 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 symbol,
+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
+\PageSep{107}
+rôle 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.
+
+\Par{The Thread School.}---M.~Andrade, in his \Title{Leçons
+de \Typo{Mecanique}{Mécanique} physique}, has modernised anthropomorphic
+mechanics. To the school 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 \emph{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.
+
+%[** TN: "A" variously italicized and not in the original]
+{\Loosen 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~$\alpha$, and the acceleration
+of the body~$A$ is recorded. $A$~is then detached,
+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~$\alpha$, and the
+\PageSep{108}
+acceleration of~$B$ is noted. The experiment is
+then renewed with both $A$~and~$B$ until the increment
+of length is~$\beta$. The four accelerations
+observed should be proportional. Here we have
+an experimental 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, which 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 measuring the increment
+of the thread. Nine times out of ten, in consequence,
+our definition will be in default; no
+\PageSep{109}
+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 acceleration.
+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 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 reaction of the bodies
+attached to its extremities.
+
+The ideas expounded by M.~Andrade are none
+the less very interesting. 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 naïve
+anthropomorphism to the present conception of
+science.
+
+We see that we end with an experiment which
+\PageSep{110}
+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
+absolute. 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.
+\PageSep{111}
+
+
+\Chapter{VII.}{Relative and Absolute Motion.}
+
+\Par{The Principle of Relative Motion.}---Sometimes
+endeavours 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 reasons: 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 depend 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 regulations for the degree of B.~ès~Sc.
+\PageSep{112}
+It is clear that the attempt has 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 positions, 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 \Foreign{à~priori} conditions? Remembering what we
+\PageSep{113}
+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 generalised 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.
+
+\Par{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 \Foreign{à~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 motion 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 mysterious in
+that, and if I have not been subject to the action
+\PageSep{114}
+of any external force, the train has experienced an
+external impact. There can be nothing paradoxical
+in the relative motion of two bodies being
+disturbed when the motion of one or the other is
+modified by an external cause. Nor need I dwell
+on the case of relative motion referring to axes
+which rotate uniformly. If the sky were for ever
+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 absolute space, can a thing turn without turning
+with respect to something; and, on the other
+hand, how can we admit Newton's conclusion and
+believe in absolute space? But it is not sufficient
+to state that all possible solutions are equally
+unpleasant to us. We must analyse 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
+\PageSep{115}
+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 ordinary centrifugal force and compounded
+centrifugal force. Our imaginary scientists can
+thus explain everything by looking upon these two
+forces as real, and they would not see in this a
+contradiction of the generalised principle of inertia,
+for these forces would depend, the one on the
+relative positions of the different parts of the
+system, such as real attractions, 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
+realising an isolated system, the centre of gravity
+of this system would not have an approximately
+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 realised. 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
+\PageSep{116}
+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
+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 indifferently 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 complications 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.\Add{''} That
+does not prevent absolute space---that is to say,
+the point to which we must refer the earth to
+\PageSep{117}
+know if it really does turn round---from having
+no objective existence. And hence this affirmation:
+``the earth turns round,'' has no meaning,
+since it cannot be verified by experiment; since
+such an experiment not only cannot be realised or
+even dreamed of by the most daring Jules Verne,
+but cannot even be conceived of without contradiction;
+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 differential equations of the second
+order, and that so are the differences of these
+co-ordinates. This is what we have called the
+generalised principle of inertia, and the principle
+of relative motion. If the distances 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
+\PageSep{118}
+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 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 distances 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 connecting 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 requires 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 \emph{accidental} constant. Or again, let us
+\PageSep{119}
+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
+\emph{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
+\emph{accidental} constant. If, on the other hand, the
+exponent of the distance which figures in the
+expression of the attractive force is equal to~$-2$
+and not to~$-3$, it is not by chance, but because it
+is required by Newton's law. It is an \emph{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 accustomed
+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
+\PageSep{120}
+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;
+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, distances
+will be determined by differential equations
+of the second order. Would this completely
+satisfy the minds of these astronomers? I think
+not. In the first place, they would very soon see
+that in differentiating their equations so as to
+raise them to a higher order, these equations
+would become much more simple, and they would
+be especially 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
+\PageSep{121}
+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 generalised 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 derivatives, and
+something else. What is that \emph{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 something
+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
+\PageSep{122}
+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 entirely 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 indications 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.
+\PageSep{123}
+
+
+\Chapter{VIII.}{Energy and Thermo-dynamics.}
+
+\Par{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
+principle 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 \emph{kinetic energy}, or
+\Foreign{vis~viva}, and \emph{potential energy}. Every change
+that the bodies of nature can undergo is regulated
+by two experimental laws. First, the sum of the
+kinetic and potential energies is constant. This
+is the principle of the conservation of energy.
+Second, if a system of bodies is at~$A$ at the time~$t_{0}$,
+and at~$B$ at the time~$t_{1}$, it always passes from the
+first position to the second by such a path that
+the \emph{mean} value of the difference between the two
+kinds of energy in the interval of time which
+separates the two epochs $t_{0}$~and~$t_{1}$ 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
+\PageSep{124}
+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 principles 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 remain constant. This quantity
+is the sum of two terms. The first depends only on
+the position of the material points, and is independent
+of their velocities; the second is proportional
+to the squares of these velocities. This
+decomposition can only take place in one way.
+The first of these terms, which I shall call~$U$, will
+\PageSep{125}
+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~$T + U$, $\phi(T + U)$.
+But this function $\phi(T + U)$ 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~$T + U$\Typo{)}{},
+it matters not which, since this linear function may
+always be reduced to~$T + U$ by a change of unit
+and of origin\Typo{}{)}. This, then, is what we call energy.
+The first term we shall call potential energy, and
+the second kinetic energy. The definition of the
+two kinds of energy may therefore be carried
+through without any ambiguity.
+
+So it is with the definition of mass. Kinetic
+energy, or \Foreign{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 expression of the kinetic energy
+as a function of these relative velocities, the co-efficients
+of this expression will give us the masses.
+So in this simple case the fundamental ideas can
+be defined without difficulty. 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
+\PageSep{126}
+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 more reason to take as our
+definition $T + U$ rather than any other function of~$T + U$,
+when the property which characterised
+$T + U$ 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 constant, 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
+their internal state. The expression for the energy
+could be decomposed in one way only into three
+\PageSep{127}
+terms of this form. But this is not the case. Let
+us consider electrified bodies. The electro-static
+energy due to their mutual action will evidently
+depend on their charge---\ie, 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, $\phi(T + U + Q)$.
+
+If $T + U + Q$ were of the particular form that I
+have suggested above, no ambiguity would ensue.
+Among the functions $\phi(T + U + Q)$ 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:---\emph{There
+is something which remains constant.} In this
+form it, in its turn, is outside the bounds of experiment
+and reduced to a kind of tautology. It
+\PageSep{128}
+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 molecule, acted upon
+by no force, but compelled to move on a surface,
+will take as its path the geodesic line---\ie, the
+shortest path. This molecule 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. It is clear that it would be better
+to replace it by a less objectionable enunciation,
+one in which, as philosophers would say, final
+\PageSep{129}
+effects do not seem to be substituted for acting
+causes.
+
+\Par{Thermo-dynamics.}---The rôle 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,
+\Foreign{àpropos} 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 generalise, as we do
+every day, empiric data, and I shall not be so
+foolhardy as to discuss this question, 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
+\PageSep{130}
+would be reduced to a kind of inventory, to the
+ascertaining of isolated facts. It would not 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 generalisation to predict whether the fact will
+be renewed as soon as the least of these circumstances
+is changed. But every proposition may
+be generalised in an infinite number of ways.
+Among all possible generalisations 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 proclaimed
+abroad that Nature loves simplicity; but Nature
+has proved the contrary since then on more than
+one occasion. We no longer confess 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 necessity 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 laws from which it was deduced, in
+\PageSep{131}
+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 perturbations 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 rigour,
+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 experiments, 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
+reasons. Before we raise Mayer's law to the
+\PageSep{132}
+dignity of an absolute principle, a deeper discussion
+is necessary. But if we embark on this discussion
+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 generality and apply it to
+the universe, we see it vanish, so to speak, and
+nothing is left but this---\emph{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 parameters, which I shall call $x_{1}$,~$x_{2}$, $x_{3}\Add{,}~\ldots\Add{,},~x_{n}$.
+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 values of these same parameters at an
+anterior or ulterior moment. In 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
+$x_{1}$,~$x_{2}$, $x_{3}\Add{,}~\ldots\Add{,}~x_{n}$, which remain constant. If we
+say then, \emph{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
+\PageSep{133}
+have only $n - p$ 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? \emph{There is a combination of these $n - p$
+relations, 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
+\PageSep{134}
+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 absolute
+meaning, we must extend it to the whole
+universe, and then we find ourselves face to
+face with the very difficulty we have endeavoured
+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 determinist 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, that Mayer's law is a form subtle enough
+\PageSep{135}
+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 is reduced to
+mere tautology; since, in each particular case, and
+provided 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 utilise 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
+inequality. 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.
+
+\Par{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 approximately as far as almost isolated
+systems are concerned; on the other hand,
+\PageSep{136}
+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 beforehand 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 rôle
+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
+\PageSep{137}
+convenient the fundamental conventions of
+geometry refer to bodies which have nothing
+in common with those that are studied by
+geometry. They refer to the properties of solid
+bodies and to the propagation of light in a straight
+line. These are mechanical, optical experiments.
+In no way can they be regarded as geometrical
+experiments. 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 physiological 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 fundamental
+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 generalisations of experimental
+and particular principles. 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
+\PageSep{138}
+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 familiarised, 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 generalised far too much.
+They have thought that the principles were
+the whole of science, and therefore that the
+whole of science was conventional. This paradoxical
+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
+\PageSep{139}
+not rigorously true, it was only approximate.
+We introduce arbitrarily an intermediate term,~$C$,
+more or less imaginary, and $C$~is \emph{by definition} that
+which has with~$A$ \emph{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 decomposition may be carried, laws will
+always remain. We shall now enter into the
+domain of laws properly so called.
+\PageSep{140}
+
+
+\Part{IV.}{Nature.}
+
+\Chapter{IX.}{Hypotheses in Physics.}
+
+\Par{The Rôle of Experiment and Generalisation.}---Exper\-iment
+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 questioned. 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 sufficient
+merely to observe; we must use our observations,
+and for that purpose we must generalise. This
+is what has always been done, only as the recollection
+of past errors has made man more and more
+circumspect, he has observed more and more and
+generalised less and less. Every age has scoffed
+at its predecessor, accusing it of having generalised
+\PageSep{141}
+too boldly and too naïvely. Descartes used to
+commiserate the Ionians. 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 accumulation 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
+foresight. 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.''\footnote
+  {V. \Title{Past and Present}, end of Chapter~I., Book~II.\Transl}
+Carlyle was a compatriot of Bacon,
+and, like him, he wished to proclaim his worship
+of \emph{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 Lackland
+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,
+\PageSep{142}
+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 \Foreign{experimentum
+crucis}; 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 generalise. Without generalisation,
+prediction is impossible. The circumstances
+under which one has operated will never again
+be reproduced simultaneously. The fact observed
+will never be repeated. All that can be affirmed
+is that under analogous circumstances an analogous
+fact will be produced. To predict it, we must
+therefore invoke the aid of analogy---that is to say,
+even at this stage, we must generalise. However
+timid we may be, there must be interpolation.
+Experiment only gives us a certain number of
+isolated points. They must be connected by a
+continuous line, and this is a true generalisation.
+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 generalising our
+\PageSep{143}
+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, generalised.
+
+It is often said that experiments should be made
+without preconceived 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 intervene 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 \Foreign{ipso~facto}, they will
+force us to look at things under different aspects.
+This is enough to free us. He is no longer a slave
+who can choose his master.
+
+Thus, by generalisation, 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
+\PageSep{144}
+probable. However solidly founded a prediction
+may appear to us, we are never \emph{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 presents itself. But
+every experiment is long and difficult, and the
+labourers 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 predictions
+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 increasing indefinitely; the librarian
+has limited funds for his purchases, and he must,
+therefore, strain every nerve not to waste them.
+Experimental 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 utilise its riches;
+and also by showing the librarian the gaps in his
+\PageSep{145}
+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 rôle of mathematical physics. It
+must direct generalisation, 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.
+
+\Par{The Unity of Nature.}---Let us first of all observe
+that every generalisation 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 \Chg{re-act}{react} 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 argument in favour of its
+accuracy: when Fresnel himself, after having 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
+considerably; but those who do not believe that
+natural laws must be simple, are still often obliged
+\PageSep{146}
+to act as if they did believe it. They cannot
+entirely dispense with this necessity without
+making all generalisation, and therefore all science,
+impossible. It is clear that any fact can be
+generalised 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 inflexions
+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 movements 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 determinations obtained
+by these three methods, we find three numbers
+very close together, but not quite identical. This
+result might be interpreted 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.
+\PageSep{147}
+
+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; sometimes, 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 discussed, whose paths, deformed by incessant
+impacts, have the most capricious shapes, and
+plough their way through space in every direction.
+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,
+\PageSep{148}
+and the coarseness of our senses alone prevents us
+from seeing the complexity.
+
+Many phenomena obey a law of proportionality.
+But why? Because 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 proportional
+to the increment of the variable. As in reality
+our increments are not infinitely small, but only
+very small, the law of proportionality 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 perhaps only apparent. Who
+knows if it be not due to some complicated
+mechanism, to the impact of some subtle matter
+animated by irregular 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 negligible, 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
+\PageSep{149}
+the complex, and then the complex 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 generalisations. 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 generalisations by the belief in simplicity.
+We have verified a simple law in a considerable
+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
+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
+\PageSep{150}
+always be able to reason in the same fashion,
+and if a simple law has been observed 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
+rôle 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.
+
+\Par{Rôle of Hypothesis.}---Every generalisation is a
+hypothesis. Hypothesis therefore plays a necessary
+rôle, which no one has ever contested. 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
+\PageSep{151}
+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 something 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 consequence.
+
+Now, under what conditions is the use of
+hypothesis without danger? 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 mathematical physics may
+render us. By the precision which is its characteristic,
+we are compelled to formulate all the
+hypotheses that we would unhesitatingly make
+without its aid. Let us also notice that it is
+important not to multiply hypotheses indefinitely.
+If we construct a theory based upon multiple hypotheses,
+and if experiment condemns it, which of
+\PageSep{152}
+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
+determined 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 distant 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 indifferent.
+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 conclusions,
+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
+\PageSep{153}
+same conclusions 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 necessary to represent it by an~$x$ 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 artifices for
+calculation, or to assist our understanding by
+concrete images, to fix the ideas, as we say. They
+need not therefore be rejected. The hypotheses
+of the third category are real generalisations.
+They must be confirmed or invalidated by experiment.
+Whether verified or condemned, they will
+always be fruitful; but, for the reasons I have
+given, they will only be so if they are not too
+numerous.
+
+\Par{Origin of Mathematical Physics.}---Let us go
+further and study more closely the conditions
+which have assisted the development of mathematical
+physics. We recognise at the outset that
+\PageSep{154}
+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 \emph{differential equation};
+for the laws of Kepler we substitute the law of
+Newton.
+
+Next, we try to decompose the phenomena in
+space. What experiment gives us is a confused
+aggregate of facts spread over a scene of considerable
+extent. We must try to deduce the elementary
+phenomenon, which will still be localised 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 neighbouring
+point, it will immediately impart that
+\PageSep{155}
+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 phenomenon
+is the interchange of heat between two
+contiguous points. It is strictly localised 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 investigation 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 approximately, 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 independent, and may be
+added one to the other, either as vectors or as scalar
+\PageSep{156}
+quantities. The elementary phenomenon is then
+the action of an isolated body. Or suppose, again,
+it is a question of small movements, or more
+generally of small variations which obey the well-known
+law of mutual or relative independence.
+The movement observed 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 phenomena,
+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 mechanism. The law of
+great numbers will suffice. Take for example the
+propagation of heat. Each molecule radiates towards
+its neighbour---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 neighbouring
+molecules attract and repel each other, we
+need not inquire by what law. It is enough for us
+that this attraction is sensible at small distances
+only, and that the molecules are very numerous,
+that the medium is symmetrical, and we have
+\PageSep{157}
+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 complex mechanism. Evidently the
+best means of reaching the elementary phenomenon
+would be experiment. It would be necessary
+by experimental artifices to dissociate the
+complex system which nature offers for our investigations
+and carefully to study the elements as
+dissociated as possible; for example, natural white
+light would be decomposed into monochromatic
+lights by the aid of the prism, and into polarised
+lights by the aid of the polariser. 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 natural lights which are called \emph{monochromatic}
+%[** TN: "nous donnent une raie très fine, mais qui n'est pas cependant infiniment fine"]
+give us a very fine \Reword{array, but a y}{ray, but one} 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 rigorously monochromatic
+\PageSep{158}
+light. That would not be accurate.
+I assume that two rays emanate from the same
+source, that they are first polarised in planes at
+right angles, that they are then brought back
+again to the same plane of polarisation, and that
+we try to obtain interference. If the light were
+\emph{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
+\emph{integration}, and it is the province of the mathematician.
+It might be asked, why in physical
+science generalisation 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 \emph{all
+similar to each other}; and in this way differential
+equations are quite naturally introduced. It is
+\PageSep{159}
+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 divine 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 \hyperref[chapref:I]{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 physicists, that mathematical
+physics came into existence. In the natural
+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
+generalisation.
+\PageSep{160}
+
+
+\Chapter{X.}{The Theories of Modern Physics.}
+
+\Par{Significance of Physical Theories.}---The ephemeral
+nature of scientific 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 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
+\emph{bankruptcy of science}.
+
+His scepticism is superficial; he does not take
+into account the object of scientific theories and
+the part they play, or he would understand 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
+\PageSep{161}
+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 \emph{motion}, we now call \emph{electric
+current}. But these are merely names of the images
+we substituted for the real objects which Nature
+will hide for ever 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 between 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 instance) 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
+phenomena an intimate relationship which corresponds
+to a profound reality; that this relationship,
+\PageSep{162}
+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 Helmholtz, 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
+Helmholtz obtain the same equations, although
+their starting-points were to all appearance widely
+separated. I venture to say that these theories
+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
+\PageSep{163}
+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 contradiction 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 impact 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
+\PageSep{164}
+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 sensations 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 restricted meaning I have
+already mentioned, which is the sound meaning,
+then it has no meaning at all. Hypotheses of
+this kind have therefore 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 thought to be abandoned and
+definitively condemned by experiment, are suddenly
+revived from their ashes and begin a new life.
+It is because 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 \emph{electrons}. In what
+do these permanently electrified molecules differ
+\PageSep{165}
+from the electric molecules 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 established 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, Carnot's theory expressed in addition to
+true relations, other inexact relations, the \Foreign{débris}
+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
+\Chg{thermodynamics}{thermo-dynamics}. 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.
+Nor have the reasonings of Carnot perished on
+this account; they were applied to an imperfect
+conception of matter, but their form---\ie, the
+essential part of them, remained correct. What
+\PageSep{166}
+I have just said throws some light at the same
+time on the rôle of general principles, such as
+those of the principle of least action or of the
+conservation of energy. These principles are of
+very great 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 innumerable
+observations. However, from their very generality
+results a consequence to which I have called
+attention in \ChapRef{VIII}.---namely, that they are
+no longer capable of verification. As we cannot
+give a general definition of energy, the principle
+of the conservation of energy simply signifies that
+there is a \emph{something} which remains constant.
+\Pagelabel{166}%
+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 \emph{energy}. Does
+this mean that the principle has no meaning and
+vanishes into a tautology? Not at all. It means
+that the different things to which we give the
+name of \emph{energy} are connected by a true relationship;
+it affirms between them a real relation.
+But then, if this principle has 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---\ie,
+\PageSep{167}
+when we can no longer use it to predict correctly
+new phenomena. We shall be certain in such a
+case that the relation affirmed is no longer real,
+for otherwise it would be fruitful; experiment
+without directly contradicting a new extension of
+the principle will nevertheless have condemned it.
+
+\Par{Physics and Mechanism.}---Most theorists have a
+constant predilection 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 in \ChapRef{XII}.,
+\Foreign{àpropos} of Maxwell'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 number of such explanations. By
+means of a well-known theorem due to Königs,
+\PageSep{168}
+it may be shown that we can explain everything
+in an unlimited number of ways, by connections
+after the manner of Hertz, or, again, by central
+forces. No doubt it may be just as easily demonstrated
+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
+movements 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 \emph{ether} or other names, have
+from all time played so important a rôle 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 reduce 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 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
+\PageSep{169}
+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 materialising them, their
+individuality was, so to speak, emphasised---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
+relations 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 material
+molecules. We see, for instance, that the photographic
+plate experiences the consequences of a
+phenomenon of which the incandescent mass of
+a star was the scene several years before. Now,
+\PageSep{170}
+in ordinary mechanics, the state of the system
+under consideration depends 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 universe 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 \Chg{re-action}{reaction} is not true
+if applied to matter \emph{alone}, and that this can be
+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 \Chg{re-action}{reaction} of matter on something.
+\PageSep{171}
+
+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 \ChapRef{XIII}., were
+true, Newton's principle would not apply to matter
+\emph{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
+experiments have been undertaken, it is because
+we have not been certain beforehand; and indeed,
+according to current theories, the compensation
+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
+\PageSep{172}
+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 material bodies. Experiments have
+been made that should have disclosed 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 explanation was necessary, and was
+forthcoming; they always are; hypotheses 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 rôle? 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 the higher order, and that the mutual
+destruction of these terms will be rigorous and
+absolute.
+
+\Par{The Present State of Physics.}---Two opposite
+tendencies may be distinguished in the history
+\PageSep{173}
+of the development of physics. On the one hand,
+new relations are continually being discovered
+between objects which seemed destined to remain
+for ever 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 possible; but nothing
+proves this \Foreign{à~priori}, and it may be that after
+unsuccessful efforts to bend Nature to our ideal of
+unity in spite of herself, 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 yesterday.
+\PageSep{174}
+No 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 phenomena 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 concerned, there
+was not so great an advance. Electricity had just
+annexed 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 universal mechanism? No one had
+the slightest idea. As to the possibility 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.
+
+To-day, what do we see? In the first place, a
+step in advance---immense progress. The relations
+between light and electricity are now known; the
+three domains of light, electricity, and magnetism,
+formerly separated, are now one; and this annexation
+seems definitive.
+
+Nevertheless the conquest has caused us some
+\PageSep{175}
+sacrifices. Optical phenomena become particular
+cases in electric phenomena; as long as the former
+remained isolated, it was easy to explain them by
+movements 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 difficulty.
+
+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
+Newton's law that action and \Chg{re-action}{reaction} 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 \Chg{re-action}{reaction} 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 abnormal 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
+\PageSep{176}
+rotation, which had defied all Maxwell's efforts.
+This facility proves that Lorentz's theory is not a
+mere artificial combination 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, MacCullagh'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 movements 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 \Chg{re-action}{reaction}
+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.
+
+If Larmor has failed, as in my opinion he has,
+does it necessarily follow that a mechanical explanation
+is impossible? Far from it. I said
+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
+\PageSep{177}
+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 condition 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 confirmed,
+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
+Lagrange'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
+\PageSep{178}
+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 physics 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 eminently abstract
+concept of entropy which is as universal as the
+concept 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 pursued in detail; electric resistance
+resembles the viscosity of fluids; hysteresis 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 irreversibility is but apparent, that
+the elementary phenomena are reversible 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---\ie, all seems
+to progress in the same direction, and that without
+\PageSep{179}
+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 contradictions, and if it is in conformity
+with the true nature of things.
+
+Let us notice, however, the original ideas of
+M.~Gouy on the Brownian 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.\footnote
+  {Clerk-Maxwell imagined some supernatural agency at work,
+  sorting molecules in a gas of uniform temperature into (\textit{a})~those
+  possessing kinetic energy above the average, (\textit{b})~those possessing
+  kinetic energy below the average.\Transl}
+
+To resume, phenomena long known are gradually
+being better classified, 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 unexpected
+\PageSep{180}
+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 sometimes 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 important
+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 facilitate 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 majestic simplicity, but
+matter properly so called seems more and more
+complex; all we can say of it is but approximate,
+and our formulæ 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
+\PageSep{181}
+when their complexity is recognised, 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 equations
+persists. Take for instance the laws of reflection.
+Fresnel established 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 polarisation were detected everywhere.
+But it is owing to the first approximation that the
+cause of these anomalies was found in the existence
+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 properties
+of matter better we see that continuity reigns.
+\PageSep{182}
+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.
+Similarly, 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 satisfy 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 realisation of unity. This has not
+been done as quickly as was hoped fifty years ago,
+and the path predicted has not always been
+followed; but, on the whole, much ground has
+been gained.
+\PageSep{183}
+
+
+\Chapter{XI.}{The Calculus of Probabilities.}
+
+\First{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 experiment 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 generalisations. 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
+\PageSep{184}
+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 favourable 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
+$6 × 6 = 36$. The number of favourable cases is~$11$;
+the probability is~$\dfrac{11}{36}$. 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
+$\dfrac{6 × 7}{2} = 21$ different combinations. Among these
+combinations, six are favourable; the probability
+\PageSep{185}
+is~$\dfrac{6}{21}$. Now why is the first method of calculating
+the number of possible cases more legitimate than
+the second? In any case it is not the definition
+that tells us. We are therefore bound to complete
+the definition by saying, ``\ldots 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 successively adopted two conventions
+which seemed to be equally imperative in the eyes
+of common-sense, and with one convention he finds~$\frac{1}{2}$,
+and with the other~$\frac{1}{3}$. The conclusion which
+seems to follow from this is that the calculus of
+probabilities is a useless science, that the obscure
+\PageSep{186}
+instinct which we call common-sense, and to which
+we appeal for the legitimisation 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
+impossible, 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 perturbations? Here
+again the only answer is: It is very improbable.
+From this point of view all the sciences would only
+be unconscious applications of the calculus of probabilities.
+And if this calculus be condemned, then
+the whole of the sciences must also be condemned.
+I shall not dwell at length on scientific problems
+in which the intervention of the calculus of probabilities
+is more evident. In the forefront of these
+is the problem of interpolation, in which, knowing
+a certain number of values of a function, we try
+to discover the intermediary values. I may also
+mention the celebrated theory of errors of observation,
+\PageSep{187}
+to which I shall return later; the kinetic
+theory of gases, a well-known hypothesis wherein
+each gaseous molecule is supposed 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 confidence 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 \Foreign{coup}, and he
+\PageSep{188}
+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 \emph{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 \Foreign{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
+\emph{objective probability}, and it is this phenomenon
+which has to be explained. There are numerous
+Insurance Societies which apply the rules of the
+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 scepticism
+is not admissible. We may distrust, but we
+cannot condemn \Foreign{en~bloc}. Discussion is necessary.
+
+\Par[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 \emph{generality}. I said above that probability
+is the ratio of the number of favourable 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, if we take a throw of the
+\PageSep{189}
+dice in which the number of possible cases is~$36$.
+That is the first degree of generality. But if we
+ask, for instance, 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
+%[** TN: French edition uses a fraction]
+that this probability is~\Reword{$1$-$10$th}{$\frac{1}{10}$}. 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.
+\PageSep{190}
+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 distribution.
+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 probabilities 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 imperfect 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
+\PageSep{191}
+from the effects. Now, these problems are classified
+as \emph{probability of causes}, and are the most interesting
+of all from their scientific applications. I play at
+\Foreign{écarté} with a gentleman whom I know to be perfectly
+honest. What is the chance that he turns
+up the king? It is~$\frac{1}{8}$. 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 experimental
+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 divergencies 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.
+
+\Par[II. ]{Probability in Mathematics.}---The impossibility
+of squaring the circle was shown in~1885, but
+before that date all geometers considered this impossibility
+as so ``probable'' that the Académie des
+Sciences rejected without examination the, alas!\
+\PageSep{192}
+too numerous memoirs on this subject that a
+few unhappy madmen sent in every year. Was
+the Académie 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 Académie 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 probability 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 psychological. 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 $\pi$~be the root of an algebraical equation,
+why do you expect this root to be a period of
+%[** TN: sin italicized in the original]
+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
+\PageSep{193}
+what I called above objective probability has
+nothing in common with this first problem. It is
+otherwise with the second. Let us consider the
+first $10,000$ logarithms 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 hesitation that the probability is~$\frac{1}{2}$, 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 \Title{Revue générale 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 derivatives
+of the logarithm remain, in the interval considered,
+\PageSep{194}
+between certain limits. Hence our first conclusion
+is that the property is not only true of the
+logarithm but of any continuous function whatever,
+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 unconscious 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 approximately. 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~$\sin nu$? 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
+$\phi(a)\, da$; that it is therefore proportional to the
+infinitely small interval~$da$, and is equal to this
+multiplied by a function~$\phi(a)$, 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~$2\pi$,
+I may without loss of generality assume that
+\PageSep{195}
+$u$~lies between $0$~and~$2\pi$, and I shall thus be
+led to suppose that $\phi(a)$~is a periodic function
+whose period is~$2\pi$. The mean value that we
+seek is readily expressed by a simple integral,
+and it is easy to show that this integral is smaller
+than
+%[** TN: Expression displayed in the French edition]
+\[
+\frac{2\pi M_{K}}{n^{K}},
+\]
+$M_{K}$~being the maximum value of the
+$K$th~derivative of~$\phi(u)$. We see then that if the
+$K$th~derivative is finite, our mean value will
+tend towards zero when $n$~increases indefinitely,
+and that more rapidly than~$\dfrac{1}{n^{K+1}}$.
+
+%[** TN: Paragraph break in the French edition]
+The mean
+value of~$\sin nu$ when $n$~is very large is therefore
+zero. To define this value I required a convention,
+but the result remains the same \emph{whatever
+that convention may be}. I have imposed upon
+myself but slight restrictions when I assumed that
+the function~$\phi(a)$ 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 rôle of what philosophers call the
+principle of sufficient reason, and on the other hand
+of the importance of the fact that certain properties
+are common to all continuous functions.
+The study of probability in the physical sciences
+will lead us to the same result.
+
+\Par[III. ]{Probability in the Physical Sciences.}---We
+now come to the problems which are connected
+with what I have called the second degree of
+\PageSep{196}
+ignorance---namely, those in which we know the
+law but do not know the initial state of the
+system. I could multiply examples, but I shall
+take only one. What is the probable present
+distribution 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 uniform. 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---\ie, 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 certain
+region of the plane, but as they are very numerous
+this region will appear dotted with points. We
+know nothing else about the distribution 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
+\PageSep{197}
+compelled to make an arbitrary hypothesis. To
+explain the nature of this hypothesis 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
+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
+regions. Here then are two distributions, one
+real, in which the representative points are very
+numerous, very close together, but discrete like the
+molecules of matter in the 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 continuous matter may be nearly proportional
+to the number of representative points,
+or, if it is preferred, to the number of atoms which
+are contained in that region. Even that is impossible,
+\PageSep{198}
+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 longitude at the
+moment~$t$---\ie, of $\sin(at + b)$? We made at the
+outset an arbitrary convention, but if we adopt it,
+this probable value is entirely defined. Let us
+decompose the plane into elements of surface.
+Consider the value of $\sin(at + b)$ at the centre of
+each of these elements. 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~$\phi$ which defines the density of the imaginary
+matter, and as this function~$\phi$ 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
+\PageSep{199}
+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. Whatever the function~$\phi$
+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
+very small. I may give to~$\phi$ 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 instance, for supposing
+that the initial longitude might be exactly~$0°$, but
+that it could not lie between $0°$~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
+\[
+\frac{1}{n} \sum \sin(at + b),
+\]
+$n$~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
+\PageSep{200}
+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 increased. 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 $b = \dfrac{\pi}{2} - at$, the
+longitude of all the planets at the time~$t$ would be~$\dfrac{\pi}{2}$,
+and the mean value in question would be
+evidently unity. For this to be the case at the
+time~$0$, 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 realised, 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 anything
+on the subject of the probability of this or
+that present distribution. Once more we shall
+invoke the principle of sufficient reason, to which
+we must always recur. We might admit that at
+the beginning the planets were distributed almost
+\PageSep{201}
+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.
+
+\Par[IV. ]{Rouge et Noir.}---The questions raised by
+games of chance, such as roulette, are, fundamentally,
+quite analogous to those we have just
+%[** TN: "est partagé en un grand mombre de subdivisions égales"]
+treated. For example, a wheel is divided into \Reword{thirty-seven}{a large number of}
+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 division is red is obviously~$\frac{1}{2}$. The needle
+describes an angle~$\theta$, including 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 $\theta$~and~$\theta + d\theta$, but I
+can make a convention. I can suppose that this
+probability is~$\phi(\theta)\, d\theta$. As for the function~$\phi(\theta)$, I
+can choose it in an entirely arbitrary manner. I
+have nothing to guide me in my choice, but I am
+naturally induced to suppose the function to be
+continuous. Let $\epsilon$~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~$\phi(\theta)\, d\theta$, extending it on the one
+hand to all the red, and on the other hand to all
+\PageSep{202}
+the black compartments, and to compare the
+results. Consider an interval~$2\epsilon$ comprising two
+consecutive red and black compartments. Let
+$M$~and~$m$ be the maximum and minimum values of
+the function~$\phi(\theta)$ in this interval. The integral
+extended to the red compartments will be smaller
+than~$\sum M\epsilon$; extended to the black it will be greater
+than~$\sum m\epsilon$. The difference will therefore be
+smaller than $\sum (M - m)\epsilon$. But if the function~$\phi$ is
+supposed continuous, and if on the other hand the
+interval~$\epsilon$ 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~$\frac{1}{2}$. We see that
+without knowing anything of the function~$\phi$ we
+must act as if the probability were~$\frac{1}{2}$. And on
+the other hand it explains why, from the
+objective point of view, if I watch a certain
+number of \Foreign{coups}, observation will give me almost
+as many black \Foreign{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~$\frac{1}{2}$. Observation shows, it is true, that
+the series of seven consecutive reds is very rare,
+\PageSep{203}
+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.
+
+\Par[V. ]{The Probability of Causes.}---We now come to
+the problems of the probability of causes, the
+most important from the point of view 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 correspond
+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 \Foreign{à~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 produced
+by the cause~$a$ or by the cause~$b$. The effect
+has just been observed. We ask the probability
+\PageSep{204}
+that it is due to the cause~$a$. This is an \Foreign{à~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 \Foreign{à~priori}
+probability for the cause~$a$ to come into play---I
+mean the probability of this event to some one
+who had not observed the effect. To make my
+meaning clearer, I go back to the game of \Foreign{écarté}
+mentioned before. My adversary deals for the
+first time and turns up a king. What is the
+probability that he is a sharper? The formulæ
+ordinarily taught give~$\frac{8}{9}$, 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 \Foreign{à~priori} probability was unjustified, and that
+is why the conclusion of the \Foreign{à~posteriori} probability
+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 \Foreign{à~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. I require to determine an
+experimental law; this law, when discovered, can
+\PageSep{205}
+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 inflexions, and any sudden variation
+of the radius of curvature. This curve will represent
+to me the probable law, and not only will
+it give me the values of the functions intermediary
+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 combination 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 observations
+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 observations
+I had an \Foreign{à~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
+\PageSep{206}
+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 \Foreign{à~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 principle was even
+invoked in favour 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 disappear, experimental science would become
+impossible.
+
+\Par[VI. ]{The Theory of Errors.}---We are thus brought
+to consider the theory of errors which is directly
+\PageSep{207}
+connected with the problem of the probability
+of causes. Here again we find \emph{effects}---to wit,
+a certain number of irreconcilable observations,
+and we try to find the \emph{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 observation. We must calculate
+the probable \Foreign{à~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
+\Foreign{à~priori}---\ie, before any observations are made---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 between 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
+mistake, 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?
+\PageSep{208}
+Numerous proofs have been attempted, almost all
+of them crude paralogisms. But starting from
+the following hypotheses we may prove Gauss's
+law: the error is the result of a very large number
+of partial and independent 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 astronomers. This
+is no doubt because the latter, apart from the
+systematic errors to which they and the physicists
+are subject alike, have to contend with an
+extremely important source of error which is
+entirely accidental---I mean atmospheric undulations.
+So it is very curious 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 disappear.''
+
+What conclusion must we draw? Must we
+continue to use the method of least squares?
+\PageSep{209}
+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 result is so much. \emph{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 observations; 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 \emph{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 absolute.
+
+\Par[VII. ]{Conclusions.}---In the preceding lines I have
+set several problems, and have given no solution.
+\PageSep{210}
+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. Unfortunately, this principle
+is very vague and very elastic, and in the cursory
+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 hypothesis made at the outset,
+provided only that this hypothesis satisfies the
+condition of continuity.
+\PageSep{211}
+
+
+\Chapter{XII.\protect\footnotemark}{Optics And Electricity.}
+\footnotetext{This chapter is mainly taken from the prefaces of two of my
+  books---\Title{Théorie Mathématique de la lumière} (Paris: Naud, 1889),
+  and \Title{Électricité et Optique} (Paris: Naud, 1901).}
+
+\Par{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 undulations 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 mathematical 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, the 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
+\PageSep{212}
+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 rôle. 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.
+
+\Par{Maxwell's Theory.}---We all know that it was
+Maxwell who connected by a slender tie two
+branches of physics---optics and electricity---until
+\PageSep{213}
+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 revealed to us other
+relations, hitherto unsuspected, between the
+different branches of optics and the domain of
+electricity.
+
+\Pagelabel{213}%
+The first time a French reader opens Maxwell's
+book, his admiration 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 acclimatised among us? No
+doubt the education received by most enlightened
+Frenchmen predisposes them to appreciate precision
+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 lines. Starting with clearly enunciated
+hypotheses, they deduced from them all their
+consequences with mathematical rigour, 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.
+\PageSep{214}
+
+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 hypotheses 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 its opinion---which will
+no longer have anything but purely geometrical
+qualities, and the atoms of which will be mathematical
+points subject to the laws of dynamics
+alone. And yet he will try to represent to
+himself, by an unconscious contradiction, these
+invisible and colourless 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 satisfaction 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 theoretical 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
+\PageSep{215}
+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 particular case of electro-magnetic phenomena.
+From the whole theory of electricity a theory of
+light can be immediately deduced. Unfortunately
+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 ourselves 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 constructions,
+between which communication is difficult and
+sometimes impossible. Take, for instance, the
+chapter in which \Chg{electrostatic}{electro-static} attractions are
+explained by the pressures and tensions of the
+dielectric medium. This chapter might be suppressed
+without the rest of the book being
+\PageSep{216}
+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 independent 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 make 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 polarisation that there is an
+identity between optical and magnetic phenomena?
+
+We must not flatter ourselves that we have
+avoided every contradiction, 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
+popularised, 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
+\PageSep{217}
+fundamental idea consists; but for that purpose
+a short digression is necessary.
+
+\Par{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 endeavour 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 phenomenon? It
+will be, on the one hand, when we know the
+differential equations which are satisfied by the
+co-ordinates of these hypothetical 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 functions 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.
+\PageSep{218}
+
+The first of these two principles teaches us that
+the total energy is constant, and may be divided
+into two parts:\Add{---}
+
+(1) Kinetic energy, or \Foreign{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 instant~$t_{0}$ to
+the final position occupied at the instant~$t_{1}$, the
+system must describe such a path that in the
+interval of time between the instant $t_{0}$~and~$t_{1}$,
+the mean value of the action---\ie, the \emph{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\Typo{ }{ one} 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
+Lagrange. In these equations the independent
+\PageSep{219}
+variables are the co-ordinates of the hypothetical
+molecules~$m$; but I now assume that we take for
+the variables the parameters~$q$, which are directly
+accessible to experiment.
+
+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 endeavour to define
+kinetic and potential energy by the aid of
+quantities he can directly observe.\footnote
+  {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 derivatives.}
+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
+parameters that we define the initial and \Typo{fina}{final}
+positions; the principle of least action will always
+remain true.
+
+Now here again, of the whole of the paths which
+lead from one position to another, there is one and
+only one for which the mean action is a minimum.
+The principle of least action is therefore sufficient
+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.
+\PageSep{220}
+
+To form these equations we need not know the
+relations which connect the parameters~$q$ 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~$U$ as a
+function of the parameters~$q$, and that of~$T$ as a
+function of the parameters~$q$ and their derivatives---\ie,
+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 indicated, 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 \emph{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
+\emph{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
+equations formed by the aid of these two functions
+$T$~and~$U$ are in conformity with the data of the
+\PageSep{221}
+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 \Foreign{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 \Title{Électricité 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 comes to the same thing, we look on them as
+derivatives of the parameters~$q$, or as combinations
+of these parameters 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
+parameters? This choice remains arbitrary to a
+large extent, but a mechanical explanation will be
+\PageSep{222}
+possible if it is done so as to satisfy the principle of
+least action.
+
+Next, Maxwell asks: Can this choice and that of
+the two energies $T$~and~$U$ be made so that electric
+phenomena will satisfy this principle? Experiment
+shows us that the energy of an electro-magnetic
+field decomposes into electro-static and electro-dynamic
+energy. Maxwell recognised 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 parameters~$q$, and the intensity
+of the currents as derivatives of other
+parameters~$q$---under these conditions, Maxwell
+has recognised that electric phenomena \Reword{satisfies}{satisfy} 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 mechanical 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 polarisation; Neumann holds that it is
+parallel to that plane. For a long time an \Foreign{experimentum
+crucis} was sought for, which would enable
+us to decide between these two theories, but in
+\PageSep{223}
+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.
+
+\Pagelabel{223}%
+It 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's 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 perhaps come
+when physicists will no longer concern themselves
+with questions which are inaccessible to positive
+methods, and will leave 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,
+\PageSep{224}
+Maxwell 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 \Foreign{magnum opus} he
+develops no complete explanation, he has attempted
+one in an article in the \Title{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---\ie,
+what is common to all theories; everything 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 compelled 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.
+\Pagelabel{224}%
+\PageSep{225}
+
+
+\Chapter{XIII.}{Electro-Dynamics.}
+
+\First{The} history of electro-dynamics is very instructive
+from our point of view. The title of Ampère's
+immortal work is, \Title{Théorie des phénomènes electro-dynamiques,
+uniquement fondée sur expérience}. He
+therefore imagined that he had made no hypotheses;
+but as we shall not be long in recognising, he was
+mistaken; only, of these hypotheses he was quite
+unaware. On the other hand, his successors see
+them clearly enough, because their attention is
+attracted by the weak points in Ampère'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.
+
+\Par[I. ]{Ampère's Theory.}---In Ampère'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
+current is set up which passes from one to the
+\PageSep{226}
+other until the two potentials are equal. According
+to the ideas of Ampère's time, this was
+considered to be an open current; the current was
+known to pass 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 Ampère to be open
+currents---for instance, the currents of discharge
+of a condenser; he was unable to experiment 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 electric charge, and leaving~$B$ are
+set in motion along a path~$BNA$, carrying 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 \emph{through} a fixed conductor
+like a voltaic current, overcoming an ohmic resistance
+and developing heat; we say that it is
+displaced by \emph{conduction}. In the part~$BNA$ the
+electricity is \emph{carried} by a moving conductor, and is
+said to be displaced by \emph{convection}. If therefore the
+convection current is considered to be perfectly
+analogous to the conduction current, the circuit~$BNAMB$
+is closed; if on the contrary the convection
+\PageSep{227}
+current is not a ``true current,'' and, for
+instance, does not act on the magnet, there is only
+the conduction current~$AMB$, which is \emph{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 produce with appreciable intensity;
+in fact, with the means at Ampère's disposal we
+may almost say it was impossible.
+
+To sum up, Ampère 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. Experiment therefore could only show him
+the action of a closed current on a closed current---or
+more accurately, the action of a closed current
+on a portion of current, because a current can be
+made to describe a \emph{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, Ampère had no means of studying the action
+of an open current either on a closed or on another
+open current.
+
+\Par[1.\ ]{The Case of Closed Currents.}---In the case of
+the mutual action of two closed currents, experiment
+revealed to Ampère remarkably simple
+laws. The following will be useful to us in the
+sequel:---
+\PageSep{228}
+
+(1) \emph{If the intensity of the currents is kept constant},
+and if the two circuits, after having undergone any
+displacements and deformations whatever, return
+finally to their initial positions, the total work
+done by the electro-dynamical actions is zero. In
+other words, there is an \emph{electro-dynamical potential}
+of the two circuits proportional to the product of
+their intensities, and depending on the form and
+relative 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:---
+
+(\textit{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.
+
+(\textit{b}) A very short magnetic needle tends to take
+the direction of the magnetic force, and the couple
+to which it tends to reduce is proportional to the
+product of the magnetic force, the magnetic
+moment of the needle, and the sine of the dip
+of the needle.
+
+(\textit{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
+\PageSep{229}
+of magnetic force'' which passes through the
+circuit.
+
+\Par[2.\ ]{Action of a Closed Current on a Portion of
+Current.}---Ampère 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~$\alpha\beta$, the ends $\alpha$~and~$\beta$
+of which could slide along a fixed wire. In one of
+the positions of the movable wire the end~$\alpha$ rested
+on the point~$A$, and the end~$\beta$ on the point~$B$ of
+the fixed wire. The current ran from~$\alpha$ to~$\beta$---\ie,
+from~$A$ to~$B$ along the movable wire, and then
+from~$B$ to~$A$ along the fixed wire. \emph{This current
+was therefore closed.}
+
+In the second position, the movable wire
+having slipped, the points $\alpha$~and~$\beta$ were respectively
+at $A'$~and~$B'$ on the fixed wire. The current
+ran from~$\alpha$ to~$\beta$---\ie, from~$A'$ to~$B'$ on the movable
+wire, and returned 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.
+Ampère \emph{admits} that the force, apparently acting on
+the movable part~$AB$, representing the action of~$C$
+on the portion~$\alpha\beta$ of the current, remains the
+same whether an open current runs through~$\alpha\beta$,
+\PageSep{230}
+stopping at $\alpha$~and~$\beta$, or whether a closed current
+runs first to~$\beta$ and then returns to~$\alpha$ through the
+fixed portion of the circuit. This hypothesis
+seemed natural enough, and Ampère innocently
+assumed it; nevertheless the hypothesis \emph{is not a
+necessity}, for we shall presently see that Helmholtz
+rejected it. However that may be, it enabled
+Ampère, 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:\Add{---}
+
+(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 normal to the element.
+
+(2) The action of a closed solenoid on an
+element of current is zero. But the electro-dynamic
+potential has disappeared---\ie, when a
+closed and an open current of constant intensities
+return to their initial positions, the total work
+done is not zero.
+
+\Par[3.\ ]{Continuous Rotations.}---The most remarkable
+electro-dynamical experiments are those in which
+continuous rotations are produced, and which are
+called \emph{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
+\PageSep{231}
+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.
+Continuous 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 convenient to draw a distinction. The
+passage from the fixed to the movable part, or
+\Foreign{vice~versâ}, 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 connexions and therefore the
+conditions of equilibrium, so that as the position
+of equilibrium is ever eluding, so to speak, the
+system which is trying to reach it, rotation may
+take place indefinitely.
+\PageSep{232}
+
+Ampère 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 current. He concluded that the action of
+a closed on an open current, or \Foreign{vice~versâ}, 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.
+
+\Par[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. Ampère
+falls back on hypothesis. He assumes: (1)~that
+the mutual action of two elements reduces to a
+force acting along their \emph{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 Ampère
+makes two hypotheses 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
+\PageSep{233}
+current acting on an open current. Next, there
+is, properly 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 current 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 continuous 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, \emph{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
+\PageSep{234}
+of current 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 to Ampère
+it would not be true if the systems contained open
+currents. 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~$A$.
+
+\Par[5.\ ]{Induction.}---We know that the discovery of
+electro-dynamical induction followed not long after
+the immortal work of Ampère. 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 conservation of energy is
+sufficient for us to deduce the laws of induction
+from the electro-dynamical laws of Ampère. But
+on the condition, as Bertrand has shown,---that
+we make a certain number of hypotheses.
+
+The same principle again enables this deduction
+to be made in the case of open currents, although
+the result cannot be tested by experiment, since
+such currents cannot be produced.
+
+If we wish to compare this method of analysis
+with Ampère'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;
+\PageSep{235}
+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 \emph{mean} electro-motive force
+%[** TN: "induite dans le circuit C est nulle" in the French edition.]
+induced in the \Reword{current}{circuit}~$C$ 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 Ampère, 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.
+
+\Par[II. ]{Helmholtz's Theory.}---I have dwelt upon the
+consequences of Ampère'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
+Ampère---namely, that the mutual action of two
+elements of current reduces to a force along their
+join. He admits that an clement of current is not
+\PageSep{236}
+acted upon by a single force but by a force and a
+couple, and this is what gave rise to the celebrated
+polemic between Bertrand and Helmholtz.
+Helmholtz replaces Ampère's hypothesis 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 Ampère, 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 Ampère 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 current. 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~$\alpha\beta$ 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 elements
+\PageSep{237}
+$AA'$~and~$B'B$ with respect to~$C$. It is this \emph{double}
+increment which represents the work of the force
+acting upon the portion~$AB$. If, on the contrary,
+$\alpha\beta$~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 experiment, 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 contacts of the \Typo{weir}{wire}
+with the magnet were suppressed, and an open
+current 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 Ampère, 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
+\PageSep{238}
+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 the
+definition of magnetic force to have any meaning,
+the action exercised by an open current on
+an indefinite solenoid would only depend on the
+position of the extremity of that solenoid---\ie,
+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.
+
+\Par[III. ]{Difficulties raised by these Theories.}---Helmholtz's
+theory is an advance on that of Ampère;
+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 convention, 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 difficult to give up deeply-rooted habits of
+\PageSep{239}
+language and thought. There is something more.
+If we do not believe in actions at a distance,
+electro-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.
+
+\Par[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 \emph{current
+of displacement}. If, then, two conductors bearing
+positive and negative charges are placed in connection
+by means of a wire, during the discharge
+there is an open current of conduction in that
+wire; but there are produced at the same time in
+the surrounding dielectric currents of displacement
+which close this current of conduction. We
+know that Maxwell's theory leads to the explanation
+of optical phenomena 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 Maxwell's ideas received the
+confirmation of experiment. Hertz succeeded in
+producing systems of electric oscillations which
+\PageSep{240}
+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
+displacement on the galvanometer. That is true
+in a sense. What he has shown directly is that
+electro-magnetic induction is not instantaneously
+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---\emph{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.
+
+\Par[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 convection
+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.
+\PageSep{241}
+
+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''---\ie, a charged conductor in
+motion---could act on the galvanometer. But experimental
+confirmation 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. Rowland, an
+extremely skilful experimentalist, was the first to
+triumph, or to seem to triumph, over these difficulties.
+A disc received a strong \Chg{electrostatic}{electro-static}
+charge and a very high speed of rotation. An
+astatic magnetic system placed beside the disc
+underwent deviations. The experiment was made
+twice by Rowland, once in Berlin and once at Baltimore.
+It was afterwards repeated by Himstedt.
+These physicists even believed that they could
+announce that they had succeeded in making
+quantitative measurements. For twenty years
+Rowland's law was admitted without objection
+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
+recognise the lines of the metal of the electrode,
+a proof of it? The spark would then be a real
+current of induction.
+\PageSep{242}
+
+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 electrolyte would
+therefore also be a current of convection; but it
+acts on the magnetic needle. And in the same
+way for cathode rays; \Typo{Crooks}{Crookes} attributed these
+rays to very subtle matter charged with negative
+electricity and moving with very high velocity.
+He looked upon them, in other words, as currents
+of convection. Now, these cathode rays are
+deviated by the magnet. In virtue of the
+principle of action and \Chg{re-action}{reaction}, 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 collecting 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 Rowland's
+law.
+
+\Par[VI. ]{Lorentz's Theory.}---We need not go much
+further. According to Lorentz's theory, currents
+\PageSep{243}
+of conduction would themselves be true convection
+currents. Electricity would remain indissolubly
+connected with certain material particles called
+\emph{electrons}. The circulation 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 attractive. 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 aberration of light, the partial
+impulse of luminous waves, magnetic polarisation,
+and Zeeman's experiment.
+
+A few objections still remained. The phenomena
+of an electric system 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.~Crémieu, M.~Lippman 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.
+``No!'' replied the partisans of Lorentz. ``What
+we could measure in that way is not their
+\PageSep{244}
+absolute velocity, but their relative velocity \emph{with
+respect to the ether}, so that the principle of relativity
+is safe.'' Whatever there may be in these
+objections, the edifice of electro-dynamics seemed,
+at any rate in its broad lines, definitively constructed.
+Everything was presented under the
+most satisfactory aspect. The theories of Ampère
+and Helmholtz, which were made for the open
+currents that no longer existed, seem to have no
+more than purely historic interest, and the inextricable
+complications to which these theories
+led have been almost forgotten. This quiescence
+has been recently disturbed by the experiments of
+M.~Crémieu, which have contradicted, or at least
+have seemed to contradict, the results formerly
+obtained by Rowland. Numerous investigators
+have endeavoured 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 book is ready for the press and the day on
+which it is placed before the public.
+\begin{center}
+\vfill
+\footnotesize THE END.
+\vfill\vfill
+\rule{1in}{0.5pt} \\[4pt]
+\makebox[0pt][c]{\scriptsize THE WALTER SCOTT PUBLISHING CO., LIMITED, FELLING-ON-TYNE.}
+\end{center}
+
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+% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %
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+% The Project Gutenberg EBook of Science and hypothesis, by Henri Poincaré%
+%                                                                         %
+% This eBook is for the use of anyone anywhere at no cost and with        %
+% almost no restrictions whatsoever.  You may copy it, give it away or    %
+% re-use it under the terms of the Project Gutenberg License included     %
+% with this eBook or online at www.gutenberg.net                          %
+%                                                                         %
+%                                                                         %
+% Title: Science and hypothesis                                           %
+%                                                                         %
+% Author: Henri Poincaré                                                  %
+%                                                                         %
+% Release Date: August 21, 2011 [EBook #37157]                            %
+%                                                                         %
+% Language: English                                                       %
+%                                                                         %
+% Character set encoding: ISO-8859-1                                      %
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+% *** START OF THIS PROJECT GUTENBERG EBOOK SCIENCE AND HYPOTHESIS ***    %
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+%%   OCR text for this ebook was obtained on July 30, 2011, from    %%
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+\small
+\begin{PGtext}
+The Project Gutenberg EBook of Science and hypothesis, by Henri Poincaré
+
+This eBook is for the use of anyone anywhere at no cost and with
+almost no restrictions whatsoever.  You may copy it, give it away or
+re-use it under the terms of the Project Gutenberg License included
+with this eBook or online at www.gutenberg.net
+
+
+Title: Science and hypothesis
+
+Author: Henri Poincaré
+
+Release Date: August 21, 2011 [EBook #37157]
+
+Language: English
+
+Character set encoding: ISO-8859-1
+
+*** START OF THIS PROJECT GUTENBERG EBOOK SCIENCE AND HYPOTHESIS ***
+\end{PGtext}
+\end{minipage}
+\end{center}
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+Produced by Andrew D. Hwang
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+%%%%%%%%%%%%%%%%%%%%%%%%%%% FRONT MATTER %%%%%%%%%%%%%%%%%%%%%%%%%%
+\PageSep{i}
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+\ifthenelse{\boolean{ForPrinting}}{%
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+{\Large SCIENCE AND HYPOTHESIS}
+\vfill
+\cleardoublepage
+}{}% Omit half-title in screen version
+\PageSep{ii}
+%[Blank page]
+\PageSep{iii}
+\begin{center}
+\huge SCIENCE \\
+AND HYPOTHESIS
+\vfill
+
+\normalsize
+{\footnotesize BY} \\
+H. POINCARÉ, \\
+{\scriptsize MEMBER OF THE INSTITUTE OF FRANCE.}
+\vfill
+
+{\small\textsc{With a Preface by}} \\
+\textsc{J. LARMOR, D.Sc., Sec. R.S.}, \\
+{\scriptsize\textit{Lucasian Professor of Mathematics in the University of Cambridge.}}
+\vfill\vfill
+
+\textswab{London and Newcastle-on-Tyne:} \\
+THE WALTER SCOTT PUBLISHING CO., LTD \\
+{\footnotesize NEW YORK: 3 EAST 14TH STREET.} \\
+\oldstylenums{1905}.
+\end{center}
+\newpage
+\PageSep{iv}
+%[Blank page]
+\PageSep{v}
+\TableofContents
+\iffalse
+CONTENTS.
+
+                                       PAGE
+Translator's Note........................ix
+
+Introduction.............................xi
+
+Author's Preface........................xxi
+
+
+PART I.
+
+NUMBER AND MAGNITUDE.
+
+CHAPTER I.
+On the Nature of Mathematical Reasoning...1
+
+CHAPTER II.
+Mathematical Magnitude and Experiment....17
+
+
+PART II.
+
+SPACE.
+
+CHAPTER III.
+Non-Euclidean Geometries.................35
+\PageSep{vi}
+
+CHAPTER IV.
+Space and Geometry.......................51
+
+CHAPTER V.
+Experiment and Geometry..................72
+
+
+PART III.
+
+FORCE.
+
+CHAPTER VI.
+The Classical Mechanics..................89
+
+CHAPTER VII.
+Relative and Absolute Motion............111
+
+CHAPTER VIII.
+Energy and Thermo-dynamics..............123
+
+
+PART IV.
+
+NATURE.
+
+CHAPTER IX.
+Hypotheses in Physics...................140
+\PageSep{vii}
+
+CHAPTER X.
+The Theories of Modern Physics..........160
+
+CHAPTER XI.
+The Calculus of Probabilities...........183
+
+CHAPTER XII.
+Optics and Electricity..................211
+
+CHAPTER XIII.
+Electro-dynamics........................225
+\fi
+\PageSep{viii}
+%[Blank page]
+\PageSep{ix}
+
+
+\OtherChapter{Translator's Note.}
+
+\First{The} translator wishes to express his indebtedness
+to Professor Larmor, for kindly consenting
+to introduce the author of \Title{Science and Hypothesis}
+to English readers; to Dr.~F.~S. Macaulay and
+Mr.~C.~S. Jackson,~M.A., who have read the whole
+of the proofs and have greatly helped by suggestions;
+also to Professor G.~H. Bryan,~F.R.S., who
+has read the proofs of \ChapRef{VIII}., and whose
+criticisms have been most valuable.
+
+%[** TN: William John Greenstreet (1861--1930), not identified by name]
+\Signature{W.\;J.\;G.}{\textit{February} 1905.}
+\PageSep{x}
+%[Blank page]
+\PageSep{xi}
+
+
+\OtherChapter{Introduction.}
+
+\First{It} is to be hoped that, as a consequence of the
+present active scrutiny of our educational aims
+and methods, and of the resulting encouragement
+of the study of modern languages, we shall not
+remain, as a nation, so much isolated from
+ideas and tendencies in continental thought and
+literature as we have been in the past. As things
+are, however, the translation of this book is
+doubtless required; at any rate, it brings vividly
+before us an instructive point of view. Though
+some of M.~Poincaré's chapters have been collected
+from well-known treatises written several years
+ago, and indeed are sometimes in detail not quite
+up to date, besides occasionally suggesting the
+suspicion that his views may possibly have been
+modified in the interval, yet their publication in
+a compact form has excited a warm welcome in
+this country.
+
+It must be confessed that the English language
+\PageSep{xii}
+hardly lends itself as a perfect medium for the
+rendering of the delicate shades of suggestion
+and allusion characteristic of M.~Poincaré's play
+around his subject; notwithstanding the excellence
+of the translation, loss in this respect is
+inevitable.
+
+There has been of late a growing trend of
+opinion, prompted in part by general philosophical
+views, in the direction that the theoretical constructions
+of physical science are largely factitious,
+that instead of presenting a valid image of the
+relations of things on which further progress can
+be based, they are still little better than a mirage.
+The best method of abating this scepticism is to
+become acquainted with the real scope and modes
+of application of conceptions which, in the popular
+language of superficial exposition---and even in
+the unguarded and playful paradox of their
+authors, intended only for the instructed eye---often
+look bizarre enough. But much advantage
+will accrue if men of science become their own
+epistemologists, and show to the world by critical
+exposition in non-technical terms of the results
+and methods of their constructive work, that more
+than mere instinct is involved in it: the community
+has indeed a right to expect as much as
+this.
+\PageSep{xiii}
+
+It would be hard to find any one better
+qualified for this kind of exposition, either
+from the profundity of his own mathematical
+achievements, or from the extent and freshness
+of his interest in the theories of physical science,
+than the author of this book. If an appreciation
+might be ventured on as regards the later chapters,
+they are, perhaps, intended to present the stern
+logical analyst quizzing the cultivator of physical
+ideas as to what he is driving at, and whither he
+expects to go, rather than any responsible attempt
+towards a settled confession of faith. Thus, when
+M.~Poincaré allows himself for a moment to
+indulge in a process of evaporation of the
+Principle of Energy, he is content to sum up:
+``Eh bien, quelles que soient les notions nouvelles
+que les expériences futures nous donneront sur le
+monde, nous sommes sûrs d'avance qu'il y aura
+quelque chose qui demeurera constant et que nous
+pourrons appeler \Foreign{énergie}'' (\Pageref{166}), and to leave
+the matter there for his readers to think it out.
+Though hardly necessary in the original French, it
+may not now be superfluous to point out that
+independent reflection and criticism on the part
+of the reader are tacitly implied here as elsewhere.
+
+An interesting passage is the one devoted to
+\PageSep{xiv}
+Maxwell's theory of the functions of the æther,
+and the comparison of the close-knit theories of
+the classical French mathematical physicists with
+the somewhat loosely-connected \Foreign{corpus} of ideas by
+which Maxwell, the interpreter and successor of
+Faraday, has (posthumously) recast the whole
+face of physical science. How many times has
+that theory been re-written since Maxwell's day?
+and yet how little has it been altered in essence,
+except by further developments in the problem of
+moving bodies, from the form in which he left it!
+If, as M.~Poincaré remarks, the French instinct
+for precision and lucid demonstration sometimes
+finds itself ill at ease with physical theories of
+the British school, he as readily admits (\Pagerefs{223}{224}),
+and indeed fully appreciates, the advantages
+on the other side. Our own mental philosophers
+have been shocked at the point of view indicated
+by the proposition hazarded by Laplace, that a
+sufficiently developed intelligence, if it were made
+acquainted with the positions and motions of the
+atoms at any instant, could predict all future
+history: no amount of demur suffices sometimes
+to persuade them that this is not a conception
+universally entertained in physical science. It
+was not so even in Laplace's own day. From
+the point of view of the study of the evolution
+\PageSep{xv}
+of the sciences, there are few episodes more
+instructive than the collision between Laplace
+and Young with regard to the theory of capillarity.
+The precise and intricate mathematical
+analysis of Laplace, starting from fixed preconceptions
+regarding atomic forces which were
+to remain intact throughout the logical development
+of the argument, came into contrast with the
+tentative, mobile intuitions of Young; yet the
+latter was able to grasp, by sheer direct mental
+force, the fruitful though partial analogies of this
+recondite class of phenomena with more familiar
+operations of nature, and to form a direct picture
+of the way things interacted, such as could only
+have been illustrated, quite possibly damaged or
+obliterated, by premature effort to translate it
+into elaborate analytical formulas. The \Foreign{aperçus}
+of Young were apparently devoid of all cogency
+to Laplace; while Young expressed, doubtless in
+too extreme a way, his sense of the inanity of the
+array of mathematical logic of his rival. The
+subsequent history involved the Nemesis that the
+fabric of Laplace was taken down and reconstructed
+in the next generation by Poisson; while
+the modern cultivator of the subject turns, at any
+rate in England, to neither of those expositions
+for illumination, but rather finds in the partial
+\PageSep{xvi}
+and succinct indications of Young the best starting-point
+for further effort.
+
+It seems, however, hard to accept entirely
+the distinction suggested (\Pageref{213}) between the
+methods of cultivating theoretical physics in
+the two countries. To mention only two
+transcendent names which stand at the very
+front of two of the greatest developments of
+physical science of the last century, Carnot and
+Fresnel, their procedure was certainly not on the
+lines thus described. Possibly it is not devoid of
+significance that each of them attained his first
+effective recognition from the British school.
+
+It may, in fact, be maintained that the part
+played by mechanical and such-like theories---analogies
+if you will---is an essential one. The
+reader of this book will appreciate that the human
+mind has need of many instruments of comparison
+and discovery besides the unrelenting logic of the
+infinitesimal calculus. The dynamical basis which
+underlies the objects of our most frequent experience
+has now been systematised into a great
+calculus of exact thought, and traces of new real
+relationships may come out more vividly when
+considered in terms of our familiar acquaintance
+with dynamical systems than when formulated
+under the paler shadow of more analytical abstractions.
+\PageSep{xvii}
+It is even possible for a constructive
+physicist to conduct his mental operations entirely
+by dynamical images, though Helmholtz, as well
+as our author, seems to class a predilection in this
+direction as a British trait. A time arrives when,
+as in other subjects, ideas have crystallised out
+into distinctness; their exact verification and
+development then becomes a problem in mathematical
+physics. But whether the mechanical
+analogies still survive, or new terms are now
+introduced devoid of all naïve mechanical bias,
+it matters essentially little. The precise determination
+of the relations of things in the
+rational scheme of nature in which we find
+ourselves is the fundamental task, and for its
+fulfilment in any direction advantage has to be
+taken of our knowledge, even when only partial,
+of new aspects and types of relationship which
+may have become familiar perhaps in quite
+different fields. Nor can it be forgotten that the
+most fruitful and fundamental conceptions of
+abstract pure mathematics itself have often been
+suggested from these mechanical ideas of flux
+and force, where the play of intuition is our
+most powerful guide. The study of the historical
+evolution of physical theories is essential to the
+complete understanding of their import. It is in
+\PageSep{xviii}
+the mental workshop of a Fresnel, a Kelvin, or
+a Helmholtz, that profound ideas of the deep
+things of Nature are struck out and assume
+form; when pondered over and paraphrased by
+philosophers we see them react on the conduct
+of life: it is the business of criticism to polish
+them gradually to the common measure of human
+understanding. Oppressed though we are with
+the necessity of being specialists, if we are
+to know anything thoroughly in these days of
+accumulated details, we may at any rate profitably
+study the historical evolution of knowledge
+over a field wider than our own.
+
+The aspect of the subject which has here been
+dwelt on is that scientific progress, considered
+historically, is not a strictly logical process, and
+does not proceed by syllogisms. New ideas
+emerge dimly into intuition, come into consciousness
+from nobody knows where, and become
+the material on which the mind operates, forging
+them gradually into consistent doctrine, which
+can be welded on to existing domains of knowledge.
+But this process is never complete: a
+crude connection can always be pointed to by a
+logician as an indication of the imperfection of
+human constructions.
+
+If intuition plays a part which is so important,
+\PageSep{xix}
+it is surely necessary that we should possess a firm
+grasp of its limitations. In M.~Poincaré's earlier
+chapters the reader can gain very pleasantly a
+vivid idea of the various and highly complicated
+ways of docketing our perceptions of the relations
+of external things, all equally valid, that were
+open to the human race to develop. Strange to
+say, they never tried any of them; and, satisfied
+with the very remarkable practical fitness of the
+scheme of geometry and dynamics that came
+naturally to hand, did not consciously trouble
+themselves about the possible existence of others
+until recently. Still more recently has it been
+found that the good Bishop Berkeley's logical
+jibes against the Newtonian ideas of fluxions and
+limiting ratios cannot be adequately appeased in
+the rigorous mathematical conscience, until our
+apparent continuities are resolved mentally into
+discrete aggregates which we only partially
+apprehend. The irresistible impulse to atomize
+everything thus proves to be not merely a disease
+of the physicist; a deeper origin, in the nature
+of knowledge itself, is suggested.
+
+Everywhere want of absolute, exact adaptation
+can be detected, if pains are taken, between the
+various constructions that result from our mental
+activity and the impressions which give rise to
+\PageSep{xx}
+them. The bluntness of our unaided sensual
+perceptions, which are the source in part of the
+intuitions of the race, is well brought out in this
+connection by M.~Poincaré. Is there real contradiction?
+Harmony usually proves to be recovered
+by shifting our attitude to the phenomena.
+All experience leads us to interpret the totality of
+things as a consistent cosmos---undergoing evolution,
+the naturalists will say---in the large-scale
+workings of which we are interested spectators
+and explorers, while of the inner relations and
+ramifications we only apprehend dim glimpses.
+When our formulation of experience is imperfect
+or even paradoxical, we learn to attribute the
+fault to our point of view, and to expect that
+future adaptation will put it right. But Truth
+resides in a deep well, and we shall never get
+to the bottom. Only, while deriving enjoyment
+and insight from M.~Poincaré's Socratic exposition
+of the limitations of the human outlook on
+the universe, let us beware of counting limitation
+as imperfection, and drifting into an inadequate
+conception of the wonderful fabric of human
+knowledge.
+
+\Signature{J. LARMOR.}{}
+\PageSep{xxi}
+
+
+\OtherChapter{Author's Preface.}
+
+\First{To} the superficial observer scientific truth is unassailable,
+the logic of science is infallible; and if
+scientific men sometimes make mistakes, it is
+because they have not understood the rules of
+the game. Mathematical truths are derived from
+a few self-evident propositions, by a chain of
+flawless reasonings; they are imposed not only on
+us, but on Nature itself. By them the Creator is
+fettered, as it were, and His choice is limited to
+a relatively small number of solutions. A few
+experiments, therefore, will be sufficient to enable
+us to determine what choice He has made. From
+each experiment a number of consequences will
+follow by a series of mathematical deductions,
+and in this way each of them will reveal to us a
+corner of the universe. This, to the minds of most
+people, and to students who are getting their first
+ideas of physics, is the origin of certainty in
+science. This is what they take to be the rôle of
+\PageSep{xxii}
+experiment and mathematics. And thus, too, it
+was understood a hundred years ago by many
+men of science who dreamed of constructing the
+world with the aid of the smallest possible amount
+of material borrowed from experiment.
+
+But upon more mature reflection the position
+held by hypothesis was seen; it was recognised that
+it is as necessary to the experimenter as it is to the
+mathematician. And then the doubt arose if all
+these constructions are built on solid foundations.
+The conclusion was drawn that a breath would
+bring them to the ground. This sceptical attitude
+does not escape the charge of superficiality. To
+doubt everything or to believe everything are two
+equally convenient solutions; both dispense with
+the necessity of reflection.
+
+Instead of a summary condemnation we should
+examine with the utmost care the rôle of hypothesis;
+we shall then recognise not only that it is
+necessary, but that in most cases it is legitimate.
+We shall also see that there are several kinds of
+hypotheses; that some are verifiable, and when
+once confirmed by experiment become truths of
+great fertility; that others may be useful to us in
+fixing our ideas; and finally, that others are
+hypotheses only in appearance, and reduce to
+definitions or to conventions in disguise. The
+\PageSep{xxiii}
+latter are to be met with especially in mathematics
+and in the sciences to which it is applied. From
+them, indeed, the sciences derive their rigour;
+such conventions are the result of the unrestricted
+activity of the mind, which in this domain recognises
+no obstacle. For here the mind may affirm
+because it lays down its own laws; but let us
+clearly understand that while these laws are
+imposed on \emph{our} science, which otherwise could
+not exist, they are not imposed on Nature. Are
+they then arbitrary? No; for if they were, they
+would not be fertile. Experience leaves us our
+freedom of choice, but it guides us by helping us to
+discern the most convenient path to follow. Our
+laws are therefore like those of an absolute
+monarch, who is wise and consults his council of
+state. Some people have been struck by this
+characteristic of free convention which may be
+recognised in certain fundamental principles of
+the sciences. Some have set no limits to their
+generalisations, and at the same time they have
+forgotten that there is a difference between liberty
+and the purely arbitrary. So that they are compelled
+to end in what is called \emph{nominalism}; they
+have asked if the \Foreign{savant} is not the dupe of his
+own definitions, and if the world he thinks he has
+discovered is not simply the creation of his own
+\PageSep{xxiv}
+caprice.\footnote
+  {Cf.\ M.~le~Roy: ``Science et Philosophie,'' \Title{Revue de Métaphysique
+  et de Morale}, 1901.}
+Under these conditions science would
+retain its certainty, but would not attain its object,
+and would become powerless. Now, we daily see
+what science is doing for us. This could not be
+unless it taught us something about reality; the
+aim of science is not things themselves, as the
+dogmatists in their simplicity imagine, but the
+relations between things; outside those relations
+there is no reality knowable.
+
+Such is the conclusion to which we are led; but
+to reach that conclusion we must pass in review
+the series of sciences from arithmetic and
+geometry to mechanics and experimental physics.
+What is the nature of mathematical reasoning?
+Is it really deductive, as is commonly supposed?
+Careful analysis shows us that it is nothing of the
+kind; that it participates to some extent in the
+nature of inductive reasoning, and for that reason
+it is fruitful. But none the less does it retain its
+character of absolute rigour; and this is what
+must first be shown.
+
+When we know more of this instrument which
+is placed in the hands of the investigator by
+mathematics, we have then to analyse another
+fundamental idea, that of mathematical magnitude.
+\PageSep{xxv}
+Do we find it in nature, or have we ourselves
+introduced it? And if the latter be the
+case, are we not running a risk of coming to
+incorrect conclusions all round? Comparing the
+rough data of our senses with that extremely complex
+and subtle conception which mathematicians
+call magnitude, we are compelled to recognise a
+divergence. The framework into which we wish
+to make everything fit is one of our own construction;
+but we did not construct it at random, we
+constructed it by measurement so to speak; and
+that is why we can fit the facts into it without
+altering their essential qualities.
+
+Space is another framework which we impose
+on the world. Whence are the first principles of
+geometry derived? Are they imposed on us by
+logic? Lobatschewsky, by inventing non-Euclidean
+geometries, has shown that this is not the case.
+Is space revealed to us by our senses? No; for
+the space revealed to us by our senses is absolutely
+different from the space of geometry. Is geometry
+derived from experience? Careful discussion will
+give the answer---no! We therefore conclude that
+the principles of geometry are only conventions;
+but these conventions are not arbitrary, and if
+transported into another world (which I shall
+call the non-Euclidean world, and which I shall
+\PageSep{xxvi}
+endeavour to describe), we shall find ourselves
+compelled to adopt more of them.
+
+In mechanics we shall be led to analogous conclusions,
+and we shall see that the principles of
+this science, although more directly based on
+experience, still share the conventional character
+of the geometrical postulates. So far, nominalism
+triumphs; but we now come to the physical
+sciences, properly so called, and here the scene
+changes. We meet with hypotheses of another
+kind, and we fully grasp how fruitful they are.
+No doubt at the outset theories seem unsound,
+and the history of science shows us how ephemeral
+they are; but they do not entirely perish, and of
+each of them some traces still remain. It is these
+traces which we must try to discover, because in
+them and in them alone is the true reality.
+
+The method of the physical sciences is based
+upon the induction which leads us to expect the
+recurrence of a phenomenon when the circumstances
+which give rise to it are repeated. If all
+the circumstances could be simultaneously reproduced,
+this principle could be fearlessly applied;
+but this never happens; some of the circumstances
+will always be missing. Are we absolutely certain
+that they are unimportant? Evidently not! It
+may be probable, but it cannot be rigorously
+\PageSep{xxvii}
+certain. Hence the importance of the rôle that is
+played in the physical sciences by the law of
+probability. The calculus of probabilities is therefore
+not merely a recreation, or a guide to the
+baccarat player; and we must thoroughly examine
+the principles on which it is based. In this connection
+I have but very incomplete results to lay
+before the reader, for the vague instinct which
+enables us to determine probability almost defies
+analysis. After a study of the conditions under
+which the work of the physicist is carried on, I
+have thought it best to show him at work. For
+this purpose I have taken instances from the
+history of optics and of electricity. We shall thus
+see how the ideas of Fresnel and Maxwell took
+their rise, and what unconscious hypotheses were
+made by Ampère and the other founders of
+electro-dynamics.
+\PageSep{xxviii}
+%[Blank page]
+\PageSep{1}
+\MainMatter
+%[** TN: Commented text is printed by the \Part macro]
+% SCIENCE AND HYPOTHESIS.
+
+
+\Part{I.}{Number and Magnitude.}
+
+\Chapter[Nature of Mathematical Reasoning.]{I.}{On the Nature of Mathematical Reasoning.}
+
+\Section{I.}
+
+\First{The} very possibility of mathematical science seems
+an insoluble contradiction. If this science is only
+deductive in appearance, from whence is derived
+that perfect rigour 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 mathematics 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
+\PageSep{2}
+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 necessity, there is still one
+resource left to us: we may class them among
+\Foreign{à~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 reasoning 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 demonstration; reasoning can
+only give us immediately evident truths borrowed
+from direct intuition; it would only be an intermediary
+parasite. Should we not therefore have
+reason for asking if the syllogistic 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 generalising some
+proposition already known. Does the mathematical
+method proceed from the particular to the general,
+and, if so, how can it be called deductive?
+\PageSep{3}
+
+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, properly so called,
+retain the analytical character, and \Foreign{ipso facto}, lose
+their power.
+
+\Section{II.}
+
+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~$x + 1$---\ie, the adding of unity
+to a given number~$x$. These definitions, whatever
+they may be, do not enter into the subsequent
+reasoning. I next define the numbers $2$,~$3$,~$4$ by
+the equalities\Chg{:---}{}
+%[** TN: Numbered eqns displayed in the French, but not in the English transl.]
+\[
+\Tag{(1)} 1 + 1 = 2;\qquad
+\Tag{(2)} 2 + 1 = 3;\qquad
+\Tag{(3)} 3 + 1 = 4\Chg{,}{;}
+\]
+and in
+\PageSep{4}
+the same way I define the operation~$x + 2$ by the
+relation\Chg{;}{}
+\[
+\Tag{(4)}
+x + 2 = (x + 1) + 1.
+\]
+
+Given this, we have\Chg{:---}{}
+\begin{alignat*}{2}
+      2 + 2 &= (2 + 1) + 1\Chg{;}{,}\ &&\text{(def.~4)\Chg{.}{;}} \\
+(2 + 1) + 1 &= 3 + 1\Add{,}  &&\text{(def.~2)\Chg{.}{;}} \\
+      3 + 1 &= 4\Add{,}      &&\text{(def.~3)\Chg{.}{;}} \\
+\text{whence } 2 + 2 &= 4\Add{,}&&\quad\QED
+\end{alignat*}
+
+It cannot be denied that this reasoning is purely
+analytical. But if we ask a mathematician, he will
+reply: ``This is not a demonstration properly so
+called; it is a verification.'' We have confined
+ourselves to bringing together one or other of two
+purely conventional definitions, and we have verified
+their identity; nothing new has been learned.
+\emph{Verification} differs from proof precisely because it
+is analytical, and because it leads to nothing. It
+leads to nothing because the conclusion is nothing
+but the premisses translated into another language.
+A real proof, on the other hand, is fruitful, because
+the conclusion is in a sense more general than the
+premisses. The equality $2 + 2 = 4$ can be verified
+because it is particular. Each individual enunciation
+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.
+\PageSep{5}
+
+\Section{III.}
+
+Let us now see the geometer at work, and try
+%[** TN: "surprise" is correct: "...cherchons à surprendre ses procédés."]
+to surprise some of his methods. The task is
+not without difficulty; it is not enough to open a
+book at random and to analyse any proof we may
+come across. First of all, geometry must be excluded,
+or the question becomes complicated by
+difficult problems relating to the rôle of the
+postulates, the nature and the origin of the idea
+of space. For analogous reasons we cannot
+avail ourselves of the infinitesimal calculus. We
+must seek mathematical thought where it has
+remained pure---\ie, in Arithmetic. 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 analyse
+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 rigour. We may not impute this to
+them as a crime; they have obeyed a necessity.
+Beginners are not prepared for real mathematical
+rigour; 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
+\PageSep{6}
+which was trodden slowly by the founders of the
+science.
+
+Why is so long a preparation necessary to
+habituate oneself to this perfect rigour, 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.
+
+\Subsection{Definition of Addition.}
+
+I assume that the operation~$x + 1$ 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~$x + a$, which
+consists in adding the number~$a$ to any given
+number~$x$. Suppose that we have defined the
+operation
+%[** TN: Displayed eqns as in the French; all are inline in the English transl.]
+\[
+x + (a - 1);
+\]
+the operation~$x + a$ will be
+defined by the equality\Chg{:}{}
+\[
+\Tag{(1)}
+x + a = \bigl[x + (a - 1)\bigr] + 1.
+\]
+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
+successively and ``by recurrence'' the operations
+$x + 2$, $x + 3$,~etc. This definition deserves a moment's
+\PageSep{7}
+attention; it is of a particular nature which
+distinguishes it even at this stage from the purely
+logical definition; the equality~(1), in fact, contains
+an infinite number of distinct definitions, each
+having only one meaning when we know the
+meaning of its predecessor.
+
+\Subsection{Properties of Addition.}
+
+\Par{Associative.}---I say that
+%[** TN: Displayed eqns as in the French; all are inline in the English transl.]
+\[
+a + (b + c) = (a + b) + c;
+\]
+in
+fact, the theorem is true for $c = 1$. It may then be
+written
+\[
+a + (b + 1) = (a + b) + 1;
+\]
+which, remembering
+the difference of notation, is nothing but the equality~(1)
+by which I have just defined addition. Assume
+the theorem true for $c = \gamma$, I say that it will be true for
+$c = \gamma + 1$. Let
+\[
+(a + b) + \gamma = a + (b + \gamma)\Chg{,}{;}
+\]
+it follows that
+\[
+\bigl[(a + b) + \gamma\bigr] + 1 = \bigl[a + (b + \gamma)\bigr] + 1;
+\]
+or by def.~(1)\Chg{---}{,}
+\[
+(a + b) + (\gamma + 1)
+  = a + (b + \gamma + 1)
+  = a + \bigl[b + (\gamma + 1)\bigr]\Chg{,}{;}
+\]
+which shows by a series of purely analytical deductions
+that the theorem is true for $\gamma + 1$. Being
+true for $c = 1$, we see that it is successively true for
+$c = 2$, $c = 3$,~etc.
+
+\Par{Commutative.}---(1)~I say that
+%[** TN: Displayed eqns as in the French; all are inline in the English transl.]
+\[
+a + 1 = 1 + a.
+\]
+The
+theorem is evidently true for $a = 1$; we can \emph{verify}
+by purely analytical reasoning that if it is true for
+$a = \gamma$ it will be true for $a = \gamma + 1$.\footnote
+  {For $(\gamma + 1) + 1 = (1 + \gamma) + 1 = 1 + (\gamma + 1)$.\Transl}
+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
+\[
+a + b = b + a.
+\]
+The theorem has just
+\PageSep{8}
+been shown to hold good for $b = 1$, and it may be
+verified analytically that if it is true for $b = \beta$, it
+will be true for $b = \beta + 1$. The proposition is thus
+established by recurrence.
+
+\Subsection{Definition of Multiplication.}
+
+We shall define multiplication by the equalities\Chg{:}{}
+\begin{gather*}
+\Tag{(1)}
+a × 1 = a\Chg{.}{;} \\
+\Tag{(2)}
+a × b = \bigl[a × (b - 1)\bigr] + a.
+\end{gather*}
+Both of
+these include an infinite number of definitions;
+having defined~$a × 1$, it enables us to define in
+succession $a × 2$, $a × 3$, and so on.
+
+\Subsection{Properties of Multiplication.}
+
+\Par{Distributive.}---I say that
+%[** TN: Displayed eqns as in the French; all are inline in the English transl.]
+\[
+(a + b) × c = (a × c) + (b × c).
+\]
+We can verify analytically that the theorem
+is true for $c = 1$; then if it is true for $c = \gamma$, it will be
+true for $c = \gamma + 1$. The proposition is then proved
+by recurrence.
+
+\Par{Commutative.}---(1) I say that
+\[
+a × 1 = 1 × a.
+\]
+The
+theorem is obvious for $a = 1$. We can verify
+analytically that if it is true for $a = \alpha$, it will be
+true for $a = \alpha + 1$.
+
+(2)~I say that
+\[
+a × b = b × a.
+\]
+The theorem has
+just been proved for $b = 1$. We can verify analytically
+that if it be true for $b = \beta$ it will be true for
+$b = \beta + 1$.
+
+\Section{IV.}
+
+This monotonous series of reasonings may now
+be laid aside; but their very monotony brings
+vividly to light the process, which is uniform,
+\PageSep{9}
+and is met again at every step. The process is
+proof by recurrence. 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 instrument of transformation which lends
+itself to many more different combinations than
+the simple syllogism; but it is still a purely analytical
+instrument, and is incapable of teaching us
+anything new. If mathematics had no other instrument,
+it would immediately be arrested in its
+development; but it has recourse anew to the
+same process---\ie, 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 therefore mathematical reasoning
+\Foreign{par excellence}, and we must examine it closer.
+
+\Section{V.}
+
+The essential characteristic of reasoning by recurrence
+is that it contains, condensed, so to
+speak, in a single formula, an infinite number of
+syllogisms. We shall see this more clearly if we
+enunciate the syllogisms one after another. They
+follow one another, if one may use the expression,
+in a cascade. The following are the hypothetical
+\PageSep{10}
+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 unending 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 analytical
+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 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
+\PageSep{11}
+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 ourselves very far from the infinitesimal
+analysis, but the idea of mathematical infinity is
+already playing a preponderating part, and without
+it there would be no science at all, because there
+would be nothing general.
+\PageSep{12}
+
+\Section{VI.}
+
+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 another, 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. Experiment 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 contradiction would be sufficient, it
+would always enable us to develop as many
+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 principle 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 \Foreign{à~priori}
+\PageSep{13}
+synthetic intuition. On the other hand, we
+cannot see in it a convention as in the case of the
+postulates of geometry.
+
+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 repetition 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 observations.
+
+It cannot escape our notice that here is a
+striking analogy with the usual processes of
+induction. But an essential difference exists.
+Induction 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---\ie,
+proof by recurrence---is, on the contrary,
+necessarily imposed on us, because it is only the
+affirmation of a property of the mind itself.
+\PageSep{14}
+
+\Section{VII.}
+
+Mathematicians, as I have said before, always
+endeavour to generalise the propositions they have
+obtained. To seek no further example, we have
+just shown the equality\Chg{,}{}
+%[** TN: Displayed eqns as in the French; all are inline in the English transl.]
+\[
+a + 1 = 1 + a,
+\]
+and we then
+used it to establish the equality\Chg{,}{}
+\[
+a + b = b + a,
+\]
+which
+is obviously more general. Mathematics 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 parallel 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
+\[
+\Tag{(1)}
+a + 2 = 2 + a,
+\]
+we need
+only apply the rule
+\[
+a + 1 = 1 + a\Chg{,}{}
+\]
+twice, and write
+\[
+\Tag{(2)}
+a + 2 = a + 1 + 1 = 1 + a + 1 = 1 + 1 + a = 2 + a.
+\]
+
+The equality thus deduced by purely analytical
+means is not, however, 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
+\PageSep{15}
+the equality~(2) are merely more complicated
+combinations than the two sides of the equality~(1),
+and analysis only serves to separate the elements
+which enter into these combinations and to
+study their relations.
+
+Mathematicians therefore proceed ``by construction,''
+they ``construct'' more complicated combinations.
+When they analyse these combinations,
+these aggregates, so to speak, into their primitive
+elements, 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 ``construction,'' 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 accurately,
+there must be some advantage in considering the
+construction rather than the elements themselves.
+What can this advantage be? Why reason on a
+polygon, for instance, which is always decomposable
+into triangles, and not on elementary
+triangles? It is because there are properties of
+\PageSep{16}
+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 particular 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.
+\PageSep{17}
+
+
+\Chapter[Mathematical Magnitude.]{II.}{Mathematical Magnitude and Experiment.}
+
+\First{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 be 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 disengaged mathematics from all extraneous
+elements, and he is in a position to answer
+our question:---``Tell me exactly what this continuum
+is, about which mathematicians reason.''
+Many analysts who reflect on their art have
+already done so---M.~Tannery, for instance, in
+his \Title{Introduction à la théorie 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
+\PageSep{18}
+again, and so on indefinitely. We thus get an
+unlimited number of terms, 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 arranged 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
+intimate 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 formula. 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 rigour. 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 contented with this definition are the
+dupes of words, that the nature of each of these
+sets should be precisely indicated, that it should
+\PageSep{19}
+be explained 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
+preceding or succeeding these or those other sets;
+this alone should therefore intervene in the definition.
+So we need not concern ourselves with the
+manner in which the sets are intercalated, and
+no one will doubt the possibility of the operation
+if he only remembers that ``possible'' in the
+language of geometers simply means exempt from
+contradiction. But our definition is not yet complete,
+and we come back to it after this rather long
+digression.
+
+\Par{Definition of Incommensurables.}---The mathematicians
+of the Berlin school, and Kronecker
+in particular, have devoted themselves to constructing
+this continuous scale of irrational and
+fractional numbers without using any other
+materials than the integer. The mathematical
+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 incommensurable
+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.
+\PageSep{20}
+
+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 Kronecker 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 numbers 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 example,
+if the first class comprises all the numbers greater
+than~$2$, and if, in the second, are all the numbers
+\PageSep{21}
+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 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~$\sqrt{2}$ 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 symbol. 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 conceive as infinitely
+divisible---\ie, as a continuum?
+
+\Par{The Physical Continuum.}---We are next led to ask
+\PageSep{22}
+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, according to which sensation is proportional
+to the logarithm of the stimulus. But if we
+examine the experiments by which the endeavour
+has been made to establish this law, we shall be
+led to a diametrically opposite conclusion. It has,
+for instance, been observed that a weight~$A$ of $10$~grammes
+and a weight~$B$ of $11$~grammes produced
+identical sensations, that the weight~$B$ could no
+longer be distinguished from a weight~$C$ of $12$~grammes,
+but that the weight~$A$ was readily
+distinguished from the weight~$C$. Thus the rough
+results of the experiments may be expressed by
+the following relations\Chg{:}{}
+%[** TN: Not displayed in the English translation]
+\[
+A = B,\qquad B = C,\qquad A < C,
+\]
+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 conclude 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~$B$,
+\PageSep{23}
+and $B$~from~$C$, and that if we have not been
+aware of this, it is due to the imperfections of our
+senses.
+
+\Par{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?
+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 distinguish 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 contradiction 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
+powerful enough to decompose the physical continuum
+into discrete elements, 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
+\PageSep{24}
+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 collection 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
+\PageSep{25}
+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 numbers.
+If, then, we intercalate new sets according to the
+laws of incommensurable numbers, we obtain
+what may be called a continuum of the second
+order.
+
+\Par{Second Stage.}---We have only taken our first
+step. We have explained 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 physical 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; without 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
+\PageSep{26}
+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---\ie,
+the lines traced by the geometer should only
+give us points, the co-ordinates of which are
+rational numbers. 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
+(\Foreign{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 recognise the conception of Kronecker,
+\PageSep{27}
+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 continuum
+properly so called.
+
+\Par{Summary.}---To sum up, the mind has the faculty
+of creating symbols, and it is thus that it has constructed
+the mathematical continuum, which is
+only a particular system of symbols. The only
+limit 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 contradiction,---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.
+
+\Par{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
+\PageSep{28}
+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 $n$~intermediary
+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 interval~$CD$, the interval~$AD$ will by
+definition be the sum of the intervals $AB$~and~$AC$.
+This definition is very largely, but not altogether,
+arbitrary. It must satisfy certain conditions---the
+commutative and associative 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.
+
+\Par{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
+Reymond.
+\PageSep{29}
+
+We know that mathematicians distinguish
+between infinitesimals of different orders, and that
+infinitesimals of the second order are infinitely
+small, not only absolutely so, but also in relation
+to those of the first order. It is not difficult to
+imagine infinitesimals of fractional or even of
+irrational order, and here once more we find the
+mathematical continuum which has been dealt
+with in the preceding 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~$1 + \epsilon$, however
+small~$\epsilon$ 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 mathematical continuum, are
+\PageSep{30}
+we protected from contradictions analogous to
+those which gave it birth? No, and the following
+is an instance:---
+
+He is a \Foreign{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---\ie, 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 common 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 endeavouring 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.
+
+\Par{The Physical Continuum of several Dimensions.}---We
+have discussed above the physical continuum
+as it is derived from the immediate evidence of our
+\PageSep{31}
+senses---or, if the reader prefers, from the rough
+results of Fechner's experiments; I have shown
+that these results are summed up in the contradictory
+formulæ\Chg{:---}{}
+\[
+A = B,\qquad B = C,\qquad A < C.
+\]
+
+Let us now see how this notion is generalised,
+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 them, or we cannot; just as in
+Fechner'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 \emph{element}. It will be in a measure analogous to
+the \emph{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 neighbours, and it is thus
+surrounded by a kind of fog. If the astronomical
+comparison may be allowed, our ``elements''
+would be like nebulæ, 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 consecutive
+elements such that each cannot be distinguished
+from its predecessor. This \emph{linear} series is to the
+\emph{line} of the mathematician what the isolated \emph{element}
+was to the point.
+\PageSep{32}
+
+Before going further, I must explain what is
+meant by a \emph{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 \emph{cut}. By means of this
+cut, the continuum~$C$ will be \emph{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 distinct 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~$B$), \emph{unless
+one of the elements of this series is indistinguishable
+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 distinguishable
+the one from the other (and therefore
+forming neither one continuum nor several continuums),
+we shall call~$C$ a continuum \emph{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
+\PageSep{33}
+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.
+
+\Par{The Mathematical Continuum of Several Dimensions.}---The
+conception 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 defined by a system of $n$~distinct
+magnitudes which we call its co-ordinates.
+
+The magnitudes need not always be measurable;
+there is, for instance, one branch of geometry
+independent of the measure of magnitudes, 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~$AB$ is equal to or
+twice the arc~$BC$. This branch is called \emph{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 unskilful draughtsman, with the result that
+\PageSep{34}
+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.
+\PageSep{35}
+
+
+\Part{II.}{Space.}
+
+\Chapter{III.}{Non-Euclidean Geometries.}
+
+\First{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 \Foreign{à~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
+\PageSep{36}
+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 admitted 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,
+%[** TN: Correct ("Hongrois") in the French edition]
+two scientists, a Russian and a \Reword{Bulgarian}{Hungarian}, 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 Académic des Sciences receives only
+about one or two new demonstrations 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: \Title{Ueber
+die Hypothesen welche der Geometrie 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.
+
+\Par{The Geometry of Lobatschewsky.}---If it were
+possible to deduce Euclid's postulate from the
+several axioms, it is evident that by rejecting
+the postulate and retaining the other axioms we
+\PageSep{37}
+should be led to contradictory consequences. It
+would be, therefore, impossible to found on those
+premisses a coherent geometry. Now, this is
+precisely 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 impossible 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 difference 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. Lobatschewsky's propositions
+have no relation to those of Euclid,
+but they are none the less logically interconnected.
+
+\Par{Riemann's Geometry.}---Let us imagine to ourselves
+a world only peopled with beings of no
+thickness, and suppose these ``infinitely flat''
+\PageSep{38}
+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 endow 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
+spherical, 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 construct? 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 \emph{unbounded}, since on a sphere one may
+always walk forward without ever being brought
+to a stop, and yet it will be \emph{finite}; the end will
+never be found, but the complete tour can be
+made. Well, Riemann's geometry is spherical
+geometry extended to three dimensions. To construct
+it, the German mathematician had first of
+all to throw overboard, not only Euclid's postulate
+\PageSep{39}
+but also the first axiom that \emph{only one line can pass
+through two points}. On a sphere, through two
+given points, we can \emph{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 Lobatschewsky. 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 Lobatschewsky.
+Let us add that Riemann's space is
+finite, although unbounded in the sense which we
+have above attached to these words.
+
+\Par{Surfaces with Constant Curvature.}---One objection,
+however, remains possible. There is no contradiction
+between the theorems of Lobatschewsky and
+Riemann; but however numerous are the other
+consequences that these geometers have deduced
+\PageSep{40}
+from their hypotheses, 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
+Riemann'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 different 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
+lines of which remain of a constant length. Such
+\PageSep{41}
+a movement would appear absurd, on the other
+hand, to animals without thickness living on a
+surface of variable curvature. These surfaces of
+constant curvature are of two kinds. The
+curvature of some is \emph{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 \emph{negative}. Beltrami has shown
+that the geometry of these surfaces is identical
+with that of Lobatschewsky. Thus the two-dimensional
+geometries of Riemann and Lobatschewsky
+are connected with Euclidean geometry.
+
+\Par{Interpretation of Non-Euclidean Geometries.}---Thus
+vanishes the objection so far as two-dimensional
+geometries are concerned. It would be easy to
+extend Beltrami's reasoning to three-dimensional
+geometries, and minds which do not recoil before
+space of four dimensions 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
+columns, and corresponding each to each, just as
+in ordinary dictionaries the words in two languages
+which have the same signification correspond to
+one another:---
+\Dict{Space}{\raggedright The portion of space situated
+above the fundamental
+plane.}
+\PageSep{42}
+\Dict{Plane}{\raggedright Sphere cutting orthogonally
+the fundamental plane.}
+\Dict{Line}{\raggedright Circle cutting orthogonally
+the fundamental plane.}
+\Dict{Sphere}{Sphere.}
+\Dict{Circle}{Circle.}
+\Dict{Angle}{Angle.}
+\Dict{Distance between
+two points}{Logarithm of the anharmonic
+ratio of these two points
+and of the intersection
+of the fundamental plane
+with the circle passing
+through these two points
+and cutting it orthogonally.}
+\Dict{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. \emph{We shall then
+obtain the theorems of ordinary geometry.} For
+instance, Lobatschewsky's theorem: ``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 will be less than two right angles.'' Thus,
+however far the consequences of Lobatschewsky's
+hypotheses are carried, they will never lead to a
+\PageSep{43}
+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.
+
+\Par{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 which are common to the geometries
+of Euclid, Lobatschewsky, and Riemann. These
+propositions must be based on premisses that
+\PageSep{44}
+geometers admit without enunciation. It is interesting
+to try and extract them from the classical
+proofs.
+
+John Stuart Mill asserted\footnote
+  {\Title{Logic}, c.~viii., cf.\ Definitions, §5--6.\Transl}
+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, either in
+itself, or with the propositions previously 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 concealed in this definition.
+It is true we might change it, and that would be
+preferable, but then we should have to enunciate
+the axiom explicitly. Other definitions may give
+rise to no less important reflections, such as, for
+example, that of the equality of two figures. Two
+\PageSep{45}
+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
+properties of natural solids which do not much
+differ from those of the ideal solids, all of whose
+dimensions are invariable. However, imperfect as
+it may be, this definition implies an axiom. The
+possibility 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 \Foreign{à~priori} intuition would be. Moreover,
+when we study the definitions and the proofs
+of geometry, we see that we are compelled 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 invariable 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
+\PageSep{46}
+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 perpendicular
+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.
+
+\Par{The Fourth Geometry.}---Among these explicit
+axioms there is one which seems to me to deserve
+some attention, because when we abandon it we
+can construct a fourth geometry as coherent as
+those of Euclid, Lobatschewsky, and Riemann.
+To prove that we can always draw a perpendicular
+at a point~$A$ to a straight line~$AB$, we consider a
+straight line~$AC$ movable about the point~$A$, and
+initially identical with the fixed straight line~$AB$.
+We then can make it turn about the point~$A$ until
+it lies in~$AB$ produced. Thus we assume two
+propositions---first, that such a rotation is possible,
+and then that it may continue until the two lines
+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 of these
+theorems, and I shall not choose the least remarkable
+of them. \emph{A real straight line may be perpendicular
+to itself.}
+
+\Par{Lie's Theorem.}---The number of axioms implicitly
+\PageSep{47}
+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 premisses 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.
+
+\emph{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.
+
+\Par{Riemann's Geometries.}---However, this result
+seems contradicted by Riemann, for that scientist
+constructs an infinite number of geometries, and
+that to which his name is usually attached is only
+a particular 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.
+\PageSep{48}
+These geometries of Riemann, so interesting on
+various grounds, can never be, therefore, purely
+analytical, and would not lend themselves to
+proofs analogous to those of Euclid.
+
+\Par{On the Nature of Axioms.}---Most mathematicians
+regard Lobatschewsky'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
+geometry. Now, to discuss this view we must
+first of all ask ourselves, what is the nature of
+geometrical axioms? Are they synthetic \Foreign{à~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, let us take a true synthetic
+\Foreign{à~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~$n + 1$, provided it is
+\PageSep{49}
+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 foundation 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 unsurmountable.
+If geometry were an experimental science, it would
+not be an exact science. It would be subjected to
+\PageSep{5O}
+continual revision. Nay, it would from that day
+forth be proved to be erroneous, for we know that
+no rigorously invariable solid exists. \emph{The geometrical
+axioms are therefore neither synthetic \Foreign{à~priori}
+intuitions nor experimental facts.} They are conventions.
+Our choice among all possible conventions
+is \emph{guided} by experimental facts; but it remains
+\emph{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 approximate. In other words,
+\emph{the axioms of geometry} (I do not speak of those of
+arithmetic) \emph{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 degree 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.
+\PageSep{51}
+
+
+\Chapter{IV.}{Space and Geometry.}
+
+\First{Let} us begin with a little paradox. Beings whose
+minds were made as ours, and with senses like
+ours, but without any preliminary education,
+might receive from a suitably-chosen external
+world impressions which would lead them to
+construct a geometry other than that of Euclid,
+and to localise 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
+suddenly 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.
+
+\Par{Geometrical Space and Representative Space.}---It is
+often said that the images we form of external
+objects are localised 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
+\PageSep{52}
+geometers, having all the properties of that space.
+To all clear-headed men who think in this way,
+the preceding statement 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 properly so called?
+I mean of that space which is the object of
+geometry, 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 dimensions; 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 \emph{representative
+space}.
+
+\Par{Visual Space.}---First of all let us consider a
+purely visual impression, 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 other hand, the image is enclosed within
+a limited framework; and there is a no less
+important difference: \emph{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 rôle. The yellow spot can in no
+way be regarded as identical with a point on the
+\PageSep{53}
+edge of the retina. Not only does the same object
+produce on it much brighter impressions, but in
+the whole of the \emph{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
+accommodation 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 sensations which have
+given us the concept of the two first dimensions.
+The third dimension will therefore not appear to us
+as playing the same rôle as the two others. What
+may be called \emph{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 mathematical language, they will be functions
+of three independent variables. But let us
+look at the matter a little closer. The third
+\PageSep{54}
+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 convergence $A$~and~$B$
+are indistinguishable, the two sensations of
+accommodation $A'$~and~$B'$ which accompany them
+respectively will also be indistinguishable. But
+that is, so to speak, an experimental fact. Nothing
+prevents us \Foreign{à~priori} from assuming the contrary,
+and if the contrary takes place, if these two
+muscular sensations both vary independently, 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 \emph{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 indications which enable us to
+appreciate distances would cease to be connected
+by a constant relation. A being educating his
+\PageSep{55}
+senses in such a world would no doubt attribute
+four dimensions to complete visual space.
+
+\Par{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 everybody knows, which accompany all our
+movements, and which we usually call muscular
+sensations. The corresponding framework constitutes
+what may be called \emph{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 \emph{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 \emph{direction} of each movement, 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 accompanied 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 analyse my sensations. What I do see is that
+the sensations which correspond to movements in
+\PageSep{56}
+the same direction are connected in my mind by a
+simple \emph{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 association is extremely complex,
+for the contraction of the same muscle may correspond,
+according to the position of the limbs,
+to very different movements of direction. Moreover,
+it is evidently acquired; it is like all
+associations of ideas, the result of a \emph{habit}. This
+habit itself is the result of a very large number of
+\emph{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.
+
+\Par{Characteristics of Representative Space.}---Thus representative
+space in its triple form---visual,
+tactile, and motor---differs essentially from geometrical
+space. It is neither homogeneous nor
+isotropic; 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 ``localise'' them. Now, has
+that any meaning, and if so what is that meaning?
+Does it mean that we \emph{represent} to ourselves external
+objects in geometrical space? Our representations
+are only the reproduction of our sensations;
+they cannot therefore be arranged in the
+same framework---that is to say, in representative
+\PageSep{57}
+space. It is also just as impossible for us to represent
+to ourselves external objects in geometrical
+space, as it is impossible for a painter to paint on
+a flat surface objects with their three dimensions.
+Representative 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 \emph{represent} to ourselves
+external bodies in geometrical space, but we
+\emph{reason} about these bodies as if they were situated
+in geometrical space. When it is said, on the
+other hand, that we ``localise'' such an object in
+such a point of space, what does it mean? \emph{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.
+
+\Par{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
+\PageSep{58}
+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. \emph{None of our sensations, if isolated,
+could have brought us to the concept 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 impressions are subject to change; but among
+the changes that we ascertain, 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, \emph{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 impressions by making movements which
+would confront us with the movable object in
+the same \emph{relative} situation. We thus \emph{correct} the
+modification which was produced, and we re-establish
+the initial state by an inverse modification.
+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 accompanied by
+\PageSep{59}
+muscular sensations. But that does not mean
+that we represent them to ourselves in geometrical
+space. So what characterises change of position,
+what distinguishes it from change of state, is that
+it can always be \emph{corrected} by this means. It may
+therefore happen that we pass from the aggregate
+of impressions~$A$ to the aggregate~$B$ in two different
+ways. First, involuntarily and without experiencing
+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
+\emph{a series of sensations}; but a \emph{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.
+
+\Par{Conditions of Compensation.}---How is such a
+compensation possible in such a way that two
+\PageSep{60}
+changes, otherwise mutually independent, may be
+reciprocally corrected? A mind \emph{already familiar
+with geometry} would reason as follows:---If there
+is to be compensation, the different parts of the
+external object on the one hand, and the different
+organs of our senses on the other, must be in the
+same \emph{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
+\Foreign{à~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.
+
+\Par{Solid Bodies and Geometry.}---Among surrounding
+objects there are some which frequently experience
+displacements that may be thus corrected by a
+\emph{correlative} movement of our own body---namely,
+\emph{solid bodies}. The other objects, whose form is variable,
+\PageSep{61}
+only in exceptional circumstances undergo
+similar displacement (change of position without
+change of form). When the displacement of a
+body takes place with deformation, we can no
+longer by appropriate movements place the organs
+of our body in the same \emph{relative} situation with
+respect to this body; we can no longer, therefore,
+reconstruct the primitive aggregate 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
+modification 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.
+
+\emph{If, then, there were no solid bodies in nature there
+would be no geometry.}
+
+Another remark deserves a moment's attention.
+Suppose a solid body to occupy successively the
+positions $\alpha$~and~$\beta$; in the first position it will give
+us an aggregate of impressions~$A$, and in the second
+position the aggregate of impressions~$B$. Now let
+\PageSep{62}
+there be a second solid body, of qualities entirely
+different from the first---of different colour, for
+instance. Assume it to pass from the position~$\alpha$,
+where it gives us the aggregate of impressions~$A'$ to
+the position~$\beta$, 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 \emph{in themselves} have in general
+nothing in common. Yet we consider both
+these changes as displacements; and, further, we
+consider them the \emph{same} displacement. How can
+this be? It is simply because they may be both
+corrected by the \emph{same} correlative movement of our
+body. ``Correlative movement,'' therefore, constitutes
+the \emph{sole connection} between two phenomena
+which otherwise we should never have dreamed of
+connecting.
+
+On the other hand, our body, thanks to the
+number of its articulations 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 \Foreign{en bloc}---\ie, without any variation of
+their relative positions, as in the case of a solid
+body.
+\PageSep{63}
+
+To sum up:\Add{---}
+
+1. In the first place, we distinguish two categories
+of phenomena:---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 correlative 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.
+\emph{The laws of these phenomena are the object of
+geometry.}
+
+\Par{Law of Homogeneity.}---The first of these laws
+is the law of homogeneity. Suppose that by an
+external change we pass from the aggregate of
+impressions~$A$ to the aggregate~$B$, and that then
+this change~$\alpha$ is corrected by a correlative
+voluntary movement~$\beta$, so that we are brought
+back to the aggregate~$A$. Suppose now that
+another external change~$\alpha'$ brings us again from
+the aggregate~$A$ to the aggregate~$B$. Experiment
+then shows us that this change~$\alpha'$, like the change~$\alpha$,
+may be corrected by a voluntary correlative
+movement~$\beta'$, and that this movement~$\beta'$ corresponds
+\PageSep{64}
+to the same muscular sensations as the
+movement~$\beta$ which corrected~$\alpha$.
+
+This fact is usually enunciated as follows:---\emph{Space
+is homogeneous and isotropic.} We may also say that a
+movement which is once produced 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 importance 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.''
+
+\Par{The Non-Euclidean World.}---If geometrical space
+were a framework imposed on \emph{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, therefore, 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
+\PageSep{65}
+may thus conceive that beings whose education
+has 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
+$R$~be the radius of the sphere, and $r$~the distance
+of the point considered from the centre, the absolute
+temperature will be proportional to~$R^{2} - r^{2}$.
+Further, I shall suppose that in this world all bodies
+have the same co-efficient of dilatation, so that the
+linear dilatation of any body is proportional to its
+absolute 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 circumference
+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
+surface 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
+\PageSep{66}
+sphere. If to us geometry is only the study of the
+laws according to which invariable solids move, to
+these imaginary beings it will be the study of the
+laws of motion of solids \emph{deformed by the differences
+of temperature} 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 refraction is inversely
+proportional to~$R^{2} - r^{2}$. Under these conditions it
+is clear that the rays of light will no longer be
+rectilinear but circular. To justify what has 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 a
+\PageSep{67}
+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 neighbourhood 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 according 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,
+nevertheless 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 curvature 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 position,'' which may be corrected
+\PageSep{68}
+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 \emph{this will be non-Euclidean geometry}.
+So that beings like ourselves, educated in
+such a world, will not have the same geometry as
+ours.
+
+\Par{The World of Four Dimensions.}---Just as we have
+pictured to ourselves a non-Euclidean world, so we
+may picture a world of four dimensions.
+
+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 \emph{perspectives}.
+But as eye and objects are movable, 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 perspective~$B'$ are accompanied
+by the same muscular sensations, we
+connect them as we do other operations of the
+same nature. Then when we study the laws
+\PageSep{69}
+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 is
+from the properties of this group that we derive
+the idea of geometrical 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 \emph{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
+perspectives, since they are of only three dimensions.
+Imagine that the different perspectives of
+one and the same object to 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 transitions
+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.
+\PageSep{70}
+In this there is nothing that we cannot represent
+to ourselves, and, moreover, these sensations are
+those which a being would experience 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.
+
+\Par{Conclusions.}---It is seen that experiment plays a
+considerable rôle 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 approximative
+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, absolutely
+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 \emph{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
+\PageSep{71}
+truest, but what is the most convenient geometry.
+It will be noticed that my description of these
+fantastic worlds has required no language other
+than that of ordinary geometry. 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.
+\PageSep{72}
+
+
+\Chapter{V.}{Experiment and Geometry.}
+
+\ParSkip1. 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.
+However 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~$\pi$. What have we done? We
+have made an experiment on the properties of the
+matter with which this \emph{roundness} has been realised,
+and of which the measure we used is made.
+
+\Par[3.\ ]{Geometry and Astronomy.}---The same question
+may also be asked in another way. If Lobatschewsky's
+geometry is true, the parallax of a very
+distant star will be finite. If Riemann'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
+%[** TN: "...les trois géométries" in the French edition]
+us to decide between the \Reword{two}{three} geometries. But
+\PageSep{73}
+what we call a straight line in astronomy is simply
+the path of a ray of light. If, therefore, 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, therefore, 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 contradict the non-Euclidean hypothesis?
+I think that such a question cannot be seriously
+asked. To me it is exactly equivalent to the following,
+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
+existence of these lengths, would directly contradict
+this hypothesis, that 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 possesses the property~$B$; and, finally, I
+\PageSep{74}
+assume that in both Euclidean and non-Euclidean
+space the straight line is the only line that possesses
+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 endeavour
+to find out if it does, or does not, possess the property~$B$.
+But \emph{it is not so}. There exists no
+property which can, like this property~$A$, be an
+absolute criterion enabling us to recognise 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 property which in either Euclidean or
+non-Euclidean space belongs to the straight line,
+and belongs to it alone. But how can we ascertain
+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 difficulty a little farther
+off. In reality, the property that I have just
+enunciated is not a property of the straight line
+\PageSep{75}
+alone; it is a property of the straight line and of
+distance. For it to serve as an absolute 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 Euclidean system, and which cannot be interpreted
+in the system of Lobatschewsky. As I
+am well aware that this challenge will never be
+accepted, I may conclude that no experiment will
+ever be in contradiction 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 only 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 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,
+\PageSep{76}
+and the conditions which define the absolute position
+of the system and its absolute orientation in
+space. The law of the phenomena 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
+distances at the initial moment, but will in no
+way depend on the absolute initial position of
+the system and of its absolute initial orientation.
+This is what we shall call, for the sake of
+abbreviation, \emph{the law of relativity}.
+
+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 recognised
+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
+\PageSep{77}
+be still in agreement 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 rigour, 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 considered 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 independent of all
+interpretation by experiments. If the law is true
+in the Euclidean interpretation, it will be also true
+in the non-Euclidean interpretation. Allow me
+to make a short digression on this point. I have
+spoken above of the data which define the position
+\PageSep{78}
+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 experiments---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 means of astronomical observations, but which
+\PageSep{79}
+could be done by 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 that
+from this fact Newton concluded the existence of
+absolute space. I myself cannot accept this way
+of looking at it. I shall explain why in Part~III.,
+but for the moment it is not my intention to
+discuss this difficulty. I must therefore resign
+myself, in the enunciation 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 rational
+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 one equation with
+several unknowns; but when I have made enough
+experiments I shall have enough equations to
+calculate all my unknowns.'' If I know the height
+of the main-mast, that is not sufficient to enable
+\PageSep{80}
+me to calculate the age of the captain. When
+you have measured every fragment of wood in a
+ship you will have many equations, 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, however 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. When
+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 body, I am
+\PageSep{81}
+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 knowledge 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 $\alpha\ \beta\ \gamma$. I now suppose
+that we find that we can bring into contact $\Chg{\alpha\ \beta\ \gamma}{\alpha\beta\gamma}$
+with~$ago$; by that I mean $\alpha$~with~$a$, and at the
+same time $\beta$~with~$g$, and $\gamma$~with~$o$. Then we can
+successively bring into contact $\alpha\beta\gamma$ with $bgo$, $cgo$,
+$dgo$, $ego$, $fgo$, then with $aho$, $bho$, $cho$, $dho$, $eho$, $fho$;
+and then $\alpha\gamma$ successively with $ab$, $bc$, $cd$, $de$, $ef$, $fa$.
+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
+geometry). They therefore suffice to prove that
+these bodies move according to the Euclidean
+group; or at least that they do not move according
+\PageSep{82}
+to the Lobatschewskian group. That they may
+be compatible with the Euclidean group is easily
+seen; for we might make them so if the body~$\alpha\beta\gamma$
+were an invariable solid of our ordinary
+geometry in the shape of a right-angled triangle,
+and if the points $abcdefgh$ were the vertices of
+a 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~$\alpha\beta\gamma$ successively to~$ago$,
+$bgo$, $cgo$, $dgo$, $ego$, $fgo$, $aho$, $bho$, $cho$, $dho$, $eho$, $fho$,
+and then that we can apply~$\alpha\beta$ (and no longer~$\alpha\gamma$)
+successively to~$ab$, $bc$, $cd$, $de$, $ef$, and~$fa$. These are
+observations that could be made if non-Euclidean
+geometry were true. If the bodies~$\alpha\beta\gamma$, $oabcdefgh$
+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 therefore 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
+according to the Euclidean group. And so, without
+making any hypothesis 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
+\PageSep{83}
+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 experiment proving that space is
+Euclidean, and the second an experiment proving
+that space is non-Euclidean; in fact, it might be
+imagined (note that I use the word \emph{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 construct it if he would only take
+the trouble. But you must not conclude, however,
+that space is non-Euclidean. In the same way,
+just as ordinary solid bodies would continue
+to exist when the mechanic had constructed 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~$R$, 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 \Typo{negligeable}{negligible}, and which would behave
+as ordinary invariable solids; and, on the other
+hand, we might have very dilatable bodies, which
+would behave as non-Euclidean solids. We
+might have two double pyramids~$oabcdefgh$ and
+\PageSep{84}
+$o'a'b'c'd'e'f'g'h'$, and two triangles $\alpha\beta\gamma$~and~$\alpha'\beta'\gamma'$.
+The first double pyramid would be rectilinear, and
+the second curvilinear. The triangle~$\alpha\beta\gamma$ 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~$\alpha'\beta'\gamma'$.
+
+And then the experiment would seem to show---first,
+that Euclidean geometry is true, and then
+that it is false. Hence, \emph{experiments have reference
+not to space but to bodies}.
+
+\Subsection{Supplement.}
+
+\ParSkip8. 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 \Title{Revue
+de Métaphysique et de Morale} and in the \Title{Monist}.
+When we say that space has three dimensions,
+what do we mean? We have seen the importance
+of these ``internal changes'' which are revealed to
+us by our muscular sensations. They may serve
+to characterise 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
+\PageSep{85}
+initial and final attitudes $A$~and~$B$ remaining the
+same, the intermediary attitudes of the corresponding
+sensations may differ). How then can
+we recognise the equivalence of these two series?
+Because they may serve to compensate for the same
+external change, or more generally, because, when
+it is a question of compensation 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 recognise it? I mean that \emph{although I have
+not moved} (my muscular sense tells me this), my
+finger, which just now touched the object~$A$, is
+\PageSep{86}
+now touching the object~$B$. I might have 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 \Foreign{à~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 furnished 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
+\PageSep{87}
+each point a class. Yet it may be said that what
+we get from this experiment 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 continuum
+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 \Foreign{à~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
+affirmative. If we recur to what we have called
+displacements, the aggregate 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 characterise 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 neighbouring
+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.
+\PageSep{88}
+Which shall we choose to form a kind of
+standard by which to compare natural phenomena?
+And when this group is chosen, which
+of the sub-groups shall we take to characterise a
+point in space? Experiment has guided us by
+showing us what choice adapts itself best to the
+properties of our body; but there its rôle ends.
+\PageSep{89}
+
+
+\Part{III.}{Force.}
+
+\Chapter{VI.}{The Classical Mechanics.}
+
+\First{The} English teach mechanics as an experimental
+science; on the Continent it is taught always more
+or less as a deductive and \Foreign{à~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 predecessors 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
+compelled 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 reasoning, what is
+convention, and what is hypothesis. This is not
+all.
+\PageSep{90}
+
+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.''\footnote
+  {\Title{Revue de Métaphysique et de Morale}, t.~vi., pp.~1--13, January,
+  1898.}
+
+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 become
+more complicated, but it still would be possible.
+
+Thus, absolute space, absolute time, and even
+geometry are not conditions which are imposed on
+mechanics. All these things no more existed
+before mechanics than the French language can
+be logically said to have existed before the truths
+which are expressed in French. We might
+endeavour to enunciate the fundamental law of
+mechanics in a language independent of all these
+\PageSep{91}
+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 \Title{Leçons de Mécanique
+physique}. Of course the enunciation of these laws
+would become much more complicated, because all
+these conventions 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.
+
+\Par{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
+\Foreign{à~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 \Foreign{à~priori} truth, an
+experimental fact? Have there ever been experiments
+\PageSep{92}
+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 mechanics 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 consequences 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
+neighbouring bodies, and on their velocities.
+Mathematicians would say that the movements
+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
+generalisation 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
+\Foreign{à~priori}; other laws would be just as compatible
+with the principle of sufficient reason. If a body
+is not acted upon by a force, instead of supposing
+\PageSep{93}
+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
+the law of inertia: what will be the natural
+generalisation? 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 neighbouring bodies; in the second case,
+that the variation of the acceleration of a body
+depends only on the position of the body and of
+neighbouring 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 analogous 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, I 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
+position 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.
+\PageSep{94}
+
+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 the 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 generalised
+law of inertia only by opposing it to a contrary
+hypothesis.
+
+Has this generalised law of inertia been verified
+by experiment, and can it be so verified?
+When Newton wrote the \Title{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
+\PageSep{95}
+planet is entirely determined by its initial position
+and initial velocity; this, indeed, is what our
+generalised 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 principles 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 attribute 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
+astronomy 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?
+\PageSep{96}
+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 positions after
+a certain time. We ought then to find that they
+would resume 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 \emph{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 velocities 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
+\PageSep{97}
+that their $3n$~co-ordinates satisfy a system of $3n$~differential
+equations of the fourth order (and
+not of the second, as required by the law of
+inertia). We know that by introducing $3n$~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 confirmed nor contradicted by
+experiment.
+
+\Par{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
+\PageSep{98}
+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 quotient of the force by the acceleration?
+These difficulties are insurmountable.
+
+When we say force is the cause of motion, we
+are talking metaphysics; and this definition, if we
+had to be content with it, would be absolutely
+fruitless, would lead to absolutely nothing. For a
+definition to be of any use it must tell us how to
+measure force; and that is quite sufficient, 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 opposite 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 another. 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 opposite directions.
+It is this definition which we try to materialise, as
+it were, when we measure a force with a dynamometer
+or with a balance. Two forces, $F$~and~$F'$,
+\PageSep{99}
+which I suppose, for simplicity, to be acting
+vertically upwards, are respectively applied to two
+bodies, $C$~and~$C'$. I attach a body weighing~$P$
+first to~$C$ and then to~$C'$; if 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 rigour; this
+rigour does not exist. What I say of weight
+would apply equally to the force of the spring of
+a dynamometer, which would vary according 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~$R$
+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 because they
+are in equilibrium. From these three equalities
+\PageSep{100}
+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; \emph{hence this
+principle can no longer be regarded as an experimental
+law but only as a definition}.
+
+To recognise 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 constant in
+magnitude and direction. But this third rule is
+an experimental law. It is only approximately
+true: \emph{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 definition. 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 acceleration
+\PageSep{101}
+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~$B$ arc respectively in 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, $C$,~$D$,~\ldots. 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 \emph{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
+\PageSep{102}
+attraction of two bodies is proportional to their
+masses; if $r$~is their distance apart, $m$~and~$m'$ their
+masses, $k$~a constant, then their attraction will be~$kmm'/r^{2}$.
+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
+%[** TN: "mais sans que l'on eût f = kmm'"]
+masses, that it is equal to~$f/r^{2}$ \Reword{and not to~$kmm'$}{but without having $f = kmm'$}.
+If it were so, we should nevertheless, by observing
+the \emph{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 assertion? 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 acceleration 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 principle
+\PageSep{103}
+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, \emph{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 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 system
+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. \emph{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 conclusion
+is obvious; the motion of the centre of gravity
+of the universe will be for ever 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:
+\emph{Masses are co-efficients which it is found convenient to
+introduce into calculations.}
+
+We could reconstruct our mechanics by giving
+to our masses different values. The new mechanics
+would be in contradiction neither with
+\PageSep{104}
+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 \emph{would not be so simple}.
+Let us clearly understand this. It would be only
+the first terms which would be less simple---\ie,
+those we already know through experiment;
+perhaps the small masses could be slightly altered
+without the \emph{complete} equations gaining or losing
+in simplicity.
+
+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 experiment.'' From
+what we have just seen these fears would appear
+to be groundless. The principles of dynamics
+appeared to us first as experimental truths, but
+we have been compelled to use them as definitions.
+It is \emph{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
+\PageSep{105}
+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 \emph{perfectly} isolated, perfectly abstracted from
+all external action; but there are systems which
+are \emph{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. We then find that
+the motion of its centre of gravity is \emph{nearly} uniform
+and rectilinear in conformity with Newton's Third
+Law. This is an experimental fact, which cannot
+be invalidated by a more accurate 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.
+\emph{Thus is explained how experiment may serve as a basis
+for the principles of mechanics, and yet will never
+invalidate them.}
+
+\Par{Anthropomorphic Mechanics.}---It will be said that
+Kirchoff has only followed the general tendency of
+mathematicians towards nominalism; 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 primitive, 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
+\PageSep{106}
+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 us 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 conductor 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 experience more fatigue in
+lifting a weight of $100$~lb.\ 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 definitively 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 symbol,
+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
+\PageSep{107}
+rôle 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.
+
+\Par{The Thread School.}---M.~Andrade, in his \Title{Leçons
+de \Typo{Mecanique}{Mécanique} physique}, has modernised anthropomorphic
+mechanics. To the school 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 \emph{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.
+
+%[** TN: "A" variously italicized and not in the original]
+{\Loosen 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~$\alpha$, and the acceleration
+of the body~$A$ is recorded. $A$~is then detached,
+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~$\alpha$, and the
+\PageSep{108}
+acceleration of~$B$ is noted. The experiment is
+then renewed with both $A$~and~$B$ until the increment
+of length is~$\beta$. The four accelerations
+observed should be proportional. Here we have
+an experimental 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, which 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 measuring the increment
+of the thread. Nine times out of ten, in consequence,
+our definition will be in default; no
+\PageSep{109}
+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 acceleration.
+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 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 reaction of the bodies
+attached to its extremities.
+
+The ideas expounded by M.~Andrade are none
+the less very interesting. 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 naïve
+anthropomorphism to the present conception of
+science.
+
+We see that we end with an experiment which
+\PageSep{110}
+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
+absolute. 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.
+\PageSep{111}
+
+
+\Chapter{VII.}{Relative and Absolute Motion.}
+
+\Par{The Principle of Relative Motion.}---Sometimes
+endeavours 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 reasons: 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 depend 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 regulations for the degree of B.~ès~Sc.
+\PageSep{112}
+It is clear that the attempt has 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 positions, 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 \Foreign{à~priori} conditions? Remembering what we
+\PageSep{113}
+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 generalised 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.
+
+\Par{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 \Foreign{à~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 motion 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 mysterious in
+that, and if I have not been subject to the action
+\PageSep{114}
+of any external force, the train has experienced an
+external impact. There can be nothing paradoxical
+in the relative motion of two bodies being
+disturbed when the motion of one or the other is
+modified by an external cause. Nor need I dwell
+on the case of relative motion referring to axes
+which rotate uniformly. If the sky were for ever
+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 absolute space, can a thing turn without turning
+with respect to something; and, on the other
+hand, how can we admit Newton's conclusion and
+believe in absolute space? But it is not sufficient
+to state that all possible solutions are equally
+unpleasant to us. We must analyse 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
+\PageSep{115}
+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 ordinary centrifugal force and compounded
+centrifugal force. Our imaginary scientists can
+thus explain everything by looking upon these two
+forces as real, and they would not see in this a
+contradiction of the generalised principle of inertia,
+for these forces would depend, the one on the
+relative positions of the different parts of the
+system, such as real attractions, 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
+realising an isolated system, the centre of gravity
+of this system would not have an approximately
+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 realised. 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
+\PageSep{116}
+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
+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 indifferently 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 complications 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.\Add{''} That
+does not prevent absolute space---that is to say,
+the point to which we must refer the earth to
+\PageSep{117}
+know if it really does turn round---from having
+no objective existence. And hence this affirmation:
+``the earth turns round,'' has no meaning,
+since it cannot be verified by experiment; since
+such an experiment not only cannot be realised or
+even dreamed of by the most daring Jules Verne,
+but cannot even be conceived of without contradiction;
+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 differential equations of the second
+order, and that so are the differences of these
+co-ordinates. This is what we have called the
+generalised principle of inertia, and the principle
+of relative motion. If the distances 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
+\PageSep{118}
+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 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 distances 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 connecting 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 requires 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 \emph{accidental} constant. Or again, let us
+\PageSep{119}
+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
+\emph{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
+\emph{accidental} constant. If, on the other hand, the
+exponent of the distance which figures in the
+expression of the attractive force is equal to~$-2$
+and not to~$-3$, it is not by chance, but because it
+is required by Newton's law. It is an \emph{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 accustomed
+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
+\PageSep{120}
+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;
+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, distances
+will be determined by differential equations
+of the second order. Would this completely
+satisfy the minds of these astronomers? I think
+not. In the first place, they would very soon see
+that in differentiating their equations so as to
+raise them to a higher order, these equations
+would become much more simple, and they would
+be especially 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
+\PageSep{121}
+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 generalised 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 derivatives, and
+something else. What is that \emph{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 something
+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
+\PageSep{122}
+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 entirely 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 indications 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.
+\PageSep{123}
+
+
+\Chapter{VIII.}{Energy and Thermo-dynamics.}
+
+\Par{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
+principle 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 \emph{kinetic energy}, or
+\Foreign{vis~viva}, and \emph{potential energy}. Every change
+that the bodies of nature can undergo is regulated
+by two experimental laws. First, the sum of the
+kinetic and potential energies is constant. This
+is the principle of the conservation of energy.
+Second, if a system of bodies is at~$A$ at the time~$t_{0}$,
+and at~$B$ at the time~$t_{1}$, it always passes from the
+first position to the second by such a path that
+the \emph{mean} value of the difference between the two
+kinds of energy in the interval of time which
+separates the two epochs $t_{0}$~and~$t_{1}$ 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
+\PageSep{124}
+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 principles 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 remain constant. This quantity
+is the sum of two terms. The first depends only on
+the position of the material points, and is independent
+of their velocities; the second is proportional
+to the squares of these velocities. This
+decomposition can only take place in one way.
+The first of these terms, which I shall call~$U$, will
+\PageSep{125}
+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~$T + U$, $\phi(T + U)$.
+But this function $\phi(T + U)$ 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~$T + U$\Typo{)}{},
+it matters not which, since this linear function may
+always be reduced to~$T + U$ by a change of unit
+and of origin\Typo{}{)}. This, then, is what we call energy.
+The first term we shall call potential energy, and
+the second kinetic energy. The definition of the
+two kinds of energy may therefore be carried
+through without any ambiguity.
+
+So it is with the definition of mass. Kinetic
+energy, or \Foreign{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 expression of the kinetic energy
+as a function of these relative velocities, the co-efficients
+of this expression will give us the masses.
+So in this simple case the fundamental ideas can
+be defined without difficulty. 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
+\PageSep{126}
+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 more reason to take as our
+definition $T + U$ rather than any other function of~$T + U$,
+when the property which characterised
+$T + U$ 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 constant, 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
+their internal state. The expression for the energy
+could be decomposed in one way only into three
+\PageSep{127}
+terms of this form. But this is not the case. Let
+us consider electrified bodies. The electro-static
+energy due to their mutual action will evidently
+depend on their charge---\ie, 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, $\phi(T + U + Q)$.
+
+If $T + U + Q$ were of the particular form that I
+have suggested above, no ambiguity would ensue.
+Among the functions $\phi(T + U + Q)$ 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:---\emph{There
+is something which remains constant.} In this
+form it, in its turn, is outside the bounds of experiment
+and reduced to a kind of tautology. It
+\PageSep{128}
+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 molecule, acted upon
+by no force, but compelled to move on a surface,
+will take as its path the geodesic line---\ie, the
+shortest path. This molecule 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. It is clear that it would be better
+to replace it by a less objectionable enunciation,
+one in which, as philosophers would say, final
+\PageSep{129}
+effects do not seem to be substituted for acting
+causes.
+
+\Par{Thermo-dynamics.}---The rôle 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,
+\Foreign{àpropos} 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 generalise, as we do
+every day, empiric data, and I shall not be so
+foolhardy as to discuss this question, 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
+\PageSep{130}
+would be reduced to a kind of inventory, to the
+ascertaining of isolated facts. It would not 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 generalisation to predict whether the fact will
+be renewed as soon as the least of these circumstances
+is changed. But every proposition may
+be generalised in an infinite number of ways.
+Among all possible generalisations 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 proclaimed
+abroad that Nature loves simplicity; but Nature
+has proved the contrary since then on more than
+one occasion. We no longer confess 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 necessity 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 laws from which it was deduced, in
+\PageSep{131}
+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 perturbations 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 rigour,
+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 experiments, 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
+reasons. Before we raise Mayer's law to the
+\PageSep{132}
+dignity of an absolute principle, a deeper discussion
+is necessary. But if we embark on this discussion
+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 generality and apply it to
+the universe, we see it vanish, so to speak, and
+nothing is left but this---\emph{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 parameters, which I shall call $x_{1}$,~$x_{2}$, $x_{3}\Add{,}~\ldots\Add{,},~x_{n}$.
+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 values of these same parameters at an
+anterior or ulterior moment. In 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
+$x_{1}$,~$x_{2}$, $x_{3}\Add{,}~\ldots\Add{,}~x_{n}$, which remain constant. If we
+say then, \emph{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
+\PageSep{133}
+have only $n - p$ 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? \emph{There is a combination of these $n - p$
+relations, 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
+\PageSep{134}
+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 absolute
+meaning, we must extend it to the whole
+universe, and then we find ourselves face to
+face with the very difficulty we have endeavoured
+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 determinist 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, that Mayer's law is a form subtle enough
+\PageSep{135}
+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 is reduced to
+mere tautology; since, in each particular case, and
+provided 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 utilise 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
+inequality. 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.
+
+\Par{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 approximately as far as almost isolated
+systems are concerned; on the other hand,
+\PageSep{136}
+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 beforehand 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 rôle
+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
+\PageSep{137}
+convenient the fundamental conventions of
+geometry refer to bodies which have nothing
+in common with those that are studied by
+geometry. They refer to the properties of solid
+bodies and to the propagation of light in a straight
+line. These are mechanical, optical experiments.
+In no way can they be regarded as geometrical
+experiments. 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 physiological 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 fundamental
+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 generalisations of experimental
+and particular principles. 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
+\PageSep{138}
+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 familiarised, 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 generalised far too much.
+They have thought that the principles were
+the whole of science, and therefore that the
+whole of science was conventional. This paradoxical
+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
+\PageSep{139}
+not rigorously true, it was only approximate.
+We introduce arbitrarily an intermediate term,~$C$,
+more or less imaginary, and $C$~is \emph{by definition} that
+which has with~$A$ \emph{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 decomposition may be carried, laws will
+always remain. We shall now enter into the
+domain of laws properly so called.
+\PageSep{140}
+
+
+\Part{IV.}{Nature.}
+
+\Chapter{IX.}{Hypotheses in Physics.}
+
+\Par{The Rôle of Experiment and Generalisation.}---Exper\-iment
+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 questioned. 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 sufficient
+merely to observe; we must use our observations,
+and for that purpose we must generalise. This
+is what has always been done, only as the recollection
+of past errors has made man more and more
+circumspect, he has observed more and more and
+generalised less and less. Every age has scoffed
+at its predecessor, accusing it of having generalised
+\PageSep{141}
+too boldly and too naïvely. Descartes used to
+commiserate the Ionians. 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 accumulation 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
+foresight. 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.''\footnote
+  {V. \Title{Past and Present}, end of Chapter~I., Book~II.\Transl}
+Carlyle was a compatriot of Bacon,
+and, like him, he wished to proclaim his worship
+of \emph{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 Lackland
+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,
+\PageSep{142}
+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 \Foreign{experimentum
+crucis}; 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 generalise. Without generalisation,
+prediction is impossible. The circumstances
+under which one has operated will never again
+be reproduced simultaneously. The fact observed
+will never be repeated. All that can be affirmed
+is that under analogous circumstances an analogous
+fact will be produced. To predict it, we must
+therefore invoke the aid of analogy---that is to say,
+even at this stage, we must generalise. However
+timid we may be, there must be interpolation.
+Experiment only gives us a certain number of
+isolated points. They must be connected by a
+continuous line, and this is a true generalisation.
+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 generalising our
+\PageSep{143}
+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, generalised.
+
+It is often said that experiments should be made
+without preconceived 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 intervene 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 \Foreign{ipso~facto}, they will
+force us to look at things under different aspects.
+This is enough to free us. He is no longer a slave
+who can choose his master.
+
+Thus, by generalisation, 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
+\PageSep{144}
+probable. However solidly founded a prediction
+may appear to us, we are never \emph{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 presents itself. But
+every experiment is long and difficult, and the
+labourers 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 predictions
+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 increasing indefinitely; the librarian
+has limited funds for his purchases, and he must,
+therefore, strain every nerve not to waste them.
+Experimental 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 utilise its riches;
+and also by showing the librarian the gaps in his
+\PageSep{145}
+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 rôle of mathematical physics. It
+must direct generalisation, 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.
+
+\Par{The Unity of Nature.}---Let us first of all observe
+that every generalisation 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 \Chg{re-act}{react} 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 argument in favour of its
+accuracy: when Fresnel himself, after having 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
+considerably; but those who do not believe that
+natural laws must be simple, are still often obliged
+\PageSep{146}
+to act as if they did believe it. They cannot
+entirely dispense with this necessity without
+making all generalisation, and therefore all science,
+impossible. It is clear that any fact can be
+generalised 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 inflexions
+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 movements 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 determinations obtained
+by these three methods, we find three numbers
+very close together, but not quite identical. This
+result might be interpreted 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.
+\PageSep{147}
+
+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; sometimes, 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 discussed, whose paths, deformed by incessant
+impacts, have the most capricious shapes, and
+plough their way through space in every direction.
+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,
+\PageSep{148}
+and the coarseness of our senses alone prevents us
+from seeing the complexity.
+
+Many phenomena obey a law of proportionality.
+But why? Because 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 proportional
+to the increment of the variable. As in reality
+our increments are not infinitely small, but only
+very small, the law of proportionality 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 perhaps only apparent. Who
+knows if it be not due to some complicated
+mechanism, to the impact of some subtle matter
+animated by irregular 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 negligible, 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
+\PageSep{149}
+the complex, and then the complex 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 generalisations. 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 generalisations by the belief in simplicity.
+We have verified a simple law in a considerable
+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
+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
+\PageSep{150}
+always be able to reason in the same fashion,
+and if a simple law has been observed 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
+rôle 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.
+
+\Par{Rôle of Hypothesis.}---Every generalisation is a
+hypothesis. Hypothesis therefore plays a necessary
+rôle, which no one has ever contested. 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
+\PageSep{151}
+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 something 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 consequence.
+
+Now, under what conditions is the use of
+hypothesis without danger? 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 mathematical physics may
+render us. By the precision which is its characteristic,
+we are compelled to formulate all the
+hypotheses that we would unhesitatingly make
+without its aid. Let us also notice that it is
+important not to multiply hypotheses indefinitely.
+If we construct a theory based upon multiple hypotheses,
+and if experiment condemns it, which of
+\PageSep{152}
+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
+determined 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 distant 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 indifferent.
+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 conclusions,
+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
+\PageSep{153}
+same conclusions 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 necessary to represent it by an~$x$ 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 artifices for
+calculation, or to assist our understanding by
+concrete images, to fix the ideas, as we say. They
+need not therefore be rejected. The hypotheses
+of the third category are real generalisations.
+They must be confirmed or invalidated by experiment.
+Whether verified or condemned, they will
+always be fruitful; but, for the reasons I have
+given, they will only be so if they are not too
+numerous.
+
+\Par{Origin of Mathematical Physics.}---Let us go
+further and study more closely the conditions
+which have assisted the development of mathematical
+physics. We recognise at the outset that
+\PageSep{154}
+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 \emph{differential equation};
+for the laws of Kepler we substitute the law of
+Newton.
+
+Next, we try to decompose the phenomena in
+space. What experiment gives us is a confused
+aggregate of facts spread over a scene of considerable
+extent. We must try to deduce the elementary
+phenomenon, which will still be localised 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 neighbouring
+point, it will immediately impart that
+\PageSep{155}
+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 phenomenon
+is the interchange of heat between two
+contiguous points. It is strictly localised 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 investigation 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 approximately, 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 independent, and may be
+added one to the other, either as vectors or as scalar
+\PageSep{156}
+quantities. The elementary phenomenon is then
+the action of an isolated body. Or suppose, again,
+it is a question of small movements, or more
+generally of small variations which obey the well-known
+law of mutual or relative independence.
+The movement observed 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 phenomena,
+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 mechanism. The law of
+great numbers will suffice. Take for example the
+propagation of heat. Each molecule radiates towards
+its neighbour---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 neighbouring
+molecules attract and repel each other, we
+need not inquire by what law. It is enough for us
+that this attraction is sensible at small distances
+only, and that the molecules are very numerous,
+that the medium is symmetrical, and we have
+\PageSep{157}
+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 complex mechanism. Evidently the
+best means of reaching the elementary phenomenon
+would be experiment. It would be necessary
+by experimental artifices to dissociate the
+complex system which nature offers for our investigations
+and carefully to study the elements as
+dissociated as possible; for example, natural white
+light would be decomposed into monochromatic
+lights by the aid of the prism, and into polarised
+lights by the aid of the polariser. 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 natural lights which are called \emph{monochromatic}
+%[** TN: "nous donnent une raie très fine, mais qui n'est pas cependant infiniment fine"]
+give us a very fine \Reword{array, but a y}{ray, but one} 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 rigorously monochromatic
+\PageSep{158}
+light. That would not be accurate.
+I assume that two rays emanate from the same
+source, that they are first polarised in planes at
+right angles, that they are then brought back
+again to the same plane of polarisation, and that
+we try to obtain interference. If the light were
+\emph{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
+\emph{integration}, and it is the province of the mathematician.
+It might be asked, why in physical
+science generalisation 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 \emph{all
+similar to each other}; and in this way differential
+equations are quite naturally introduced. It is
+\PageSep{159}
+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 divine 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 \hyperref[chapref:I]{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 physicists, that mathematical
+physics came into existence. In the natural
+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
+generalisation.
+\PageSep{160}
+
+
+\Chapter{X.}{The Theories of Modern Physics.}
+
+\Par{Significance of Physical Theories.}---The ephemeral
+nature of scientific 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 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
+\emph{bankruptcy of science}.
+
+His scepticism is superficial; he does not take
+into account the object of scientific theories and
+the part they play, or he would understand 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
+\PageSep{161}
+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 \emph{motion}, we now call \emph{electric
+current}. But these are merely names of the images
+we substituted for the real objects which Nature
+will hide for ever 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 between 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 instance) 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
+phenomena an intimate relationship which corresponds
+to a profound reality; that this relationship,
+\PageSep{162}
+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 Helmholtz, 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
+Helmholtz obtain the same equations, although
+their starting-points were to all appearance widely
+separated. I venture to say that these theories
+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
+\PageSep{163}
+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 contradiction 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 impact 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
+\PageSep{164}
+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 sensations 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 restricted meaning I have
+already mentioned, which is the sound meaning,
+then it has no meaning at all. Hypotheses of
+this kind have therefore 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 thought to be abandoned and
+definitively condemned by experiment, are suddenly
+revived from their ashes and begin a new life.
+It is because 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 \emph{electrons}. In what
+do these permanently electrified molecules differ
+\PageSep{165}
+from the electric molecules 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 established 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, Carnot's theory expressed in addition to
+true relations, other inexact relations, the \Foreign{débris}
+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
+\Chg{thermodynamics}{thermo-dynamics}. 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.
+Nor have the reasonings of Carnot perished on
+this account; they were applied to an imperfect
+conception of matter, but their form---\ie, the
+essential part of them, remained correct. What
+\PageSep{166}
+I have just said throws some light at the same
+time on the rôle of general principles, such as
+those of the principle of least action or of the
+conservation of energy. These principles are of
+very great 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 innumerable
+observations. However, from their very generality
+results a consequence to which I have called
+attention in \ChapRef{VIII}.---namely, that they are
+no longer capable of verification. As we cannot
+give a general definition of energy, the principle
+of the conservation of energy simply signifies that
+there is a \emph{something} which remains constant.
+\Pagelabel{166}%
+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 \emph{energy}. Does
+this mean that the principle has no meaning and
+vanishes into a tautology? Not at all. It means
+that the different things to which we give the
+name of \emph{energy} are connected by a true relationship;
+it affirms between them a real relation.
+But then, if this principle has 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---\ie,
+\PageSep{167}
+when we can no longer use it to predict correctly
+new phenomena. We shall be certain in such a
+case that the relation affirmed is no longer real,
+for otherwise it would be fruitful; experiment
+without directly contradicting a new extension of
+the principle will nevertheless have condemned it.
+
+\Par{Physics and Mechanism.}---Most theorists have a
+constant predilection 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 in \ChapRef{XII}.,
+\Foreign{àpropos} of Maxwell'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 number of such explanations. By
+means of a well-known theorem due to Königs,
+\PageSep{168}
+it may be shown that we can explain everything
+in an unlimited number of ways, by connections
+after the manner of Hertz, or, again, by central
+forces. No doubt it may be just as easily demonstrated
+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
+movements 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 \emph{ether} or other names, have
+from all time played so important a rôle 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 reduce 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 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
+\PageSep{169}
+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 materialising them, their
+individuality was, so to speak, emphasised---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
+relations 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 material
+molecules. We see, for instance, that the photographic
+plate experiences the consequences of a
+phenomenon of which the incandescent mass of
+a star was the scene several years before. Now,
+\PageSep{170}
+in ordinary mechanics, the state of the system
+under consideration depends 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 universe 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 \Chg{re-action}{reaction} is not true
+if applied to matter \emph{alone}, and that this can be
+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 \Chg{re-action}{reaction} of matter on something.
+\PageSep{171}
+
+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 \ChapRef{XIII}., were
+true, Newton's principle would not apply to matter
+\emph{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
+experiments have been undertaken, it is because
+we have not been certain beforehand; and indeed,
+according to current theories, the compensation
+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
+\PageSep{172}
+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 material bodies. Experiments have
+been made that should have disclosed 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 explanation was necessary, and was
+forthcoming; they always are; hypotheses 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 rôle? 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 the higher order, and that the mutual
+destruction of these terms will be rigorous and
+absolute.
+
+\Par{The Present State of Physics.}---Two opposite
+tendencies may be distinguished in the history
+\PageSep{173}
+of the development of physics. On the one hand,
+new relations are continually being discovered
+between objects which seemed destined to remain
+for ever 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 possible; but nothing
+proves this \Foreign{à~priori}, and it may be that after
+unsuccessful efforts to bend Nature to our ideal of
+unity in spite of herself, 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 yesterday.
+\PageSep{174}
+No 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 phenomena 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 concerned, there
+was not so great an advance. Electricity had just
+annexed 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 universal mechanism? No one had
+the slightest idea. As to the possibility 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.
+
+To-day, what do we see? In the first place, a
+step in advance---immense progress. The relations
+between light and electricity are now known; the
+three domains of light, electricity, and magnetism,
+formerly separated, are now one; and this annexation
+seems definitive.
+
+Nevertheless the conquest has caused us some
+\PageSep{175}
+sacrifices. Optical phenomena become particular
+cases in electric phenomena; as long as the former
+remained isolated, it was easy to explain them by
+movements 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 difficulty.
+
+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
+Newton's law that action and \Chg{re-action}{reaction} 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 \Chg{re-action}{reaction} 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 abnormal 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
+\PageSep{176}
+rotation, which had defied all Maxwell's efforts.
+This facility proves that Lorentz's theory is not a
+mere artificial combination 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, MacCullagh'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 movements 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 \Chg{re-action}{reaction}
+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.
+
+If Larmor has failed, as in my opinion he has,
+does it necessarily follow that a mechanical explanation
+is impossible? Far from it. I said
+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
+\PageSep{177}
+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 condition 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 confirmed,
+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
+Lagrange'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
+\PageSep{178}
+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 physics 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 eminently abstract
+concept of entropy which is as universal as the
+concept 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 pursued in detail; electric resistance
+resembles the viscosity of fluids; hysteresis 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 irreversibility is but apparent, that
+the elementary phenomena are reversible 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---\ie, all seems
+to progress in the same direction, and that without
+\PageSep{179}
+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 contradictions, and if it is in conformity
+with the true nature of things.
+
+Let us notice, however, the original ideas of
+M.~Gouy on the Brownian 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.\footnote
+  {Clerk-Maxwell imagined some supernatural agency at work,
+  sorting molecules in a gas of uniform temperature into (\textit{a})~those
+  possessing kinetic energy above the average, (\textit{b})~those possessing
+  kinetic energy below the average.\Transl}
+
+To resume, phenomena long known are gradually
+being better classified, 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 unexpected
+\PageSep{180}
+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 sometimes 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 important
+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 facilitate 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 majestic simplicity, but
+matter properly so called seems more and more
+complex; all we can say of it is but approximate,
+and our formulæ 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
+\PageSep{181}
+when their complexity is recognised, 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 equations
+persists. Take for instance the laws of reflection.
+Fresnel established 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 polarisation were detected everywhere.
+But it is owing to the first approximation that the
+cause of these anomalies was found in the existence
+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 properties
+of matter better we see that continuity reigns.
+\PageSep{182}
+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.
+Similarly, 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 satisfy 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 realisation of unity. This has not
+been done as quickly as was hoped fifty years ago,
+and the path predicted has not always been
+followed; but, on the whole, much ground has
+been gained.
+\PageSep{183}
+
+
+\Chapter{XI.}{The Calculus of Probabilities.}
+
+\First{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 experiment 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 generalisations. 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
+\PageSep{184}
+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 favourable 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
+$6 × 6 = 36$. The number of favourable cases is~$11$;
+the probability is~$\dfrac{11}{36}$. 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
+$\dfrac{6 × 7}{2} = 21$ different combinations. Among these
+combinations, six are favourable; the probability
+\PageSep{185}
+is~$\dfrac{6}{21}$. Now why is the first method of calculating
+the number of possible cases more legitimate than
+the second? In any case it is not the definition
+that tells us. We are therefore bound to complete
+the definition by saying, ``\ldots 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 successively adopted two conventions
+which seemed to be equally imperative in the eyes
+of common-sense, and with one convention he finds~$\frac{1}{2}$,
+and with the other~$\frac{1}{3}$. The conclusion which
+seems to follow from this is that the calculus of
+probabilities is a useless science, that the obscure
+\PageSep{186}
+instinct which we call common-sense, and to which
+we appeal for the legitimisation 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
+impossible, 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 perturbations? Here
+again the only answer is: It is very improbable.
+From this point of view all the sciences would only
+be unconscious applications of the calculus of probabilities.
+And if this calculus be condemned, then
+the whole of the sciences must also be condemned.
+I shall not dwell at length on scientific problems
+in which the intervention of the calculus of probabilities
+is more evident. In the forefront of these
+is the problem of interpolation, in which, knowing
+a certain number of values of a function, we try
+to discover the intermediary values. I may also
+mention the celebrated theory of errors of observation,
+\PageSep{187}
+to which I shall return later; the kinetic
+theory of gases, a well-known hypothesis wherein
+each gaseous molecule is supposed 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 confidence 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 \Foreign{coup}, and he
+\PageSep{188}
+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 \emph{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 \Foreign{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
+\emph{objective probability}, and it is this phenomenon
+which has to be explained. There are numerous
+Insurance Societies which apply the rules of the
+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 scepticism
+is not admissible. We may distrust, but we
+cannot condemn \Foreign{en~bloc}. Discussion is necessary.
+
+\Par[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 \emph{generality}. I said above that probability
+is the ratio of the number of favourable 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, if we take a throw of the
+\PageSep{189}
+dice in which the number of possible cases is~$36$.
+That is the first degree of generality. But if we
+ask, for instance, 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
+%[** TN: French edition uses a fraction]
+that this probability is~\Reword{$1$-$10$th}{$\frac{1}{10}$}. 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.
+\PageSep{190}
+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 distribution.
+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 probabilities 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 imperfect 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
+\PageSep{191}
+from the effects. Now, these problems are classified
+as \emph{probability of causes}, and are the most interesting
+of all from their scientific applications. I play at
+\Foreign{écarté} with a gentleman whom I know to be perfectly
+honest. What is the chance that he turns
+up the king? It is~$\frac{1}{8}$. 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 experimental
+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 divergencies 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.
+
+\Par[II. ]{Probability in Mathematics.}---The impossibility
+of squaring the circle was shown in~1885, but
+before that date all geometers considered this impossibility
+as so ``probable'' that the Académie des
+Sciences rejected without examination the, alas!\
+\PageSep{192}
+too numerous memoirs on this subject that a
+few unhappy madmen sent in every year. Was
+the Académie 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 Académie 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 probability 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 psychological. 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 $\pi$~be the root of an algebraical equation,
+why do you expect this root to be a period of
+%[** TN: sin italicized in the original]
+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
+\PageSep{193}
+what I called above objective probability has
+nothing in common with this first problem. It is
+otherwise with the second. Let us consider the
+first $10,000$ logarithms 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 hesitation that the probability is~$\frac{1}{2}$, 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 \Title{Revue générale 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 derivatives
+of the logarithm remain, in the interval considered,
+\PageSep{194}
+between certain limits. Hence our first conclusion
+is that the property is not only true of the
+logarithm but of any continuous function whatever,
+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 unconscious 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 approximately. 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~$\sin nu$? 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
+$\phi(a)\, da$; that it is therefore proportional to the
+infinitely small interval~$da$, and is equal to this
+multiplied by a function~$\phi(a)$, 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~$2\pi$,
+I may without loss of generality assume that
+\PageSep{195}
+$u$~lies between $0$~and~$2\pi$, and I shall thus be
+led to suppose that $\phi(a)$~is a periodic function
+whose period is~$2\pi$. The mean value that we
+seek is readily expressed by a simple integral,
+and it is easy to show that this integral is smaller
+than
+%[** TN: Expression displayed in the French edition]
+\[
+\frac{2\pi M_{K}}{n^{K}},
+\]
+$M_{K}$~being the maximum value of the
+$K$th~derivative of~$\phi(u)$. We see then that if the
+$K$th~derivative is finite, our mean value will
+tend towards zero when $n$~increases indefinitely,
+and that more rapidly than~$\dfrac{1}{n^{K+1}}$.
+
+%[** TN: Paragraph break in the French edition]
+The mean
+value of~$\sin nu$ when $n$~is very large is therefore
+zero. To define this value I required a convention,
+but the result remains the same \emph{whatever
+that convention may be}. I have imposed upon
+myself but slight restrictions when I assumed that
+the function~$\phi(a)$ 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 rôle of what philosophers call the
+principle of sufficient reason, and on the other hand
+of the importance of the fact that certain properties
+are common to all continuous functions.
+The study of probability in the physical sciences
+will lead us to the same result.
+
+\Par[III. ]{Probability in the Physical Sciences.}---We
+now come to the problems which are connected
+with what I have called the second degree of
+\PageSep{196}
+ignorance---namely, those in which we know the
+law but do not know the initial state of the
+system. I could multiply examples, but I shall
+take only one. What is the probable present
+distribution 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 uniform. 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---\ie, 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 certain
+region of the plane, but as they are very numerous
+this region will appear dotted with points. We
+know nothing else about the distribution 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
+\PageSep{197}
+compelled to make an arbitrary hypothesis. To
+explain the nature of this hypothesis 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
+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
+regions. Here then are two distributions, one
+real, in which the representative points are very
+numerous, very close together, but discrete like the
+molecules of matter in the 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 continuous matter may be nearly proportional
+to the number of representative points,
+or, if it is preferred, to the number of atoms which
+are contained in that region. Even that is impossible,
+\PageSep{198}
+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 longitude at the
+moment~$t$---\ie, of $\sin(at + b)$? We made at the
+outset an arbitrary convention, but if we adopt it,
+this probable value is entirely defined. Let us
+decompose the plane into elements of surface.
+Consider the value of $\sin(at + b)$ at the centre of
+each of these elements. 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~$\phi$ which defines the density of the imaginary
+matter, and as this function~$\phi$ 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
+\PageSep{199}
+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. Whatever the function~$\phi$
+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
+very small. I may give to~$\phi$ 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 instance, for supposing
+that the initial longitude might be exactly~$0°$, but
+that it could not lie between $0°$~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
+\[
+\frac{1}{n} \sum \sin(at + b),
+\]
+$n$~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
+\PageSep{200}
+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 increased. 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 $b = \dfrac{\pi}{2} - at$, the
+longitude of all the planets at the time~$t$ would be~$\dfrac{\pi}{2}$,
+and the mean value in question would be
+evidently unity. For this to be the case at the
+time~$0$, 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 realised, 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 anything
+on the subject of the probability of this or
+that present distribution. Once more we shall
+invoke the principle of sufficient reason, to which
+we must always recur. We might admit that at
+the beginning the planets were distributed almost
+\PageSep{201}
+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.
+
+\Par[IV. ]{Rouge et Noir.}---The questions raised by
+games of chance, such as roulette, are, fundamentally,
+quite analogous to those we have just
+%[** TN: "est partagé en un grand mombre de subdivisions égales"]
+treated. For example, a wheel is divided into \Reword{thirty-seven}{a large number of}
+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 division is red is obviously~$\frac{1}{2}$. The needle
+describes an angle~$\theta$, including 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 $\theta$~and~$\theta + d\theta$, but I
+can make a convention. I can suppose that this
+probability is~$\phi(\theta)\, d\theta$. As for the function~$\phi(\theta)$, I
+can choose it in an entirely arbitrary manner. I
+have nothing to guide me in my choice, but I am
+naturally induced to suppose the function to be
+continuous. Let $\epsilon$~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~$\phi(\theta)\, d\theta$, extending it on the one
+hand to all the red, and on the other hand to all
+\PageSep{202}
+the black compartments, and to compare the
+results. Consider an interval~$2\epsilon$ comprising two
+consecutive red and black compartments. Let
+$M$~and~$m$ be the maximum and minimum values of
+the function~$\phi(\theta)$ in this interval. The integral
+extended to the red compartments will be smaller
+than~$\sum M\epsilon$; extended to the black it will be greater
+than~$\sum m\epsilon$. The difference will therefore be
+smaller than $\sum (M - m)\epsilon$. But if the function~$\phi$ is
+supposed continuous, and if on the other hand the
+interval~$\epsilon$ 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~$\frac{1}{2}$. We see that
+without knowing anything of the function~$\phi$ we
+must act as if the probability were~$\frac{1}{2}$. And on
+the other hand it explains why, from the
+objective point of view, if I watch a certain
+number of \Foreign{coups}, observation will give me almost
+as many black \Foreign{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~$\frac{1}{2}$. Observation shows, it is true, that
+the series of seven consecutive reds is very rare,
+\PageSep{203}
+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.
+
+\Par[V. ]{The Probability of Causes.}---We now come to
+the problems of the probability of causes, the
+most important from the point of view 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 correspond
+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 \Foreign{à~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 produced
+by the cause~$a$ or by the cause~$b$. The effect
+has just been observed. We ask the probability
+\PageSep{204}
+that it is due to the cause~$a$. This is an \Foreign{à~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 \Foreign{à~priori}
+probability for the cause~$a$ to come into play---I
+mean the probability of this event to some one
+who had not observed the effect. To make my
+meaning clearer, I go back to the game of \Foreign{écarté}
+mentioned before. My adversary deals for the
+first time and turns up a king. What is the
+probability that he is a sharper? The formulæ
+ordinarily taught give~$\frac{8}{9}$, 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 \Foreign{à~priori} probability was unjustified, and that
+is why the conclusion of the \Foreign{à~posteriori} probability
+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 \Foreign{à~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. I require to determine an
+experimental law; this law, when discovered, can
+\PageSep{205}
+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 inflexions, and any sudden variation
+of the radius of curvature. This curve will represent
+to me the probable law, and not only will
+it give me the values of the functions intermediary
+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 combination 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 observations
+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 observations
+I had an \Foreign{à~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
+\PageSep{206}
+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 \Foreign{à~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 principle was even
+invoked in favour 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 disappear, experimental science would become
+impossible.
+
+\Par[VI. ]{The Theory of Errors.}---We are thus brought
+to consider the theory of errors which is directly
+\PageSep{207}
+connected with the problem of the probability
+of causes. Here again we find \emph{effects}---to wit,
+a certain number of irreconcilable observations,
+and we try to find the \emph{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 observation. We must calculate
+the probable \Foreign{à~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
+\Foreign{à~priori}---\ie, before any observations are made---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 between 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
+mistake, 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?
+\PageSep{208}
+Numerous proofs have been attempted, almost all
+of them crude paralogisms. But starting from
+the following hypotheses we may prove Gauss's
+law: the error is the result of a very large number
+of partial and independent 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 astronomers. This
+is no doubt because the latter, apart from the
+systematic errors to which they and the physicists
+are subject alike, have to contend with an
+extremely important source of error which is
+entirely accidental---I mean atmospheric undulations.
+So it is very curious 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 disappear.''
+
+What conclusion must we draw? Must we
+continue to use the method of least squares?
+\PageSep{209}
+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 result is so much. \emph{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 observations; 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 \emph{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 absolute.
+
+\Par[VII. ]{Conclusions.}---In the preceding lines I have
+set several problems, and have given no solution.
+\PageSep{210}
+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. Unfortunately, this principle
+is very vague and very elastic, and in the cursory
+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 hypothesis made at the outset,
+provided only that this hypothesis satisfies the
+condition of continuity.
+\PageSep{211}
+
+
+\Chapter{XII.\protect\footnotemark}{Optics And Electricity.}
+\footnotetext{This chapter is mainly taken from the prefaces of two of my
+  books---\Title{Théorie Mathématique de la lumière} (Paris: Naud, 1889),
+  and \Title{Électricité et Optique} (Paris: Naud, 1901).}
+
+\Par{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 undulations 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 mathematical 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, the 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
+\PageSep{212}
+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 rôle. 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.
+
+\Par{Maxwell's Theory.}---We all know that it was
+Maxwell who connected by a slender tie two
+branches of physics---optics and electricity---until
+\PageSep{213}
+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 revealed to us other
+relations, hitherto unsuspected, between the
+different branches of optics and the domain of
+electricity.
+
+\Pagelabel{213}%
+The first time a French reader opens Maxwell's
+book, his admiration 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 acclimatised among us? No
+doubt the education received by most enlightened
+Frenchmen predisposes them to appreciate precision
+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 lines. Starting with clearly enunciated
+hypotheses, they deduced from them all their
+consequences with mathematical rigour, 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.
+\PageSep{214}
+
+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 hypotheses 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 its opinion---which will
+no longer have anything but purely geometrical
+qualities, and the atoms of which will be mathematical
+points subject to the laws of dynamics
+alone. And yet he will try to represent to
+himself, by an unconscious contradiction, these
+invisible and colourless 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 satisfaction 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 theoretical 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
+\PageSep{215}
+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 particular case of electro-magnetic phenomena.
+From the whole theory of electricity a theory of
+light can be immediately deduced. Unfortunately
+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 ourselves 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 constructions,
+between which communication is difficult and
+sometimes impossible. Take, for instance, the
+chapter in which \Chg{electrostatic}{electro-static} attractions are
+explained by the pressures and tensions of the
+dielectric medium. This chapter might be suppressed
+without the rest of the book being
+\PageSep{216}
+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 independent 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 make 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 polarisation that there is an
+identity between optical and magnetic phenomena?
+
+We must not flatter ourselves that we have
+avoided every contradiction, 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
+popularised, 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
+\PageSep{217}
+fundamental idea consists; but for that purpose
+a short digression is necessary.
+
+\Par{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 endeavour 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 phenomenon? It
+will be, on the one hand, when we know the
+differential equations which are satisfied by the
+co-ordinates of these hypothetical 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 functions 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.
+\PageSep{218}
+
+The first of these two principles teaches us that
+the total energy is constant, and may be divided
+into two parts:\Add{---}
+
+(1) Kinetic energy, or \Foreign{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 instant~$t_{0}$ to
+the final position occupied at the instant~$t_{1}$, the
+system must describe such a path that in the
+interval of time between the instant $t_{0}$~and~$t_{1}$,
+the mean value of the action---\ie, the \emph{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\Typo{ }{ one} 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
+Lagrange. In these equations the independent
+\PageSep{219}
+variables are the co-ordinates of the hypothetical
+molecules~$m$; but I now assume that we take for
+the variables the parameters~$q$, which are directly
+accessible to experiment.
+
+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 endeavour to define
+kinetic and potential energy by the aid of
+quantities he can directly observe.\footnote
+  {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 derivatives.}
+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
+parameters that we define the initial and \Typo{fina}{final}
+positions; the principle of least action will always
+remain true.
+
+Now here again, of the whole of the paths which
+lead from one position to another, there is one and
+only one for which the mean action is a minimum.
+The principle of least action is therefore sufficient
+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.
+\PageSep{220}
+
+To form these equations we need not know the
+relations which connect the parameters~$q$ 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~$U$ as a
+function of the parameters~$q$, and that of~$T$ as a
+function of the parameters~$q$ and their derivatives---\ie,
+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 indicated, 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 \emph{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
+\emph{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
+equations formed by the aid of these two functions
+$T$~and~$U$ are in conformity with the data of the
+\PageSep{221}
+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 \Foreign{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 \Title{Électricité 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 comes to the same thing, we look on them as
+derivatives of the parameters~$q$, or as combinations
+of these parameters 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
+parameters? This choice remains arbitrary to a
+large extent, but a mechanical explanation will be
+\PageSep{222}
+possible if it is done so as to satisfy the principle of
+least action.
+
+Next, Maxwell asks: Can this choice and that of
+the two energies $T$~and~$U$ be made so that electric
+phenomena will satisfy this principle? Experiment
+shows us that the energy of an electro-magnetic
+field decomposes into electro-static and electro-dynamic
+energy. Maxwell recognised 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 parameters~$q$, and the intensity
+of the currents as derivatives of other
+parameters~$q$---under these conditions, Maxwell
+has recognised that electric phenomena \Reword{satisfies}{satisfy} 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 mechanical 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 polarisation; Neumann holds that it is
+parallel to that plane. For a long time an \Foreign{experimentum
+crucis} was sought for, which would enable
+us to decide between these two theories, but in
+\PageSep{223}
+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.
+
+\Pagelabel{223}%
+It 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's 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 perhaps come
+when physicists will no longer concern themselves
+with questions which are inaccessible to positive
+methods, and will leave 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,
+\PageSep{224}
+Maxwell 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 \Foreign{magnum opus} he
+develops no complete explanation, he has attempted
+one in an article in the \Title{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---\ie,
+what is common to all theories; everything 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 compelled 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.
+\Pagelabel{224}%
+\PageSep{225}
+
+
+\Chapter{XIII.}{Electro-Dynamics.}
+
+\First{The} history of electro-dynamics is very instructive
+from our point of view. The title of Ampère's
+immortal work is, \Title{Théorie des phénomènes electro-dynamiques,
+uniquement fondée sur expérience}. He
+therefore imagined that he had made no hypotheses;
+but as we shall not be long in recognising, he was
+mistaken; only, of these hypotheses he was quite
+unaware. On the other hand, his successors see
+them clearly enough, because their attention is
+attracted by the weak points in Ampère'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.
+
+\Par[I. ]{Ampère's Theory.}---In Ampère'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
+current is set up which passes from one to the
+\PageSep{226}
+other until the two potentials are equal. According
+to the ideas of Ampère's time, this was
+considered to be an open current; the current was
+known to pass 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 Ampère to be open
+currents---for instance, the currents of discharge
+of a condenser; he was unable to experiment 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 electric charge, and leaving~$B$ are
+set in motion along a path~$BNA$, carrying 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 \emph{through} a fixed conductor
+like a voltaic current, overcoming an ohmic resistance
+and developing heat; we say that it is
+displaced by \emph{conduction}. In the part~$BNA$ the
+electricity is \emph{carried} by a moving conductor, and is
+said to be displaced by \emph{convection}. If therefore the
+convection current is considered to be perfectly
+analogous to the conduction current, the circuit~$BNAMB$
+is closed; if on the contrary the convection
+\PageSep{227}
+current is not a ``true current,'' and, for
+instance, does not act on the magnet, there is only
+the conduction current~$AMB$, which is \emph{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 produce with appreciable intensity;
+in fact, with the means at Ampère's disposal we
+may almost say it was impossible.
+
+To sum up, Ampère 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. Experiment therefore could only show him
+the action of a closed current on a closed current---or
+more accurately, the action of a closed current
+on a portion of current, because a current can be
+made to describe a \emph{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, Ampère had no means of studying the action
+of an open current either on a closed or on another
+open current.
+
+\Par[1.\ ]{The Case of Closed Currents.}---In the case of
+the mutual action of two closed currents, experiment
+revealed to Ampère remarkably simple
+laws. The following will be useful to us in the
+sequel:---
+\PageSep{228}
+
+(1) \emph{If the intensity of the currents is kept constant},
+and if the two circuits, after having undergone any
+displacements and deformations whatever, return
+finally to their initial positions, the total work
+done by the electro-dynamical actions is zero. In
+other words, there is an \emph{electro-dynamical potential}
+of the two circuits proportional to the product of
+their intensities, and depending on the form and
+relative 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:---
+
+(\textit{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.
+
+(\textit{b}) A very short magnetic needle tends to take
+the direction of the magnetic force, and the couple
+to which it tends to reduce is proportional to the
+product of the magnetic force, the magnetic
+moment of the needle, and the sine of the dip
+of the needle.
+
+(\textit{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
+\PageSep{229}
+of magnetic force'' which passes through the
+circuit.
+
+\Par[2.\ ]{Action of a Closed Current on a Portion of
+Current.}---Ampère 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~$\alpha\beta$, the ends $\alpha$~and~$\beta$
+of which could slide along a fixed wire. In one of
+the positions of the movable wire the end~$\alpha$ rested
+on the point~$A$, and the end~$\beta$ on the point~$B$ of
+the fixed wire. The current ran from~$\alpha$ to~$\beta$---\ie,
+from~$A$ to~$B$ along the movable wire, and then
+from~$B$ to~$A$ along the fixed wire. \emph{This current
+was therefore closed.}
+
+In the second position, the movable wire
+having slipped, the points $\alpha$~and~$\beta$ were respectively
+at $A'$~and~$B'$ on the fixed wire. The current
+ran from~$\alpha$ to~$\beta$---\ie, from~$A'$ to~$B'$ on the movable
+wire, and returned 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.
+Ampère \emph{admits} that the force, apparently acting on
+the movable part~$AB$, representing the action of~$C$
+on the portion~$\alpha\beta$ of the current, remains the
+same whether an open current runs through~$\alpha\beta$,
+\PageSep{230}
+stopping at $\alpha$~and~$\beta$, or whether a closed current
+runs first to~$\beta$ and then returns to~$\alpha$ through the
+fixed portion of the circuit. This hypothesis
+seemed natural enough, and Ampère innocently
+assumed it; nevertheless the hypothesis \emph{is not a
+necessity}, for we shall presently see that Helmholtz
+rejected it. However that may be, it enabled
+Ampère, 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:\Add{---}
+
+(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 normal to the element.
+
+(2) The action of a closed solenoid on an
+element of current is zero. But the electro-dynamic
+potential has disappeared---\ie, when a
+closed and an open current of constant intensities
+return to their initial positions, the total work
+done is not zero.
+
+\Par[3.\ ]{Continuous Rotations.}---The most remarkable
+electro-dynamical experiments are those in which
+continuous rotations are produced, and which are
+called \emph{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
+\PageSep{231}
+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.
+Continuous 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 convenient to draw a distinction. The
+passage from the fixed to the movable part, or
+\Foreign{vice~versâ}, 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 connexions and therefore the
+conditions of equilibrium, so that as the position
+of equilibrium is ever eluding, so to speak, the
+system which is trying to reach it, rotation may
+take place indefinitely.
+\PageSep{232}
+
+Ampère 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 current. He concluded that the action of
+a closed on an open current, or \Foreign{vice~versâ}, 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.
+
+\Par[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. Ampère
+falls back on hypothesis. He assumes: (1)~that
+the mutual action of two elements reduces to a
+force acting along their \emph{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 Ampère
+makes two hypotheses 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
+\PageSep{233}
+current acting on an open current. Next, there
+is, properly 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 current 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 continuous 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, \emph{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
+\PageSep{234}
+of current 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 to Ampère
+it would not be true if the systems contained open
+currents. 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~$A$.
+
+\Par[5.\ ]{Induction.}---We know that the discovery of
+electro-dynamical induction followed not long after
+the immortal work of Ampère. 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 conservation of energy is
+sufficient for us to deduce the laws of induction
+from the electro-dynamical laws of Ampère. But
+on the condition, as Bertrand has shown,---that
+we make a certain number of hypotheses.
+
+The same principle again enables this deduction
+to be made in the case of open currents, although
+the result cannot be tested by experiment, since
+such currents cannot be produced.
+
+If we wish to compare this method of analysis
+with Ampère'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;
+\PageSep{235}
+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 \emph{mean} electro-motive force
+%[** TN: "induite dans le circuit C est nulle" in the French edition.]
+induced in the \Reword{current}{circuit}~$C$ 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 Ampère, 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.
+
+\Par[II. ]{Helmholtz's Theory.}---I have dwelt upon the
+consequences of Ampère'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
+Ampère---namely, that the mutual action of two
+elements of current reduces to a force along their
+join. He admits that an clement of current is not
+\PageSep{236}
+acted upon by a single force but by a force and a
+couple, and this is what gave rise to the celebrated
+polemic between Bertrand and Helmholtz.
+Helmholtz replaces Ampère's hypothesis 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 Ampère, 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 Ampère 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 current. 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~$\alpha\beta$ 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 elements
+\PageSep{237}
+$AA'$~and~$B'B$ with respect to~$C$. It is this \emph{double}
+increment which represents the work of the force
+acting upon the portion~$AB$. If, on the contrary,
+$\alpha\beta$~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 experiment, 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 contacts of the \Typo{weir}{wire}
+with the magnet were suppressed, and an open
+current 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 Ampère, 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
+\PageSep{238}
+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 the
+definition of magnetic force to have any meaning,
+the action exercised by an open current on
+an indefinite solenoid would only depend on the
+position of the extremity of that solenoid---\ie,
+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.
+
+\Par[III. ]{Difficulties raised by these Theories.}---Helmholtz's
+theory is an advance on that of Ampère;
+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 convention, 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 difficult to give up deeply-rooted habits of
+\PageSep{239}
+language and thought. There is something more.
+If we do not believe in actions at a distance,
+electro-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.
+
+\Par[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 \emph{current
+of displacement}. If, then, two conductors bearing
+positive and negative charges are placed in connection
+by means of a wire, during the discharge
+there is an open current of conduction in that
+wire; but there are produced at the same time in
+the surrounding dielectric currents of displacement
+which close this current of conduction. We
+know that Maxwell's theory leads to the explanation
+of optical phenomena 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 Maxwell's ideas received the
+confirmation of experiment. Hertz succeeded in
+producing systems of electric oscillations which
+\PageSep{240}
+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
+displacement on the galvanometer. That is true
+in a sense. What he has shown directly is that
+electro-magnetic induction is not instantaneously
+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---\emph{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.
+
+\Par[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 convection
+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.
+\PageSep{241}
+
+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''---\ie, a charged conductor in
+motion---could act on the galvanometer. But experimental
+confirmation 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. Rowland, an
+extremely skilful experimentalist, was the first to
+triumph, or to seem to triumph, over these difficulties.
+A disc received a strong \Chg{electrostatic}{electro-static}
+charge and a very high speed of rotation. An
+astatic magnetic system placed beside the disc
+underwent deviations. The experiment was made
+twice by Rowland, once in Berlin and once at Baltimore.
+It was afterwards repeated by Himstedt.
+These physicists even believed that they could
+announce that they had succeeded in making
+quantitative measurements. For twenty years
+Rowland's law was admitted without objection
+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
+recognise the lines of the metal of the electrode,
+a proof of it? The spark would then be a real
+current of induction.
+\PageSep{242}
+
+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 electrolyte would
+therefore also be a current of convection; but it
+acts on the magnetic needle. And in the same
+way for cathode rays; \Typo{Crooks}{Crookes} attributed these
+rays to very subtle matter charged with negative
+electricity and moving with very high velocity.
+He looked upon them, in other words, as currents
+of convection. Now, these cathode rays are
+deviated by the magnet. In virtue of the
+principle of action and \Chg{re-action}{reaction}, 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 collecting 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 Rowland's
+law.
+
+\Par[VI. ]{Lorentz's Theory.}---We need not go much
+further. According to Lorentz's theory, currents
+\PageSep{243}
+of conduction would themselves be true convection
+currents. Electricity would remain indissolubly
+connected with certain material particles called
+\emph{electrons}. The circulation 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 attractive. 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 aberration of light, the partial
+impulse of luminous waves, magnetic polarisation,
+and Zeeman's experiment.
+
+A few objections still remained. The phenomena
+of an electric system 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.~Crémieu, M.~Lippman 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.
+``No!'' replied the partisans of Lorentz. ``What
+we could measure in that way is not their
+\PageSep{244}
+absolute velocity, but their relative velocity \emph{with
+respect to the ether}, so that the principle of relativity
+is safe.'' Whatever there may be in these
+objections, the edifice of electro-dynamics seemed,
+at any rate in its broad lines, definitively constructed.
+Everything was presented under the
+most satisfactory aspect. The theories of Ampère
+and Helmholtz, which were made for the open
+currents that no longer existed, seem to have no
+more than purely historic interest, and the inextricable
+complications to which these theories
+led have been almost forgotten. This quiescence
+has been recently disturbed by the experiments of
+M.~Crémieu, which have contradicted, or at least
+have seemed to contradict, the results formerly
+obtained by Rowland. Numerous investigators
+have endeavoured 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 book is ready for the press and the day on
+which it is placed before the public.
+\begin{center}
+\vfill
+\footnotesize THE END.
+\vfill\vfill
+\rule{1in}{0.5pt} \\[4pt]
+\makebox[0pt][c]{\scriptsize THE WALTER SCOTT PUBLISHING CO., LIMITED, FELLING-ON-TYNE.}
+\end{center}
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+Output written on 37157-t.pdf (306 pages, 994390 bytes).
+PDF statistics:
+ 1752 PDF objects out of 2073 (max. 8388607)
+ 453 named destinations out of 1000 (max. 131072)
+ 230 words of extra memory for PDF output out of 10000 (max. 10000000)
+
diff --git a/37157-t/old/37157-t.zip b/37157-t/old/37157-t.zip
new file mode 100644
index 0000000..554f9f9
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+++ b/37157-t/old/37157-t.zip