LIHKAKY OF THK l.MVI-KSITY OF CAUFOKMA. HYSICS DEPART. MI-. NT GIR" OP MISS ROSE WHITINii. Accession No. September, 1896. Class No. :S4<-:- ife^^i^ggS frttSdR THE INTERNATIONAL SCIENTIFIC SERIES. VOLUME XXXVIII. THE INTERNATIONAL SCIENTIFIC SERIES. EACH BOOK COMPLETE IN ONE VOLUME, 12MO, AND BOUND IN CLOTH. I. FOEMS OF W ATEE : a Familiar Exposition of the Origin and Phenomena of Glaciers. By J. TYNDALL, LL. D., F. E. S. With 25 Illustrations. $1.50, II. PHYSICS AND POLITICS ; Or, Thoughts on the Application of the Principles of "Natural Selection" and "Inheritance" to Political Society. By WALTER BAGEHOT. $1.50. III. FOODS. By EDWARD SMITH, M. D., LL. B., F. E. S. With numer- ous Illustrations. $1.75. IV. MIND AND BODY: The Theories of their Eolation. By ALEX- ANDER BAIN, LL. D. With Four Illustrations. 1.50. V. THE STUDY OF SOCIOLOGY. By HERBERT SPENCER. $1.50. VI. THE NEW CHEMISTRY. By Professor J. P. COOKE, of Harvard University. With 31 Illustrations. $2.00. VII. ON THE CONSEEVATION OF ENEEGY. By BALFOTTR STEWART, M. A., LL. D., F. E. S. With 14 Hlustrations. $1.50. VIII. ANIMAL LOCOMOTION ; or, Walking, Swimming, and Flying. By J. B. PETTIGREW, M. D., F.E.S., etc. With 130 Illustra- tions. $1.75. IX. RESPONSIBILITY IN MENTAL DISEASE. By HENRY MAUDS- LEY, M.D. $1.50. X. THE SCIENCE OF LAW. By Professor SHELDON AMOS. $1.75, XI. ANIMAL MECHANISM : A Treatise on Terrestrial and Aerial Locomotion. By Professor E. J. MAREY. With 117 Illustra- tions. $1.75. New York: D. APPLETON & CO., 1, 3, & 5 Bond Street 2 The International /Scientific Series. (Continued.) XII. THE HISTORY OF THE CONFLICT BETWEEN RELIGION AND SCIENCE. By J. W. DRAPER, M. D., LL. D. $1.75. XIH. THE DOCTRINE OF DESCENT AND DARWINISM. By Professor OSCAK SCHMIDT (Strasburg University). With 26 Illustrations. $1.50. XIV. THE CHEMICAL EFFECTS OF LIGHT AND PHOTOG- RAPHY. By Dr. HERMANN VOGEL (Polytechnic Academy of Berlin). Translation thoroughly revised. With 100 Hlus- trations. $2.00. XV. FUNGI : Their Nature, Influences, Uses, etc. By M. C. COOKE, M. A., LL. D. Edited by the Rev. M. J. BERKELEY, M. A., F. L. S. With 109 Illustrations. $1.50. XVI. THE LIFE AND GROWTH OF LANGUAGE. By Professor WILLIAM DWIGHT WHITNEY, of Yale College. $1.50. XVII. MONEY AND THE MECHANISM OF EXCHANGE. By W. STANLEY UEVONS, M. A., F. R. S. $1.75. XVIII. THE NATURE OF LIGHT, with a General Account of Physi- cal Optics. By Dr. EUGENE LOMMEL. With 188 Illustrations and a Table of Spectra in Chromo-lithography. $2.00. XIX. ANIMAL PARASITES AND MESSMATES. By Monsieur VAN BENEDEN. With 83 Illustrations. $1.50. XX. FERMENTATION. By Professor SCIIUTZENBERQER. With 2S Illustrations. $1.50. XXI. THE FIVE SENSES OF MAN. By Professor BERNSTEIN. With 91 Illustrations. $1.75. XXII. THE THEORY OF SOUND IN ITS RELATION TO MUSIC. By Professor PIETRO BLASERNA. With numerous Illustra- tions. $1.50. New York: D. APPLETON & CO., 1, 3, & 5 Bond Street. The International Scientific Series. (Continued.) XXIII. STUDIES IN SPECTRUM ANALYSIS. By J. LOCKYER, F. E. S. With Six Photographic Illustrations of Spectra, and numerous Engravings on Wood. $2.50. XXIV. A HISTOEY OF THE GEOWTH OF THE STEAM-EK- GINE. By Professor E. H. THUBSTON. With 163 Illustra- tions. $2.50. XXV. EDUCATION AS A SCIENCE. By ALEXANDER BAIN, LL. D. $1.75. XXVI. STUDENTS' TEXT-BOOK OF COLOE, OR, MODERN CHRO- MATICS. With Applications to Art and Industry. By Pro- fessor OGDEN N. EOOD, Columbia College. New edition. With 130 Illustrations. $2.00. XXVII. THE HUMAN SPECIES. By Professor A. DE QUATREFAGEB, Membre de 1'Institut. $2.00. XXVIII. THE CEAYFISH : an Introduction to the Study of Zoology. By T. H. HUXLEY, F. E. S. With 82 Illustrations. $1.75. XXIX. THE ATOMIC THEOEY. By Professor A. WURTZ. Trans- lated by E. CLEMINSHAW, F. C. S. $1.50. XXX. ANIMAL LIFE AS AFFECTED BY THE NATUEAL CONDITIONS OF EXISTENCE. By KARL SEMPER. With Two Maps and 106 Woodcuts. $2.00. XXXI. SIGHT : An Exposition of the Principles of Monocular and Bi- nocular Vision. By JOSEPH LE CONTE, LL. D. With 132 Illustrations. $1.50. XXXII. GENEEAL PHYSIOLOGY OF MUSCLES AND NEEVES. By Professor J. EOSENTHAL. With 75 Illustrations. $1.50. XXXIII. ILLUSIONS: A Psychological Study. By JAMES SULLY. fcl.50, XXXIV. THE SUN. By C. A. YOUNG, Professor of Astronomy in the College of New Jersey. With numerous Illustrations. $2.00. New York : D. APPLETON & CO., 1, 3, & 5 Bond Street. 4 The International Scientific Series. (Continued.) XXXV. VOLCANOES : What they Are and what they Teach. Bj JOHN W. JUDD, F. R. S., Professor of Geology in the RoyaJ School of Mines. With 96 Illustrations. $2.00. XXXVI. SUICIDE: An Essay in Comparative Moral Statistics. By HENRY MORSELLI, M. D., Professor of Psychological Medi- cine, Royal University, Turin. $1.75. XXXVII. THE FORMATION OF VEGETABLE MOULD, THROUGH THE ACTION OF WORMS. With Observations on their Habits. By CHARLES DARWIN, LL. D., F. R. S., author of "On the Origin of Species," etc., etc. With Illustrations. $1.50. XXXVIII. THE CONCEPTS AND THEORIES OF MODERN PHYS- ICS. By J. B. STALLO. $1.75. XXXIX. THE BRAIN AND ITS FUNCTIONS. By J. LTJYB. $1.50. XL. MYTH AND SCIENCE. An Essay. By TITO VIGNOLI. $1.50. XLI. DISEASES OF MEMORY : An Essay in the Positive Psy- chology. By TH. RIBOT, author of " Heredity." From the French, by WILLIAM HUNTINGTON SMITH. $1.50. XLII. ANTS, BEES, AND WASPS. A Record of Observations on the Habits of the Social Hymenoptera. By Sir JOHN LUB. BOCK, Bart., F. R. S., D. C. L., LL. D., etc. $2.00. XLIII. SCIENCE OF POLITICS. By SHELDON AMOS, author of " Science of Law." $1.75. XLIV. ANIMAL INTELLIGENCE. By GEORGE J. ROMANES. $1.75. XLV. MAN BEFOEE METALS. By N. JOLT, Correspondent of the Institute. With 148 Illustrations. $1.75. XLVI. THE OEGANS OF SPEECH AND THEIR APPLICA- TION IN THE FORMATION OF ARTICULATE SOUNDS. By GEOEG HERMANN VON MEYER, Professor in Ordinary of Anatomy at the University of Zurich. With 47 Woodcuts. $1.75. New York : D. APPLETON & CO., 1, 3, & 5 Bond Street. THE INTERNATIONAL SCIENTIFIC SERIES. THE CONCEPTS AND THEORIES OF MODERN PHYSICS. % BY J. B. STALLO. NEW YORK: B. APPLETON AND COMPANY, 1, 3, AND 5 BOND STREET. 1884, PHYSICS DEPT. COPYRIGHT, 1881, 1884, BY D. APPLETON AND COMPANY. OF DR. WALTER STALLO. summa tamen omnia constant. " Inficitur autem intellects humanus ex intuitu eorum, quae in artibus mecJianicis fiunt, in quibus corpora per compositiones aut separationes utplurimum alterantur ; ut cogitet, simile quiddam etiam in naturd rerum universali fieri. Unde fiuxit commentum illud elementorum deque illorum concursu, ad constituenda corpora naturalia" BAOO VEEUL., Nov. OBG., LIB. I, APH. 66. U1TI7ERSIT7 PEEFACE. THE following pages are designed as a contribution, not to physics, nor, certainly, to metaphysics, but to the theory of cognition. Their contents are the result of a somewhat careful study of the true relation of the physical sciences to the general progress of human knowledge. It is the common opinion of contempo- rary physicists that there was a total breach of continu- ity in the line of this progress at the point where the thoughts of men were turned from ancient and mediae- val traditions respecting the phenomena of nature and their significance to the order and sequence of these phenomena as disclosed by their own observations and experiments, and that the structure of what may, for want of a better name, still be called philosophy now rests upon foundations wholly different from those upon which it stood before the days of Galilei and Bacon. According to this view, Bacon's demand (in the pref- ace to his Novum Organum) " that the whole work of the mind be undertaken anew " ut opus mentis uni- versum de Integra resumatur has been thoroughly complied with, and Newton's admonition to the physi- cists, " to beware of metaphysics," has been effectually heeded. The belief is that modern physical science has not only made its escape from the cloudy regions of metaphysical speculation, and discarded its methods 8 PREFACE. of reasoning, but that it lias likewise emancipated itself from the control of its fundamental assumptions. It is my conviction that this belief is but partially conform- able. to the fact, and that the prevailing misconceptions in regard to the true logical and psychological premisses of science are prolific of errors, whose reaction upon the character and tendencies of modern thought becomes more apparent from day to day. The shallow and scio- listic materialism I allude, of course, not to its sup- posed ethical but to its purely intellectual aspects which for a time threatened to blight the soil and poi- son the atmosphere even of the old highlands of thought on the continent of Europe, claims to be a presentation of conclusions from the facts and principles established in the. several departments of physical science. It is part of my endeavor to meet this claim by an examina- tion of the fundamental concepts and general theories .of that department of physical science which is, in a sense, the basis and support of all its other departments the department of physics. It will be seen at once, upon a most cursory glance at any one of the chapters of this little book, that it is in no wise intended as an open or covert advocacy of a return to metaphysical methods and aims ; but that, on the contrary, its ten- dency is throughout to eliminate from science its latent metaphysical elements, to foster and not to repress the spirit of experimental investigation, and to accredit in- stead of discrediting the great endeavor of scientific research to gain a sure foothold on solid empirical ground, where the real data of experience may be re- duced without ontological prepossessions. An attentive perusal of these pages will make it clear, I think, that this endeavor is continually thwarted by the insidious intrusion into the meditations of the man of science PREFACE. 9 of the old metaphysical spirit. This fact having been established, it was incumbent on me to ascertain, if possible, its causes and, within the narrow limits at my command, to develop its consequences. In the per- formance of this task it became necessary inasmuch as I wrote for a class of readers with whom, unfortu- nately, familiarity with the laws of thought is a some- what rare accomplishment to make an excursion into the domain of logic, and to enter upon a brief discus- sion of the theory of conception. This discussion is, of necessity, very perfunctory, but I venture to hope that it will not prove wholly devoid of interest even to those who are thoroughly familiar with the subject. Fur- thermore, the atomo-mechanical theory, which is sup- posed to be the only and all-sufficient basis of the sci- ence of physics, has become complicated with, or, rather, has led to, certain remarkable speculations as to the nature and properties of space ; and this necessitated another excursion into the field of mathematics, for the purpose of examining the validity of the doctrines of what is generally known as transcendental geometry with its hypotheses of non-homaloidal space and of space of more than three dimensions. What is here presented is not, of course, a new the- ory of the universe, or a novel system of philosophy. I have undertaken, not to solve all or any of the problems of cognition, but simply to show that some of them are in need of being stated anew so as to be rationalized, if not deepened. It is an old truth, which, however, is too often lost sight of, that many of the questions of science and philosophy remain unanswered, not by rea- son of the insufficiency of our knowledge, but because the questions themselves are founded on erroneous as- sumptions and require answers in irrational or impos- 10 PREFACE. sible terms. The utter anarchy which notoriously pre- vails in the discussion of ultimate scientific questions, so called, indicates that a determination of the proper attitude of scientific inquiry toward its objects is the most pressing intellectual need of our time, as it is an indispensable prerequisite of real intellectual progress at all times. And such a determination, however par- tial, is in itself a decided advance in the direction of our legitimate cognitive aspirations. " Rightly to pro- pose a problem," says "Whewell, " is 110 inconsiderable step to its solution." In the language of Kant : "Ifis ist sohon ein grosser und noethiger Beweis der Kluglieit und Einsicht zu wissen, was man vernuenftiger Weise fragen solle" And in the pithy phrase of Bacon : " Prudens interrogatio quasi dimidium scientiae" My views respecting the actual state of physical science and the value of many of the current theoretical interpretations of scientific facts are, no doubt, at vari- ance with the tenets of many distinguished scientific men. That I have, nevertheless, given fearless expres- sion to them will not, I hope, be construed as a want of appreciation of the merits of those to whose labors mod- ern culture owes its life, and the pursuit of knowledge in the interest of that culture its practical success. And, if it should be regarded as evidence of presumption, I desire to say that there are suggestions, in many of the utterances of the men of science here referred to, of a growing sense of the questionability of some of the ele- ments of their scientific faith. I have taken frequent occasion, in the progress of my discussion, to point to these suggestions, to the end of showing that my thoughts are, after all, but the inevitable outcome of the tendencies of modern science, and are, therefore, rather "partus temporis quam ingenii" PREFACE. 11 I deem it important to have it understood, at the outset, that this treatise is in no sense a further exposi- tion of the doctrines of a book (" The Philosophy of Nature," Boston, Crosby & Nichols, 1848) which I published more than a third of a century ago. That book was written while I was under the spell of Hegel's ontological reveries at a time when I was barely of age and still seriously affected with the metaphysical malady which seems to be one of the unavoidable dis- orders of intellectual infancy. The labor expended in writing it was not, perhaps, wholly wasted, and there are things in it of which I am not ashamed, even at this day ; but I sincerely regret its publication, which is in some degree atoned for, I hope, by the contents of the present volume. It ought to be added that parts of the seventh and eleventh chapters of this book, and a few sentences in the other chapters, were published in " The Popular Science Monthly " in October, November, and Decem- ber, 1873, and January, 1874. J. B. STALLO. CINCINNATI, September 1, 1881. CONTENTS. PAG3 INTRODUCTION AND PREFACE TO THE SECOND EDITION i CHAPTER I. Introductory 15 CHAPTER II. First Principles of the Mechanical Theory of the Universe . . 26 CHAPTER III. The Proposition that the Elementary Units of Mass are equal . 30 CHAPTER IV. The Proposition that the Elementary Units of Mass are absolutely hard and inelastic 40 CHAPTER V. The Proposition that the Elementary Units of Mass are absolutely inert 52 CHAPTER VI. The Proposition that all Potential Energy is in Reality kinetic? Evolution of the Doctrine of the Conservation of Energy . . 66 CHAPTER VII. The Theory of the Atomic Constitution of Matter .... 84 CHAPTER VIII. The Kinetic Theory of Gases. Conditions of the Validity of Scien- tific Hypotheses 104 14 CONTENTS. CHAPTER IX. PAGE The Relation of Thoughts to Things. The Formation of Concepts. Metaphysical Theories 129 CHAPTER X. Character and Origin of the Mechanical Theory. Its Exemplifica- tion of the First and Second Radical Errors of Metaphysics . 148 CHAPTER XI. Character and Origin of the Mechanical Theory (continued). Its Exemplification of the Third Radical Error of Metaphysics . 171 CHAPTER XII. Character and Origin of the Mechanical Theory (continued). Its Exemplification of the Fourth Radical Error of Metaphysics . 183 CHAPTER XIII. The Theory of the Absolute Finitude of the World and of Space. The Assumption of an Absolute Maximum of Material Existence as a Necessary Complement to the Assumption of the Atom as its Absolute Minimum. Ontology in Mathematics. The Reifi- cation of Space. Modern Transcendental Geometry. Non- homaloidal (Spherical and Pseudo-spherical) Space . . . 207 CHAPTER XIY. Metageometrical Space in the Light of Modern Analysis. Rie- mann's Essay . . . . ... . . .248 CHAPTER XV. Cosmological and Cosmogenetic Speculations. The Nebular Hy- pothesis 270 CHAPTER XVI. Conclusion , 294 IWTEODUOTIOS" PREFACE TO THE SECOND EDITION. THE present new edition of this volume is sub- stantially a reprint of the first. I have made a brief addition to the fifteenth chapter, taking note of G. H. Darwin's investigations respecting the influence of the solar tides upon the rotation of the planets, as bearing upon the anomalies observed in the periods of revolu- tion of the satellites of Mars ; but beyond this I have confined myself to the correction of a number of typo- graphical errors, and of a few verbal inaccuracies. It was at one time my intention to expand some of the early chapters, by extending the considerations therein presented to other physical and chemical details, and to supplement the ninth, tenth, eleventh, and twelfth chap- ters, by a discussion of certain general topics, such as the concepts of substance and cause as commonly dealt with in physical science. But I found that it was im- practicable to carry this intention into effect without transcending the limits, if not of my subject, at least of the space at my command. And there being nothing in the English and American criticisms of the book, so far as they have fallen under my notice, which, in my judgment, requires a serious revision of any part of its contents, I have concluded to republish it in its origi- ii CONCEPTS OF MODERN PHYSICS. nal form. Nevertheless, I avail myself of this oppor- tunity to reply to some of the criticisms alluded to, chiefly to the end of removing a curious misapprehen- sion which is common to nearly all my critics, but inci- dentally also of meeting certain objections that are more or less founded upon or connected with it. I. The misapprehension I speak of is very surprising, in view of the explicit declaration, contained in the very first sentence of my preface, that the book is " de- signed as a contribution not to physics, nor certainly to metaphysics, but to the theory of cognition." Not- withstanding this declaration, most of my critics assume it to be my purpose to expose the shortcomings and de- fects of particular physical theories as devices for the colligation of facts, or as instruments of research, and suppose that my endeavor is simply, as one of my crit- ics expresses it, " to pick flaws in these theories," or, in the language of another critic, " to crassify and develop contradictions " between them, to " set facts by the ears," and " bump friendly heads together " in short, in the spirit of a sort of scientific pyrrhonism, to dis- credit the familiar methods of physical science, if not to invalidate its results. And they complain that I fail to apprehend what one of them is pleased to term the " laboratory function " of a physical theory or hypothe- sis, and to appreciate the distinction between a " work- ing hypothesis " and a theory advanced with the claim of its final validity or truth. Now, the fact is that, for the purposes of the inquiry to which my book is devoted, I am not directly con- cerned with the "laboratory function" of "working hypotheses " or physical theories at all. My object is to INTRODUCTION TO THE SECOND EDITION. m consider current physical theories and the assumptions which underlie them in the light of the modern theory of cognition a theory which has taken its rise in very recent times, and is founded upon the investigation, by scientific methods analogous to those employed in the physical sciences, of the laws governing the evolution of thought and speech. Among the important truths developed by the sciences of comparative linguistics and psychology are such as these : that the thoughts of men at any particular period are limited and controlled by the forms of their expression, viz., by language (using this term in its most comprehensive sense) ; that the lan- guage spoken and " thought in " by a given generation is to a certain extent a record of the intellectual activity of preceding generations, and thus embodies and serves to perpetuate its errors as well as its truths ; that this is the fact hinted at, if not accurately expressed, in the old observation according to which every distinct form or system of speech involves a distinct metaphysical theory ; that the metaphysical systems in vogue at any particular epoch, despite their apparent differences and antagonisms, on proper analysis are found to be charac- terized by certain common features in which the latent metaphysics of the language in which such systems have originated, or are presented, are brought to view ; that philosophers as well as ordinary men are subject to the thralldom of the intellectual prepossessions em- bodied in their speech as well as in the other inherited forms of their mental and physical organizations, and are unable to emancipate themselves from this thralldom otherwise than by slow and gradual advances, in con- formity to the law of continuity which governs all pro- cesses of evolution whatever. It being my belief that all this applies to the votaries of science as well as to iv CONCEPTS OF MODERN PHYSICS. the devotees of metaphysics or ontology, I sought to en- force this belief by an examination of the general con- cepts and theories of modern physics. According to the opinion of contemporary men of science, these con- cepts and theories are sinTply generalizations of the data of experience, and are thus not only independent of the old a priori notions of metaphysics, but destructive of them. But, although the founders of modern physi- cal science at the outset of their labors were animated by a spirit of declared hostility to the teachings of med- ieval scholasticism a fact which is nowhere more con- spicuous than in the writings of Descartes neverthe- less, when they entered upon the theoretical discussion of the results of their experiments and observations, they unconsciously proceeded upon the old assumptions of the very ontology which they openly repudiated. That ontology founded upon the inveterate habit of searching for " essences " by the interpretation of words and the analysis of the concepts underlying them, before the relations of words to thoughts and of thoughts to things were properly understood was char- acterized by three great errors : its hypostasis of con- cepts (notwithstanding the protest of the nominalists against the reification of universals); its disregard of the twofold relativity of all physical phenomena ; and its confusion of the order of intellectual apprehension with the order of nature. These errors gave rise to a number of cardinal doctrines respecting the "substance of things," among which were the assertion of its exist- ence as a distinct thing or real entity, apart from its properties ; the further assertion of its absolute perma- nence and immutability ; and, finally, the assertion of the absolute solidity and inertia of its parts and their incapacity to act upon each other otherwise than by con- INTRODUCTION TO THE SECOND EDITION. v tact.* And all these doctrines lie at the base, not only of Cartesian physics and metaphysics, but of the scientific creed of the great majority of the physicists of the present day. The eminent physicist and physiologist who declares that " before the differential equations of the world-formula could be formed " [i. e., before the ultimate, true, and exhaustive theory of the universe could be constructed], " all processes of nature must be reduced to the motions of a substratum substantially homogeneous, and therefore totally destitute of quality, of that which appears .to us as heterogeneous matter in other words, all quality must be explained by the ar- rangement and motion of such a substratum," f and the equally distinguished physicist and mathematician who enters upon the attempt at a solution of the problem thus stated by endeavoring to deduce the phenomenal diversities and changes of the universe from imaginary vortical motions of the undistinguishable parts of an as- sumed universal, homogeneous, continuous and incom- pressible fluid, are both as truly instinct with the spirit of the old scientia entis guatenus entis as the most ar- dent disciple of the Stagirite in the times of Erigena or Aquinas. The physicist who insists upon impact theories of gravitation, cohesion, or chemical affinity, * In this connection it may be worth while to direct the attention of our modern " Baconian " physicists to the fact that the proposition, ac- cording to which there can be no physical action without contact, is one of the fundamental doctrines of Aristotle. See the references in Zeller^s Philosophic der Griechen, second ed., J7 2 , p. 2Q8. f " Ehe die Differentialgleichungen der Weltformel angesetzt wcrden koennten, muessten alle Naturvorgaenge auf Bewegungen eines substan- tiell unterschiedslosen, mithin eigenschaf tslosen, Substrates dessen zurueck- gefuehrt sein, was uns als verschiedenartige Materie erscheint, mit andera Worten, alle Qualitaet muesste aus Anordnung und Bewegung solchen Substrates erklaert sein." Du Bois-Reymond, Ueber die Grenzen des Naturerkennens, 2. ed.,p. 5. vi CONCEPTS OF MODERN PHYSICS. has the same intellectual blood in his veins which coursed in those of the old disputants about " first mat- ter " or " substantial forms." When the Professor of Physics in the University of Edinburgh teaches that matter is absolutely passive, dead,* that all physical ac- tion is action by contact, that nothing is real which is not indestructible,f etc., he stands as unmistakably upon scholastic ontological ground as did Descartes or any of his ecclesiastical contemporaries. The proposition of the modern kinematist, that the true explanation of the phenomena of heat, light, electricity, magnetism, etc., consists in their reduction to the elements of matter and motion, differs in little else than its phraseology from the metaphysical theorem that all the " secondary quali- ties " of the universal substance are mere specifications or derivatives of its " primary qualities." Such being the theme of the little book which is now before the reader, it is readily seen that it became incumbent on me to establish two main propositions. The first of these is the proposition just stated and briefly illustrated, that the general principles of the atomo-mechanical theory, which is said to be the basis of modern physics, are substantially identical with the cardinal doctrines of ontological metaphysics ; and the second, that the fundamental errors of ontology become apparent in proportion to the advance of physical sci- ence, inasmuch as the four great assumptions which the atomo-mechanical theory necessarily involves (viz., those of the absolute equality, inertia, and rigidity of the ele- mentary atoms and of the essentially kinetic character of all physical energy) are distinctly and irreconcilably * Cf. The Unseen Universe, 104 > Tait, On some Recent Advances in Physical Science, p. 24, etc. f On Some Recent Advances, etc., pp. 14-17. INTRODUCTION TO THE SECOND EDITION. v ii at war with the widest and most trustworthy experi- ential inductions of physics, chemistry, and astronomy. In the natural order of my discussion, the establishment of the first proposition would have preceded that of the second ; and, when I first undertook to write the book, I actually began to conform to that order by a brief re- view of the history of physics from the time of Des- cartes. A very cursory examination of the Principles, the Discourse on Method, and parts of the Meteors, of Descartes, is sufficient to show that, notwithstanding his own sincere belief that his philosophical and scientific doctrines were thoroughly subversive of Aristotelianism and scholasticism, he was at bottom as thorough a scho- lastic ontologist as the doctors of the Sorbonne whose favors and indulgence he openly courted while he pri- vately despised them. And an equally cursory glance at the history of theoretical physics since Descartes's day reveals the fact that the persistent denial of the possibility of actio in distans and the incessant renewal of the attempts at kinetic explanations of gravity, co- hesion, affinity, electric and magnetic attraction, etc., are neither more nor less than recrudescences of ontological Cartesianism. In the progress of my work I became apprehensive, however, lest my scientific readers should be deterred from perusing my pages by the quasi-meta- physical aspect of the introductory parts. So I con- cluded to reverse the order of the argument, and to present the second propositon first. It is not improba- ble that this was unwise. For, on the one hand, it has , led to the strange consequence that the task of review- ing the book has generally been assigned to specialists, who not only abhor metaphysics, but regard as meta- physical everything which does not present itself in the guise of a differential equation or of an atomic formula 2 viii CONCEPTS OF MODERN PHYSICS. so that the editor of Mind, for instance, has enthroned in the critical chair a learned gentleman who declares that he does not venture an opinion on what he terms " the more strictly metaphysical part " of the book, and the whole tenor of whose discussion shows that he is unaware even of the existence of the science to which I attempted to make a contribution, although this sci- ence is repeatedly referred to in the very number of Mind, in which his criticism appears, under the name of epistemology ; and, on the other hand, it has brought upon me the charge of being a mere destructionist who fails to understand the laboratory functions of physical theories, and who is intent upon subverting these the- ories without offering or suggesting others to replace them. If the foregoing exposition has properly served to define the purpose and scope of my inquiry, the reader will have no difficulty in seeing that the strictures of my critics upon my supposed ignorance of the distinc- tion between working hypotheses and ultimate theories, and of the " radically different tests of logical cano- nicity " to be applied to them, as well as their animad- versions upon my inordinate stickling for " exiguous con- sistency " between the parts of a theory in preference to " its flexile and serviceable adaptability to facts of many and diverse orders," etc., etc., are as irrelevant as they are unfounded. The tone in which these strictures are presented, as though they were in any sense refuta- tions of the general argument of my book, affords curi- ous evidence of the confusion which runs riot among the theoretical speculations of modern physicists. For purposes of practical scientific research the chief value of hypotheses or provisional theories lies, no doubt, in their capacity to effect a momentary fusion of experi- INTRODUCTION TO THE SECOND EDITION. i x mental results ; and the physicist or chemist in his labo- ratory may, with a certain degree of safety, either wholly ignore the question of their ultimate validity, or content himself with placing it in the distant perspective. The primary relation of these hypotheses or theories to sci- entific research is analogous to that of language to the ordinary operations of thought : they serve to fix and record the results of experiment and observation. But, generally speaking, hypotheses are more than mere ar- bitrary and artificial devices for the enchainment and classification of facts. They are in most cases guesses at the ultimate truth suggested by the analogies of ex- perience, and are primarily used as working hypotheses only in the sense that they afford a basis for further ex- periment and observation whereby their ultimate valid- ity is to be established or overthrown. In the progress of the various attempts at their verification they are al- most always modified and transformed, so as to bring them into conformity with the facts. And not unfre- quently these transformations are different in different departments of scientific investigation, in each of which the hypotheses are tested by different methods, and confronted with different orders of facts. The result is, that in many cases not only serious discrepancies but radical inconsistencies are developed between the sev- eral forms into which the hypotheses are forced on dif- ferent lines of research. A very good illustration of this is afforded by the hypothetical aether, which has played a part more or less conspicuous in physical as- tronomy, in ordinary physics, and in chemistry. By the astronomers, this sether was originally regarded as a fluid of extreme tenuity and mobility, offering no sen- sible resistance to the movements of celestial bodies ; and the question of its continuity or discontinuity was X CONCEPTS OF MODERN PHYSICS. not seriously mooted. Its main function in modern astronomy has been to serve as a basis for hydrodynam- ical theories of gravitation. In physics, this fluid for some time appeared in several roles in connection with the " imponderables," some physicists going so far as to identify it with one or more of them. But since the promulgation of the kinetic theories of these imponder- ables, and especially of the dynamical theories of heat, it has been in requisition chiefly in optics as a substratum for luminar undulations. And here, to account for the dispersion of light, physicists came to insist upon its atomic or molecular structure, finding it necessary to assume that the particles of the aether were separated by finite intervals bearing a sensible ratio to the length of a luminar wave. Moreover, they had to endow it with an enormous elasticity, so that its resistance to de- formation far exceeded that of the most rigid-elastic bodies. But presently, in other departments of phys- ics, the admission of the molecular or atomic constitu- tion of the aether led to consequences subversive of a number of well-ascertained facts, to some of which I have referred in my seventh chapter. And in chemis- try, too, it was found impossible to concede the enor- mous elasticity of the aether without depriving it of those properties upon which its serviceableness in the construction of chemical theories mainly depended. Furthermore, the exigencies of the atomo-mechanical theory have led distinguished mathematicians and physi- cists to attempt a substitution of peculiar forms of vor- tical motion in a universal, homogeneous, incompressi- ble, and continuous material medium, which (unless the attribute of impenetrability is to be dismissed from the concept of matter) must, of course, be identical with the all-pervading aether. INTRODUCTION TO THE SECOND EDITION. x i Now, when two hypotheses are radically inconsist- ent with each other, one or both of them must eventu- ally be discarded. It is true that in certain cases hypotheses which have proved to be untenable, may continue to be serviceable as " working hypotheses," in a secondary sense, as mere devices for holding together facts which have been collected by means of or with reference to them. So long as they are employed for this purpose alone, and with the clear understanding that they are not propounded for any other, there can be no serious objection to their use. But it is other- wise w r hen the specialist seeks to obtrude his own par- ticular hypothetical figment as a finality upon science generally, and to make it the basis of assertions respect- ing the ultimate constitution of things, and the uni- versal order of nature. It must not be forgotten that the several departments of science are simply arbitrary divisions of science at large, and that their extent and limits are representative of nothing more than the ne- cessities and conveniences of the division of labor. In these several departments the same physical object may be considered under different aspects. The physicist may study its molecular relations, while the chemist de- termines its atomic constitution. But when they both deal with the same element or agent, it can not have one set of properties in physics, and another set contra- dictory of them in chemistry. If the physicist and chemist alike assume the existence of ultimate atoms absolutely invariable in bulk and weight, this atom can not be a cube or oblate spheroid for physical, and a sphere for chemical purposes. And a group of constant atoms can not be an aggregate of extended and abso- lutely inert and impenetrable masses in a crucible or retort, and a system of mere centers of force as part of xii CONCEPTS OF MODERN PHYSICS. a magnet or of a Clamond battery. The universal aether can not be soft and mobile to please the chemist, and rigid-elastic to satisfy the physicist ; it can not be continuous at the command of Sir William Thomson, and discontinuous on the suggestion of Cauchy or Fresnel.* I am told that scientific men are fully aware of the provisional and tentative character of their theories, and * Since writing the above I have met with an admirable passage re- lating to the same subject in an article contributed by the eminent physi- cist, G. A. Him, to the forty-third volume of the Memoir cs dc V Academic Royale de Belgique, in which he says : " Lorsqu'on voit 1'assurance avec laquelle s'affirment aujourd'hui les doctrines qui rapportent aux seuls mouvements de 1'atome 1'ensemble, 1'universalite des phenomenes, on est en droit de s'attendre a ce qu'il y ait unanimite aussi sur les qualites qu'on assigne a cet etre unique, fon- dement de toute existence. Or des le premier examen des systemes par- ticuliers proposes, on eprouve la plus 4trange deception ; on s'aper9oit que 1'atome du chimiste, celui du physicien, celui du metaphysicicn, celui du mathematicien . . . n'ont absolument de commun que le nom ! Re- sultat inevitable du morcellement actuel de nos sciences, chacun, dans son easier, se fabrique un atome qui satisfait aux exigences des phe- nomenes qu'il etudie, sans s'inquieter le moms du monde des exigences propres aux phenomenes du easier voisin. Le metaphysicien bannit les principes de 1'attraction, de la repulsion, comme des reves : le mathema- ticien, qui analyse les lois de Pelasticite", celles de la propagation de la lumiere, les admet implicitement, sans meme les nommer (jamais on n'cxpliquera la propagation longitudinale des vibrations transversales ad- mises en optique, sans rendre les atomes vibrants solidaires par quelque chose d'autre que les atomes eux-me'mes). Le chimiste ne peut expliquer le groupement des atomes dans ses molecules souvent si compliquees, sans attribuer a ses atomes des qualit4s specifiques qui les distinguent : pour le physicien et le metaphysicien, partisans des doctrines modcrncs, f atome est, au contraire, toujours et partout le meme. Que dis-je ! on n'est pas plus d'accord dans une seule et meme science sur les qualite's de 1'atome. Chacun le construit a sa guise pour 1'expli cation de tel phe- nomene restreint dont il s'occupe en particulier." (Recherches experi- mentales sur la relation qui existe entre la resistance de Vair et sa tempera- ture, p. 68.) INTRODUCTION TO THE SECOND EDITION. xiii carefully refrain from making assertions respecting their ultimate validity. This is, no doubt, true as to a great number of minor hypotheses ; but it is far from being true as to those theories and concepts which are the subject of my discussion. If any proof of this were necessary, it would be afforded by the violence with which my " assaults " upon the atomic theory and the kinetic theory of gases, and upon the current misinter- pretation of the principle of inertia which Kes at the base of the constantly recurring phrase of dead matter, etc., have been repelled. So thoroughly axiomatic have the doctrines of the absolutely independent and passive existence of matter, and of the constitution of bodies as aggregates of absolutely constant physical units, become in the minds of modern physicists, that they not only regard them as the indispensable foundations of the whole structure of physical science, but do not hesitate to use them as supports for professorial chairs of meta- physics, from which they promulgate doctrines like those set forth in the " Unseen Universe," or of pulpits from which they preach dogmatic theology. And no thoughtful reader of Du Bois-Heymond's " Grenzen des Naturerkennens" or his " Sieben Weltraethsel" can be in any doubt as to the degree of certitude ascribed by the highest scientific authorities to the fundamental principles of the atomo-mechanical theory. There are, of course, men of science who do not share this confi- \ dence in the absolute and final truth of the theories in j question. But the great majority of them deem it a I sufficient reply to all suggestions respecting their rela- | tive or provisional value to point with pride to the enormous success which has attended physical research since their adoption. In view of this it may be well to i j examine for a moment to what this success is really due. xiv CONCEPTS OF MODERN PHYSICS. All knowledge results from the establishment of relations between phenomena. And systematic knowl- edge, science, results from the establishment of definite relations between phenomena distinctly and clearly ap- prehended. One of the great vices of the old meta- physical mode of reasoning was, that it operated with vague and abstract concepts, without direct reference to distinct phenomena. Hence the conclusions of meta- physics were as confused, indefinite, and fallacious as its premisses. In contrast to this, the empirical method, proceeding from the observation of particular facts, and basing its cautious and slow progress upon the identifi- cation of new phenomena with those previously and familiarly known, avoids this vagueness and confusion. It is in constant sight, and under perpetual control of facts, which act as ever-present checks upon the vaga- ries of speculation. In a certain sense, the empirical method of science conforms to the natural methods of ordinary thought and speech. It is a well-known say- ing of Aristotle that all thought is a succession of pict- ures ; and it is a theorem of modern linguistics that forms of speech are a succession of metaphors. The difference between the concepts and theories embodied in the words of a language, and the concepts and theo- ries of science, lies in this, that the former are grounded upon superficial, partial, and often fanciful analogies ; whereas the latter are based upon wide generalizations, or upon classifications according to resemblances and correspondences evidencing essential and truly signifi- cant relations, such as those of origin, structure, func- tional equivalence, and the like. Now, both the forma- tion of concepts and words in the evolution of ordinary thought and speech, and the construction of physical theories, are liable to the great structural fallacy which INTRODUCTION TO THE SECOND EDITION. xv I have pointed out in the eleventh chapter, and which is due to the fact that, by comparing and partially or . wholly identifying phenomena that are new and strange with those that are old and familiar, the first crude and superficial apprehensions of fact are made both the starting-points and the goals of scientific explanation. It is this fallacy which lies at the bottom of the demand of Professor Lodge and Mr. MacAlister (to which I shall have to refer hereafter), "to see the traces" by which the sun's pull upon the planets is exerted. Scientific theorists, no less than ordinary men, incessantly lose sight of the circumstance that, while it is true that the advance of knowledge depends upon the progressive identification of phenomena, so that the most compre- hensive theory, the most exhaustive analysis, and the profonndest insight can never yield anything more than ultimate facts in which sense it has been well said that all science is a classification of mysteries ; never- theless, it does not follow that the ultimate fact, which lies at the point of convergence of the several lines of identification, is necessarily the fact first observed in the infancy of human intelligence ; but that, on the contrary, the highest generalizations and the profound- est theories lead to typical facts which are generally so obscure and difficult of apprehension, that they are brought to light only on the later stages of intellectual development. But this very fallacy attending the ha- bitual explanation of unfamiliar facts, by reference to, and identification with, old and familiar facts, is to a certain extent inseparable from the practical serviceable- ness of ordinary physical theories. It is the indispensa- ble prerequisite of that incessant play of the "scientific imagination," upon which Professor Tyndall discourses with so much eloquence. The scientific imagination. xvi CONCEPTS OF MODERN PHYSICS. no less than the ordinary phantasy, is restricted to the elements furnished by sensible experience, which it may group and combine, but can not create. Its play, there- fore, is vivid, and, as an aid to the prosecution of phys- ical research (which depends largely upon the pursuit of the analogies of experience), effective in proportion to our familiarity with these elements. This very cir- cumstance, however, in most instances, seriously im- pairs the ultimate value of the theories founded upon it. The vivid play of the " scientific imagination " de- pends upon the identification of phenomena with famil- iar facts ; the truth of a theory, on the contrary, upon their identification with recondite, unfamiliar facts, so that the " sensuous picturability," as it has been called, of a physical explanation is generally in inverse ratio to its truth, and the success which has attended the use of a particular hypothesis in the progress of a special scientific inquiry, is by no means a proper measure of its real scientific value. I have already said that the complaints of my critics of my failure to do justice to "working hypotheses" are not only irrelevant, but also unfounded. Indeed, I am wholly at a loss to see what justification there is in the pages of my book for the charge of the reviewer in the New York "Nation," that I fail to comprehend that physical theories are " mere formal, explanatory, didactic devices " ; that " atomism is a symbolical, and later a graphic, system, which might almost be visual- ized illustratively for specific purposes " (so that, as it would seem, there are, in the opinion of our critic, graphic systems that can not be " illustratively visual- ized "), etc., etc. In the seventh and eighth chapters of this volume (pp. 85 and 105-116) the reader will meet with a somewhat careful exposition of the nature and INTRODUCTION TO THE SECOND EDITION. xvii function of a scientific hypothesis ; while the concluding chapter (pp. 295-299) contains an elaborate discussion of the necessity and value of provisional explanatory devices. And on page 101 he will find this language respecting the atomic theory : " The foregoing considerations do not, of course, detract from the merits of the atomic hypothesis as a graphic or expository device as an aid to the repre- sentative faculty in ' realizing ' the phases of chemical or physical transformation. It is a fact beyond dis- pute that chemistry owes a great part of its practical advance to its use, and that the structural formulae founded upon it have enabled the chemist, not merely to trace the connection and mutual dependence of the various stages in the metamorphosis of 6 elements ' and * compounds ' so called, but in many cases (such as that of the hydrocarbon series in organic chemistry) suc- cessfully to anticipate the results of experimental re- search." I hope that no presumption, either of ignorance or of a want of literary taste, arises from the fact that I failed to anticipate the peculiar phraseology in which my critic sees fit to paraphrase my own words, when he speaks of the "flexile and serviceable adaptability [of a theory] to many and diverse orders," notwithstanding their want of " exiguous consistency," and of the " radi- cally different tests of canonicity " (whatever that may mean) to be applied to " a material, a working, and an ultimate hypothesis." * * As exemplifying the humors of modern criticism, it may be men- tioned that my critic in " The Nation," in animadverting upon my treat- ment of the doctrines of pangeometry, has this passage : "... Although he cites Stumpf, he has not grasped the psychological problem involved in a geometry without parallels." From this the reader will, no doubt, infer that my critic had read xviii CONCEPTS OF MODERN PHYSICS. II. I NOW come to the consideration of the objections which have been urged against the premisses and con- clusions of my general argument. The persons by whom these objections are made may be distributed into two classes : Those belonging to the first class ad- mit that the atomo-mechanical theory is generally held to be the basis of modern physics; but they deny, either that the four propositions which I have enumer- ated as necessarily involved in that theory notably the first, second, and fourth are essential parts or con- sequences of that theory, or that the facts established by modern physical research, and the generalizations based upon these facts, are in conflict with them. The objectors of the second class, on the contrary, dispute the general thesis that modern physics profess to be founded on the theory in question. In reference to the objectors of both these classes, but especially those of the first, it is to be said, at the outset, that the professed antagonism of science to meta- physical speculation has led the majority of scientific specialists to assume that the methods and results of empirical research are wholly independent of the con- trol of the laws of thought. They either silently ig- nore, or openly repudiate, the simplest canons of logic, including the laws of non-contradiction and excluded Stumpf s book, and was familiar with its contents. Now, the fact is, that the book in question treats of a subject wholly foreign to the pan- geometrical question, and its author does not make the most distant reference to any of the "problems involved in a geometry without parallels," or to any topic the discussion of which could throw the faint- est light on such a " problem." No one would be more astonished than Dr. Stumpf, if he saw the article in " The Nation," and found himself cited as an authority on the subject of a " geometry without parallels." INTRODUCTION TO THE SECOND EDITION. XIX middle, and resent, with the utmost vehemence, every application of the rule of consistency to their hypotheses and theories. They are apparently without the least suspicion that not only the theoretical evaluation of the data of experience, but experience itself, is impossible, except on the assumption of definite, universal, and inexorable laws of cognition, and that, but for this assumption, the framing of hypotheses and theories is utterly senseless and vain ; and they regard an exami- nation of their hypotheses and theories, in the light of these laws, as an impertinent intrusion of " a priori principles and methods " into the domains of empirical science. Persons of this cast of mind find no difficulty in holding that atoms are absolutely inert, and, at the same time, asserting that these atoms are perfectly elastic; or in maintaining that the physical universe, in its last analysis, resolves itself into "dead" matter and motion, and yet denying that all physical energy is in reality kinetic ; or in proclaiming that all phenomenal differences in the objective world are ultimately due to the various motions of absolutely simple material units, and, nevertheless, to repudiate the proposition that these units are equal. An admirable illustration of all this is afforded by Mr. Donald MacAlister, who, curiously enough, writes the review of my book for Mind, a journal which, ac- cording to the announcement of its title-page, is a " Quarterly Review of Psychology and Philosophy." Mr. MacAlister denies the first two propositions of the atomo-mechanical theory, i. e., those asserting the equal- ity of the elements of mass, and their rigidity. He asks, "on what grounds" these .propositions are ad- vanced " as essential doctrines of modern physics," and points out that the number of scientific men whose xx CONCEPTS OF MODERN PHYSICS. writings I quote in support of them is, after all, com- paratively insignificant. He evidently thinks that the only ground upon which. I assert the propositions in question to be essential parts of the atomo-mechanical theory (not of modern physics ', as my critic has it) is, that they are broached by certain prominent scientific men. " It would be easy," he says, " to name half a dozen, with Maxwell and Thomson at their head, who theorize as if the exact opposite were the truth." Now, Mr. MacAlister will, no doubt, be very much aston- ished when I tell him that this accords precisely with what I say, viz., that these propositions are not held by the great body of scientific men who theorize in the presence of the facts, for the simple reason that the facts are inconsistent with them. What I maintain is, that the majority of physicists hold a general doctrine which I designate as the atomo-mechanical theory, from which these propositions inevitably follow ; but that, when they construct their special theories, either by generalizing the facts of experience, or by framing hy- potheses to account for them, they are constrained to dis- card and repudiate that doctrine. The authorities ad- duced in support of the several " quadrilateral " propo- sitions were cited simply for the purpose of showing that these propositions are recognized as necessary cor- ollaries of the atomo-mechanical theory by clear-headed men of science who do not share the delusion that sci- ence requires, not merely the substitution of empirical research for the old attempts at reaching physical truths by an analysis of ontological concepts, but also the re- jection of the canons of logic. I may say here, incidentally, that the number of citations from the writings of eminent men of science in support of the propositions just referred to, might have INTRODUCTION TO THE SECOND EDITION. xxi been extended. Among those, for instance, wlio assert the equality of the elements of mass, is Professor Emil Du Bois-Reymond, as appears from the passage I have already quoted in this introduction for another purpose. And in France, not only the well-known explicit decla- rations of M. Marignac, but also the ceaseless recurrence among French physicists and chemists of the attempt to exhibit the chemical elements as compounds or allo- tropic forms of some single element, afford striking proof of the logical constraint, resulting from the atomo-mechanical theory, to reduce the primordial atoms to uniformity and equality.* While Mr. MacAlister denies that modern physicists assent to the first and second propositions of the atomo- mechanical theory, he admits that they insist upon the third and fourth. But here he contests my assertion, that these propositions are in conflict with the teachings of the sciences of physics and astronomy. He concedes, indeed, that thus far no attempt to account for gravita- tion, without assuming action at a distance, has met with success ; but he contends that there is promise of such success in the vortex-atom theory of Sir William Thomson, my criticism of which he deems wholly in- adequate and inconclusive, because it is, as he asserts, ontological and based on a priori considerations. Mr. MacAlister's treatment of this subject is so characteristic of the confusion of modern theorists, who insist upon reducing all physical action to impact, that it is, per- haps, not improper to examine it at some length. * One of the curious animadversions of Mr. MacAlister is, that I quote some of the declarations of scientific men respecting the equality of the elements of mass from magazine articles. I am happy to inform him that he may now read the opinions of Professor Wundt in a bulky volume vol. ii cf his " Logik " which has just left the press. xxii CONCEPTS OF MODERN PHYSICS. Mr. MacAlister enforces and illustrates his claim, according to which no theory of gravitation is valid which does not account for it upon the principles of impact or pressure, by the following quotation from a lecture of Professor Lodge : * " If a man explained the action of a horse on a cart by saying that there was an attraction between them, varying as some high direct power of the distance, he would not be saying other than the truth the facts may be so expressed ; but he would be felt to be giving a wretchedly lame explanation, and any one who simply pointed out the traces, would be going much more to the root of the matter. Similarly with the attraction of a magnet for another magnetic pole. To say that there is an attraction as the inverse cube of the distance between them, is true, but it is not the whole truth ; and we should be obliged to any one who will point out the traces, for traces we feel sure there are." The passage here adduced is followed, in Professor Lodge's lecture, by others which Mr. MacAlister does not quote. The lecturer proceeds to say, among other things, that " A pull resolves itself into a push ; to pull a thing toward you, you have to put your finger behind it, and push ; a horse is said to pull a cart, but he is really pushing at the collar ; an en- gine pushes a truck by means of a hook and eye, and so on. There is still the further very important question as to why the parts hang together, and why, when you push one part, the rest follows. Cohesion is a very striking fact, and an explanation of it is much to he desired. I shall have a little more to Bay about it later ; at present we have nothing more than an indication of the direction in which an explanation seems possible." In another place in the same lecture : " Metaphysical arguments, in so far as they have any weight or validity whatever, are unconscious appeals to experience; a person endeavors to find out whether a certain condition of things * " Nature," vol. xxvii, p. 3C4. INTRODUCTION TO THE SECOND EDITION. xxiii is by Lim conceivable, and, if it is not conceivable, he has some prima facie ground for asserting that it probably does not exist. ... If a highly-developed mind, or set of minds, find a doctrine about some comparatively simple and fundamental matter abso- lutely unthinkable, it is an evidence, and is accepted as good evi- dence, that the unthinkable state of things has no existence ; the argument being that, if it did exist, either it, or something not wholly unlike it, would have come within the range of experi- ence. We have no further evidence than this for the statement that two straight lines can not inclose a space, or that three an- gles of a triangle are equal to two right angles." And thereupon, toward the end of his lecture, Pro- fessor Lodge indicates that the explanation of cohesion, as well as of gravity, is to be looked for in the vortex- atom theory of Sir "William Thomson. Now, what is the gist of all this reasoning ? Why is it necessary, in the opinion of Professor Lodge and Mr. MacAlister, to " point out the traces " in account- ing for gravitation? Obviously for no other reason than this, that every true account of a physical phe- nomenon is, in its nature, an exhibition of its identity with some familiar fact of experience. In the language of Newton, the cause to which it is referred must be a vera causa. Actio in distans, according to the claim of Professor Lodge and Mr. MacAlister, is not a fact of familiar experience indeed, it is not, in any proper sense, a fact at all while a pull, by means of a con- tinuous line, the parts of which cohere, or rather a push, by means of a continuous rod, is such a fact. In discussing this, I will not stop to inquire whether it be true or not that distant action is not a familiar fact; or, indeed, whether we have any experience of physical action which, on close examination, does not resolve itself into actio in distans. I will content my- self with inquiring whether or not the elements of the xxiv CONCEPTS OF MODERN PHYSICS. vortex-atom theory are familiar, or even possible, facts of experience. For, if they ,are not, clearly that the- ory is obnoxious to the same criticism which is said to invalidate the assumption of actio in distans. The medium in which the vortex-movements arise is, according to Professor Lodge's own express state- ment,* " a perfectly homogeneous, incompressible, con- tinuous body, incapable of being resolved into simple elements or atoms ; it is, in fact, continuous, not mo- lecular." And, after making this statement, Professor Lodge adds, " There is no other ~body of which we can say this, and hence the properties of the cether must he somewhat different from those of ordinary matter" It appears, then, that the whole vortex-atom theory, which is offered to us as a substitute for the " meta- physical theory " of actio in distans, rests upon the hypothesis of the existence of a material medium which is utterly unknown to experience, and which has prop- erties' somewhat^ different from those of ordinary mat- ter. Hence this theory, instead of being, as is claimed, a reduction of an unfamiliar fact of experience to a familiar fact, is, on the contrary, a reduction of a fact which is perfectly familiar, to a fact which is not only unfamiliar, but wholly unknown, unobserved, and un- observable. Furthermore, the alleged vortical motion of, or rather in, the assumed sethereal medium is, as I have shown on pages 43 and 44 of this volume, impossible, because * " Nature," vol. xxvii, p. 305. f Somewhat different ! The real import of this " somewhat " is, that the medium in question is not, in any intelligible sense, material at all, having none of the properties of matter. All the properties of matter depend upon differences and changes, but the hypothetical aether here defined is not only destitute of differences, but incapable of difference and change. INTRODUCTION TO THE SECOND EDITION. xxv "motion in a perfectly homogeneous, incompressible, and therefore continuous fluid, is not sensible motion." Mr. MacAlister again attempts to break the force of this simple and obvious reflection, by stigmatizing it as " ontological," and (in his ignorance of the meanings of ordinary logical and psychological terms, construing the word sensible as denoting simply what can not be felt) admonishes me that, on proper occasion, "the impact of a vortex-atom stick on a vortex-atom head may quite well give rise to the idea of ' sensible motion ' in the head." But, if he will refer once more to Professor Lodge's lecture, he will find that his authority, in one of the passages just quoted, distinctly sets up the cri- terion of conceivability as a test of probable or possible existence ; and even contends that by this test alone can we establish the truth of a geometrical proposition. What is not thinkable can not, according to the doc- trine of Professor Lodge, be true. In the light of this doctrine let us examine, then, whether or not motion in an absolutely homogeneous, continuous, and incompress- ible medium, is thinkable. Observe : it is not some thing or body distinct from the medium whose motion is to be conceived ; it is motion of and in the medium itself. Now, the con- ception of real (i. e., sensible) motion of necessity in- volves three distinct elements, viz. : a moving thing or body ; a place from which it moves ; and a place to vMch it moves. And these elements or data are pre- requisites to the conception, and, therefore (according to Professor Lodge's own criteria), to the possible re- ality of motion. They must not only be discriminated by some mark or attribute from the medium in which the motion occurs, but they must be so discriminated before the occurrence ef the motion, and independently xxvi CONCEPTS OF MODERN PHYSICS. of it ; their identification can not be the result of the motion itself. But all this is expressly negatived or precluded by the hypothesis of the absolute homoge- neity, continuity, and incompressibility of the medium, and of the total absence of any thing or physical entity which could serve as the identifiable substratum of mo- tion, or as an object or point of reference. It is mani- fest, therefore, upon the considerations presented by my critic and Professor Lodge themselves that, wherever the vortex-atom theory may land us, it certainly does not land us anywhere in the region of physics, or in the domain of verce causce. And I may add that, inas- much as the hypothetical undifferentiated and undiffer- entiable medium is clearly an involuntary reification of the old ontological concept pure being, the theory un- der discussion has all the attributes of an inapprehen- sible metaphysical phantom. It ought to be noticed, perhaps, in passing, that the argument just presented, from my critic's own premisses, against the possibility of actual motion in the hypotheti- cal medium, is a valid argument, although one of these premisses the doctrine of Professor Lodge that incon- ceivability, or " unthinkableness," of a particular state of things by a " highly-developed " mind, or group of minds, is proof of its impossibility is unsound. As I have shown, in the ninth chapter, inconceivability is proof of impossibility only in case the concept at- tempted to be formed requires the union of contra- dictory attributes. The attempt to conceive motion as taking place in the medium referred to obviously pre- sents that case : in asserting that its parts are capable of real motion, it impliedly invests the medium with attributes of which it is expressly deprived by the nega- tions by which alone it is defined. INTRODUCTION TO THE SECOND EDITION. xxvii Having disposed of the objections of my critic in Mind, so far as, in my judgment, they are deserving of notice, I proceed to consider another objection which has been urged by certain specialists : the denial of the necessity of attributing elasticity to the ultimate atoms of matter. It is said that all that is required to be as- signed to the ultimate atoms is, " an insuperable repul- sive force." The first obvious reflection respecting this claim is, that an indefinite, insuperable force, if it be a real force of repulsion, and not merely a passive resist- ance to intrusion into the space occupied by the atom (i. e., impenetrability), is inconsistent with the conserva- tion of energy. If, on the other hand, it is a definite mechanical function, it is clear that such a force is but another name for elasticity, and that the proposed sub- stitution of a new name for the old one does not, in the least degree, affect the validity of my argument. There are physicists, however, who imagine that the fact of resilience in cases of atomic impact is compatible with the theory of the absolute rigidity of the atoms ; and, in support of this proposition, they invoke the authority of Dr. G. Luebeck * and O. E. Meyer, f But, on exam- ination of Dr. Luebeck's article, it appears that he recognizes the necessity of attributing elasticity to the atoms, if they are regarded as bodies, and seeks to avoid this necessity by contending that they are not bodies, asserting that, in view of this fact, it follows from the formulae respecting the impact- of both elastic and in- elastic bodies, in conjunction with the principle of the conservation of vis viva, that the impact of atoms must result in resilience. Stated in simple words, his argu- * " Ablcitung dcs elastischen Stosses zweicr Atome aus mechanischen Principien," Schloemilch's Zeitschrift f. M. u. P., xxii, 126. f Die kinetische Thcorie dcr G ise, third edition, p. 237, scq. xxviii CONCEPTS OF MODERN PHYSICS. ment is that, inasmuch, as in the case of the impact of ordinary rigid bodies consisting of parts, the translatory motion of these bodies may be converted into molecular motion, and inasmuch as such conversion is impossible in the case of simple atoms, therefore it is necessary, under the coercion of the principle of the conservation of vis viva, to frame a concept of the atom which in- cludes the attribute of resilience. It is hardly necessary to observe that this is not a refutation but an enforce- ment of the doctrine according to which the ultimate material constituents of a body must have the property of elasticity, because without it their collision would be destructive of energy that the absolute inertia and rigidity of these constituents is inconsistent with the conservation of energy. This evidently has not wholly escaped the apprehension of O. E. Meyer ; for, though he refers to Dr. Luebeck's article, he observes that, in his opinion, the simplest and most probable supposition is that, " at the moment of the impact of absolutely rigid bodies, their rigidity and the impossibility of compressing them suddenly, give rise to a repellent force" * Of this strange theory it is sufficient to re- mark that, if projectile forces can suddenly and spon- taneously arise upon the mere contact of hard bodies from the impossibility of compressing them, we have sources of physical energy which will very materially extend the catalogue of Sir William Thomson. The last objection which I propose to consider is very strenuously urged, not only by opponents but * ... bci dem Zusammcnstoss absolut barter Koerper, aus dercn Ilaerte und dcr Unmoeglichkeit sie zusammenzudruecken, im Moraentc dcs Stosses ploetzlich cine zurueckstossende Kraft entsteht. Die letztere Annahme halte icb fuer die einfacbste und wahrscheinlicbste. Loc. cit., p. 239. gv itfir INTRODUCTION TO THE SECOND EDITION. xxix also by adherents of the atomo-mfcSamcal theory, and consists in the denial of my statement that, in the light of that theory, all potential energy is in reality kinetic. Those who urge this objection do not attempt to show that there is any escape from this proposition, if all the phenomena of the world are ultimately resolvable into matter and motion, and if matter is absolutely inert, dead. Nevertheless, they say that the proposition in question is the ill-considered and hastily-formed opinion of Professor P. Gr. Tait, for which physicists generally are not to be held responsible. Now, in the first place, the opinion of Professor Tait is neither ill-considered nor hastily formed. That it is his deliberate opinion appears sufficiently from the fact that lie has very recently repeated and enforced it by new considerations in the article on " Mechanics " contributed by him to the ninth edition of the " Ency- clopaedia Britannica." And, in the second place, he is by no means alone in that opinion. One of the most noted physicists and chemists in the United States is Professor G. F. Barker, of the University of Pennsyl- vania, who, in the address delivered by him as retiring President of the American Association for the Advance- ment of Science, in 1880, said : * " As defined hitherto, energy is either motion or position ; is kinetic or potential. Energy of position derives its value obvi- ously from the fact that, in virtue of attraction, it may become energy of motion. But attraction implies action at a distance ; and action at a distance implies that matter may act where it is not. This, of course, is impossible ; and hence action at a dis- tance, and with it attraction and potential energy, are disappear- ing from the language of science. . . . Now, as Preston has sug- gested, if we regard the sether as a gas, defined by the kinetic theory that its molecules move in straight lines, but with an * "Popular Science Monthly," October, 1SSO, pp. 766, 1Q1. xxx CONCEPTS OF MODERN PHYSICS. enormous length of free path, it is obvious that this aether may be clearly conceived of as the source of all ordinary matter. It is an enormous store-house of energy, which is continually passing to and from ordinary matter, precisely as we know it to do in the case of radiant transmission. Before so simple a conception as this, both potential energy and action at a distance are easily given up. All energy is kinetic energy, the energy of motion." And in England we have Herbert Spencer, who, though not a professional physicist, is regarded by a great number of eminent men of science as the highest authority on matters like that now under discussion. While he is generally at war with Professor Tait, he is in full agreement with him here. In the " Appendix " to the fourth edition of his " First Principles," * Mr. Spencer, replying to certain criticisms of Professor Birks, says : " Now, the tacit implication here is, that I accept the doc- trine of potential energy. ... In the first place, I have to ask on what authority Professor Birks assumes that I hold the doc- trine of potential energy in the way in which it is held by those named ? . . . In the chapter on ' The Continuity of Motion,' I have, at considerable length, given reasons for regarding the con- ception of potential energy as an illegitimate one ; and have dis- tinctly stated that I am at issue with scientific friends on the matter. . . . Let me add that my rejection of this doctrine is not without other warrant than my own. Since the issue of the last edition of this work, . . . Mr. James Croll, no mean author- ity as a mathematician and a physicist, has published, in the 'Philosophical Magazine ' for October, 1876, page 241, a paper in which he shows, I think conclusively, that the commonly accepted view of potential energy can not be sustained, but that energy invariably remains actual." Although, as I have said, the denial of the essen- tially kinetic character of physical energy proceeds, not only from those who reject the atomo mechanical * Pp. 583, 584. INTRODUCTION TO THE SECOND EDITION. xxxi theory, but likewise from many who recognize it as the true basis of modern physics, yet this denial, on the whole, is the point of divergence between two distinct schools of physicists. "With few exceptions, the adher- ents of both schools agree in regarding the ultimate constituents of the material world as absolutely indivisi- ble, inert atoms of invariable mass and volume, whose constancy in number (in conjunction with that of mass and volume) constitutes the conservation or indestructi- bility of matter. In order to account for the differ- ences and changes in the material world, this inert ele- ment, matter, obviously requires to be supplemented by another element; and this element, according to the adherents of the atomo-mechanical theory, is motion, whereas, in the view of its opponents, it is force. This force is assumed to be an independent, substantial en- tity, which is not a property of matter, nor essentially related to matter, otherwise than by its power to act upon it. Force, it is said, resides, not in the atoms, but in the space between them. The most distinguished representative of this school is G. A. Him, the well- known author of the Theorie Mecanique de la Chaleur, in whose writings the independent substantiality of force is stated and illustrated in a variety of ways, of which the following may serve as an example : * " The question," says M. Hirn, " brought to its simplest ex- pression, reduces itself to the inquiry, Whether force resides only in the atom, or outside of it ? ... Is force in the material atom, or in the space which separates two atoms? ... By the aid of the data of the mechanical theory of heat I have demonstrated that matter can not be regarded as infinitely divisible ; that the atom of the chemists is not an entity of pure convention, and simply * Consequences philosophiques ct metaphysiques de la Thermodyna- miquc, p. 61, seq. 3 xxxii CONCEPTS OF MODERN PHYSICS. an explicative device, but that it exists really, that its volume is absolutely unalterable, and that, consequently, it is not elastic. Force, therefore, is not in the atom; it is in the apace which separates the atoms from each other." In the light of this doctrine gravitation, cohesion, affinity, etc., are, of course, not due to any real actio in distans exerted by the atoms upon each other, but to the forces which continuously fill the spaces between them. To what remarkable consequences this theory leads, may be seen from the article of Dr. James Croll,* already referred to, in the quotation from the Appendix to the " First Principles" of Herbert Spencer, in which it is said : "The fact that gravity increases inversely as the square of the distance may be regarded as evidence of the truth of the views advocated by Faraday, Waterston, and others, that it is a force pervading space external to bodies, and that on the mutual ap- proach of the bodies the force is not increased, as is generally supposed, but the bodies merely pass into a place where the force Exists with greater intensity. . . . When a stone, projected up- ward, recedes from the earth, its vis viva is transferred to space, and exists there as gravity. When the stone approaches the earth, the force existing in space is transferred back to the body, and reappears as vis viva." I cite these passages for the purpose, not of entering upon a lengthy discussion of the doctrine set forth in them, but simply of indicating its character. There are two suggestions respecting it, however, which I de- sire to make in passing, without dwelling upon them. The first is, that the substantialization of force here presented rests upon an oblivion of the fact that all force is essentially a stress an action between two bodies. The other suggestion is that, if the increase of * Croll, On the Transformation of Gravity, Phil Mag. [F], vol. ii, p. 252. INTRODUCTION TO THE SECOND EDITION. xxxiii the velocity of falling bodies is due to the circumstance that they pass into places " where the force exists with greater intensity " if bodies projected against gravity transfer their vis viva to space where it is left as force, and take it up again, reconverting it into vis viva as they return to the same points, it follows that each point in space has an infinite store of forces of all de- grees of intensity, so as to supply the countless bodies gravitating toward each other in all directions and from all distances, each with the amount of gravitative force due to it at that point.* It is proper to observe here that, while the substan- tialization of force is, in my judgment, an ontological error, nevertheless the objections urged by M. Hirn and others to the assertion that the phenomena of the uni- verse can be reduced to the motions of inert atoms, appear to me worthy of the most serious attention. M. Him has recently, since the appearance of the first edi- * The blindness of eminent physicists to some of the most obvious consequences of their own theories is marvelous. On page 54 of this volume I have cited the twenty-first query appended by Newton to his " Opticks," in which he suggests an explanation of gravity on the suppo- sition that the attracting and attracted bodies are surrounded by sethercal media increasing in density from the centers of such bodies outward. Now, nothing seems to be more manifest than that this theory requires a separate and distinct medium for each body, and that a single medium surrounding all bodies can not possibly be so constituted as to increase in density from the center of each body outward. A similar observation applies to many of the assumptions respecting the constitution of the aether, which have been made to account for the phenomena of cohesion, chemical affinity, etc., including those of M. Cauchy in his Memoirc sur la dispersion de la lumierc. Mr. Walter R. Browne, in a very thoughtful essay on Action at a Distance (Phil. Mag., December, 1880), has pointed out that the several kinetic theories of gravitation, cohesion, magnetism, etc., require a different and distinct " gravity-gas," " cohesion-gas," " mag- netism-gas," etc., for each of the phenomena sought to be explained by the theory of impact. xxxiv CONCEPTS OF MODEKN PHYSICS. tion of tliis volume, published an article on the results of a series of experiments made by him to determine the relation between the resistance of the air and its tem- perature,* in which he arrives at the conclusion, on ex- perimental grounds, that the kinetic theory of gases, and "all explicative hypotheses which pretend to ac- count for the phenomena of heat by the motions of ponderable atoms," are untenable and must be aban- doned. He shows, by an analysis and development of the formulae of Clausius himself, that, according to the kinetic theory of gases, the resistance of air (or any other gas) to the motion of a body is not merely a me- diate or indirect function of its temperature (i. e., by reason of the effect of variations of temperature upon the density of the gas), but that it is an immediate or direct function of the temperature otherwise expressed, that in gases of constant density the resistance neces- sarily varies with the temperature, if the kinetic theory be true ; whereas, on the contrary, it is the unquestion- able result of four distinct lines of experiment that, in fact, the resistance does not vary with the temperature so long as the density remains the same. To leave no doubt as to the comprehensiveness of his proposition that "the pressure and temperature of a gas do not consist in the motions of material atoms," M. Hirn adds : " I say whatever may oe the nature of the motion. For the rectilinear and translatory motions postulated in the particular theory we have discussed [the kinetic theory of Clausius], we may substitute any other im- aginable motion: vibratory motion, gyratory motion (molecular vortices), . . . the moment these motions simulate a repulsive force they modify in the same way the law of resistance in gases and vapors." * Memoires de V Academic Romaic dc Bclgiquc, torn, xliii. INTRODUCTION TO THE SECOND EDITION. xxxv III, A CRITICAL observation has been made by persons who correctly apprehend my purpose, and are more or less in sympathy with it, that should not, perhaps, be passed without notice. The observation is, that the nature of my inquiry is, after all, metaphysical, and that I am wrong in identifying metaphysics with on- tology. The question thus raised relates exclusively to the proper use of terms. It is not denied that, up to a very recent period, the speculations commonly des- ignated as metaphysical have proceeded upon onto- logical assumptions. But it is asserted that there is a " new metaphysic " which has discarded these as- sumptions, and which, though abandoning the old lines of purely speculative reasoning, and resorting to methods of investigation that have been accredited by their results in special departments of science, nev- ertheless aims at a solution of many, if not all, of the old problems of metaphysics, or at least at a de- termination of the limits within which a solution is possible. I can not but think that this attempt to retain an old name for an essentially new thing is unfortunate, because it leads to intolerable confusion. A very sig- nal illustration of this is afforded by the fate of Kant. It was the distinct and avowed purpose of the Critique of Pure Reason to demonstrate the utter futility of metaphysical speculation as it existed in Kant's time ; and yet, by reason of his retention of the term " meta- physics " to denote the investigation of the laws which, in his view, were not results but conditions of experi- ence, he is constantly (and not wholly without justice) xxxvi CONCEPTS OF MODERN PHYSICS. i assigned to the ranks of the ontological metaphysi- cians.* But, while I am at issue on this point with the modern vindicators of the name and authority of met- aphysics, I am in entire accord with them when they maintain that physical science can not be permanently constructed from the crude results of experiment by the mere aid of mathematical operations. There is a cur- rent saying that every problem in physics abuts upon a problem in metaphysics. The real import of this say- ing, I take it, is that the ultimate interpretation of the data of experiment and observation involves questions which can not be answered by the balance or retort, or * My antagonism to metaphysics has produced an impression in the minds of some of my readers that I am at bottom an adherent of the " positivistic " philosophy of Comte ; and passages are cited from Comte's writings which are analogous to certain propositions in my ninth chapter. There is no doubt that Comte has the great merit of having brought some of the fallacies of metaphysical reasoning into clear relief ; and, if it be true that he was unacquainted with the writings of Kant, some of his statements are very remarkable. It has sometimes appeared to me that his English critics, in protesting against the imputation of Comtism, have failed to do him full justice in this respect. But it is proper to say that I am not, so far as I am aware, indebted to Comte for any of the elements of my discussion, and that the propositions which are in substan- tial accord with his writings are simply part of the modern theory of Cog- nition, whose fundamental principles are so familiar to those who are ac- quainted with that theory, that no one, at this day, thinks of citing author- ity for them. The difficulty with the French adherents of Comte is that, with few exceptions, they are strangers to the logical and epistemological discussions that have been carried on in Germany since the days of Kant and Fries, and therefore credit their master with discoveries which, if he made them independently, were cnfoncements de portes ouvertes. As to those doctrines whereof Comte is the original promulgator, and the body of which constitutes what is generally known as positivism, it is hardly necessary to inform the intelligent reader that I dissent from them in toto. I may add that I have not looked into any of Comte's writings for more than twenty years. INTRODUCTION TO THE SECOND EDITION, xxxvii by a mathematical formula, and that this interpretation is possible only in the light of a general theory of cog- nition. The immediate results of experiment can not become elements of any sort of reasoning, not except- ing mathematical reasoning, without being transformed into concepts. And the scope of mathematical reason- ing is limited to the determination of the quantitative relations between phenomena or the concepts represent- ing them. The declaration of Professor Tait * that " physical science is to be based entirely upon experi- ment and mathematical deductions from experiment," ignores the nature and limits, both of experimental in- quiry and of mathematical operations. A satisfactory and exhaustive discussion of this topic is impossible here, and must be reserved for a future occasion ; but I may, to some extent, anticipate this discussion by indi- cating some of its elements. Physical experiments, as well as mathematical de- ductions from them, are based on certain assumptions respecting the nature of space and the universality of the laws of causality, constancy, and continuity. As to the importance of the assumptions respecting the nature of space, and their bearing upon the interpretation of the data of experiment and observation, it is sufficient to point to the questions hinted at by Professor Tait himself in the passage I have quoted on page 211 of this volume. Without attempting to add anything to what I have said on this subject in the thirteenth and fourteenth chapters, I proceed to a cursory notice of the laws of causality, constancy, and continuity, and of the use made of them in modern physics. The law of causality is essentially a law of the cor- respondence and equivalence of changes, its root being * On Sonic Recent AdvauccsJ|^Jgasi8si^cicncc, p. 6. xxxviii CONCEPTS OF MODERN PHYSICS. the essential relativity and mutual dependence of all physical phenomena. It imports that, whenever a change is observed in a given phenomenon or set of phenomena, we are constrained, by reason of the relations of these phenomena to other phenomena upon which their ex- istence depends, to look for an equivalent change in these other phenomena. The question of cause never arises, except where there is change ; and the cause de- manded is likewise a change. This, of course, at once presents the question, What are the criteria of the equivalence required? a question which can not be examined here, and must be dismissed with the simple remark that these criteria depend upon the nature of the changes whose correspondence and equivalence are under consideration. If this account of the nature and origin of the law of causality is correct, it is readily seen that the miscon- ceptions of this law by the most distinguished modern physicists are as fatal and prolific of error as those of the ontologists. " The final aim of theoretical physical science," says Helmholtz, " is to find the ultimate un- changeable causes of the processes in nature." * These ultimate unchangeable causes, in the view of the ma- jority of physicists, are simple and unchangeable things or substances, forces, and qualities. " Whenever," says Isenkrahe (after citing the words of Helmholtz just re- ferred to), "natural science succeeds in reducing all changes in the material world to unchangeable qualities of an immutable substance, it has arrived, in my opin- ion, at the goal of its endeavors." f Similarly Du * Das endliclic Ziel dcr tJicoretischen NaturwisscnscJiaft ist also, die letztcn imveraenderlichcn Ursachcn der Vorgaenge in der Natur aufzusuchen. Helmholtz, Ueber die Erhaltung der Kraft, p. 2. f Kommt die Naturforschung einmal dazu, doss sie alle Veracnderungcn INTRODUCTION TO THE SECOND EDITION. xxxix Bois-Reymond : " Our demand for causality is satisfied only when we conceive matter as being at rest and uni- formly distributed in infinite space an infinite time ago." * It is evident that the background of all these statements is the ontological reification of the concept cause, analogous to the pangeometrical reification of space. The law of constancy is nothing more than the law of causality in the aspect of the equivalence and recipro- cal correspondence of the phenomena determining each other, which, in mechanics, is exemplified in the law of the equality of action and reaction. The necessity of finding a basis for the determination of this equivalence involves the requirement of an ideal or conceptual con- stant amid phenomenal variation. Here, again, the tendency to ontological reification has led the modern physicist to assert the constancy of absolutely unchange- able finite material elements and of determinate physi- cal forces. The law of continuity is another aspect of the law of causality, resulting from the fact that all physical changes occur in space and time which are necessarily conceived as continuous. To say that space is discon- tinuous would be to say that between two nearest spa- tial points there is a spatial interval ; and similarly, to say that time is discontinuous would be to say that be- tween two nearest points of time there is an interval of dcr Koerperwelt auf unvcracndcrliche Qualitacten einer stcts sick gleichblei- Uenden Substanz znrueckfueliren kann, dann ist sie meiner Meinung nach am Ziel Hires Strebcns angelangt. Isenkrahe, Das Raethsel der Schwer- kraft, p. 139. * Unser Causalitactsbeduerfniss fuelilt sich nur befriedigt, wenn wir uns vor unendlicher Zcit die Matcrie ruhcnd und glcichmaessig im unend- lichen Raum verthdlt denkcn. Emil Du Bois-Reymond, Die Sieben Welt- raethsel, p. 10. x l CONCEPTS OF MODEKN PHYSICS. time. The vague apprehension of this law, in conjunc- tion with the assumption of the permanent existence of a constant number of discrete atoms of invariable mass and volume, and of a constant sum of forces or motions, has given rise to the standing protest against the possi- bility of actio in distans, and to the various theories respecting the existence of continuous sethereal media, or of forces, between the atoms, in order to account for the continuity of their interaction. A question of exceeding interest and importance is the old question whether the law of causality, with its several specifications in physical science, including the laws of constancy and continuity, the law of the con- servation of energy, the law of least action, the law of motion under least constraint, etc., are purely inductions from experience, or are, on the contrary, conditions pre- cedent to experience, and therefore in some sense inde- pendent of it. The tendency among modern physicists and mathematicians is decidedly toward the purely em- pirical view, according to which the only warrant for the assumption of the universality of these laws lies in the fact that thus far no experiential instance has been adduced against them. Helmholtz,* among others, emphasizes this view. But physicists, and especially mathematicians, are puzzled by the circumstance that not only has the law of causality always been applied before any experiential induction was thought of, but that all the other laws above mentioned were announced long before they were precisely formulated and experi- entially verified, or had been recognized as subjective psychological laws before they were applied to the ob- * Fuer die Anwendung des Causalitaetsgcsetzes liaben wir Jceine wcitere JBuergschaft ah seinen Erfolg. Helmholtz, Die Thatsachen in der Wakr- nehmung, p. 41. INTRODUCTION TO THE SECOND EDITION. x li jective world. This is strikingly illustrated by the law of least action and Gauss's law of motion under least constraint. The law of least action is stated with sin- gular distinctness by Leibnitz ; * and Gauss observes in relation to the law of motion under least constraint that " it is remarkable that free motions, when they are in- consistent with the conditions of the system, are modi- fied by Kature in precisely the same way in which the calculating mathematician corrects quantitative results that are necessarily interdependent, by the method of least squares." f It is my belief that the mysteries here presented will be cleared up by a re-examination, upon new grounds, of the nature of the old antithesis between truths a priori and cognitions a posteriori. Although it must have become clear, by this time, to those with whom I shall deem it profitable hereafter to have any discussion, that I am not intent on subvert- ing the science of physics or on questioning the facts it has brought to light and the legitimate generalizations based upon them, and, furthermore, that I do not pro- pose to deprive the physicist of his laboratory hypoth- eses, and hence am under no obligation to supply him with new ones, the question is not unnatural, what con- cern the physicist has with the discussions in this vol- ume. The simple answer to this is, that physical theo- ries are not merely instrumentalities for the discovery and classification of facts in furtherance of the practical purposes of life, but that they also serve as a basis for the various attempts at a solution of the great questions * Semper scilicet cst in rebus principium determination^ quod a maxi- ma minimove petcndum est, ut nempe maximus prcestetur ejfcctus minimo ut sic dicam sumtu. Leibnitii, Opp. ed. Erdmann, p. 147. f Ueber cin neues allgemeines Grundgesetz der Mechanik, Gauss' Werkc, V, p. 28. xlii CONCEPTS OF MODERN PHYSICS. which always have been and always will be the cardinal problems of human thought. And before these ques- tions can be properly submitted to the arbitrament of the physicist, it is necessary that he should have a clear insight into the nature of the concepts and theories with which he operates, and into their relations to the phenomena which they represent. When Du Bois- Reymond, in his Sieben Weltraethsel, declares that all questions relating to the "essence" of matter, force, and motion, and the origin of sensation, are necessarily transcendent, while those concerning the nature and origin of life and thought are not, he assumes the abso- lute objective reality of atoms independently of the in- tellect ; and it certainly is of the greatest importance to ascertain whether, and in what sense, this assumption is true. When the physiological psychologist investigates the laws governing the correspondence between physi- cal and psychological phenomena, he is confronted at the outset with what one of my critics terms the " resid- ual but tedious problem of idealism " with which, in his opinion, I unnecessarily and impertinently weary the reader. The proper statement of that problem, at least, if not its solution, is not " residual," but prelimi- nary to any fruitful discussion of that large number of ultimate questions which are covered by a mass of wordy rubbish to which the physicists have made as large a contribution as the ontological metaphysi- cians. Irrespective of this, however, if physics as a science are not to fall into utter disrepute, it is time to evoke some order from the confusion which prevails among the very first principles, theories, and definitions of theoretical physics. When Professor Tait, in conjunc- tion with Professor Stewart, announces that " matter INTRODUCTION TO THE SECOND EDITION. xliii is simply passive," * and then, in connection with Sir William Thomson, declares that " matter has an innate power of resisting external influences," f it is hardly impertinent to inquire how these statements are to be reconciled. "When Professor Du Bois-Reymond, in a passage heretofore quoted, insists upon the necessity of " reducing all the processes of nature to motions of a substantial, indifferent substratum wholly destitute of quality" \ having declared shortly before in the same lecture that " resolution of all changes in the material world into motions of atoms caused In/ tlmr constant central forces would be the completion of natural sci- ence," * we are in a perplexity from which we have the right to be relieved. There is, moreover, a lesson to be learned, by the ordinary physicist, from the discussions to whicli I have invited his attention, as to the limits within which the arbitrary construction of hypotheses and theories, with- out regard to the validity of the assumptions on which they rest, is useful and proper. This is beginning to be understood by thoughtful physicists, who suggest, for example, that the exhibition of the equivalence of physical and chemical phenomena would lose nothing in clearness if the atomic hypothesis were wholly dis- carded. An analogous suggestion respecting the basis of statics has recently been made by Professor Horace Lamb, 1 who insists that certain assumptions in the current theories relating to the transmission of force, such as that of the absolute rigidity of the bodies by * The Unseen Universe, 104. f Thomson and Tait, Treatise on Natural Philosophy, vol. i, 216. \ Ueber die Grenzen des Naturerkennens, p. 5. * 75., p. 2. 1 Phil. Mag. [V], vol. xv, p. 187. xliv CONCEPTS OF MODERN PHYSICS. means of which this transmission is effected, are not helps but hindrances to a proper apprehension of the laws of equilibrium and motion, and that the science of statics ought to be constituted by means of the prin- ciples of linear and angular momentum, under the definition that " two sets of forces are equivalent when, and only when, they produce the same effect on the linear and angular momentum of any material system to which they are applied." The same spirit is evinced in Kirchhoff's Lectures on Mathematical Physics, to which I have already had occasion to refer in another connection. And there is evidence, in the best scien- tific literature of the day, of a growing tendency to divest the edifice of physical science as far as possible of the hypothetical scaffolding, which not only ob- structs the view of its fair proportions, but masks the real principles of its construction, by which its strength and permanence are assured.* * Before permitting this edition to go to press, it is proper to apolo- gize for an omission in the fifth chapter which I have not deemed it necessary to supply, because it does not affect the validity of my reason- ing, and the correction could not be made without serious disturbance of the text. In speaking (pp. 61, 62) of Mr. Adams's revision of the calcu- lations of Laplace respecting the gradual diminution of the excentricity of the earth's orbit due to the disturbing action of the other planets, it ought to have been mentioned that the revision of Mr. Adams has since been revised in turn by Professor John N. Stockwell, of Cleveland, the distinguished author of the Theory of the Moon's Motion. I regret this omission the more, because Professor Stockwell is a citizen of my own State. THE CONCEl^TD THEORIES OF MODERN PHYSICS CHAPTEE I. INTEODUCTOEY. MODEEN physical science aims at a mechanical inter- pretation of all the phenomena of the universe. It seeks to explain these phenomena by reducing them to the elements of mass and motion and exhibiting their diversities and changes as mere differences and varia- tions in the distribution and aggregation of ultimate and invariable bodies or particles in space. Naturally the supremacy of mechanics became conspicuous first in the domains of those sciences which deal with the visible motions of palpable masses in astronomy and molar physics ; but its recognition is now all but uni- versal in all the physical sciences, including, not only molecular physics and chemistry, but also such depart- ments of scientific inquiry as are conversant about the phenomena of organic life. It is said that the theoretical no less than the prac- tical progress of the natural sciences, during the last 16 CONCEPTS OF MODERN PHYSICS. three centuries, is an achievement of mechanics which, besides devising the instruments of successful scientific research, has also supplied its principles and methods. It is, indeed, incontestable that the attempt at a con- sistent application of mechanical principles marks a new epoch in the history of science. The founders of mod- ern physics proceeded upon the tacit if not upon the declared assumption that all true explanations of natural phenomena are mechanical explanations. That this did not at once find articulate expression is due, partly to the fact that principles are wont to assert themselves, in thought as in action, before they are distinctly appre- hended, and partly to the circumstance that science, for a long time, was constrained to flourish under the shad- ows of metaphysics and theology. But it was not long after the days of Stevinus, Fermat and Galilei before the doctrine that all physical action is mechanical was stated in terms. Even during the life of Galilei a year before his death Descartes announced that " all variations of matter, or all diversity of its forms, de- pends on motion." * And nine years before the appearance of Newton's Principia Thomas Hobbes declared that " change [i. e., physical change] is of necessity nothing else than motion of the parts of the body changed,"! at the same time adding that " there can be no cause of motion in a body but in another body contiguous and moved." J Leibnitz was even more emphatic, asserting that the doctrine in question is not merely an experiential induction, but a self-evi- * " Omnis materiae variatio sive omnium ejus formarum diversitas pcndet a motu." Cartes. Princ. Phil, ii, 23. f " Necesse est ut mutatio aliud non sit praeter partium corporis mu- tati motum." Hobbes, Philos. prima, pars secunda, ix, 9. \ " Causa motus nulla essc potest in corporc nisi contiguo ct moto." Ib. INTRODUCTORY. 17 dent truth. " Everything in nature," he said, " is ef- fected mechanically a principle which can be made certain by reason alone, and never by experiments, however numerous they may be." * He, too, insisted that all motion is caused by impact. " A body is never moved naturally, except by another body which presses in touching it." f Similarly Huygens, the great con- temporary of Leibnitz and Newton, said that " in true philosophy the causes of all natural effects are, and in his judgment must be, conceived mechanically, unless we are to renounce all hope of understanding anything in physics." ^ And in the first comprehensive treatise on physics ever published, that of Musschenbroek, it is put forth as an axiom that " no change is induced in bodies whose cause is not motion." * The most definite statement, however, of the prop- osition that the true aim and object of all physical science is a reduction of the phenomena of nature to a coherent mechanical system is found in- the scientific * " Tout se fait mecaniquement dans la nature, principe qu'on peut rendre certain par la seule raison et jamais par les experiences, quelque nombre qu'on en fasse." Leibnitz, Nouveaux Essais, Opp. ed. Erdmann, p. 383. f " Un corps n'est jamais mu naturcllemcnt quo par un autre corps qui le presse en le touchant." 5me lettre a Clarke, Erdmann, p. 767. Hence Wolff, the dogmatic expounder of the Leibnitian philosophy: " Corpus non agit in alterum nisi dum in ipsum impingit." Wolff, Cos- mologia gen., 129. \ "... in vera philosophia, in qua omnium effectuum causae conci- piuntur per rationes mcchanicas : id quod meo judicio fieri debet nii' vclimus omnem spem abjicere aliquid in physicis intelligendi." Ilugc- nii Opp. reliqua, Amst., 1728, vol. i (Tract, de lumine), p. 2. * " Nulla autem corporibus inducitur mutatio, cujus causa non fuerit motus, sive excitatus, sive minutus, aut suffocatus ; omne enim incre- mentum vel decrementum, generatio, corruptio, vel qualiscunque alteratio, quae in corporibus contingit, a motu pendet." P. v. Musschcnbroek, In- trod. ad. philos. naturalem, vol. i, cap. 1, 18 (ed. Patav., 1768). 18 CONCEPTS OF MODERN PHYSICS. writings published during the second half of the pres- ent century, since the discoveries made in organic chem- istry by the aid of the atomic theory, the revelations of the spectroscope, the establishment of the doctrine of the conservation of energy, and the promulgation of the mechanical theory of heat with its complement, the kinetic theory of gases. Thus Kirchhoff, one of the founders of the theory of spectral analysis, said in 1865: "The highest object at which the natural sci- ences are constrained to aim, but which they will never reach, is the determination of the forces which are present in nature, and of the state of matter at any given moment in one word, the reduction of all the phenomena of nature to mechanics." * To the same effect Helmholtz, in his inaugural address delivered be- fore the meeting of the association of physicians and naturalists at Innspruck, in 1869 : " The object of the natural sciences is to find the motions upon which all other changes are based, and their corresponding mo- tive forces to resolve themselves, therefore, into me- chanics." f No less pointed are the words of Clerk Maxwell : " When a physical phenomenon," he writes, " can be completely described as a change in the con- figuration, and motion of a material system, the dynami- cal explanation of that phenomenon is said to be com- * " Das hoechste Ziel, welches die Naturwisscnschaften zu erstreben haben, aber niemals erreichen werden, ist die Ermittelung der Kracfte, welche in der Natur vorhanden sind und des Zustandes in dem die Ma- tcrie in einem Augenblick sieh befindet, mit einem Worte, die Zurucck- f uehrung aller Naturerscheinungen auf die Mechanik." Kirchhoff, Uebcr das Ziel der Naturwissenschaftcn. Prorcctoratsrede, Heidelberg, 1865. S. 9, 24. f " Das Endzicl der Naturwissenschaf ten ist, die alien andern Veraen- derungen zu Grunde liegenden Bewegungcn und deren Triebkraefte zu finden, also sich in Mechanik aufzuloesen." Helmholtz, Populaerwis- senschaftlichc Vortraegc, i, 93. INTRODUCTORY. 19 plete. We can not conceive any further explanation to be either necessary, desirable, or possible, for as soon as we know what is meant by the words configuration, mass and force, we see that the ideas which they repre- sent are so elementary that they can not be explained by means of anything else." * Citations like these, from the writings of eminent physicists, might be multiplied almost indefinitely. And, if we turn from the physicists to the physiologists, we meet with declarations equally explicit. "Every analysis," said Ludwig in 1852, " of the animal organism has thus far brought to light a limited number of chem- ical atoms, the presence of the light- (heat-) bearing aether and of the electric fluids. These data lead to the inference that all the phenomena of animal life are consequences of the simple attractions and repul- sions resulting from the concurrence of these element- ary substances." f In a similar strain Wundt, writing twenty-five years later : " The view that has now be- come dominant [in physiology], and is ordinarily desig- nated as the mechanical or physical view, has its origin in the causal conception long prevalent in the kindred departments of natural science, which regards nature as a single chain of causes and effects wherein the ulti- mate laws of causal action are the laws of mechanics. * Clerk Maxwell, " On the Dynamical Evidence of the Molecular Con- stitution of Bodies." " Nature," March 4 and 11, 1875. f " So oft nun eine Zergliederung der leistungserzengenden Einrich- tungen des thierischen Koerpers geschah, so oft stiess man schliesslich auf cine begrenzte Zahl chemisher Atome, die Gegenwart des Licht- (Wacrme-) Aethers und diejenige der electrischen Fluessigkeiten. Dic- ser Erfahrung entsprechend zieht man den Schluss, dass alle vom thier- ishen Koerper ausgehenden Ersheinungen eine Folge der einfachen An- ziehungen und Abstossungen sein moechten, welche an jenen elementaren Wesen bei einem Zusammentreffen derselben beobachtet werden." Lud- wig, Lehrbuch der Physiologic des Menschen, Band i, Einleitung, p. 2. 20 CONCEPTS OF MODERN PHYSICS. Physiology thus appears as a branch of applied phys- ics, its problem being a reduction of vital phenomena to general physical laws, and thus ultimately to the fundamental laws of mechanics." * Still more broadly, Haeckel: "The general theory of evolution . . . as* sumes that in nature there is a great, unital, continuous and everlasting process of development, and that all natural phenomena without exception, from the motion of the celestial bodies and the fall of the rolling stone up to the growth of the plant and the consciousness of man, are subject to the same great law of causation that they are ultimately to be reduced to atomic me- chanics." f This theory, Haeckel declares, " is the only scientific theory which affords a rational explanation of the universe, and satisfies the craving of the intellect for causal connections, inasmuch as it links all the phe- nomena of nature as parts of a great unital process of development and as a series of mechanical causes and * " Die jetzt zur Herrschaft gelangte Auffassung dagegen, die man als die physikalishe oder mechanistische zu bczcichnen pflegt r ist aus der in den verwandten Zweigen der Naturwissenschaft schon laenger zur Gel- tung gckommenen causalen Naturansicht entsprungen, welche die Natur als einen einzigen Zusammenhang von Ursachen und Wirkungen ansieht, wobei als letzte Gesetze, nach denen die natuerlichen Ursachen wirken, sich stets die Grundgesetze der Mcchanik ergebcn. Die Physiologic er- sheint daher als ein Zweig der angewandten Naturlchre. Ihre Aufgabe erkennt sie darin, die Lebenserscheinungen auf die allgemcinen Natur- gesetze, also schliesslieh auf die Grundgesetze des Mechanik, zurueckzu- fuchrcn." Wundt, Lchrbuch der Physiologic des Menschen, 4te Au- flage, p. 2. f "Die allgemeine Entwickelungslehre . . . nimmt an, dass in der ganzen Natur ein grosser, einheitlicher, ununterbrochener und ewiger Entwickelungsvorgang stattfindct, und dass alle Naturerscheinungcn ohne Ausnahme, von der Bewegung der Himmelskoerper und dem Fall des rollenden Steins bis zum Wachsen der Pflanze und zum Bewusstsein des Menschen, nach einem und demselbcn grosscn Causal-Gesetze erfolgen, dass alle schliesslieh auf Mcchanik der Atorae zurucckzufuehrcn sind." Haeckel, Freie Wissenschaft und freie Lehre, pp. 9, 10. INTRODUCTORY. 21 effects." * In the same sense Huxley speaks of " that purely mechanical view toward which modern physi- ology is striving." f A very lucid and thorough exposition of the aims of modern physical science is contained in the following passage taken from a recent lecture of Emil Du Bois- Reymond equally distinguished as a physicist and physiologist: "Natural science more accurately ex- pressed, scientific cognition of nature, or cognition of the material world by the aid and in the sense of the- oretical physical science is a reduction of the changes in the material world to motions of atoms caused by central forces independent of time or a resolution of the phenomena of nature into atomic mechanics. It is a fact of psychological experience that, whenever such a reduction is successfully effected, our craving for causality is, for the time being, wholly satisfied. The propositions of mechanics are reducible to mathematical form, and carry within them the same apodictic cer- tainty which belongs to the propositions of mathemat- ics. "When the changes in the material world have been reduced to a constant sum of potential and kinetic energy inherent in a constant mass of matter, there is nothing left in these changes for explanation. " The assertion of Kant, in the preface to the ' Met- aphysical Rudiments of Natural Science,' that i in every department of physical science there is only so much science, properly so called, as there is mathematics,' is * " Der Monismus, die universale Entwickelungstheorie, oder die mo- nistische Progenesistheorie ist die einzige wissenschaftliche Theorie, welche das Weltganze vernunftgcmass erklaert, und das Causalitaets- beduerfniss unserer menschlichcn Vermmft befriedigt, indem sie alle Natur-Erscheiimngen als Theile eines einbcitlichcn grossen Entwickelungs- Processes in mechanischen Causal-Zusammenhang bringt." Ibid., p. 11. f Lay Sermons, Addresses and Reviews (Appletons' ed.), p. 331. 22 CONCEPTS OF MODERN PHYSICS. to be sharpened by substituting ' mechanics of atoms ' for ' mathematics.' This was evidently his own mean- ing when he denied the name ' science ' to chemistry. It is not a little remarkable that in our time chemistry, since it has been constrained, by the discovery of sub- stitution, to abandon the old electro-chemical dualism, has seemingly taken a retrograde step in its advance toward science in this sense. The resolution of all changes in the material world into 'motions of atoms caused l)y their constant central forces would be the completion of natural science" * * " Naturerkenncn genaucr gcsagt, naturwissenschaf tlichcs Erkcn- nen oder Erkennen der Koerperwelt mit Huelfe und im Sinne der theo- retischen Naturwissenschaft 1st Zurueckfuehren der Veraenderungen in der Koerperwelt auf Bewegungen von Atomen die durch deren von der Zeit unabhaengige Centralkraefte bewirkt wcrdcn, oder Aufloesung der Naturvorgaenge in Mechanik der Atome. Es ist psychologische Erfahr- ungsthatsache, dass wo solche Aufloesung gelingt, unser Causalitaetsbe- duerfniss vorlaeufig sich bef riedigt fuehlt. Die Saclze der Mechanik sind mathematisch darstellbar, und tragen in sich dieselbe apodiktische Gewiss- heit, wie die Saetze der Mathematik. Indem die Veraenderungen in der Koerperwelt auf eine constante Summe potentieller und kinetischer Encr- gie, welche einer constanten Menge von Materie anhaf tet, zurueckgef uehrt werden, bleibt in diesen Veraenderungen selber nichts zu erklacrcn uebrig. " Kant's Behauptung in der Vorrede zu den * Metaphysischen An- fangsgruenden der Naturwissenschaft,' ' dass in jeder besonderen Na- turlehre nur so vicl eigentliche Wissenschaft angetroffen werden koenne, als darin Mathematik anzutreffcn sei,' ist also vielmehr noch dahin zu verschaerfen, dass fuer Mathematik Mechanik der Atome gesetzt wird. Sichtlich diess meintc er selber als er der Chemie den Namen einer Wis- senschaft absprach, und sie unter die Experimentallehren verwics. Es ist nicht wenig mcrkwuerdig dass in unserer Zeit die Chemie indem sie durch die Entdeckung der Substitution gezwungen wurde den electro- chemischen Dualismus aufzugeben, sich von dcm Ziel, eine Wissenschaft in diesem Sinne zu werden, scheinbar wieder weitcr entfernt hat. Denken wir uns alle Veraenderungen in der Koerperwelt in Bewegungen von Atomen aufgeloest, die durch deren constante Centralkraefte bewirkt werden, so waere das Weltall naturwissenschaftlich erkannt." Emil Du Bois-Rcymond, " Ueber die Grenzen des Naturerkcnnens," p. 2 scq. INTRODUCTORY. 23 With few exceptions, scientific men of the present day hold the proposition, that all physical action is me- chanical, to be axiomatic, if not in the sense of being self-evident, at least in the sense of being an induction from all past scientific experience. And they deem the validity of the mechanical explanation of the phe- nomena of nature to be, not only unquestionable, but ab- solute, exclusive, and final. They believe that this validity is not conditioned, either by the present state of human intelligence, or by the nature and extent of the phenomena which present themselves as objects of scientific investigation. Thoughtful men like Du Bois- Reymond have at times suggested that it is not unlim- ited ; but the only limits assigned to it are those of the general capacity of the human intellect. Although they concede that there is a class of phenomena those of organic life which, under their characteristic as- pect, are wholly irreducible by the mere aid of mechan- ical principles, it is, nevertheless, insisted that these principles constitute the only intellectual solvent that can be applied to them, and that the residue which re- sists the solution is to be relegated for ever to that end- less array of facts which are proof against all the re- agents of scientific cognition. It is claimed that, if it is impossible theoretically to construct a living organism out of molecules or atoms, and mechanical forces under the guidance of the principle of the conservation of en- ergy, the laws of electric or magnetic coercion, the first and second laws of thermo-dynamics, etc., the attempt to frame a theory of life in harmony with the laws con- trolling ordinary material action must be utterly aban- doned. Such a claim ought not, in my judgment, to be admitted without a careful examination of the grounds upon which it is made. It is my purpose, 24 CONCEPTS OF MODERN PHYSICS. therefore, in the following pages to inquire whether or not the validity of the mechanical theory of the uni- verse in its present form, and with its ordinary assump- tions, is indeed absolute within the bounds of human intelligence, and to this end, if possible, to ascertain the nature of this theory as well as its logical and psycho- logical origin. Obviously the first question presenting itself in the course of an examination into its validity is whether it is consistent with itself and with the facts for the explanation of which it is propounded. Our initial problem, then, will be that of finding an answer to this question. Of TH1 -U1TI7BIISIT CHAPTER II. FIRST PRINCIPLES OF THE MECHANICAL THEORY OF THE UNIVERSE. THE mechanical theory of the universe undertakes to account for all physical phenomena by describing them as variances in the structure or configuration of material systems. It strives to apprehend all phenom- enal diversities in the material world as varieties in the grouping of primordial units of mass, to recognize all phenomenal changes as movements of unchangeable elements, and thus to exhibit all apparent qualitative heterogeneity as mere quantitative difference. In the light of this theory the ultimates of scientific analysis are mass * and motion, which are assumed to be essen- tially disparate. Mass, it is said, exists independently of motion and is indifferent to it. It is the same whether it be in motion or at rest. Motion may be transferred from one mass to another without destroy- ing the identity of either. The prime postulate of all science is that there is some constant amid all phenomenal variations. Science is possible only on the hypothesis that all change is in its nature transformation. "Without this hypothesis it * It is hardly necessary to say that I purposely designate mass, and not (as is usual) matter, as the correlate of motion. When a body is di- vested, in thought, of all those qualities which, according to the teachings of modern science, are in their nature phases of motion, the residue is not matter, but mass. 4 26 CONCEPTS OF MODERN PHYSICS. could discharge neither of its two great functions those of determining, from the present state of things, the past on the one hand and the future on the other, by exhibiting the one as its necessary antecedent, and the other as its equally necessary consequent. It is evident that the computations of science would be utterly frus- trated by the sudden disappearance of one or moro of its elements, or the unbidden intrusion of new elements. If, therefore, scientific analysis yields mass and motion as its absolutely irreducible elementary terms if these terms underlie all possible transformations it follows that both are quantitatively invariable. Accordingly the mechanical theory of the universe postulates the conservation of both mass and motion. Mass may be transformed by an aggregation or segregation of parts ; but amid all these transformations it persistently remains the same. Similarly motion may be distributed among a greater or less number of units of mass ; it may be transferred from one unit of mass to any number of units, its velocity being reduced in proportion to the number of units to which the transference takes place ; nevertheless the sum of the motions of the several units is always equal to the motion of the single unit. It may be changed in direction and form ; rectilinear mo- tion may become curvilinear, translatory motion may be broken up into vibratory motion, molar motion may be converted into molecular agitation ; yet, during all these changes, it is never increased, diminished, or lost. The conservation of mass (or, as it is generally but in- accurately termed, the conservation or indestructibility of matter) has long been a standing axiom of physical science. The conservation of motion (i. e., the conser- vation of energy, which, as will hereafter appear, is, ac- cording to the mechanical theory, the same thing), though GENERAL PRINCIPLES OF MECHANICAL THEORY. 27 but recently formulated as a distinct scientific principle, is now universally regarded as of equal evidence and axiomatic dignity with its older counterpart. Indeed, while chemistry is said to be founded on the conserva- tion of matter,* the recent progress of theoretical phys- ics has consisted mainly in the solution of the problem of reconstituting it on the basis of the conservation of energy. The science of physics, in addition to the gen- eral laws of dynamics and their application to the in- teraction of solid, liquid, and gaseous bodies, embraces the theory of those agents which were formerly desig- nated as imponderables light, heat, electricity, mag- netism., etc. ; and all these are now treated as forms of motion, as different manifestations of the same funda- mental energy, and as controlled by laws which are simple corollaries from the law of its conservation. The only apparent exception is the second law of ther- modynamics, a reduction of which, however, to the principle of least action, or rather Hamilton's extension of it, the principle of varying action, has been attempted by Boltzmann and Clausius, while others (among them Rankine, Szily, and Eddy) have sought to derive it di- rectly from the principle of the conservation of energy. It is thus seen that the theory according to which the cause of all phenomenal change and variety in na- ture is motion, and all apparent qualitative diversity is in reality quantitative difference, involves three propo- sitions, which may be stated as follows : I. The primary elements of all natural phenomena * It is gradually coming to be understood that the conservation of energy is as important a principle in chemistry as that of the conserva- tion of mass ; but as yet chemical notation takes account of masses only and makes no exhibition of the quantities of energy gained or lost in any given chemical transformation. 28 CONCEPTS OF MODERN PHYSICS. the ultimate* of scientific analysis are mass and motion. II. Mass and motion are disparate. Mass is in- different to motion, which may ~be imparted to it, and of which it may be divested, by a transference of motion from one mass to another. Mass remains the same, whether at rest or in motion. III. Both mass and motion are constant. Among the corollaries from the first and second of these propositions there are two which are as obvious as they are important : the inertia and the homogeneity of mass. Mass and motion being radically disparate, it is evident that mass can not be motion or the cause of motion it is inert. And mass in itself can not be heterogeneous, for heterogeneity is difference, and all difference is caused by motion. The propositions above set forth lie at the base of the whole mechanical theory. They command uni- versal assent among physicists of the present day, and are to be regarded as the fundamental axioms of mod- ern physical science. In addition to these propositions, however, there is the assumption, generally prevalent among physicists and chemists, of the molecular or atomic constitution of bodies, according to which mass is not continuous, but discrete, being an aggregate of unchangeable, and, in that sense at least, simple units. This assumption leads to four other propositions, which, in conjunction with the principle of the conservation of both mass and motion, may be said to constitute the foundations of the atomo-mechanical theory. They are these : 1. The elementary units of mass, being simple, are in all respects equal. This is manifestly nothing more than an assertion of the homogeneity of mass in con- GENERAL PRINCIPLES OF MECHANICAL THEORY. 29 formity with the hypothesis of its molecular or atomic constitution. 2. The elementary units of mass are absolutely hard and inelastic a necessary consequence of their sim- plicity, which precludes all motion of parts, and, there- fore, all change of figure. 3. The elementary units of mass are absolutely in- ert and therefore purely passive / hence there can be no mutual action between them, other than mutual dis- placement caused by impulses from without. 4. All potential energy, so called, is in reality kinetic. Mass and motion being fundamentally dispa- rate and inconvertible, and mass being absolutely inert, whatever be its position, motion can not originate in, or be caused by, anything but motion. Energy due to mere position, therefore, is impossible. It is necessary to take up these propositions sever- ally in their order, and to ascertain whether, and to what extent, they are consistent with, and serve as the explanation of, the facts of scientific experience. CHAPTEE III. THE PEOPOSITION THAT THE ELEMENTAKY UNITS OF MASS AKE EQUAL. IF all the diversities in nature are caused by motion, it follows that mass, the substratum of this motion, is fundamentally homogeneous. This is so evident that, in the first distinct announcements of the mechanical theory, the two propositions the principle and its cor- ollary appeared side by side. Thus the statement of Descartes cited in the first chapter * is accompanied by the declaration that " the matter which exists in the world is everywhere one and the same." f It is true that Descartes did not assert the absolute equality of single material elements, because he recognized but two primary properties of matter, extension and mobility, and therefore denied its atomic constitution. But, when in time the hypothesis of the atomic or molecular structure of matter became one of the cardinal doctrines of modern physical science, the postulate of the funda- mental homogeneity of mass necessarily assumed the form of an assertion of the absolute equality of its primor- dial units. For reasons to be discussed presently, physi- cists, and especially chemists, of our day evince a disposi- tion to ignore this essential feature of the mechanical * Supra, p. 16. f " Maleria ilaque in toto universo una et cadcm existit" Cart. Princ. Phil., ii, 23. EQUALITY OF ELEMENTS OF MASS. 31 theory ; but, among those who understand that all scien- tific theories must at last be brought to the test of con- sistency, it has rarely failed to meet with direct or im- plied recognition. " Chemistry," says Professor Wundt, " still refers the divergent qualities of matter to an origi- nal qualitative difference between the atoms. But the whole tendency of physical atomism is to derive all the qualitative properties of matter from the forms of atomic motion. Thus the atoms themselves remain as elements utterly devoid of quality " * Of the same im- port are the words of Herbert Spencer : " The proper- ties of the different elements result from differences of arrangement, arising by the compounding and recom- pounding of ultimate homogeneous units" \ Even in the writings of distinguished chemists there is no lack of utterances bearing testimony to the pressure of the logi- cal necessity which constrains the modern physicist to insist upon the fundamental equality of the material elements. " It is conceivable," says Thomas Graham, " that the various kinds of matter now recognized as different elementary substances may possess one and the same ultimate or atomic molecule existing in differ- c5 ent conditions of movement. The essential unity of matter is an hypothesis in harmony with the equal action of gravity upon all bodies. We know the anxiety with which this point was investigated by Newton and the care he took to ascertain that every kind of substance, * Die abweichenden Eigenschaftcn der Materie verlegt die Chemie noch jetzt in cine urspruengliche qualitative Verschiedenheit der Atome. Nun geht offenbar die ganze Entwickelung der physikalischen Atomistik darauf aus, alle qualitativen Eigenschaften der Materie aus den Beweg- ungsformen der Atome abzuleiten. Die Atome selbst bleiben so noth- wendig als volkommen qualitaetslose Elemente zurueck. " Die Theorie der Materie," Deutsche Rundschau, December, 1875, p. 381. f Contemporary Review, June, 1872. 32 CONCEPTS OF MODERN PHYSICS. 1 metals, stones, woods, grain, salts, animal substances,' etc., are similarly accelerated in falling, and are there- fore equally heavy. " In the condition of gas, matter is deprived of numerous and varying properties with which it ap- pears invested when in the form of a liquid or solid. The gas exhibits only a few grand and simple features. These again may all be dependent upon atomic or mo- lecular mobility. Let us imagine one kind of substance only to exist ponderable matter ; and further, that mat- ter is divisible into ultimate atoms, uniform in size and weight. We shall then have one substance and a common atom. With the atom at rest the uniformity of matter would be perfect. But the atom possesses always more or less motion, due, it must be assumed, to a primordial impulse. This motion gives rise to volume. The more rapid the movement the greater the space occupied by the atom, somewhat as the orbit of a planet widens with the degree of projectile veloci- ty. Matter is thus made to differ only in being lighter or denser matter. The specific motion of an atom being inalienable, light matter is no longer convertible into heavy matter. In short, matter of different density forms different substances different inconvertible ele- ments as they have been considered. "But, further, these more or less mobile,* or light and heavy forms of matter, have a singular relation connected with equality of volume. Equal volumes of two of them can coalesce together, unite their move- ment, and form a new atomic group, retaining the whole, the half, or some simple proportion of the original movement and consequent volume. This is chemical combination. It is directly an affair of vol- ume, and only indirectly connected with weight. EQUALITY OF ELEMENTS OF MASS. 33 Combining weights are different, because the densi- ties, atomic and molecular, are different." * Views analogous to those of Graham are held by C. K. A. "Wright, who suggests " that there is but one kind of primordial matter, all so-called elements and com- pounds being, as it were, allotropic modifications of this matter, differing from one another in the amount of energy latent per unit of mass."f And although Prout's conjecture, that the several chemical elements are really compounds or allotropic forms of hydrogen, has been definitively abandoned (even by Dumas and others who at divers times sought to revive it), it having been shown that the hypothesis, according to which the atomic weights of all the elements are exact multiples of that of hydrogen, is untenable, yet attention has late- ly been drawn to the fact that there seem to be spec- troscopic indications of the predominance of a few gase- ous elements, such as hydrogen and nitrogen, in certain nebulae which appear to represent the earlier stages of planetary or stellar development, and of a gradual in- crease of metallic and other substances in more ad- vanced forms in other words, of a progressive differen- tiation of matter, a gradual advance from homogeneity to heterogeneity, on the successive stages of planetary or stellar evolution.^ Now, while the absolute equality of the primordial units of mass is thus an essential part of the very foun- dations of the mechanical theory, the whole modern science of chemistry is based upon a principle directly * " Speculative Ideas respecting the Constitution of Matter," Phil. Mag., 4th ser., vol. xxvii, p. 81 seq. f Chemical News, October 31, 1873. \ Cf. F. W."Clarke, " Evolution and the Spectroscope," Popular Sci- ence Monthly, January, 1873, p. 320 seq, Lockyer's recent investiga- tions have brought these views into great prominence. 34 CONCEPTS OF MODERN PHYSICS. subversive of it a principle of which it has recently been said .that " it holds the same place in chemistry that the law of gravitation does in astronomy." * This principle is known as the law of Avogadro or Ampere. It im- ports that equal volumes of all substances, when in the gaseous state and under like conditions of pressure and temperature, contain the same number of molecules whence it follows that the weights of the molecules are proportional to the specific gravities of the gases ; that, therefore, these being different, the weights of the molecules are different also ; and, inasmuch as the mole- cules of certain elementary substances are monatomic (i. e., consist of but one atom each), while the molecules of various other substances contain the same number of atoms, that the ultimate atoms of such substances are of different weights. The law of Avogadro, though, like all physical theories, an hypothesis, is believed to be the only hy- pothesis which is competent to account for the well- known variation of the volume of a gas inversely as the pressure (law of Boyle or Mariotte) and directly as the absolute temperature (law of Charles) as well as for the combination of gases in simple volumetric pro- portions (law of Gay-Lussac) ; and it has served as the basis of innumerable deductions respecting the forma- tion and transformation of chemical compounds which have thus far met with unfailing experimental verifica- tion. That this cardinal principle of modem theoretical chemistry is in utter and irreconcilable conflict with the first proposition of the atomo-mechanical theory is ap- parent at a glance. No reconciliation, certainly, is pos- sible on the hypothesis suggested by Graham. For * J. P. Cookc, The New Chemistry, p. 13. EQUALITY OF ELEMENTS OF MASS. 35 tliat accounts for differences of density by attributing to equal primordial atoms unequal volumes resulting from their occupancy of unequal spaces by virtue of differences in the velocities of movement with which the several kinds of atoms are supposed to be inalien- ably endowed. It accounts for inequalities in the vol- umes of equal masses, not for inequalities of mass in equal volumes, and can not serve as an explanation of the latter, unless it is supplemented by the further as- sumption to which, indeed, it lends little, if any, aid that some, if not all, of the molecules are compounds or aggregates of different degrees of complexity. Two masses or molecules of equal volumes can be of different densities or weights only if the number of units con- tained in one is different from the number of units in the other. But Avogadro's law constrains the chemist to assume that the molecules of various elementary sub- stances, notwithstanding the diversity of their weights, consist of the same number of atoms. Thus hydrogen and chlorine, whose molecular weights are two and seventy-one respectively, are both held to be diatomic, i. e., their molecules are held to consist of two atoms each. In the case of monads, or univalent elements, such as those just mentioned, the reasoning upon which this assumption rests is very simple. One volume of hydrogen combines with one volume of chlorine, form- ing two volumes of hydrochloric acid. Each volume of this compound, according to Avogadro's law, contains as many molecules as either volume of the constituent simple elements before combination ; the two volumes of the compound, therefore, contain twice as many molecules as either volume of the constituents. But, in each molecule of the compound, both hydrogen and chlorine are present, whence it follows that each mole- 36 CONCEPTS OF MODERN PHYSICS. cule of hydrogen, as well as eacli molecule of chlorine, must have contributed at least one atom to each mole- cule of hydrochloric acid, and thus mast have consisted of at least two atoms. The argument in the case of dyads (such as oxygen, sulphur, selenium, etc.), and other elements of still higher quantivalence, though somewhat less simple, is equally cogent upon the basis of Avogadro's law. It may be said that the law in question determines only the minimum number of atoms in each molecule, leaving the maximum indeterminate, so that, after all, the molecule of greater weight may be of correspond- ingly greater complexity. But here we encounter an obstacle presented by a branch of the atomic theory in physics the science of thermo-dynamics. Modern sci- ence regards heat as a form of energy as consisting in an agitation of the molecules or atoms whereof bodies are composed ; and, in the case of gaseous bodies at least, it discriminates between that part .of this energy which is exhibited in the form of temperature, attribut- ing it to translatory motions of the molecules, or rather of their centers of mass, and another part the internal energy, so called which is supposed to be dependent upon oscillatory or rotatory motions of their component atoms. It has been shown, experimentally, that the ratio of the specific heat of a gas at constant pressure to that at constant volume* falls short of the value assigned to it by the theory upon the supposition that all the heat imparted to a gaseous body is expended in producing a translatory motion of the molecules, the * The " specific heat " (i. e. the heat required to raise the tempera- ture of a unit of mass of any substance one degree) of a gas at constant pressure under which it expands, is necessarily greater than that at con- stant volume, because in the former case part of the heat is expended in the mechanical work of expansion. EQUALITY OF ELEMENTS OF MASS. 37 effect being expansion, or increased pressure, or both ; and this difference is accounted for by the assumption that part of the heat is converted into intramolecular agitation, i. e., into motions of the particles within the molecule which do not affect its position or action as a whole. Now, it is readily seen and has been shown by Clausius, Boltzmann, Maxwell, and others, that the en- ergy thus converted into intramolecular or interatomic agitation must increase as the complexity of the molec- ular constitution increases ; it would become enormous, therefore, if a molecule consisted of a number of atoms so great as to be sufficient to account for the differences between the molecular weights of the elements. The molecular weight of chlorine, for example, is 35*5 times as great as that of hydrogen ; and if these weights are in proportion to the number of atoms contained in each molecule, it becomes necessary to assume even grant- ing that hydrogen is strictly diatomic that each chlo- rine molecule is composed of no less than seventy-one atoms. But, if this assumption were valid, nearly all the heat imparted to chlorine would be absorbed, i. e., converted into internal energy, and its calculated spe- cific heat would far exceed the amount ascertained by actual experiment. There are thus difficulties not of a speculative, but of a purely physical and chemical nature, which render the indefinite multiplication of atoms within the mole- cule, so as to account for the diversity of molecular weights, wholly inadmissible. Several elementary sub- stances are known to conform to Avogadro's law only on the supposition that they are monatomic. Among them is mercury, whose molecular weight coincides with its atomic weight as established by all the chemi- cal tests applicable to it, including that of Dulong and 38 CONCEPTS OF MODERN PHYSICS. Petit's law. And it has been demonstrated by Kundt and Warburg* that the ratio of the specific heat of mercurial vapor at constant pressure to that at constant volume, as ascertained by experiment, is precisely equal to its value calculated upon the basis of the absolute simplicity of the mercurial molecule and of the non- absorption of any part of the heat in intramolecular action. In view of all this there seems to be no escape from the conclusion that the claim, according to which mod- ern physical science is throughout a partial and pro- gressive solution of the problem of reducing all physi- cal phenomena to a system of atomic mechanics, is very imperfectly, if at all, countenanced by the actual con- stitution of theoretical chemistry that this science, which is peculiarly conversant about atoms and their motions, is founded upon propositions destructive of the very basis upon which alone a consistent super- structure of atomic mechanics can be reared. And there appears to be little ground for the hope that these propositions may be speedily abandoned ; for, in the opinion of the most distinguished chemists of the day, such an abandonment w^ould throw the mass of chemical facts, laboriously ascertained by experiment and observation (induced, partly at least, by the propo- sitions in question) into a state of hopeless prescientific confusion. In reference to the speculations of those who seek to deduce the specific differences between the ultimate units of mass from differences between their supposed inalienable velocities of motion or amounts of latent energy, it is to be said, not only that they fail to afford a solution of the difficulties of theoretical chemistry in * Pogg. Ann., vol. clvii, p. 353. EQUALITY OF ELEMENTS OF MASS. 39 the presence of the inexorable demands of the mechan- ical theory, but also that the attribution of inalienable energy or motion to a given mass is repugnant to the fundamental postulate of the absolute indifference of mass to motion. Helmholtz and others have investi- gated the conditions of vortex motion in a perfectly homogeneous, incompressible and frictionless fluid, which (as Maxwell has shown) is of necessity continu- ous and can not be molecular or atomic. If these con- ditions could be realized, we should have constant but undistinguishable volumes of a permanently homoge- neous fluid, so called, endowed with constant quantities of inalienable motion. But no energy or motion can inhere essentially in distinct and separate masses (mole- cules or atoms) if, as the mechanical theory assumes, mass and motion are disparate if mass is indifferent to motion so as to remain the same whether in motion or at rest, and if motion is transferable from one mass to another. This is one of the points distinctly insisted upon by Sir Isaac Newton, the greatest among the founders of the mechanical theory. Newton distin- guishes between two kinds of force the force of iner- tia (vis inertiae\ and impressed force (vis impressd). The former alone according to him is vis insita, i. e., inheres in matter ; while of the latter he expressly says that "this. force consists in action alone and does not abide in the body after action." * * " Consislit liacc vis in actione sold, neque post aclioncm permanet in corpore" Phil. Nat. Priuc. Math., def. iv (ed. Le Sueur et Jacquier, vol. i, p. 4), CHAPTER IY. THE PROPOSITION THAT THE ELEMENTARY UNITS OF MASS ARE ABSOLUTELY HARD AND INELASTIC. FROM the essential disparity of mass and motion and the simplicity of the elementary units of mass it follows that these units are perfectly hard and inelastic. Elas- ticity involves motion of parts and can not, therefore, be an attribute of truly simple atoms. u The concept < elastic atom,' " justly observes Professor "Wittwer, " is a contradiction in terms, because elasticity presupposes parts the distances between which can be increased and diminished." * The early founders of the mechanical theory re- garded the absolute hardness of the component par- ticles of matter as an essential feature of the original order of nature. " It seems probable to me," says Sir Isaac Newton, " that God in the beginning formed mat- ter in solid, massy, hard, impenetrable, movable parti- cles of such sizes and figures, and with such other prop- erties and in such proportion to space as most conduced to the end for which he formed them ; and that these primitive particles being solids are incomparably harder than any porous bodies compounded of them ; even so * " Der Begriff ' elastisches Atom ' ist eine contradictio in adjcctis, da die Elasticitaet immer wieder Theile voraussetzt, die sich einander naeh- ern, die sich von einander entfernen koennen." Beitraege zur Molecu- larphysik, Schloemilch's Zcitschrift fuer Math, und Phys., vol. xv, p. 114. RIGIDITY OF ELEMENTS OF MASS. 41 very hard as never to wear or break in pieces ; no or- ^ dinary power being able to divide what God himself made one in the first creation." * Strangely enough, while the requirement, by the \\ mechanical theory, of the absolute rigi(lity of the ele- mentary units of mass is no less imperative than that of their absolute simplicity, it meets with an equally signal denial in modern physics. The most conspicuous among the hypotheses which have been devised since the general adoption of the modern theories of heat, light, electricity and magnetism, and the establishment of the doctrine of the conservation of energy, in order to afford consistent ground for the mechanical inter- pretation of physical phenomena, is that known as the kinetic theory of gases. In the light of this theory a gaseous body is a swarm of innumerable solid particles incessantly moving about with different velocities in rectilinear paths of all conceivable directions, the veloci- ties and directions being changed by mutual encounters at intervals which are short in comparison with ordi- nary standards of duration, but indefinitely long as com- pared with the duration of the encounters. It is readi- ly seen that these motions would soon come to an end if the particles were wholly inelastic, or imperfectly elastic. For in that case there would be loss of motion at every encounter. The assumed perpetuity of the motion of the particles, therefore, leads to the necessity of asserting their perfect elasticity. And this neces- sity results, not merely from the peculiar exigencies of the kinetic theory of gases, but also from the principle of the conservation of energy in its general application to the ultimate constituents of sensible masses, if these constituents are supposed to be in motion. In the case * Oplicks, fourth ed., p. 375. 42 CONCEPTS OF MODERN PHYSICS. of the collision of ordinary inelastic or partially elastic bodies there is a loss of motion which is accounted for by the conversion of the motion thus lost into an agita- tion of the minute parts composing the colliding bodies. But in atoms or molecules destitute of parts no such conversion is possible, and hence we are constrained to assume that the ultimate molecules of a gaseous body are absolutely elastic. The necessity of attributing perfect elasticity to the elementary molecules or atoms in view of the kinetic theory of gases has been expressly recognized by all its founders. " Gases," says Kroenig,* "consist of atoms which behave like solid, perfectly elastic spheres mov- ing with definite velocities in void space." This state- ment is adopted by Clausius f and emphasized by Max- well, the first part of whose essay, " Illustration of the Dynamical Theory of Gases," is a treatise " on the mo- tions and collisions of perfectly elastic spheres." J And the highest scientific authorities are equally explicit in declaring that the hypothesis of the atomic or molecular constitution of matter is in conflict with the doctrine of the conservation of energy, unless the atoms or mole- cules are assumed to be perfectly elastic. "We are forbidden," says Sir William Thomson,* " by the mod- ern theory of the conservation of energy to assume in- elasticity or anything short of perfect elasticity of the ultimate molecules, whether of ultra-mundane or mun- dane matter." Naturally, eminent advocates of the kinetic hy- pothesis have taxed their ingenuity in the search of * Pogg. Ann., vol. xcix, p. 316. f 75., vol. c, p. 353. \ Phil. Mag., 4th ser., vol. xix, p. 19. * Ib., vol. xlv, p. 321. RIGIDITY OF ELEMENTS OF MASS. 4.3 methods for the extrication of the mechanical theory from the dilemma in which it is thus involved. The most notable effort thus far made is that of Sir William Thomson, in the form of a conjecture suggested by the researches of Helmholtz,* respecting the properties of rotational motion in an absolutely homogeneous, in- compressible, perfect fluid, to which reference has al- ready been made in the preceding chapter. Thomson imagines the omnipresence of this fluid, and supposes that atoms are in fact vortex-rings formed by rotational movements within it. Such rings would be permanent, of invariable volume due to an invariable quantity of motion, though susceptible of a great variety of form ; and some of their features, such as their modes of im- plication, would be indestructible ; they would be ca- pable of being knotted on themselves or linked with other vortex-rings, but could never be unknotted or untied ; finally they would be incapable of interpenetra- tion or coalescence, and their mutual approaches would result in rebounds similar to the resilience of perfectly elastic bodies. While we willingly yield our homage to the sagaci- ty displayed in this attempt to relieve the mechanical theory from one of its most fatal embarrassments, it is to be feared that its success is altogether illusory. For, it seems to be evident that motion in a perfectly homo- geneous, incompressible and therefore continuous fluid is not sensible motion. All partition of such a fluid is purely ideal ; in spite of the displacement of any por- tion of it by another portion, a given space would at any moment present the same quantity of substance absolutely indistinguishable from that present there a * Cf. Crelle-Borchardt's Journal filer reinc und angcwandte Mathe- matik, vol. Iv, p. -25. M CONCEPTS OF MODERN PHYSICS. moment before. There would be no phenomenal dif- ference or change. A fluid both destitute and incapable of difference is as impossible a vehicle of real motion as pure space ; it is as useless for the purpose of ac- counting for the phenomena of material action as the quasi-material medium without inertia of which Roger Cotes said that it was not to be distinguished from a vacuum.* Again, as Maxwell has observed,! the vortex-ring atoms moving in the hypothetical fluid would lack the essential attribute of matter : inertia. Such atoms would consist, not in the substance of the omnipresent fluid, but simply in the motions induced therein. Of these motions the persistence of both mass and energy would have to be predicated, and from them the concretions of mass, together with all the phenomena exhibited by sensible matter, would have to be derived. But that is impossible. From its very nature motion can not be the bearer of motion, nor can it, by itself, be the generator of momentum which is essentially the product of two antagonistic factors, and which would be utterly extin- guished by the suppression of either. Upon the basis of the mechanical theory, the fundamental antithesis between mass and motion, inertia and energy, can not be destroyed without an obliteration of all the distinctions which constitute the elements of our conceptions re- specting the nature of physical action. Another attempt, somewhat analogous to that of Sir William Thomson, to dispense with the necessity of en- * " Qui coelos materi& fluida rcpletos esse volunt, hanc vero non in- ertem esse statuunt, hi verbis tollunt vacuum, re ponunt. Nam cum hujusmodi materia fluida rationc nulla secerni possit ab inani spatio ; disputatio tota fit de rerum nominibus, non de naturis. Praef . in New- ton! Phil. Nat. Princ. Math.,ed. Lc Sucur & Jacquier, p. 25. f Encycl. Brit., ninth ed., s. v. Atom. RIGIDITY OF ELEMENTS OF MASS. 45 dowing the elementary atoms with the intrinsic prop- erty of elasticity has been made by A. Secchi. This distinguished physicist and astronomer also derives the resilience of the ultimate particles from their rotatory motion ; but his atoms, unlike those of Thomson, are real corpuscles separated by wide interstitial spaces, and not mere movements in a continuous and incompressible ^ethereal medium. Secchi clearly apprehends the inad- missibility of attributing elasticity to simple elementary atoms. "It is evident," he says,* "that, while it is possible to admit its existence in a compound molecule, the same thing can not be done in the case of elementary atoms. Indeed, elasticity in the received sense pre- supposes void spaces in the interior of the molecule whose form is changed by compression so as to return, afterward, to its original figure. Now, we regard the atoms as impenetrable, and not as groups of solid par- ticles ; hence they can not include void spaces which permit their dilatation and contraction. " In truth, what we call a molecule of a simple (i. e., chemically undecomposable) gas is not an elementary atom, or at least is not necessarily one. Inasmuch as this gaseous molecule is an aggregate of veritable atoms, it may well be that it has internal pores, and, generally, a number of properties which do not belong to its con- stituent atoms ; it is not absurd, therefore, to suppose it to be endowed with elasticity. Huygens has ad- mitted this hypothesis for the OBther. In his opinion the aethereal particles are composed of smaller ones ; but on closer examination it is seen that this is a mere shifting of the difficulty, and not a solution of it. We hope to be able to show that it is nowise necessary to accept such an elasticity as a primitive force, and that * L'unite des forces physiques, 2i 46 CONCEPTS OF MODERN PHYSICS. the apparent repulsion of the atoms and their reciprocal collisions can be simply referred to an appropriate mo- tion, it being sufficient for this purpose to suppose them to be in rotation. Let us prove this : " Among the beautiful theorems discovered by Poin- sot respecting the impact of bodies in rotation is found one relating to their reflection from a resisting obstacle. It teaches us that by virtue of its rotation alone a hard and inelastic body can rebound absolutely like a body perfectly elastic ; more than that : one of these bodies, thrown against a fixed obstacle, is often sent back with a velocity superior to its initial velocity. The profound mathematician shows how this phenomenon, paradoxi- cal as it seems, is due to the transformation of part of its rotatory motion into motion of translation ; whence results an increase of the velocity of the center of grav- ity. According to the ordinary theories of impact, in which no account is taken of the motion of rotation, the preceding proposition is absurd, and nevertheless it is perfectly established. Thus, by the side of cases of ordinary reflection we find the phenomena of progres- sion we might also, using the expression of Poinsot, call them negative reflections. " In negative reflection after impact, the center of gravity of the body returns with a velocity superior to that which it had at first. These questions form a wholly new and very interesting branch of mechanics ; they are easily demonstrated by compounding the two movements of rotation and translation, considered with reference to the centers of gravity, of rotation and of percussion; and we readily understand that generally it may be said : an impact, whatever it may be, can never simultaneously annihilate in a body the two mo- tions of rotation and of translation ; for, when the im- RIGIDITY OF ELEMENTS OF MASS. 47 pact is eccentric, it can destroy rotation and not trans- lation, and, when the direction of the impact passes through the center of gravity, it can annihilate transla- tion, but not rotation. Thus, the quantity of motion lost on the one side is gained on the other ; the rotation may either be reversed or simply accelerated, according to the point of the body which is struck ; whence the notion of centers of conversion. Examples of reflection succeeding the impact of bodies in rotation are found in the movements of disks and quoits, the impact of spinning-tops, etc. Billiard-players know perfectly how the rotation of the balls modifies the laws relating to the impact of elastic bodies as established in the ele- mentary treatises." * Unfortunately, the theory thus advanced finds little support in Poinsot's theorems. Secchi maintains that the impact of a rotating' body, when it is eccentric, " can destroy rotation, but not translation," and, when its direction passes through the center of gravity, " it can annihilate translation, but not rotation," so that in either case " the quantity of motion lost on the one side is gained on the other." f But, from a careful examina- tion of Poinsot's memoir, it appears that, after the colli- * The theorems to which Secchi refers are contained in the last of a series of memoirs (Questions Dynamiqucs sur la Percussion des Corps) contributed by M. Poinsot to Liouville's Journal de Mathematiques pures et appliquees, 2me serie, t. ii (1857), p. 281 seq., and t. iv (1859), p. 421 scq. This remarkable memoir was published (and probably writ- ten) by the octogenarian geometer shortly before his death ; the last in- stallment, indeed, was published after his death in the same number of Liouville's Journal which contained the addresses pronounced at his funeral by MM. Bertrand and Mathieu. t Secchi invariably speaks of loss or gain of " quantity of motion " ; but his argument requires that this should be interpreted as meaning loss or gain of energy. Whether or not this is his own meaning, I do not undertake to say. 4:8 CONCEPTS OF MODERN PHYSICS. sion of rotating inelastic bodies, their rotation, or trans- lation, or both are conserved, or the increase, diminu- tion or loss of the one is compensated for by the diminution, increase or gain of the other, only in cer- tain special cases. Poinsot shows * that, when a rotat- ing inelastic body encounters a fixed obstacle, it de- pends on the distance between the spontaneous center of rotation and the center of gravity whether the body shall be reflected with a translatory velocity greater than, equal to, or less than its initial velocity, or shall lose its translatory velocity altogether. In the first place, there are always, between the center of gravity and the center of percussion, " two points such that, if the rotating body strikes the obstacle in the line of either, its center of gravity will be reflected with an increased velocity." f I n * ne second place, " there are always, in every advancing rotating body, two points of perfect reflection, i. e., two points such that, if the body strikes an obstacle in the line of either, it will be reflected with a velocity perfectly equal to the velocity with which it is animated," J so that "the center of gravity of the body is reflected in space as though the body were perfectly elastic." But, when this occurs, the body loses, in the one case one third, and in the other two thirds of its velocity of rotation. Finally, in the third case, " if the obstacle is presented, either to the center of gravity or to the center of percussion, the velocity of translation is equally destroyed, the only difference between the two cases being that in the first case only the velocity of translation is destroyed with- out alteration of the velocity of rotation ; while in the * Liouville, Journal, etc., 2me serie, t. ii, p. 288 seq. f L. c., p. 304. 1 1* ' P- 805 - * L. c., p. 307. RIGIDITY OF ELEMENTS OF MASS. 49 second case both the velocity of translation and the velocity of rotation are annihilated." * The truth is, therefore, that in the only instances of perfect reflection specified by Poinsot there is a loss of either one third or two thirds of the rotatory motion not compensated for by any increase of translatory mo- tion, and that there are cases of impact in which both the motion of translation and that of rotation simulta- neously disappear.f That Secchi should have deemed it possible to de- volve the duty of conserving the energy of colliding atoms upon rotation as a substitute for the " occult quality " of perfect elasticity,, seems almost incredible when we come to consider the use he makes of his own theory. This theory, according to him, serves as an explanation of a number of things, among which are the formation of molecular aggregates from simple atoms, and the phenomena of gravitation. The aggregation of atoms, so as to form compound molecules, he explains thus : J " Suppose an extreme case, viz., the collision of * L. c., p. 308. f Although I have long since become utterly indifferent to questions and claims of priority, it may not be improper to say here that the fore- going pages were written before I had seen the very able pamphlet " Das Raethsel der Schwerkraft" (Braunschweig, Vieweg und Sohn, 1879), of D. C. Isenkrahe, with whom I am happy to find my*self in accord as to the validity of Secchi's attempt to deduce the property of perfect resilience from the rotation of inelastic bodies by the aid of Poinsot's exposition of the theory of rotation, although I can not, of course, accede to Isenkrahe's own theory of gravitation. There are other coincidences all the more interesting because they are, no doubt, wholly accidental between the criticisms contained in this pamphlet of Spiller's speculations and my es- timate of them which was first published in The Popular Science Month- ly, January, 1874. It is to be regretted that Isenkrahe, before publish- ing his essay, had not seen William B. Taylor's important memoir, herein- after referred to, on " Kinetic Theories of Gravitation." \ L'unite, etc., p. 51 seq. 5 50 CONCEPTS OF MODERN PHYSICS. two atoms endowed solely with translation, or, again, impinging on one another so that they can not rebound " (which would happen if rotating atoms collided in the direction of their axes of rotation). " Evidently the atoms will remain united in the same way as the bodies called c hard ' by the mechanicians, and they will form a system animated by the movement of translation re- sulting from the two other movements. This system will be able to act like a single corpuscule whose mass is double, triple or generally a multiple of that of a simple atom according as two or a greater number of atoms are thus united. Here we have an obvious in- stance of an aggregate of atoms bound to each other, not by the influence of any sort of attraction, but by simple inertia." Judging from this passage, Secchi could hardly have been ignorant of the fact that the collision of rotating inelastic bodies does not always re- sult in pseudo-elastic resilience. And in its application to the phenomena of gravitation his theory is plainly destructive of its own foundations. He seeks to ac- count for gravitation upon the assumption that the density of the sethereal medium which surrounds all ponderable bodies or molecules increases from their cen- ters outward ; * and this increase of density is said to be * This supposition is identical with that of Sir Isaac Newton, who in his letter to Boyle (Newton's Works, ed. Horsley, vol. iv, p. 385 seq.}, speculating on the " cause of gravity " said : " I will suppose aether to consist of parts differing from one another in subtilty by infinite degrees ... in such a manner that from the top of the air to the surface of the earth, and again from the surface of the earth to the centre thereof, the aether is insensibly finer and finer. Imagine now any body suspended in the air or lying on the earth, and the aether being by the hypothesis grosser in the pores which are in the upper parts of the body than in those which are in the lower parts, and that grosser aether being less apt to be lodged in those pores than the finer asther below, it will endeavor to get out and give way to the finer aether below, which can not be with- out the bodies descending to make room above for it to go into." RIGIDITY OF ELEMENTS OF MASS. 51 a consequence of the progressive conversion of rotatory into translatory motion of the sethereal particles, so that these particles are perpetually driven from the " centers of agitation" outward. "Evidently," says Secchi,* "a center of agitation, even when it is single, provided it is animated by a movement sufficiently energetic and durable, may determine the agitation of an unlimited medium, and so modify it that the density, least at its center, increases in proportion as we approach the cir- cumference." Secchi assigns no reason why there should be a perpetual increase in the translatory motions of the sethereal particles at the expense of their rotatory mo- tions why the transformation should always, or gen- erally, be from rotatory into translatory motion and not conversely ; nor does he indicate the source of that " energetic and durable " agitation at the center which is said to be productive of a continual agitation of a boundless aethereal sphere ; so that his explanation of the phenomena of gravitation is of very questionable valid- ity. But, waiving this ; surely, if the rotatory motion of the hard particles is gradually transformed into trans- latory motion, there is, by his own showing, an end to their resilience, and we are again in full presence of the unsolved problem of the reconciliation between the per- petual impact of simple, hard, and therefore inelastic atoms and the conservation of their initial energy. The difficulty, then, appears to be inherent and in- soluble. There is no method known to physical sci- ence which enables it to renounce the assumption of the perfect elasticity of the particles whereof ponderable bodies and their hypothetical imponderable envelopes are said to be composed, however clearly this assump- tion conflicts with one of the essential requirements of the mechanical theory. * L. c., p. 538. CHAPTER Y. THE PROPOSITION THAT THE ELEMENTARY UNITS OF MASS ABE ABSOLUTELY INERT. MASS and motion being mutually inconvertible, mass is absolutely inert. It can induce motion in another mass only by transferring a part or the whole of its own motion. And, inasmuch as motion can not exist by itself, but requires mass as its necessary substratum, such transference can not take place unless the masses between which it occurs are in contact. All physical action, therefore, is by impact ; action at a distance is impossible ; there are in nature no pulls, but only thrusts ; and all force is not merely (in the language of ^Newton) vis impressa, but vis a tergo. The necessity of reducing all physical action to im- pact has been a persistent tenet among physicists ever since the birth of modern physical science. And yet, here again, as in the cases discussed in the two preced- ing chapters, science rises in revolt against its own fun- damental assumptions. Its first and greatest achieve- ment was Newton's reduction of all the phenomena of celestial motion to the principle of universal gravita- tion to the principle that all bodies whatever attract each other with a force proportional directly to their masses and inversely to the squares of the distances be- tween them. That the doctrine of universal gravitation, in the INERTIA OF ELEMENTS OF MASS. 53 sense of an attraction at a distance without the inter- vention of a medium capable of propagating mechanical impulses, is at variance with the elements of the me- chanical theory was felt by no one more distinctly than by Xewton himself. At the very outset of his Prin- cipia he carefully guarded against the imputation that he looked upon gravity as an essential and inherent attribute of matter or believed the mutual attraction of bodies to be an ultimate physical fact. The force which urges bodies in their central approach was to him, as he expressly says, a purely mathematical concept in- volving no consideration of real and primary physical causes.* And, evidently apprehensive lest this dis- claimer should, after all, be lost sight of, he repeated it, in terms no less explicit, at the close of his great work. " The reason of these properties of gravity," he said, " I have not, as yet, been able to deduce ; and I frame no hypotheses." f If, after this, there were still room for doubt as to Newton's opinions respecting the nature of gravity, it would be removed by the well-known passage in his third letter to Bentley. " It is incon- ceivable that inanimate brute matter should, without the mediation of something else which is not material, operate upon and affect other matter, without mutual contact, as it must do if gravitation, in the sense of Epicurus, be essential and inherent in it. And this is the reason why I desired you would not ascribe innate gravity to me. That gravity should be innate, inherent and essential to matter, so that one body may act upon * " Mathcmaticus duntaxat est hie conccptus. Nam virium causas et scdes physicas jam non cxpendo" Princ., Def . viii. f " Rationem vero kamm gravitatis proprietatum nondumpotui dedu- ccre ; et hypotheses non jingo." Princ., Schol. Gen. ad fin. The same disclaimer is implied in the words of a scholium to the 29th Theorem, Prop. 69, Book I, of the Printipia. 54 CONCEPTS OF MODERN PHYSICS another at a distance, through a vacuum, without the mediation of anything else by and through which their action may be conveyed from one to another, is to me so great an absurdity that I believe no man, who has in philosophical matters .a competent faculty of thinking, can ever fall into it. Gravity must be caused by an agent acting constantly according to certain laws ; but, whether this agent be material or immaterial, I have left to the consideration of my readers." * There is still further evidence that Newton regarded universal gravitation as a secondary phenomenon, to be explained on the principles of ordinary impact or press- ure. In the later edition of his Opticks he pro- pounds certain " Queries " relating to the possibility of deducing some of the properties of light from the un- dulations of an all-pervading aether, and adds (Query 21) : " Is not this medium much rarer within the dense bodies of the sun, stars, planets, and comets, than in the empty celestial spaces between them ? And, in passing from them to great distances, doth it not grow denser and denser perpetually, and thereby cause the gravity of those great bodies towards one another, and of their parts towards the bodies ; every body endeavoring to go from the denser parts of the medium towards the rarer." f * Newton's Works, ed. S. Horsley, vol. iv, p. 438. Zoellner (Prin- cipien einer electrodynamischen Theorie der Materie, vol. i, preface) at- tempts to break the force of this and other passages in the writings of Newton, but, as It appears to me, wholly without avail. f Opticks, 4th cd., p. 325. The " Queries " appeared for the first time in the second edition of the Opticks, in the preface to which Newton again says : " To shew that I do not take gravity for an essential property of bodies, I have added one question concerning its cause, chus- ing to propose it by way of a question, because I am not satisfied about it for want of experiments." I have already cited in another place (SM- pra, p. 42) a similar exposition of his views in the letter to Boyle. INERTIA OF ELEMENTS OF MASS. 55 Notwithstanding these explicit declarations, New- ton's contemporaries took alarm at the apparent return of occult causes into the domain of physics. It is in- teresting to note the energy with which the philosophers and mathematicians of his day protested against the assumption of physical action at a distance. Huygens did not hesitate to say that " Newton's principle of at- traction^ appeared to him absurd." Leibnitz called it " an incorporeal and inexplicable power " ; John Ber- noulli, who sent to the Academy of Paris two essays, in which he sought to explain the movements of the plan- ets by an improved form of the Cartesian theory of vortices, denounced " the two suppositions of an at- tractive faculty and a perfect void " as u revolting to minds accustomed to receiving no principle in physics save those which are incontestable and evident." Nor did the principle of distant action find greater favor with the physicists and astronomers of a later genera- tion. Euler observed that the action of gravity must be due either to the intervention of a spirit or to that of some subtle material medium escaping the perception of our senses ; and he insisted that the latter was the only admissible alternative, although the exact demon- stration of the origin of gravitative force might be difficult or impossible.* His great rival and antagonist, D'Alembert, relegated gravity to that class of causes .productive of motion whose real nature is to us entirely unknown, in contradistinction to action by impact, of which we have a clear mechanical conception, f And, * Euler, " Thcoria motus corporum solidorum," p. 68. See also his " Lettres a une princesse d'Allemagne," No. 68. October 18, 1760. f D'Alembert, " Dynamique " (2me ed.), p. ix seq. It is well known how slowly and reluctantly the Newtonian philosophy found recognition and acceptance in France, where Cartesianism held undisputed sway al- most to the end of the eighteenth century. What the Cartesians gener- 56 CONCEPTS OF MODERN PHYSICS. in spite of the assertion of John Stuart Mill and others that the thinkers of our own time have emancipated themselves from the old prejudice against actio in dis- tans, it is easy to show that it is almost, if not quite, as prevalent now as it was two centuries ago. To cite but a few instances : Professor Challis, who has spent a number of years in the effort to establish a complete hydro-dynamical theory of attraction, says : " There is no other kind of force than pressure by contact of one body with another. This hypothesis is made on the principle of admitting no fundamental ideas that are not referable to sensation and experience. It is true that we see bodies obeying the influence of an external force, as when a body descends toward the earth by the action of gravity ; so far as the sense of sight informs us, we do not in such cases perceive either the contact or the pressure of another body. But we have also the sense of touch or of pressure by contact for in- stance, of the hand with another body and we feel in ourselves the power of causing motion by such press- ure. The consciousness of this power and the sense of touch give a distinct idea, such as all the world un- derstands and acts upon, as to how a body may be moved ; and the rule of philosophy which makes per- sonal sensation and experience the basis of scientific knowledge, as they are the basis of the knowledge that ally thought of the distant action of gravitation may be gathered from a paper read by Saurin to the Academic des Sciences in 1709, from which Edleston ("Correspondence between Newton and Cotes," p. 213) makes the following quotation : " II (Newton) airne mieux considerer la pesan- teur comme une qualite inherente dans les corps et ramener les idees taut decrie"es de qualite occulte et d'attraction." If we abandon mechan- ical principles (i. e., the principles of mechanical impact and propulsion), he continues, " nous voila rcplonges de nouveau dans les anciennes tone- bres du peripatetismc dont le ciel nous veuillc preserver." INERTIA OF ELEMENTS OF MASS. 57 regulates the common transactions of life, forbids rec- ognizing any other mode than this. When, therefore, a body is caused to move without apparent contact and pressure of another body, it must still be concluded that the pressing body, although invisible, exists, unless we are prepared to admit that there are physical operations which are, and ever will be, incomprehensible by us. This admission is incompatible with the principles of the philosophy I am advocating, which assume that the information of the senses is adequate, with the aid of mathematical reasoning, to explain phenomena of all kinds. . . . All physical force being pressure, there must be a medium by which the pressure is exerted." * With equal vigor the " assumption " of universal at- traction is reprobated as " an absurdity " by James Croll. " No principle " he contends, " will ever be generally received that stands in opposition to the old adage, i A thing can not act where it is not,' any more than it would were it to stand in opposition to that other adage, ' A thing can not act before it is or when it is not.' " f Secchi protests in almost the same words. " We have said elsewhere," he declares, " how impossi- ble it is to conceive what is called an attractive force in the strict sense of the term, that is, to imagine an ac- tive principle having its seat within the molecules and acting without a medium through an absolute void. This amounts to an admission that bodies act upon each other at a distance, that is, where they are not : an ab- surd hypothesis equally absurd in the case of enor- mous and in that of very small distances." J Friedrich * " On the Fundamental Ideas of Matter and Force in Theoretical Physics," Phil. Mag., 4th series, vol. xxxi, p. 467. f " On Certain Hypothetical Elements in the Theory of Gravitation," Phil. Mag., 4th series, vol. xxxiv, p. 450. \ L'unite, etc., p. 532 scq. 58 CONCEPTS OF MODERN PHYSICS. Mohr (who appears to be entitled to the honor of hav- ing distinctly announced the principle of the conser- vation of energy, even before Julius Robert Mayer) formulates his scientific creed in a number of " Theses," among which is this : " Gravity can not act except by the interposition of ponderable matter." * So also E. Du Bois-Reymond : " Forces acting through void space are in themselves inconceivable, nay absurd, and have become familiar concepts among physicists since New- ton's time from a misapprehension of his doctrine and against his express warning." f And finally Balfour Stewart and P. G. Tait : " Of course, the assumption of action at a distance may be made to account for any- thing ; but it is impossible (as Newton long ago pointed out in his celebrated letter to Bentley) for any one ' who has in philosophical matters a competent faculty of thinking ' for a moment to admit the possibility of such action." J The most conclusive evidence, however, of the re- pugnance between the assumption of distant attraction and the elementary concepts of mechanical action is found in the incessant renewal, by distinguished men since Newton's day, of the attempts to account for the phenomena ' of gravitation on the principles of fluid pressure or solid impact.* These attempts have recently * " Nonnisi materia ponderabili interposita attractio agere potest." Gcschichte der Erde, Appendix, p. 512. f Ueber die Grenzen des Naturerkennens, etc., p. 11. j The Unseen Universe, third ed. (1875), p. 100. * Some of these attempts are very ably discussed in a recent memoir by William B. Taylor : " Kinetic Theories of Gravitation," Smithsonian Report, 1876. This interesting essay, though quite exhaustive in the enumeration of the theories of English and French origin, might be sup- plemented by a collection of references to German articles and books on the same subject. See, e. g., Schramm, " Die allgemeine Bewegung und Materie," Wien, 1872; Aurel Anderssohn, "Die Mechanik der Gravita- INERTIA OF ELEMENTS OF MASS. 59 become reinvested with an extraordinary interest in consequence of the results of certain experiments of Professor Guthrie, who found that light bodies sus- pended near a vibrating disk were drawn toward it "as by an invisible chord" a. phenomenon which, as Sir William Thomson has pointed out, is explained by the fact that in a moving fluid the pressure is least where the average energy of motion is greatest.* In the eyes of modern physics all modes of action which appear to be propagated radially from a center are progressive oscillations in elastic media. It is natu- ral, therefore, to look for the physical cause of gravita- tion in the same direction. Numerous theories have been advanced in which gravitation is referred to the wave-motion- of an elastic interstellar and interatomic fluid similar to, or identical with, the luminiferous aether. The most noteworthy of these theories is that tion," Brcslau, 1874 (containing a photograph of the results of an ex- periment in which the effects of gravitation are simulated by a ball float- ing in water agitated by a series of radial impulses) ; " Zur Loesung des Problems ueber Sitz und Wesen der Anziehung" 47 Versammlung deutscher Natureforscher und Aerzte zu Breslau, 1874 ; Hugo Fritsch, " Theorie der Newton'schen Gravitation und des Mariotte'schen Gesetzes," Koenigsberg, 1874; Ph. Spiller, "Die Urkraft des Weltalls," Berlin, 1876, etc. It is somewhat strange that Mr. Taylor ahould have omitted all reference to Huygens's elaborate " Dissertatio de causa gravitatis " (Hugenii, Opp. Reliqua, vol. i, p. 95 seq., Amstelod., 1728), as well as to the equally elaborate theory of P. A. Secchi, to which allusion has al- ready been made in the fourth chapter. In our own country Professor Pliny Earle Chase has made large contributions to this class of literature. * Guthrie's experiments had been anticipated, without his knowledge, by Guyot, Schellbach, and others, as appears from a communication of Guthrie himself to the Philosophical Magazine (fourth series, vol. xli, p. 405 scq.). Experiments similar to those of Aurel Anderssohn were made long ago by Hooke and Huygens, both of whom showed that bodies floating on water agitated by waves were drawn toward the center of agitation. Cf. Hugenii, " Diss. de causa gravitatis," Opp. Reliqua, i, p. 99 scq. 60 CONCEPTS OF MODERN PHYSICS. of Professor Challis, who assumes that all space is filled with a vibrating aether which " is a continuous elastic medium perfectly fluid and pressing proportionally to its density." Challis, though very desirous of avoiding the cumulation of hypothetical media, and endeavoring to construe gravitative action as an incidental or resid- ual effect of the luminar and thermal vibrations (resort- ing, for this purpose, to investigations analogous to those of Daniel Bernoulli, who attempted, more than a century ago, to show that the relative motions of bodies composing a material system are compounds of simple, regular, and permanent oscillations of different kinds) is constrained at last to suggest that there may be an aether of a higher order " having the same relation to the first as that has to air, and so on ad libitum" and that " the form of gravity is due to the attractive action of a molecule of a higher order as to magnitude than the molecule of molecular attraction." I shall have occasion, in a subsequent chapter, to discuss the scientific value of theories of this sort, in which facts are explained by an indefinite number of arbitrary as- sumptions multiplied in proportion to the emergencies created by the theories themselves ; for the present it is sufficient to observe that all hydro-dynamical theories of gravitation are obnoxious to the fatal criticism of Arago : " If attraction is the result of the impulsion of a fluid, its action must employ a finite time in traversing the immense spaces which separate the celestial bod- ies," * whereas there is now no longer any reason to doubt that the action of gravity is instantaneous. If it were otherwise if gravity, like light or electricity, were propagated with a measurable velocity there would necessarily be a composition of this velocity with the * Astronomie populairc, vol. iv, p. 119. INERTIA OF ELEMENTS OF MASS. 61 angular orbital velocities of the planets resulting in their acceleration ; the apparent line of attraction would be directed to a point in advance of the real place of the sun, just as the sun's apparent position is displaced in the direction of the earth's orbital motion by the aberration of light. Such an effect, if it had any exist- ence, would have been detected long ago. There was a time when the action of gravity was supposed to be progressive. Daniel Bernoulli attributed the non-coin- cidence of the tides with the passage of the moon through the meridian to the comparative slowness of its propagation; and, at a later period, Laplace con- ceived for a moment that the gradual acceleration of the moon's mean motion (first ascertained by Halley by a comparison of modern lunar eclipses with those re- corded by Ptolemy and the Arabian astronomers) might find an explanation in the transmission of the impulse of gravity with a velocity exceeding that of light not less than eight million times. But the retardation of the tides is now known to be a consequence of the iner- tia of the water and of the obstacles which it encoun- ters in its flow; and the acceleration of the moon's motion was soon shown, by Laplace himself, to be caused, in great part at least, by the secular diminution of the eccentricity of the orbit of the earth. For this reason Laplace did not hesitate to declare that, if the action of gravity was propagated in time, its velocity must be at least fifty million times greater than that of light. It is true that the cause assigned by him for the phenomenon in question has since been found to be inadequate to its production. From a revision of the calculations of the French astronomer by Mr. Adams, some years ago, it appeared that the diminution of the eccentricity of the earth's orbit could at best account 62 CONCEPTS OF MODERN PHYSICS. for a lunar acceleration of six seconds in one century, instead of ten seconds, the amount of acceleration as- sumed by Laplace, and furthermore, that the accelera- tion amounted in fact to nearly twelve seconds. A part of the phenomenon, therefore, had to be traced to other causes ; and this has been successfully effected by show- ing its dependence upon the tidal retardation of the diurnal motion of the earth which occasions an appar- ent acceleration of the mean motion of the moon. There is thus an entire failure of analogy, in this respect, between the action of gravity and the other known modes of physical action that are referred to sethereal undulations, such as light, radiant heat and electricity, all of which are propagated with a finite velocity. There are, moreover, as Mr. Taylor has ob- served, other features of gravitation which give rise to the presumption that it is of a nature essentially differ- ent from that of other forms of radial action. The action of gravity is wholly unsusceptible of interfer- ence by intervening obstacles, or, as Jevons expresses it,* " all bodies are, as it were, absolutely transparent to it ; " its direction is in right lines between the cen- ters of the attracting masses, and is not subject to re- flection or refraction; unlike the forces of cohesion, capillarity, chemical affinity and electric or magnetic attraction, it is incapable of exhaustion, or rather satu- ration, every body attracting every other body in pro- portion to its mass; it is wholly independent of the nature, volume, or structure of the bodies between which it occurs, and its energy is unchangeable, inces- sant and inexhaustible. On the whole, it may be safely said that the undu- lations of a supposed cosmical aether can not be made * Principles of Science, vol. ii, p. 144. INERTIA OF ELEMENTS OF MASS. 63 available as a basis for a physical theory of gravitation, and that, if such a theory is to be framed, resort must be had to the analogies of the kinetic theory recently introduced into the science of thermo -dynamics. This is very frankly admitted by leading physicists of the day. " All attempts yet made," say Stewart and Tait,* " to connect gravitation with the luminiferous aether, or the medium required to explain electric and mag- netic distance-action, have completely failed, so that we are apparently driven to the impact theory as the only possible one." The only impact theory seriously discussed by modern physicists and astronomers is that of Le Sage,f which, stated in a few words, is this : Space is constantly traversed in all directions by streams of infinitely small bodies moving with an al- most infinite velocity and coming from unknown re- gions of the universe. These bodies are termed " ultramundane corpuscules." By reason of their mi- nuteness they rarely, if ever, collide, and the greater part of them find ready passage through ordinary sensible bodies, so that all parts of these bodies those in the interior as well as those on the surface are equally liable to be struck by the corpuscules, the force of the impact being thus proportional, not to the sur- faces, but to the masses of the bodies. A single body or particle would be equally battered by these corpus- * The Unseen Universe, 140. ) Arago suggests (Astr. pop., iv, p. 118) that the theory of Le Sage is simply a reproduction, in an improved form, of the systematic ideas of Fatio de Duillers (the insane and meddlesome partisan of Newton in his controversy with Leibnitz respecting the priority in the invention of the differential calculus) and Varignon, which had been communicated to Le Sage before their publication. But this is probably an error ; Vari- gnon's speculations, at least, were similar to those of Newton in the 21st Query of his " Opticks." 64: CONCEPTS OF MODERN PHYSICS. cules on all sides ; but any two bodies act as mutual screens, so that each receives a less number of impacts on the side facing the other. They are consequently driven toward each other. The motion of the. corpus- cules being rectilinear in all directions, the diminution of pressure thus resulting is inversely as the squares of the distances between the bodies affected. With all deference due to the authority of the scien- tific men by whom this theory has been countenanced, it must be said that the extravagance of its assumptions at once characterizes it as a survival of the fancies of an age in which the functions of a scientific theory were imperfectly understood. Its intellectual consanguinity with the old vortices and harmonic circulations is un- mistakable. It utterly ignores the necessity of account- ing for the origin of the enormous energy constantly expended by the supposed streams of ultramundane corpuscules ; both the agency postulated and the mode of its action are unknown to experience ; and it is doubt- ful whether its assumptions, if they could be granted, would serve as an explanation of all or any of the feat- ures of gravitation in the presence of which, as we have seen, every hydro-dynamic theory is doomed to failure. The futility of Le Sage's theory, however, is most strik- ingly exhibited by Clerk Maxwell,* who tests it by the principle of the conservation of energy. If the ultra- mundane corpuscules impinging upon sensible bodies are perfectly elastic and rebound with the same velocity with which they approach, they will " carry their energy with them into the ultramundane regions," and in that event " the corpuscules rebounding from the body in any given direction will be both in number and velocity exactly equivalent to those which are prevented from * Encyclopaedia Britannica, s. v. " Atom." INERTIA OF ELEMENTS OF MASS. 65 proceeding in that direction by being deflected by that body, whatever the shape of the body and however many bodies are in the field." In this case, therefore, there is no> gravitative action. If, on the other hand, the corpuscules are inelastic, or imperfectly elastic inasmuch as the action of gravity is supposed to be due to the comparatively small difference between the im- pacts on opposite sides of the body the energy of those impacts, at least, which balance each other, must be (partially or wholly, according to the degree of corpus- cular elasticity) converted into heat, and " the amount of heat so generated would in a few seconds raise the body, and in like manner the whole material universe, to a white heat." * Once more, then, science is in irreconcilable conflict with one of the fundamental postulates of the mechani- cal theory. Action at a distance, the impossibility of which the theory is constrained to assert, proves to be an ultimate fact inexplicable on the principles of impact and pressure of bodies in immediate contact. And this fact is the foundation of the most magnificent theoreti- cal structure which science has ever erected a founda- tion deepening with every new reach of our telescopic vision, and broadening with every further stretch of mathematical analysis. * Mr. S. Tolver Preston has recently (Phil. Mag., September and November, 1877, and February and May, 1878) proposed a modification of Le Sage's theory, in which he seeks to dispense with the ultramun- dane teature of the corpuscules, and to account for gravitation upon the postulate of the kinetic theory of gases alone. His theory is founded on the assumptions that " the range of gravity is limited," and that " the stars move in straight lines and not in orbits." In view of these assump- tions, and of my discussion of the kinetic theory of gases in a separate chapter, I do not deem it necessary to devote any space to it here. CHAPTER VI. THE PROPOSITION THAT ALL POTENTIAL ENEEGY IS IN REALITY KINETIC. EVOLUTION OF THE DOCTRINE OF THE CONSERVATION OF ENERGY. ACCORDING to the mechanical theory, motion, like mass, is indestructible and. unchangeable; it can not vanish and reappear. Any change in its rate results from its distribution among a greater or less number of units of mass. And, motion and mass being mutually inconvertible, nothing but motion can be the cause of motion. There is, therefore, no potential energy ; all energy is in reality kinetic. The close logical connection of this proposition with that discussed in the last chapter is obvious, and has not escaped the notice of leading physicists. Stewart and Tait, after giving an account of Le Sage's hypothe- sis, which, in their opinion, contains the rudiments, at least, of the only tenable physical theory of gravita- tion, proceed to say : " If Le Sage's theory, or any- thing of a similar nature, be at all a representation of the mechanism of gravitation, a fatal blow is dealt to the notion of the tranquil form of power we have called potential energy. Not that there will cease to be a profound difference in kind between it and ordi- nary kinetic energy, ~but that BOTH will be henceforth to i)e regarded as kinetic" * This declaration has re- * The Unseen Universe, 142. CONSERVATION OF ENERGY. 67 cently been repeated by Professor Tait in his lecture on Force.* The proposition here insisted upon is irrecusable by any consistent advocate of the mechanical theory. But, again, modern science peremptorily refuses its assent. It asserts that all, or nearly all, physical changes in the universe are mutual conversions of kinetic and poten- tial energies that energy is incessantly stored as virtual power and restored as actual motion. When the bob of an ordinary pendulum descends from its highest to its lowest point, its potential energy diminishes in pro- portion to the increase of its actual motion; when it rises again, its energy of motion disappears at the same rate up to its arrival at its highest point opposite the first, where it is for an instant motionless, all its energy being due to its position. And this conversion and re- conversion of the two forms of energy are typical alike of the supposed oscillations of the ultimate atoms or molecules and of the orbital swing of the large bodies composing a planetary system. A planet moving in an excentric orbit gains energy of motion as it approaches the sun and loses it again in the same proportion as it recedes from it.- The same mutual transformation is exhibited in another wide domain of physical phe- nomena : action due to chemical affinity. A lump of coal lies buried in the earth for a million years ; during all this time there is no appreciable change in its posi- tion as referred to surrounding objects, or in the relative positions of its parts it is without external or internal motion (except that which it shares with the planet of which it is a part) ; now we bring it to the surface, into the atmosphere containing oxygen and into contact with * On some Recent Advances in Physical Science, second ed., pp. 262, 263 68 CONCEPTS OF MODERN PHYSICS. a flame ; its latent power at once becomes sensible it burns, giving rise to vigorous action which manifests itself as light and heat. The tendency of modern sci- ence is to trace all physical change to a few primary forms of potential energy, chief among which are grav- ity and chemical affinity. In the opinion of modern physicists, the only plausible theory, thus far advanced, of the origin of stellar and planetary systems is that known as the nebular hypothesis; and, whether we adopt its familiar Kant-Laplacean form, or one of its more recent modifications, in either case all the molar, if not the molecular, forces of the universe are ulti- mately derived from the attraction due to the mere position of the original particles supposed to be uni- formly diffused in space. And all changes in the comparatively minute organic or inorganic forms are referred, proximately at least, in physiology as well as in physics, to the affinities of the chemical ele- ments. In truth, modern science teaches that diversity and change in the phenomena of nature are possible only on condition that energy of motion is capable of being stored as energy of position. The relatively perma- nent concretion of material forms, chemical action and reaction, crystallization, the evolution of vegetal and animal organisms all depend upon the " locking up " of kinetic action in the form of latent energy. To make this clear, and to show that the effort to abolish the distinction between kinetic and potential energy is without avail, it will be useful briefly to review the history of the doctrine of the conservation of energy. In a general sense, this doctrine is coeval with the dawn of human intelligence. It is nothing more than an application of the simple principle that nothing can CONSERVATION OF ENERGY. 69 come from or to nothing.* But the history of its de- velopment and adhibition in physical science begins with its emphatic statement in the " Principia Philosophiae " of the inventor of the system of cosmical vortices, f * It may be truly asserted that human intelligence begins and ends with the principle above stated. When all the phenomenal changes in the universe shall have been reduced to the one principle of the conser- vation of energy, the time will have come for celebrating the final con- summation of physical science in a new epic " de rerum naturd ; " and in its first chapter will again be written the words of Lucretius : "... res . , . non posse creari De nihilo, neque item genitas in nil revocari." It is not a little curious to note the unaminity and emphasis with which the early Greek philosophers gave utterance to the declaration that noth- ing could absolutely originate or perish the rudimentary form of the law of causality. Diogenes of Apollonia declared : " ovSev CK rov /ir/ vvros yiveffbai" (Diog. Laert., ix, . 57) ; Parmenides: "&s ayeverov &v KO.I av6\&pov effTiv" (Karsten, Rel., v, 58); Empedocles : "e/c rov yap uf] *6vros a^x^ov iffrl yevto-frai. " (Karsten, v, 48) ; Democritos : " /irjSeV r' e/c rov p.^ ovros yive/\ In discussing these claims, it is important, first, to verify the facts and to reduce the statements of these facts to exact expression, and then to see how far they are fused by the theory. 1. The indestructibility of matter is an unquestion- able truth. But in what sense, and upon what grounds, is this indestructibility predicated of matter ? The unan- imous answer of the atomists is: Experience teaches that all the changes to which matter is subject are but variations of form, and that amid these variations there is an unvarying constant the mass or quantity of mat- ter. The constancy of the mass is attested by the bal- ance, which shows that neither fusion nor sublimation, neither generation nor corruption, can add to or detract from the weight of a body subjected to experiment. THE ATOMIC THEORY. 37 When a pound of carbon is burned, the balance demon- strates the continuing existence of this pound in the carbonic acid, which is the product of combustion, and from which the original weight of carbon may be re- covered. The quantity of matter is measured by its weight, and this weight is unchangeable. Such is the fact, familiar to every one, and its in- terpretation equally familiar. To test the correctness of this interpretation, we may be permitted slightly to vary the method of verifying it. Instead of burning the pound of carbon, let us simply carry it to the sum- mit of a mountain, or remove it to a lower latitude ; is its weight still the same ? Relatively it is ; it will still balance the original counterpoise. But the " absolute weight" is no longer the same. This appears at once, if we give to the balance another form, taking a pendu- lum instead of a pair of scales. The pendulum on the mountain or near the equator swings more slowly than at the foot of the mountain or near the pole, for the reason that it has become specifically lighter by being farther removed from the center of the earth's attrac- tion, in conformity to the law that the attractions of bodies vary inversely as the squares of their distances. It is thus evident that the constancy, upon the ob- servation of which the assertion of the indestructibility of matter is based, is simply the constancy of a relation, and that the ordinary statement of the fact is crude and inadequate. Indeed, while it is true that the weight of a body is a measure of its mass, this is but a single case of the more general fact that the masses of bodies are inversely as the velocities imparted to them by the action of the same force, or, more generally still, in- versely as the accelerations produced in them by the same force. In the case of gravity, the forces of attrac- 88 CONCEPTS OF MODERN PHYSICS. tion are directly proportional to the masses, so that the action of these forces (weight) is the simplest measure of the relation between any two masses as such ; but, in any inquiry relating to the validity of the atomic the- ory, it is necessary to bear in mind that this weight is not the equivalent, or rather presentation, of an abso- lute substantive entity in one of the bodies (the body weighed), but the mere expression of a relation between two bodies mutually attracting each other. And it is further necessary to remember that this weight may be indefinitely reduced, without any diminution in the mass of the body weighed, by a mere change of its position in reference to the body between which and the body weighed the relation subsists. Masses find their true and only measure in the ac- tion of forces, and the persistence of the effect of this action is the simple and accurate expression of the fact which is ordinarily described as the indestructibility of matter. It is obvious that this persistence is in no sense explained or accounted for by the atomic hypothesis. It may be that such persistence is an attribute of the minute, insensible particles which are supposed to con- stitute matter, as well as of sensible masses; but, surely, the hypothetical recurrence of a fact in the atom is no explanation of the actual occurrence of the same fact in the conglomerate mass. Whatever mystery is involved in the phenomenon is as great in the case of the atom as in that of a solar or planetary sphere. Breaking a magnet into fragments, and showing that each fragment is endowed with the magnetic polarity of the integer magnet, is no explanation of the phenomenon of mag- netism. A phenomenon is not explained by being dwarfed. A fact is not transformed into a theory by being looked at through an inverted telescope. The THE ATOMIC THEORY. 89 hypothesis of ultimate indestructible atoms is not a necessary implication of the persistence of weight, and can at best account for the indestructibility of matter if it can be shown that there is an absolute limit to the compressibility of matter in other words, that there is an absolutely least volume for every determinate mass. This brings us to the consideration of that general prop- erty of matter which probably, in the minds of most men, most urgently requires the assumption of atoms its impenetrability. " Two bodies can not occupy the same space " this is the ordinary statement of the fact in question. Like the indestructibility of matter, it is claimed to be a datum of experience. " That all bodies are impene- trable," says Sir Isaac Newton, " we gather, not from reason, but from sense." * Let us see in what sense and to what extent this claim is legitimate. The proposition, according to which a space occu- pied by one body can not be occupied by another, im- plies the assumption that space is an absolute, self- measuring, objective entity, and the further assumption that there is a least space which a given body will abso- lutely fill so as to exclude any other body. A verifica- tion of this proposition by experience, therefore, must amount to proof that there is an absolute limit to the compressibility of all matter whatsoever. Now, does experience authorize us to assign such a limit ? As- suredly not. It is true that in the case of solids and liquids there are practical limits beyond which compres- sion by the mechanical means at our command is im- possible ; but even here we are met by the fact that the volumes of fluids, which effectually resist all efforts at * " Corpora omnia impcnctrabilia csse, non rationc, sed sensu colligi- mus." Phil. Nat. Princ. Math., lib. iii, reg. 3. 00 CONCEPTS OF MODERN PHYSICS. further reduction by external pressure, are readily re- duced by mere mixture. Thus, sulphuric acid and water at ordinary temperatures do not sensibly yield to pressure ; but, when they are mixed, the resulting vol- ume is materially less than the aggregate volumes of the liquids mixed. But, waiving this, as well as the phenomena which emerge in the processes of solution and chemical action, it must be said that experience does not in any manner vouch for the impenetrability of matter in all its stages of aggregation. When gases are subjected to pressure, the result is simply an increase of the expansive force in proportion to the pressure ex- erted, according to the law of Boyle or Mariotte (the modifications of and apparent exceptions to which, as exhibited in the experimental results obtained by Re- gnault and others, need not here be stated, because they do not affect the argument). A definite experi- mental limit is reached in the case of those gases only in which the pressure produces liquefaction or solidifi- cation. The most significant phenomenon, however, which experience contributes to the testimony on this subject, is the diffusion of gases. Whenever two or more gases which do not act upon each other chemical- ly are introduced into a given space, each gas diffuses itself in this space as though it were alone present there ; or, as Dalton, the reputed father of the modern atomic theory, expresses it, " Gases are mutually pas- sive, and pass into each other as into vacua." Whatever reality may correspond to the notion of the impenetrability of matter, this impenetrability is not, in the sense of the atomists, a datum of experience. Upon the whole, it would seem that the validity of the first proposition of the atomic theory is not sus- tained by the facts. Even if the assumed unchange- THE ATOMIC THEORY. 91 ability of the supposed ultimate constituent particles of matter presented itself, upon its own showing, as more than a bare reproduction of an observed fact in the form of an hypothesis, and could be dignified with the name of a generalization or of a theory, it would still be obnoxious to the criticism that it is a generalization from facts crudely observed and imperfectly appre- hended. In this connection it may be noted that the atomic theory has become next to valueless as an explanation of the impenetrability of matter, since it has been pressed into the service of the undulatory theories of radiance, and assumed the form in which it is now held by the majority of physicists, as we shall presently see. According to this form of the theory, the atoms are either mere points, wholly without extension, or their dimensions are infinitely small as compared with the distances between them, whatever be the state of aggre- gation of the substances into which they enter. In this view the resistance which a body, i. e., a system of atoms, offers to the intrusion of another body is due, not to the rigidity or unchangeability of volume of the individual atoms, but to the relation between the at- tractive and repulsive forcos with which they are sup- posed to be endowed. There are physicists holding this view who are of opinion that the atomic constitu- tion of matter is consistent with its penetrability among them M. Cauchy, who, after defining atoms as "material points without extension," uses this lan- guage : " Thus, this property of matter which we call impenetrability is explained, when we consider the atoms as material points exerting on each other attrac- tions and repulsions which vary with the distances that separate them. . . . From this it follows that, if it 92 CONCEPTS OF MODERN PHYSICS. pleased the author of Nature simply to modify the laws according to which the atoms attract or repel each other, we might instantly see the hardest bodies pene- trating each other, the smallest particles of matter oc- cupying immense spaces, or the largest masses reduc- ing themselves to the smallest volumes, the entire uni- verse concentrating itself, as it were, in a single point." * 2. The second fundamental proposition of the mod- ern atomic theory avouches the essential discontinuity of matter. The advocates of the theory affirm that there is a series of physical phenomena which are inex- plicable, unless we assume that the constituent particles of matter are separated by void interspaces. The most notable among these phenomena are the dispersion and polarization of light. The grounds upon which the assumption of a discrete molecular structure of matter is deemed indispensable for the explanation of these phenomena may be stated in a few words. According to the undulatory theory, the dispersion of light, or its separation into spectral colors, by means of refraction, is a consequence of the unequal retarda- tion experienced by the different waves, which produce the different colors, in their transmission through the refracting medium. This unequal retardation presup- * " Ainsi, cctte propriete de la matiere quc nous nommons irnpene- trabilite se trouve expliquee, quand on considere les atomes comme des points materiels qui excrcent les uns sur les autres des attractions ou repulsions variables avcc les distances qui les separent. ... II resulte encore de ce qui precede, que s'il plaisait a 1'auteur de la nature de modi- fier seulement les lois suivant lesquelles les atomes s'attirent ou se re- poussent, nous pourrions voir, & 1'instant meme, les corps les plus durs se penetrer les uns les autres,' les plus petites parcelles de matiere occuper des espaces demesures, ou les masses les plus considerables se reduire aux plus petits volumes, et 1'univers entier se conccntrer pour ainsi dire en un seul point." Sept Le9ons de Physique Generate, ed. Moigno, p. 38 seq. THE ATOMIC THEORY. 93 poses differences in the velocities with which the various colored rajs are transmitted through any medium .what- ever, and a dependence of these velocities upon the lengths of the waves. But, according to a well-estab- lished mechanical theorem, the velocities with which undulations are propagated through a continuous me- dium depend solely upon the elasticity of the medium as compared with its inertia, and are wholly independ- ent of the length and form of the waves. The correct- ness of this theorem is attested by experience in the case of sound. Sounds of every pitch travel with the same velocity. If it were otherwise, music heard at a distance would evidently become chaotic ; differences of velocity in the propagation of sound would entail a distortion of the rhythm, and, in many cases, a reversal of the order of succession. Now, differences of color are analogous to differences of pitch in sound, both reducing themselves to differences of wave-length. The lengths of the waves increase as we descend the scale of sounds from those of a higher to those of a lower pitch ; and similarly, the length of a luminar undulation increases as we descend the spectral scale, from violet to red. It follows, then, that the rays of different color, like the sounds of different pitch, should be propagated with equal velocities, and be equally re- fracted ; that, therefore, no dispersion of light should take place. This theoretical impossibility of dispersion has al- ways been recognized as one of the most formidable difficulties of the undulatory theory. In order to ob- viate it, Cauchy, at the suggestion of his friend Coriolis, entered upon a series of analytical investigations, in which he succeeded in showing that the velocities with O which the several colored rays are propagated may vary 94: CONCEPTS OF MODERN PHYSICS. according to the wave-lengths, if it be assumed that the sethereal medium of propagation, instead of being con- tinuous, consists of particles separated by sensible dis- tances. By means of a similar assumption, Fresnel has sought to remove the difficulties presented by the phenomena of polarization. In ordinary light, the different undu- lations are supposed to take place in different directions, all transverse to the course or line of propagation, while in polarized light the vibrations, though still transverse to the ray, are parallelized, so as to occur in the same plane. Soon after this hypothesis had been expanded into an elaborate theory of polarization, Poisson ob- served that, at any considerable distance from the source of the light, all transverse vibrations in a continuous elastic medium must become longitudinal. As in the case of dispersion, this objection was met by the hy- pothesis of the existence of " finite intervals " between the sethereal particles. These are the considerations, succinctly stated, which theoretical physics are supposed to bring to the support of the atomic theory. In reference to the cogency of the argument founded upon them, it is to be said, gen- erally, that evidence of the discrete molecular arrange- ment of matter is by no means proof of the alternation of unchangeable and indivisible atoms with absolute spatial voids. But it is to be feared that the argument in question is not only formally, but also materially, fallacious. It is very questionable whether the assump- tion of " finite intervals " between the particles of the luminiferous aether is competent to relieve the undula- tory - theory of light from its embarrassments. This subject, in one of its aspects, has been thoroughly dis- cussed by E. B. Hunt, in an article on the dispersion of THE ATOMIC THEORY. 95 light,* and the suggestions there made appear to me worthy of serious attention. They are briefly these : M. Cauchy brings the phenomena of dispersion within the dominion of the undulatory theory, by deducing the differences in the velocities of the several chromatic rays from the differences in the corresponding wave-lengths by means of the hypothesis of definite intervals between the particles of the light-bearing medium. He takes it for granted, therefore, that these chromatic rays are prop- agated with different velocities. But is this the fact ? Astronomy affords the means to answer this question. We experience the sensation of white light when all the chromatic rays of which it is composed strike the eye simultaneously. The light proceeding from a lu- minous body will appear colorless, even if the compo- nent rays move with unequal velocities, provided all the colored rays, which together make up white light, eoncur in their action on the retina at a given moment ; in ordinary cases it is immaterial whether these rays have left the luminous body successively or together. But it is otherwise when a luminous body becomes visi- ble suddenly, as in the case of the satellites of Jupiter, or Saturn, after their eclipses. At certain periods, more than forty-nine minutes are requisite for the transmis- sion of light from Jupiter to the earth. Now, at the moment when one of Jupiter's satellites, which has been eclipsed by that planet, emerges from the shadow, the red rays, if their velocity were the greatest, would evidently reach the eye first, the orange next, and so on through the chromatic scale, until finally the comple- ment of colors would be filled by the arrival of the vio- let ray, whose velocity is supposed to be the least. The satellite, immediately after its emersion, would appear * Silliman's Journal, 2d scries, vol. vii, p. 364 scq. 96 CONCEPTS OF MODERN PHYSICS. red, and gradually, in proportion to the arrival of the other rays, pass into white. Conversely, at the begin- ning of the eclipse, the violet rays would continue to arrive after the red and other intervening rays, and the satellite, up to the moment of its total disappearance, would gradually shade into violet. Unfortunately for Cauchy's hypothesis, the most careful observation of the eclipses in question has failed to reveal any such variations of color, either before im- mersion or after emersion, the transition between light and darkness taking place instantaneously and without chromatic gradations. Astronomy points to several other phenomena which are equally at war with the doctrine of unequal veloci- ties in the movements of the chromatic undulations. Fixed stars beyond the parallactic limit, whose light must travel more than three years before it reaches us, are subject to great periodical variations of splendor ; and yet these variations are unaccompanied by varia- tions of color. Again, the assumption of different velocities for the different chromatic rays is discoun- tenanced by the theory of aberration. Aberration is due to the fact that, in all cases where the orbit of the planet, on which the observer is stationed, forms an angle with the direction of the luminar ray, a composi- tion takes place between the motion of the light and the motion of the planet, so that the direction in which the light meets the eye is a resultant of the two component directions the direction of the ray and that of the ob- server's motion. If the several rays of color moved with different velocities there would evidently be sev- eral resultants, and each star would appear as a colored spectrum longitudinally parallel to the direction of the earth's motion. THE ATOMIC THEORY. 97 The allegation of a dependence of the velocity of the undulatory movements, which correspond to, or produce, the different colors, upon the length of the waves, is thus at variance with observed fact. The hypothesis of " finite intervals " is unavailable as a sup- plement to the undulatory theory ; other methods will have to be resorted to in order to free this theory from its difficulties.* The negative evidence here adduced against the supposition of an atomic or molecular constitution of the light-bearing medium is reenforced by positive evi- dence derived from a branch of the atomic theory itself the modern science of thermo-dynamics. Maxwell has remarked, with obvious truth, that such a medium (whose atoms or molecules are supposed to penetrate the intermolecular spaces of ordinary substances) would be nothing more nor less than a gas, though a gas of * Since the publication of Cauchy's " Memoire sur la dispersion de la lumiere " (Prag, 1836), the dependence of the dispersive powers of dif- ferent substances upon their states of aggregation and chemical composi- tion has been the subject of extensive experimental research ; and the most prominent physicists (Briot, Holtzmann, Redtenbaeher, C. Neumann, Ketteler) now look for an explanation of the phenomena of dispersion to the action of ponderable matter, or to the interaction between it and the jethcr. Cf . Briot, " Essai sur la thSorie mathematique de la lumiere " (Paris, Mallet-Bachelier, 1864), p. 89 seq. ; Redtenbaeher, "Dynamiden- system," p. 130 seq. ; Ketteler, " TJeber den Einfluss der ponderablen Molekuele auf die Dispersion des Lichts," etc. (Pogg. Ann., vol. cxl, pp. 2 seq. and 177 seq. An electro-magnetic theory of light, suggested by the proximate equality of the velocities with which light and electro- magnetic disturbances appear to be propagated through air and other media, and by the action of a magnet (observed by Faraday) in turning the plane of polarization round the direction of the luminar ray as an axis, was broached by Clerk Maxwell in 1865, and has recently been set forth at some length in his " Treatise on Electricity and Magnetism," vol. ii, pp. 383 seq. This theory is now being developed by Helmholtz, Lorentz, Fitzgerald, J. J. Thomson, and Lord Rayleigh. Y ft* mm 98 CONCEPTS OF MODERN PHYSICS. great tenuity, and that every so-called vacuum would in fact be full of this rare gas at the observed temper- ature and at the enormous pressure which the sether, in view of the functions assigned to it by the undulatory theories, must be assumed to exert. Such a gas, there- fore, must have a correspondingly enormous specific heat equal to that of any other gas at the same tem- perature and pressure, so that the specific heat of every vacuum would be incomparably greater than that of the same space filled with any other known gas. This remarkable consequence is not only without experi- mental warrant, but inasmuch as it would apply to all vacua, including the intermolecular spaces of ordinary bodies of whatever state of aggregation is in effect a fatal aggravation of a peculiar difficulty of the molecular theory which is in itself formidable to the highest de- gree. In the third chapter * I have adverted to the fact that, when a body is heated, a part only of the energy communicated to it appears in the form of temperature, i. e. (in the sense of modern theories), of progressive motions of the molecules, the other part being expended in the production of vibratory or rotatory motions of their constituent elements. According to the kinetic theory of gases, this latter part, the internal energy, so called, increases with the number of variables or degrees of freedom in each molecule, and with it, therefore, the specific heat, i. e., the ratio of the whole energy to that of translation which produces expansion or pressure, and is thus exhibited as temperature. If the mol- ecules were "material points" without internal mo- bility, or perfectly elastic and perfectly smooth spheres, the total energy would be available for the production of translatory motion, and no part of it would be con- * Supra, p. 36. THE ATOMIC THEORY. 99 verted into internal energy. But if the molecules, though perfectly elastic, are not perfect spheres as they can not be, whenever they consist of several atoms each the specific heat must at least be equal to a cer- tain minimum assigned by the theory. Now, the spe- cific heats of oxygen, nitrogen, and hydrogen (all which are diatomic, their molecules consisting of at least two atoms each), as experimentally ascertained from a com- parison of their specific heats at constant pressure and , < at constant volume, fall short of this minimum. And^ this theoretical minimum would be very materially in- creased by the addition of the specific heat due to the intermolecular aether, if this were also of atomic or molecular constitution ; the discrepancy between the theoretical postulates and the experimental data would be immeasurably widened. 3. The third proposition of the atomic hypothesis assigns to the atoms, which are said to compose the different chemical elements, determinate weights cor- responding to their equivalents of combination, and is supposed to be necessary to account for the facts whose enumeration and discussion constitute the science of chemistry. The proper verification of these facts is of great difficulty, because they have generally been ob- served through the lenses of the atomic theory, and stated in its doctrinal terms. Thus the differentiation and integration of bodies are invariably described as de- composition and composition ; the equivalents of com- bination are designated as atomic weights or volumes, and the greater part of chemical nomenclature is a sys- tematic reproduction of the assumptions of atomism. Kearly all the facts to be verified are in need of pre- paratory enucleation from the envelopes of this theory. The phenomena usually described as chemical com- 100 CONCEPTS OF MODERN PHYSICS. position and decomposition present themselves to obser- vation thus : A number of heterogeneous bodies concur in definite proportions of weight or volume ; they inter- act ; they disappear, and give rise to a new body pos- sessing properties which are neither the sum nor the mean of the properties of the bodies concurring and interacting (excepting the weight which is the aggre- gate of the weights of the interacting bodies) ; and this conversion of several bodies into one is accompanied, in most cases, by changes of volume, and in all cases by the evolution or involution of heat, or other forms of energy. Conversely, a single homogeneous body gives rise to heterogeneous bodies, between which and the body from which they originate the persistence of weight is the only relation of identity. For the sake of convenience, these phenomena may be distributed into three classes, of which the first em- braces the persistence of weight and the combination in definite proportions ; the second, the changes of volume and the evolution or involution of energy ; and the third, the emergence of a wholly new complement of chemical properties. Obviously, the atomic hypothesis is in no sense an explanation of the phenomena of the second class. It is clearly and confessedly incompetent to account for changes of volume, temperature, or latent energy. And, with the phenomena of the third class, it is apparently incompatible. For, in the light of the atomic hypothesis, chemical compositions and decompositions are in their nature nothing more than aggregations and segregations of masses whose integrity remains inviolate. But the radical change of chemical properties, which is the result of all true chemical action, and serves to distin- guish it from mere mechanical mixture or separation, THE ATOMIC THEORY. 101 evinces a thorough destruction of that integrity. It may be that the appearance of this incompatibility can be obliterated by the device of ancillary hypotheses ; but that leads to an abandonment of the simplicity of the atomic hypothesis itself, and thus to a surrender of its claims to merit as a theory. At best, then, the hypothesis of atoms of definite and different weights can be offered as an explanation of the phenomena of the first class. Does it explain them, in the sense of generalizing them, of reducing many facts to one ? Not at all ; it accounts for them, as it professed to account for the indestructibility and impenetrability of matter, by simply iterating the ob- served fact in the form of an hypothesis. It is another case (to borrow a scholastic phrase) of illustrating idem, per idem. It says: The large masses combine in defi- nitely-proportionate weights because the small masses, the atoms of which they are multiples, are of definitely- proportionate weight. It pulverizes the fact, and claims thereby to have sublimated it into a theory.* The truth is, as Sir William Thomson has observed, that " the assumption of atoms can explain no property of a body which has not previously been attributed to the atoms themselves." ^/\ The foregoing considerations do not, of course, de- tract from the merits of the atomic hypothesis as a graphic or expository device- as an aid to the repre- * That the assumption of atoms of different specific gravities is, on the basis of the atomic theory itself, simply absurd, has already been shown (supra, p. 35). According to the mechanical conception, which underlies the whole atomic hypothesis, differences of weight are differ- ences of density ; and differences of density are differences of distance between the particles contained in a given space. But, in the atom there is no multiplicity of particles and no void space ; hence differences of density or weight are impossible in the case of atoms. 102 CONCEPTS OF MODERN PHYSICS. sentative faculty in " realizing " the phases of chemical or physical transformation. It is a fact beyond dispute that chemistry owes a great part of its practical advance to its use, and that the structural formulae founded upon it have enabled the chemist, not merely to trace the connection and mutual dependence of the various stages in the metamorphosis of " elements " and " com- pounds," so called, but in many cases (such as that of the hydrocarbon series in organic chemistry) success- fully to anticipate the results of experimental research. The question, to what extent the atomic theory is still indispensable to the chemist as a " working hypothesis," is at this moment under vigorous discussion among men of the highest scientific authority, many of whom do not hesitate to indorse the declaration of Cournot (made many years ago) that " the belief in atoms is rather a hindrance than a help " * not only because, as Cournot complains, it interposes an impassable chasm between the phenomena of the inorganic and those of the organic world, but because even as a representation of the phases and results of the most ordinary chemical processes it is o^ v both inadequate and misleading. The modifications to which it has lately been found necessary to subject it, in order to meet the exigencies of the present state of chemical science modifications exemplified in the doc- trines of constant and varying atomicities or valences, of molecular or atomic enchainments, etc., with the attend- ant theories (propounded by Kekule and others) of mo- lecular impact attest the difficulties encountered in the * "En somme, pour Pharmonie generate du systeme de nos connais- sances, par consequent (autant que nous pouvons en juger) pour la plus juste perception de 1'harmonie qui certainement existe dans Pensemble des choses, la foi dans les atomes est plutot un embarras qu'un secours." Cournot, Traite de I'Enchainement des Idees Fondamentales dans les Sciences et dans 1'Histoire, i, 264 seq. THE ATOMIC THEORY. 103 attempt to bring the atomic hypothesis into conformity with the theoretical requirements of the hour. And, in proportion as the attention of the modern chemist is directed to the transference and transformation of en- ergy involved in every instance of chemical " composi- tion " and " decomposition " no less than in every case of allotropic change, its ineptitude as a figurative adumbra- tion of the real nature of chemical processes becomes more and more apparent.* I propose next to consider one of the most notable applications of the atomic hypothesis to physics the kinetic theory of gases. * As an illustration of the disfavor with which the atomic hypothesis ij coming to be regarded by distinguished chemists, I may be permitted to quote a passage from an essay by the late Sir Benjamin C. Brodie, Pro- fessor of Chemistry at Oxford : " I can not but say that I think the atomic doctrine has proved itself inadequate to deal with the complicated system of chemical fact which has been brought to light by the efforts of modern chemists. I do not think that the atomic theory has succeeded in con- structing an adequate, a worthy, or even a useful representation of those facts." " On the Mode of Representation afforded by the Chemical Calcu- lus as contrasted with the Atomic Theory." Chemical News, August, 1867, p. 72. It is but fair to add, however, that I am not in sympathy with Brodie's own theoretical scheme so far as I understand it. CHAPTER VIII. THE KINETIC THEORY OF GASES. CONDITIONS OF THE VALIDITY OF SCIENTIFIC HYPOTHESES. IN the fourth chapter * I have already given an out- line of the doctrine now generally known and accepted as the kinetic theory of gases. The assumptions of this theory are that a gaseous body consists of a great number of minute solid particles molecules or atoms in perpetual rectilinear motion, which, as a whole, is conserved by reason of the absolute elasticity of the moving particles, while the directions of the move- ments of the individual particles are incessantly changed by their mutual encounters or collisions. The colliding particles are supposed to act upon each other only with- in very small distances and for very short times before and after collision, their motion being free, and conse- quently rectilinear, in the intervals between such dis- tances and times. The durations of the rectilinear motions in free paths are, moreover, assumed to be in- definitely large as compared with the durations of the encounters and of the mutual actions. This theory was first advanced by Kroenig, f and has since been elaborated by Clausius, Maxwell, Boltz- * Supra, p. 41. f Pogg. Ann., vol. xcix, p. 315 seq. As is usual in such cases, pre- lusions of the theory have since been discovered in the writings of vari- ous older physicists cf. P. Du Bois-Reymond in Pogg. Ann., vol. cvii, p. 490 seq. THE KINETIC THEORY OF GASES. 105 maim, Stefan, Pfaundler, and other physicists of the highest note. As in the case of the atomic hypothesis generally, I propose for the present to discuss, not so much the logical warrant, as the scientific value, of the theory in question. To this end it will be necessary, however, first to ascertain the true nature and function of a scientific hypothesis not only the criteria of its value, but also the conditions of its validity. A scientific hypothesis may be defined in general terms as a provisional or tentative explanation of phys- ical phenomena.* But what is an explanation in the true scientific sense ? The answers to this question which are given by logicians and men of science, though differing in their phraseology, are essentially of the same import. Phenomena are explained by an exhibi- tion of their partial or total identity with other phe- nomena. Science is knowledge ; and all knowledge, in the language of Sir William Hamilton, f is a " unifica- tion of the multiple." " The basis of all scientific explanation," says Bain, J " consists in assimilating a fact to some other fact or facts. It is identical with the generalizing process." And " generalization is only the apprehension of the One in the Many." * Similar- ly Jevons : || " Science arises from the discovery of identity amid diversity," and A " every great advance in science consists in a great generalization pointing * Wundt has lately (Logik, i, 403) sought to distinguish hypothe- ses from " anticipations of fact " and to restrict the term " hypothesis " to a sense which, notwithstanding its etymological warrant, is at variance with ordinary as well as scientific usage. f Lectures on Metaphysics (Boston ed.), pp. 47, 48. $ Logic, ii (Inductive), chap, xii, 2. * Hamilton, 1. c., p. 48. fl Principles of Science, i, p. 1. A 76., ii, p. 281. 108 CONCEPTS OF MODERN PHYSICS. out deep and subtle resemblances." The same thing is stated by the author just quoted in another place : * " Every act of explanation consists in detecting and pointing out a resemblance between facts, or in showing that a greater or less degree of identity exists between apparently diverse phenomena." All this may be expressed in familiar language thus : When a new phenomenon presents itself to the man of science or to the ordinary observer, the question arises in the mind of either : What is it ? and this question simply means : Of what known, familiar fact is this apparently strange, hitherto unknown fact a new pres- entation of what known, familiar fact or facts is it a disguise or complication ? Or, inasmuch as the par- tial or total identity of several phenomena is the basis of classification (a class being a number of objects hav- ing one or more properties in common), it may also be said that all explanation, including explanation by hy- pothesis, is in its nature classification. Such being the essential nature of a scientific ex- planation of which an hypothesis is a probatory form, it follows that no hypothesis can be valid which does not identify the whole or a part of the phenomenon, for the explanation of which it is advanced, with some other phenomenon or phenomena previously observed. This first and fundamental canon of all hypothetical reasoning in science is formally resolvable into two propositions, the first of which is that every valid hy- pothesis must be an identification of two terms the fact to be explained and a fact by which it is explained ; and the second that the latter fact must be known to experience. Tested by the first of these propositions, all hypoth- * Principles of Science, ii, p. 166. THE KINETIC THEORY OF GASES. 1QT eses are futile which merely substitute .an assumption for a fact, and thus, in the language of the schoolmen, explain obscurum per obscurius, or (the assumption be- ing simply the statement of the fact itself in another form the " fact over again") illustrate idem per idetn. And the futility of such hypotheses goes to the verge of mischievous puerility when they replace a single fact by a number of arbitrary assumptions, among which is the fact itself. Some of the uses made of the atomic hypothesis, both in physics and chemistry, which have been discussed in the last chapter, afford conspicuous examples of this class of bootless assumptions ; and similar instances abound among the mathematical for- mulae that are not infrequently paraded as physical theories. These formulae are in many cases simply re- sults of a series of transformations of an equation which embodies an hypothesis whose elements are neither more nor less than the elements of the phe- nomenon to be accounted for, the sole merit of the emerging formula being that it is not in conflict with the initial one.* * I hope not to be misunderstood as disparaging the services for which physical science is indebted to mathematics. These services es- pecially those rendered by modem analysis are incalculable. But there are mathematicians who imagine that they have compassed a solution of all the mysteries involved in a case of physical action when they have reduced it to the form of a differential expression preceded by a group of integral signs. Even when their equations are integrable they should bear in mind that the operations of mathematics are essentially deduc- tive, and, while they may extend, can never deepen a physical theory. Granting that mathematics are much more than Ka^dp/j-ara tyvxns, and tnat their office in the investigation of the causes of natural phenomena is far more important than the purely regulative functions of formal logic in science generally conceding that the application of mathematics to physics has not only brought to ligLt the significance of many experi- mental results, but has often been a trustworthy guide to successful re- 108 CONCEPTS OF MODERN PHYSICS. In order to comply with the first condition of its validity, an hypothesis must bring the fact to be ex- plained into relation with some other fact or facts by identifying the whole or a part of the former with the whole or a part of the latter. In this sense it has been well said that a valid hypothesis reduces the number of the uncomprehended elements of a phenomenon by at ]east one.* In the same sense it is sometimes said that search nevertheless some of our prominent physicists and mathemati- cians might still read with profit the ninety-sixth aphorism in the first book of Bacon's Novum Organon : " Naturalis Philosophia adhuc sin- cera non invenitur, sed infecta et corrupta ; in Aristotelis schol& per logicam ; in Platonis schola per theologiam naturalem ; in secunda schola Platonis, Procli et aliorum per Mathcmaticam, quae philosophiam natu- ralem terminare, non gcncrare aut procrcarc debet" As to the value of the class of formulae referred to in the text it may not be inappropri- ate to cite the words of Cournot (De 1'Enchainement, etc., i, p. 249) : " Tant qu'un calcul ne fait que rendre ce que 1'on a tire de 1'observation pour 1'introduire dans les elements du calcul a vrai dire il n'ajoute rien aux donnecs de 1'observation." To the same effect are the admirable reflections of M. Poinsot (Theorie Nouvelle de la Rotation des Corps, ed. 1851, p. 79) : " Ce qui a pu faire illusion a quelques esprits sur cette cspece de force qu'ils supposent aux formules de 1'analyse, c'est qu'on en retire, avec assez de facilite, des verites deja connues, et qu'on y a, pour ainsi dire, soi-meme introduces, et il semble alors que 1'analyse nous donne ce qu'elle ne fait que nous rendre dans un autre langage. Quand un theoreme est connu, on n'a qu'a 1'exprimer par des equations ; si le theoreme est vrai, chacunc d'elle ne peut manquer d'etre exacte, aussi bien que les transformers qu'on en peut deduire ; et si Ton arrive ainsi & quelque formule evidente, ou bien etablie d'ailleurs, on n'a qu'a prendre cette expression comme un point de depart, & revenir sur ses pas, et le calcul seul parait avoir conduit comme de lui-meme au theoreme dont il s'agit. Mais c'est en cela que le lecteur est trompe." * " Der Yerstand hat das Beduerf niss jede Erscheinung zu erklaeren d. h. dieselbe als das Resultat bekannter Kraefte oder Erscheinungen be- grifflich abzuleiten. . . . Es geht hieraus hervor, dass jede Hypothese nur bekannte Kraefte oder Erscheinungen zur Erklacrung annehmen darf, in- dem die Annahme einer bisher unbekannten Kraft nur die Qualitaet des zu erklaerenden Phaenomen's aendern, abernicht die Zahl der unerklaer- ten Momente reduciren kann. Soil cine Hypothese nicht vollkommen un- THE KINETIC THEORY OF GASES. 109 every true theory or hypothesis is in effect a simplifica- tion of the data of experience an assertion which must be understood, however, with due regard to the second proposition to be discussed presently, i. e., with the proviso that the theory be not a mere asylum ignoran- tice, of the kind denoted by the schoolmen as a prin- cipium expressivum, such as the explanation of the phenomena of life by reference to a vital principle, or of certain chemical processes by catalytic action. True scientific explanations are generally complicated in form, not only because most phenomena, on proper analysis, prove to be complex, but because the simplest fact is not the effect of a single cause, but the product of a great and often indeterminate multiplicity of agencies the outcome of the concurrence of numerous condi- tions. The Newtonian theory of planetary motion is much more intricate than that of Kepler, according to which every planet is conducted along its path by an angelus rector ; and the account given by modern celes- tial mechanics of the precession of the equinoxes is far less simple than the announcement that among the great periods originally established by the Author of the uni- verse was the Hipparchian cycle. The old brocard, simplex veri judicium, is to be taken with many grains of allowance before it can be trusted as a safe rule in determining the validity or value of scientific doctrines. I now come to the second requirement of the valid- ity of an hypothesis : that the explanatory phenomenon (i. e., that with which the phenomenon to be explained is identified) must be a datum of experience. This nuctz und demgemaess die Verstandosarbeit, welche sie zur Befriedigung eines Bcduerfnisses erzeugte, keine zwecklose sein, so muss jede Hypothese die ZaJd dcr unbcgriffcnen Momcnte ciner Erscheinung mindcstem um eine crniedrigen" Zoellner, Natur dcr Kometen, p. 189 scq. HO CONCEPTS OF MODERN PHYSICS. proposition is in. substance equivalent to that part of Newton's first regula philosophandi,* in which he in- sists that the cause assigned for the explanation of nat- ural things must be a vera causa a term which he does not expressly define in the Principia, but whose import may be gathered from the following passage of his Opticks f : "To tell us that every species of things is endowed with an occult specific quality by which it acts and produces manifest effects is to tell us nothing. But to derive two or three general principles of motion from phenomena and afterward to tell us how the properties and actions of all corporeal things follow from these manifest principles would be a very great step in philosophy, though the causes of those principles were not yet discovered." The requirement in question has long been the sub- ject of animated discussion by J. S. Mill, Whewell, and others ; but it will be found, I think, that, after making due allowance for necessary implications, there is little real disagreement among thinkers. The recent state- ment of G. II. Lewes J that " an explanation to be valid must be expressed in terms of phenomena already observed," and the counter-statement of Jevons * that " agreement with fact (i. e., the fact to be explained) is the one sole and sufficient test of a true hypothesis," are both far too broad, and are, indeed, modified by Lewes and Jevons themselves in the progress of the discussion ; but the claim of Mr. Lewes is nevertheless true in the sense that no explanation is real unless it is an identification of experiential data. The confusion which, as in so many other cases of scientific contro- versy, is at the bottom of the seeming disagreement * Phil. Nat. Princ. Math., lib. iii. f Fourth edition, p. 377. \ Problems of Life and Mind, ii, 7. tt Princ. of Scienca, ii, 138. THE KINETIC THEORY OF GASES. HI between the contending parties, arises from a disregard of the circumstance that the identification of two phe- nomena may be both partial and indirect that it may be effected by showing that the phenomena have some known feature in common oil condition that the exist- ence, in one or both of the phenomena, of some other feature not yet directly observed, and perhaps incapable of direct observation, be assumed. The aptest illustra- tion of this is the much-debated undulatory theory of light. This hypothesis identifies light with other forms of radiance, and even with sound, by showing that all these phenomena have the element of vibration or undulation (which is well known to experience) in common, on the assumption of an all-pervading ma- terial medium, of a kind wholly unknown to experi- ence, as the bearer of the laminar undulations. In this case, as in all similar cases, the identity lies, not in the fictitious element, the aether, but in the real element, the undulation. It consists, not in the agent, but in the law of its action. And it is obvious that every hypothesis which establishes coincidences between phe- nomena in particulars that are purely fictitious is wholly vain, because it is in no sense an identification of phe- nomena. It is worse than vain: it is meaningless a mere collection of words or symbols without compre- hensive import. As Jevons expresses it : * " No hy- pothesis can be so much as framed in the mind, unless it be more or less conformable to experience. As the material of our ideas is undoubtedly derived from sen- sation so we can not figure to ourselves any existence or agent but as endowed with some of the properties of matter. All that the mind can do in the creation of new existences is to alter combinations, or by anal- * Princ. of Science, ii, 141. 112 CONCEPTS OF MODERN PHYSICS. ogy to alter the intensity of sensuous properties." J. S. Mill is, therefore, clearly wrong when he says * that, " an hypothesis being a mere supposition, there can be no other limits to hypotheses than those of the human imagination," and that " we may, if we please, imagine, by way of accounting for an effect, some cause of a kind utterly unknown and acting according to a law altogether fictitious." The unsoundness of the latter part of this proposition is evidently felt by Mill himself, for he adds at the end of the next sentence that " there is probably no hypothesis in the history of science in which both the agent itself and the law of its operation were fictitious." There certainly is no such hypothesis at least none which has in any way subserved the interests of science. An hypothesis may involve not only one but sev- eral fictitious assumptions, provided they bring into re- lief, or point to the probability, or at least possibility, of an agreement between phenomena in a particular that is real and observable. This is especially legiti- mate when the agreement thus brought to light is not between two, but a greater number of phenomena, and still more so when the agreement is not merely in one but in several real particulars between diverse phe- nomena, so that, in the language of Whewell,f " the hy- potheses which were assumed for one class of cases are found to explain another of a different nature a con- silience of induction." An instance of this is afforded by the hypothesis just referred to of the luminiferous sether, which was at first believed also to explain the retardation of comets. But, while the probability of the truth of an hypothesis is in direct ratio to the number * Logic, 8th ed., p. 394. f History of the Inductive Sciences (Am. ed.), ii, 186 THE KINETIC THEORY OF GASES. H3 of phenomena .thus brought into relation, it is in the inverse ratio of the number of such fictions, or, more accurately, its improbability increases geometrically while the series of independent fictions expands arith- metically.* This finds illustration again in the undu- latory theory of light. The multitude of fictitious as- sumptions embodied in this hypothesis, in conjunction with the failure of the consiliences by which it appeared at first to be distinguished, can hardly be looked upon otherwise than as a standing impeachment of its validity in its present form. However ready we may be to ac- cede to the demands of the theorist when he asks us to grant that all space is pervaded, and all sensible matter is penetrated, by an adamantine solid exerting at each point in space an elastic force 1,148,000,000,000 times that of air at the earth's surface, and a pressure upon * " En general," says Cournot (De 1'Enchainement, etc., i, 103), " une theorie scientifique quelconque, imaginee pour rclier un certain nombre de faits donnes par 1'observation, peut etre assimilee a la courbe que 1'on trace d'apres une loi geomctrique, en s'imposant la condition de la faire passer par un certain nombre de points donnes d'avance. Le jugement que la raison porte sur la valeur intrinseque de cette theorie cst un jugement probable, une induction dont la probabilite tient d'une part a la simplicite de la formule theorique, d'autre part au nombre des faits ou des groupes des faits qu'ellc relie, le meme groupe devant com- prendre tous les faits qui s'expliquent deja les uns par les autres, inde- pendamment de 1'hypothese theorique. S'ilfaut compliquer la formule d mcsnre que de nouveaux faits se revelent d ^observation elle devient de moins en moins probable en taut que loi de la Nature ; ce n'est bientot plus qu'un echafaudage artificiel qui croule enfin lorsque, par un surcroit de complication, elle perd meme 1'utilite d'un systeme artificiel, celle d'aider le travail de la pens6e et de diriger les recherches. Si au contraire les faits acquis a 1'observation posterieurement a la construction de 1'hy- pothese sont relies par elle aussi bien que les faits qui ont servi a la construire, si surtout des faits prevus comme consequences de 1'hypothese resolvent des observations posterieures une confirmation eclatante, la probabilite de 1'hypothese peut aller jusqu'a ne laisser aucuue place au doute dans un esprit eclaire." 114- CONCEPTS OF MODERN PHYSICS. the square inch of 17,000,000,000,000 pounds * a solid which, at the same time, wholly eludes our senses, is utterly impalpable and offers no appreciable resistance to the motions of ordinary bodies we are appalled when we are told that the alleged existence of this ad- amantine medium, the aether, does not, after all, explain the observed irregularities in the periods of comets; that, furthermore, not only is the supposed luminif erous aether unavailable as a medium for the origination and propagation of dielectric phenomena, so that for these a distinct all-pervading electriferous aether must be as- sumed,t but that it is very questionable whether the as- sumption of a single aethereal medium is competent to account for all the known facts in optics (as, for in- stance, the non-interference of two rays originally polar- ized in different planes when they have been brought to the same plane of polarization, and certain phenom- ena of double refraction, in view of which it is neces- sary to suppose that the rigidity of the medium varies with the direction of the strain a supposition discoun- tenanced by the facts relating to the intensities of re- flected light), and that for the adequate explanation of the phenomena of light it is " necessary to consider what we term the aether as consisting of two media, each possessed of equal and enormous self -repulsion or elasticity, and both existing in equal quantities through- out space, whose vibrations take place in perpendicular planes, the two media being mutually indifferent, neither attracting nor repelling." J In this endless superfeta- * Cf. Herschel, Familiar Lectures, etc., p. 282 ; F. De Wrede (Presi- dent Royal Academy of Sciences in Stockholm), address, Phil. Mag., 4th ser., vol. xliv, p. 82. f W. A. Norton, on Molecular Physics, Phil. Mag., 4th ser., vol. xxiii, p. 193. Hudson, on Wave Theories of Liyht, Ileat, and Electricity, Phil. THE KINETIC THEORY OF GASES. 115 tion of sethereal media upon space and ordinary matter, there are ominous suggestions of the three kinds of sethereal substances postulated by Leibnitz and Oartesius alike as a basis for their vortical systems. There is an impulsive whirl in our thoughts, at least, when we are called upon, in the interests of the received form of the undulatory theory, not only to reject all the pre- sumptions arising from our common observation and all the analogies of experience, but to cumulate hy- potheses and aethers indefinitely. And we are but par- tially reassured by the circumstance that the theory in question, besides accounting for the phenomena of op- tics which had been observed at the time of its pro- mulgation, has the great merit of successful prevision, having led to the prediction of a number of facts sub- sequently discovered. These predictions, certainly, have not only been numerous, but several of them, such as Hamilton's announcement of conical refraction (after- ward verified by Lloyd) and Fresnel's forecast (from Mag. (iv), vol. xliv, p. 210 scq. In this article the author also points out the crudeness of the subsidiary hypotheses which have been framed to obviate other difficulties of the undulatory theory, among which are those discussed in the last chapter. " Waves of sound," he says, " in our atmosphere are 10,000 times as long as the waves of light, and their velocity of propagation about 850,000 times less, and, even when air has been raised to a temperature at which waves of red light are propagated from matter, the velocity of sound-waves is only increased to about double what it was at zero centigrade. Even their velocity through glass is 55,000 times less than the speed of the aethereal undu- lations, and the extreme slowness of change of temperature in the con- duction of heat (as contrasted with the rapidity with which the vibrations of the aether exhaust themselves, becoming insensible almost instantly when the action of the existing cause ceases) marks distinctly the essential difference between molecular and aethereal vibrations. It ap- pears to me, therefore, a very crude hypothesis to imagine a combina- tion of gethereo-molecular vibrations as accounting for the very minute difference in the retardation of doubly refracted rays in crystals." 116 CONCEPTS OF MODERN PHYSICS. the imaginary form of an algebraic formula) of circular polarization after two internal reflections in a rhomb, are very striking. But, although anticipations of this sort justly serve to accredit an hypothesis, they are, as Mill has shown,* by no means absolute tests of their truth. Using the word " cause " in the sense in which it is commonly understood, an effect may be due to any one of several causes, and may, therefore, in many cases be accounted for by any one of several conflicting hypotheses, as becomes evident to the most cursory glance at the history of science. When an hypothesis successfully explains a number of phenomena with reference to which it was constructed, it is not strange that it should also explain others connected with them that are subsequently discovered. There are few discarded physical theories that could not boast the prevision of phenomena to which they pointed and which were afterward observed ; among them are the one-fluid theory of electricity and the corpuscular theory of light. There are, of course, other conditions of the validity of an hypothesis to which I have not yet adverted. Among them are those specified by Sir "W. Hamilton, Mill, Bain, and others, such as that the hypothesis must not be contradictory of itself or in conflict with the known laws of nature (which latter requirement is, how- ever, somewhat doubtful, inasmuch as the laws in ques- tion may be incomplete inductions from past experience to be supplemented by the very elements postulated by * Logic, p. 356. Long before Mill, Leibnitz observed that success in explaining (or predicting) facts is no proof of the validity of an hy- pothesis, inasmuch as right conclusions may be drawn from wrong prem- isses as Leibnitz expresses it, "comme le vrai pcut etre tire du faux." Cf. Nouveaux Essais, chap, xvii, sec. 5 Lcibnitii, opp. ed. Erdmann, p. 397. THE KINETIC THEORY OF GASES. H7 the hypothesis) ; that it must be of a nature to admit of deductive inferences, etc. Upon all these it is not nec- essary, in view of my present purpose, to dilate. The two conditions which I have sought to enforce and illus- trate are, in my judgment, sufficient tests of the validity and merits of the kinetic theory of gases. The fundamental fact to be accounted for by this theory is that gases are bodies which, at constant tern- (. ^ peratures and in the absence of external pressure, ex- ? pand at even rate. From this fact the two great empir- ical laws, so called, expressive of those physical proper- ties of a gas which are directly attested by experience, are the necessary and immediate consequences, being, indeed, nothing more than partial and complementary statements of it. The limitation of gaseous volume be- ing produced by pressure alone the cohibition of the bulk of a gas being due solely to pressure it follows that it must be proportional to it ; in other words, that the volume of a gas must be inversely as the pressure ; and this is the law of Boyle or Mariotte. Again : tem- perature is measured by the uniform expansion of a column of gas (in the air-thermometer) ; hence, if all gases expand equally, temperature is proportional to the volume of a gas and conversely ; this is the law of Charles* * One of the strangest incidents in the history of physics is the grave discussion of the question respecting the true law of gaseous expansion. "According to Gay-Lussac," says Balfour Stewart (Treatise on Heat, p. 60), " the augmentation of volume which a gas receives when the tem- perature increases 1 is a certain fixed proportion of its initial volume at C. ; while, according to Dalton, a gas at any temperature increases in volume for a rise of 1 by a constant fraction of its volume at that tem- perature. . . . The dilatation of gases has since been investigated by Rudberg, Dulong and Petit, Magnus and Regnault, and the result of their labors leaves little doubt that Gay-Lussac's method of expressing the law is much nearer the truth than Dalton's." Inasmuch as the experiments 118 CONCEPTS OF MODERN PHYSICS. The foregoing real definition (i. e., exhibition of the properties) of a gas applies only to ideal or perfect gases. In actual experience we meet with no gas which, in the absence of pressure, expands with abso- lute uniformity ; and for that reason we do not know experimentally of any gas behaving in strict conformity to the laws of Boyle and Charles. Moreover, we are unable directly to observe a gas which is wholly free from pressure ; the datum of experience is simply that gases expand (other things being equal) in proportion to the diminution of the pressure to which they are subjected. But in the case of many gases those which of Rudberg and others were necessarily made on the supposition that the coefficient of expansion was the same for all gases (the question relating, not to the expansion of some particular gas, but of gases generally), and, as the standard temperatures were those of the air-thermometer, it would have been surprising, indeed, if the result had been confirmatory of Dai- ton's view. A thermometer is graduated by dividing a given length of a tube of even bore into equal parts. It is clear, therefore, that the incre- ment of volume resulting from the expansion of the air in such a tube through one degree is a fixed part of a constant volume initially assumed, and not of a constantly increasing volume ; and the same thing is, of course, true of any other gas if it expands at the same rate. Dalton's form of the law of expansion would yield the following remarkable series of equal ratios in which the first represents the rate of expansion of air in the thermometer, and the others stand for the rate (or rather rates) of expansion of the gas under examination (a being the linear expansion of the air in the thermometer, v its initial volume, a the corresponding ex- pansion in the gas under examination, v its initial volume) :- = ,= ii it V V a a a a -, = r-7 = = -r-, etc., etc. The attempts at an v + a' v + 2a v + 3a' v + 4a' experimental solution of the question here referred to are suggestive, by the way, of a doubt as to the correctness of the prevailing systems of ther- mometry, which are founded on the assumption of equalities of volume- ratios in which one of the terms is constant while the other is variable, i. e., of fractions which have the same numerator, but different denomina- tors. These suggestions are but imperfectly met by the reflection that the bores of our thermomctrical tubes are very small. TJZTIVBRSITT THE KINETIC are either wholly incoercible, or coercible (i. e., reduci- ble to the liquid or solid state) with great difficulty, and of nearly all gases at very high temperatures the devi- ation from uniformity of expansion is very slight. Now, how does the kinetic theory of gases explain the experiential fact or facts just stated ? It professes to explain them on the basis of at least three arbitrary assumptions, not one of which is a datum of experience, viz. : / z- 1. That a gas is composed of solid particles which are indestructible and of constant mass and volume. %. That these constituent particles are absolutely elastic: 3. That these particles are in perpetual motiorif and, except at very small distances, in no wise act upon each other, so that their motions are absolutely free and therefore rectilinear. I refrain from adding a foarth assumption that of the absolute equality of the particles, in mass at leas because it is claimed (though unjustifiably) to be a ollary from the other assumptions , The first of these assumptions has been sufficiently considered in the last chapter. The second assumption asserts the absolute elasticity of the constituent solid particles. What is the import and scope of tiiife assump- tion ? The elasticity of a solid body is that property by means of which it occupies, and tends to occupy, por- tions of space of determinate volume and figure ? and therefore reacts against any force or stress producing, or tending to produce, an alteration of such volume or figure with a counter-force or stress which, in the case of perfect elasticity, is exactly proportional to the alter- ing force. Now, it is seen at once that the property ' the fact thus assumed in the constituent solid includes 'to** 120 CONCEPTS OF MODERN PHYSICS. <>~ j thejTery fact to be accounted for in the gas. A perfect ' gas reacts against a stress tending to reduce its volume with a spring proportional to the stress ; and for this reason gases are defined as elastic fluids. This resili- ence of the gas against diminution of volume is obvi- ously a simpler fact than the rebound of a solid against both diminution and increase of volume, in addition to the reaction against a change of figure. The resistance to several kinds of change implies a greater number of forces, and is therefore a more complex phenomenon, than the resistance to one kind of change.* It thus appears that the presupposition of absolute elasticity in the solids, whose aggregate is said to con- stitute a gas, is a flagrant violation of the first condition of the validity of an hypothesis the condition which requires a reduction of the number of unrelated ele- men ts in the fact to be explained, and therefore forbids a mere reproduction of this fact in the form of an as- sumption, and a fortiori a substitution of several arbi- trary assumptions for one fact. Manifestly the expla- nation offered by the kinetic hypothesis, in so far as its second assumption lands us in the very phenomenon from which it starts, the phenomenon of resilience, is (like the explanation of impenetrability, or of the com- * It may be said that the greater simplicity of the properties of a gas is purely conceptual. The identification of concepts with facts is un- doubtedly the great fundamental error of speculation ; but we are now dealing with the conceptual elements of the hypothesis under discussion. The opinion that a solid of constant volume (or, more accurately ex- pressed, of variable volume, expanding or contracting to a fixed volume proprio motu) is a simpler thing than a uniformly expanding body is certainly not based upon any fact of experience, but is a mere prejudice of the intellect akin to the notion that a body at rest is a simpler phenom- enon than a body in uniform motion, and generally that rest is simpler than motion. This prejudice has its root in our habitual oblivion of the essential relativity of all phenomena, which will be discussed hereafter. THE KINETIC THEORY OF GASES. 121 bination of elements in definite proportions by the atomic theory) simply the illustration of idem per idem, and the very reverse of a scientific procedure. It is a mere versatio in loco movement without progress. It is utterly vain ; or rather, inasmuch as it complicates the phenomenon which it professes to explicate, it is worse than vain a complete inversion of the order of intelligence, a resolution of identity into difference, a dispersion of the One into the Many, an unraveling of the Simple into the Complex, an interpretation of the Known in terms of the Unknown, an elucidation of the Evident by the Mysterious, a reduction of an os- tensible and real fact to a baseless and shadowy phan- tom.* Waiving the question already discussed, whether or not the assumed absolute solidity and constancy of vol- ume of the supposed constituent particles are consistent (in the light of the mechanical theory generally) with their absolute elasticity, I proceed to consider the third assumption of the kinetic hypothesis. This assumption is an unavoidable supplement to the initial theoretical * All theorists who attempt to account for a physical fact by a multi- plication of arbitrary assumptions in which the fact itself is reproduced are liable to Aristotle's acute animadversion upon the Platonic doctrine of ideas their endeavors are as nugatory as those of a person who, for the purpose of facilitating the operation of counting, begins by multiply- ing his numbers of Sc ras tSeas alrtas T&e/j.evoi Trp&Tov /j.ev fody except from its relation to other bodies, it is nevertheless not to ~be concluded that these things (rest and motion) are nothing in themselves l>ut a mere relation established by the intellect, and that there is nothing inherent in the ~bodies themselves which corresponds to our ideas of rest and motion. For, although we are unable to know quantity otherwise than by comparison, yet, when the things with which we instituted the comparison are gone, there is still left in the body \\\Q fundamentum quantitatis, as it were ; for, if it were extended or contracted, such extension or contraction would have to be taken as a true change. Thus, if but one body existed, we should have to say that it was either in motion or at rest, inasmuch as it could not be taken as being both or neither. Whence I conclude that rest and motion are not merely ideal things, horn from comparison alone, so that there would * Omne corpus, etiam sine respectu ad alia corpora, vel quiescit vel movetur, hoc est, vel absolute quiescit, vel absolute movetur." Jb., p. 30 (cap. ii, axioma 7). 11 194 CONCEPTS OF MODERN PHYSICS. ~be nothing Inherent in the ~body corresponding to them, but that it may be justly asked in respect to a solitary' body whether it is in motion or at rest. . . . Inasmuch, therefore, as we can justly ask respecting a single body itself, without reference to other bodies, or under the supposition that they are annihilated, whether it is at rest or in motion, we must necessarily take one or the other alternative. But what this rest or motion will be, in view of the fact that there is here no change of place with respect to other bodies, we can not even think without admitting an absolute space in which our body occupies some given space whence it can pass to other places." * Accordingly Euler most strenuously insists on the necessity of postulating an absolute, im- movable space. " Whoever denies absolute space," he says, " falls into the gravest perplexities. Since he is constrained to reject absolute rest and motion as empty sounds without sense, he is not only constrained also to reject the laws of motion, but to affirm that there are no laws of motion. For, if the question which has brought us to this point, What will be the condition of a solitary body detached from its connection with other bodies ? is absurd, then those things also which are in- duced in this body by the action of others become un- certain and indeterminable, and thus everything will have to be taken as happening fortuitously and without any reason." f That the basis of all this reasoning is purely onto- logical is plain. And, when the thinkers of the eigh- teenth century became alive to the fallacies of ontologi- cal speculation, the unsoundness of Euler's " axiom," that rest and motion are substantial attributive entities independent of all relation, could hardly escape their * Thcoria motAs, etc., p. 81. f 75., p. 32. FOURTH METAPHYSICAL ERROR. 195 notice. Nevertheless, they were unable to emancipate themselves wholly from Euler's ontological preposses- sions. They did not at once avoid his dilemma by re- pudiating it as unfounded by denying that motion and rest can not be real without being absolute but they attempted to reconcile the absolute reality of rest and motion with their phenomenal relativity by postu- lating an absolutely quiescent point or center in space to which the positions of all bodies could be referred. Foremost among those who made this attempt was Kant.* In the seventh chapter of his " Natural His- * It is remarkable how many of the scientific discoveries, specula- tions and fancies of the present day are anticipated or at least foreshad- owed in the writings of Kant. Some of them are enumerated by Zoellner (Natur der Kometen, p. 455 seq.) among them the constitution and motion of the system of fixed stars ; the nebular origin of planetary and stellar systems ; the origin, constitution and rotation of Saturn's rings and the conditions of their stability ; the non-coincidence of the moon's center of gravity with her center of figure ; the physical constitution of the comets ; the retarding effect of the tides upon the rotation of the earth; the theory of the winds, and Dove's law. Fritz Schultze has shown (Kant and Darwin, Jena, 1875) that Kant was one of the pre- cursors of Darwin. In this connection it is curious to note a coincidence (no doubt wholly accidental) in the example resorted to both by Kant and A. R. Wallace for the purpose of illustrating " adaptation by general law." The case put by both is that of the channel of a river which, in the view of the teleologists, as Wallace says (Contributions to the Theory of Natural Selection, p. 276 seq.\ "must have been designed, it answers its purpose so effectually," or, as Kant expresses it, must have been scooped out by God himself. (" Wenn man die physisch-thcolo- gischen Verfasser hoert, so wird man dahin gebracht, sich vorzustellen, ihre Lanfrinnen waeren alle von Gott ausgehoehlt." Beweisgrund zu einer Demonstration des Dasein's Gottes, Kant's Werke, i, p. 232.) Even of the vagaries of modern transcendental geometry there are sug- gestions in Kant's essays, Yon der wahren Schaetzung der lebendigen Kraefte, Werke v, p. 5, and Von dem ersten Grunde des Untcrschiedes der Gegenden im Raume, ?'&., p. 293 a fact which is not likely to con- duce to the edification of those who, like J. K. Becker, Tobias, Wcissen- born, Krausc, etc., have raised the Kantian standard in defense of Eu- 196 CONCEPTS OF MODERN PHYSICS. toiy of the Heavens " the same work in which, nearly fifty years before Laplace, he gave the first outlines of the Nebular Hypothesis he sought to show that in the universe there is somewhere a great central body whose center of gravity is the cardinal point of reference for the motions of all bodies whatever. " If in the im- measurable space," he says,* " wherein all the suns of the milky way have been formed, a point is assumed round which, from whatever cause, the first formative action of nature had its play, then at that point a body of the largest mass and of the greatest attractions, must have been formed. This body must have become able to compel all systems which were in process of forma- tion in the enormous surrounding sphere to gravitate toward it as their center, so as to constitute an entire system, similar to the solar and planetary system which was evolved on a small scale out of elementary mat- ter." A suggestion similar to that of Kant has recently been made by Professor C. Neumann, who enforces the necessity of assuming the existence, at a definite and permanent point in space, of an absolutely rigid body, to whose center of figure or attraction all motions are to be referred, by physical considerations. The drift of his reasoning appears in the following extracts from his inaugural lecture On the Principles of the Galileo- klidean space. It is probably not without significance that in the sec- ond edition of his Critique of Pure Reason Kant omits the third para- graph of the first section of the Transcendental Aesthetics, in which he had enforced the necessity of assuming the a priori character of the idea of space by the argument that without this assumption the proposi- tions of geometry would cease to be true apodictically, and that " all that could be said of the dimensions of space would be that thus far no space had been found which had more than three dimensions." * Naturgeschichte des Himmels, Werke, vol. vi, p. 152. FOURTH METAPHYSICAL ERROR. 197 Newtonian Theory : * " The principles of the Galileo- JSTewtonian theories consist in two laws the law of inertia proclaimed by Galileo, and the law of attraction added by Newton. ... A material point, when once set in motion, free from the action of an extraneous force, and wholly left to itself, continues to move in a straight line so as to describe equal spaces in equal times. Such is Galileo's law of inertia. It is impossi- ble that this proposition should stand in its present form as the corner-stone of a scientific edifice, as the starting- point of mathematical deductions. For it is perfectly unintelligible, inasmuch as we do not_know what is meant by l motion in a straight line,' or, rather, inas- much as we do know that the words ' motion in a straight line ' are susceptible of various interpretations. A mo- tion, for instance, which is rectilinear as seen from the earth, would be curvilinear as seen from the sun, and would be represented by a different curve as often as we change our point of observation to Jupiter, to Sat- urn, or another celestial body. In short, every motion which is rectilinear with reference to one celestial body will appear curvilinear with reference to another celes- tial body. . . . " The words of Galileo, according to which a mate- rial point left to itself proceeds in a straight line, appear to us, therefore, as words without meaning as express- ing a proposition which, to become intelligible, is in need of a definite background. There must he given in the universe some special l)ody as the basis of our com- parison, as the object in reference to which all motions are to he estimated ; and only when such a body is given shall we be able to attach to those words a defi- * Ueber die Principien der Galileo-Newton'schen Theoric, Leipzig, B. G. Teubner, 1870. 198 CONCEPTS OF MODERN PHYSICS. nite meaning. Now, what body is it which is to occupy this eminent position ? Or, are there several such bod- ies ? Are the motions near the earth to be referred to the terrestrial globe, perhaps, and those near the sun to the solar sphere ? . . . " Unfortunately, neither Galileo nor Newton gives us a definite answer to this question. But, if we care- fully examine the theoretical structure which they erect- ed, and which has since been continually enlarged, its foundations can no longer remain hidden. We readily see that all actual or imaginable motions in the universe must be referred to one and the same body. Where this body is, and what are the reasons for assigning to it this eminent, and, as it were, sovereign position, these are questions to which there is no answer. " It will ~be necessary ', therefore, to establish the prop- osition^ as the first principle of the Galileo-Newtonian theory, that in some unknown place of the universe there is an unknown body a body absolutely rigid and un- changeable for all time in its figure and dimensions. I may be permitted to call this body ' THE BODY ALPHA.' It would then be necessary to add that the motion of a body would import, not its change of place in reference to the earth or sun, but its change of position in refer- ence to the body Alpha. " From this point of view the law of Galileo is seen to have a definite meaning. This meaning presents it- self as a second principle, which is, that a material point left to itself progresses in a straight line proceeds, therefore, in a course which is rectilinear in reference to the body Alpha." After thus showing, or attempting to show, that the reality of motion necessitates its reference to a rigid body unchangeable in its position in space, Neumann FOURTH METAPHYSICAL ERROR. 199 seeks to verify this assumption by asking himself the question, what consequences would ensue, on the hy- pothesis of the mere relativity of motion, if all bodies but one were annihilated. " Let us suppose," he says, " that among the stars there is one which consists of fluid matter, and which, like our earth, is in rotatory motion round an axis passing through its center. In consequence of this motion, by virtue of the centrifugal forces developed by it, this star will have the form of an ellipsoid. What form, now, I ask, will this star as- sume if suddenly all other celestial bodies are annihi- lated ? " These centrifugal forces depend solely upon the state of the star itself ; they are wholly independent of the other celestial bodies. These forces, therefore, as well as the ellipsoidal form, will persist, irrespective of the continued existence or disappearance of the other bodies. But, if motion is defined as something relative -as a relative change of place of two points the an- swer is very different. If, on this assumption, we sup- pose all other celestial bodies to be annihilated, nothing remains but the material points of which the star in question itself consists. But, then, these points do not change their relative positions, and are therefore at rest. It follows that the star must be at rest at the moment when the annihilation of the other bodies takes place, and therefore must assume the spherical form taken by all bodies in a state of rest. A contradiction so intoler- able can be avoided only by abandoning the assumption of the relativity of motion, and conceiving motion as absolute, so that thus we are again led to the principle of the body Alpha." Now, what answer can be made to this reasoning of Professor Neumann ? None, if we grant the admissibil- 200 CONCEPTS OF MODERN PHYSICS. ity of the hypothesis of the annihilation of all bodies in space but one, and the admissibility of the further as- sumption that an absolutely rigid body with an abso- lutely fixed place in the universe is possible. But such a concession is forbidden by the universal principle of relativity. In the first place, the annihilation of all bod- ies but one would not only destroy the motion of this one remaining body and bring it to rest, as Professor mann sees, but would also destroy its very existence and bring it to naught, as he does not see. A body can not survive the system of relations in which alone it has its being ; its presence or position in space is no more possible without reference to other bodies than its change of position or presence is possible without such refer- ence. As has been abundantly shown, all properties of a body which constitute the elements of its distinguish- able presence in space are in their nature relations and imply terms beyond the body itself. In the second place the absolute fixity in space at- tributed to the body Alpha is impossible under the known conditions of reality. The fixity of a point in space involves the permanence of its distances from at least four other fixed points not in the same plane. But the fixity of these several points again depends on the constancy of their distances from other fixed points, and so on ad infinitum. In short, the fixity of position of any body in space is possible only on the supposition of the absolute finitude of the universe ; and this leads to the theory of the essential curvature of space, and the other theories of modern transcendental geometry, which will be discussed hereafter. There is but one issue from the perplexities of Euler, and that is through the proposition that the reality of rest and motion, far from presupposing that FOURTH METAPHYSICAL ERROR. 201 they are absolute, depends upon their relativity. The source of these perplexities is readily discovered. It is to be found in the old metaphysical doctrine, that the Real is not only distinct from, but the exact opposite of, the Phenomenal. Phenomenalities are the deliver- ances of sense ; and these are said to be contradictory of each other, and therefore delusive. Now, the truth is that there is no physical reality which is not phe- nomenal. The only test of physical reality is sensible experience. And the assertion, that the testimony of the senses is delusive, in the sense in which this asser- tion is made by the metaphysicians, is groundless. The testimony of the senses is conflicting only because the momentary deliverance of each sense is fragmentary and requires control and rectification, either by other deliverances of the same sense, or by the deliverances of the other senses. When the traveler in the desert sees before him a lake which continually recedes and finally disappears, proving to be the effect of mirage, it is said that he is deceived by his senses, inasmuch as the supposed body of water was a mere appearance without reality. But the senses were not deceptive. The lake was as real as the image. The deception lay in the erroneous inferences of the traveler, who did not take into account all the facts, forgetting (or being ignorant of) the refraction of the rays proceeding from the real object, whereby their direction and the appar- ent position of the object were changed. The true distinction between the Apparent and the Heal is that the former is a partial deliverance of sense which is mistaken for the whole deliverance. The deception or illusion results from the circumstance that the senses are not properly and exhaustively interrogated and that their whole story is not heard. 202 CONCEPTS OF MODERN PHYSICS. The coercive power of the prevailing ontological notions of Euler's time over the clear intellect of the great mathematician is most strikingly exhibited in his statement that without the assumption of absolute space and motion there could be no laws of motion, so that all the phenomena of physical action would become uncertain and indeterminable. If this argument were well founded, the same consequence would follow, a fortiori, from his repeated admissions in the first chap- ter of his book, to the effect that we have no actual knowledge of rest and motion, except that derived from bodies at rest or in motion in reference to other bodies. Euler's proposition can have no other meaning than this, that the laws of motion can not be established or verified unless we know its absolute direction and its absolute rate. But such knowledge is by his own showing unattainable. It follows, therefore, that the establishment and verification of the laws of motion are impossible. And yet no one knew better than Euler himself that all experimental ascertainment and verifi- cation of dynamical laws, like all acts of cognition, de- pend upon the insulation of phenomena ; that they can be effected only by disentangling the effects of certain forces from the effects of other forces (determinable aliunde, i. e., by their other effects) with which they are complicated a proceeding which, in many cases, is facilitated by the circumstance that these latter effects are inappreciably small. Surely the verification of the law of inertia by the inhabitants of our planet does not depend upon their knowledge, at any moment, of the exact rate of its angular velocity of motion round the sun! And the validity of the Newtonian theory of celestial motion is not to be drawn in question because its author suggests that the center of gravity of our FOURTH METAPHYSICAL ERROR. 203 solar system moves in some elliptic orbit whose ele- ments are not only unknown, but will probably never be discovered ! As well might it be contended that the mathematical theorems respecting the properties of the ellipse are of doubtful validity, since no such curve is accurately described by any celestial body or can be exactly traced by a human hand ! Although in particular operations of thought we may be constrained, for the moment, to treat the Com- plex as simple, the Variable as constant, the Transitory as permanent, and thus in a sense to view phenomena " sub quadojin specie dbsoluti" * nevertheless there is no truth in the old ontological maxirn that the true na- ture of things can be discovered only by divesting them of their relations that to be truly known they must be known as they are in themselves, in their absolute essence. Such knowledge is impossible, all cognition being founded upon a recognition of relations; and this impossibility nowhere stands out in stronger relief than in the exposition, by Newton and Euler, of the reality of rest and motion under the conditions of their determin ability. It follows, of course, from the essential relativity o? rest and motion, that the old ontological disjunction between them falls, and that in a double sense rest dif- fers from motion, in the language of Euler, " as one motion differs from another," f or, as modern mathe- maticians and physicists express it, that " rest is but a special case of motion." j And it follows, furthermore, " De natura rationis cst res sub quadam aeternitatis specie perci- pere." Spinoza, Eth., Pars, ii, Prop, xliv, Coroll. 2. t " Neque motus a quiete aliter differt, atque alius motus ab alio." Theoria motus, etc., p. 8. \ " Die Ruhe ist nur ein be?ondcrer Fall der Bewegung." Kirchhoff, Vorlesungen uebcr math. Physik, p. 32. 204 CONCEPTS OF MODERN PHYSICS. tliat rest is not the logically or cosmologically primum of material existence that it is not the natural and original state of the universe which requires no ex- planation while its motion, or that of its parts, is to be accounted for. What requires, and is susceptible of, explanation is always a change from a given state of relative rest or motion of a finite material system ; and the explanation always consists in the exhibition of an equivalent change in another material system. The question respecting the origin of motion in the uni- verse as a whole, therefore, admits of no answer, be- cause it is a question without intelligible meaning. The same considerations which evince the relativity of motion also attest the relativity of its conceptual elements, space and time. As to space, this is at once apparent. And of time. " the great independent vari- able " whose supposed constant flow is said to be the ultimate measure of all things, it is sufficient to observe that it is itself measured by the recurrence of certain relative positions of objects or points in space, and that the periods of this recurrence are variable, depending upon variable physical conditions. This is as true of the data of our modern time-keepers, the clock and chro- nometer, as of those of the clepsydra and hour-glass of the ancients, all of which are subject to variations of friction, temperature, changes in the intensity of gravi- tation, according to the latitude of the places of ob- servation, and so on. And it is equally true of the records of the great celestial time-keepers, the sun and the stars. After we have reduced our apparent solar day to the mean solar day, and this, again, to the side- real day, we find that the interval between any two transits of the equinoctial points is not constant, but becomes irregular in consequence of nutation, of the FOURTH METAPHYSICAL ERROR. 205 precession of the equinoxes, and of numerous other secular perturbations and variations due to the mutual attraction of the heavenly bodies. The constancy of the efflux of time, like that of the spatial positions which serve as the basis for our determination of the rates and amounts of physical motion, is purely concept- ual. The relativity of mass has repeatedly been adverted to in the preceding chapters. It has been shown that the measure of mass is the reciprocal of the amount of acceleration produced in a body by a given force, while force, in turn, is measured by the acceleration produced in a given mass. It is readily seen that the concept mass might be expanded, so as to assign the measure of mass, not to mechanical motion alone, but to physi- cal action generally, including heat and chemical affin- ity. This would lead to an equivalence of masses dif- fering with the nature of the agency selected as the basis of the comparison. Thermally equivalent masses would be the reciprocals of the specific heats of masses as now determined ; and chemically equivalent masses would be the atomic weights, so called. It is important to note that the determination of masses on the basis of gravitation, in preference to their valuation on the basis of thermal, chemical or other physical action, is a mere matter of convenience, and is not in any proper sense founded on the nature of things. But, apart from this, and looking to the ordinary method of determining the mass of a body by its weight, the relativity of mass is equally manifest. The weight of a body is a function, not of its own mass alone, but also of that of the body or bodies by which it is attracted, and of the distance between them. A body whose weight, as ascertained by the spring-bal- 206 CONCEPTS OF MODERN PHYSICS. ance or pendulum, is a pound on the surface of the earth, would weigh but two ounces on the moon, less than one fourth of an ounce on several of the smaller planets, about six ounces on Mars, two and one half pounds on Jupiter, and more than twenty-seven pounds on the sun. And while the fall of bodies, in vacuo, near the surface of the earth amounts to about sixteen feet (more or less, according to the latitude) during the first second, their corresponding fall near the sur- face of the sun is more than four hundred and thirty- five feet. The thoughtlessness with which it is assumed by some of the most eminent physicists that matter is composed of particles which have an absolute primor- dial weight persisting in all positions and under all cir- cumstances, is one of the most remarkable facts in the history of science. "The absolute weight of atoms," says Professor Redtenbacher,* " is unknown " his meaning being, as is evident from the context, and from the whole tenor of his discussion, that our ignorance ' of this absolute weight is due solely to the practical impossibility of insulating an atom, and of contriving instruments delicate enough to weigh it. There is nothing absolute or unconditioned in the world of objective reality. As there is no absolute standard of quality, so there is no absolute measure of duration, nor is there an absolute system of coordinates in space to which the positions of bodies and their changes can be referred. A physical ens per se and a physical constant are alike impossible, for all physical existence resolves itself into action and reaction, and action imports change. * Dynamidensystcrn (Mannheim, Bassermann, 1857), p. 14. CHAPTER XIII. THE THEORY OF THE ABSOLUTE FINITUDE OF THE WORLD AND OF SPACE. THE ASSUMPTION OF AN ABSOLUTE MAXIMUM OF MATERIAL EXISTENCE AS A NECESSARY COMPLEMENT TO THE ASSUMPTION OF THE ATOM AS ITS ABSOLUTE MINIMUM. ONTOLOGY IN MATHEMAT- ICS. THE REIFICATION OF SPACE. MODERN TRANS- CENDENTAL GEOMETRY. NON-HOMALOIEAL (SPHERICAL AND PSEUDO-SPHERICAL) SPACE. IT was shown in the last chapter, that the theory, according to which space and motion are real only on condition of being absolute, involved the assumption of the existence of an absolutely fixed point of reference in space, and that this again of necessity led to the doctrine of the absolute finitude of the universe. Although the connection between this doctrine and the prevalent ontological theorems respecting space and motion has not hitherto, so far as I am aware, been pointed out, the doctrine itself has been variously suggested in the interests of cosmological speculations founded upon the atomo-mechanical theory, as a means of escape from certain inevitable consequences of this theory with which those speculations are found at last to conflict. And it has recently been urged by eminent mathematicians upon considerations respecting the true nature of space and the real character of space-relations. It is readily seen that the assertion of the absolute 208 CONCEPTS OF MODERN PHYSICS. finitude of the material universe is a logically integral part of the general assertion that whatever is real is absolute, and that the assumption of an absolute maxi- mum of material existence is a necessary complement of the assumption of its absolute minimum, the atom. The first explicit announcement of a scientific belief in this maximum appears to have been made by C. F. Gauss, in one of his letters to Schumacher,* in which he discusses the attempts of his Transylvanian friend Bolyai and of the Russian geometer Lobatschewsky to found a geometrical system which should be indepen- dent of the Euclidean axioms in regard to parallels. The hints thrown out by Gauss in the letters just referred to, as well as in various parts of his other writings,f have, within the last twenty years, been fruitful of a discussion respecting the nature of space, the founda- tions of geometry and the origin and import of geomet- rical axioms, which has already produced an extensive and rapidly increasing literature.^ The first effective * Gauss, Briefwechsel mit Schumacher, vol. ii, pp. 268-271. f Cf. " Disquisitiones generates circa seriem infinitam ! + + ," etc. (Comm. recent. Soc. Gott., ii, 1811-'13); " Theoria residuorum bi- quadraticorum Commentatio secunda" (ib., vii, 1828-'32). To those who are familiar with Herbart's theory that our idea of spatial extension is a psychological elaborate of qualitative data, i. e., of sensations which are in themselves without extension, it will not appear improbable that Gauss's mathematical transcendentalism was to some extent due to the speculations of his colleague in the philosophical faculty of Goettingen, although Gauss habitually professed great contempt for the Herbartian system just as Descartes was influenced by the teachings of his antago- nist Gassendi. The connection of Gauss's metageometrical or (to use the expression of Lobatschewsky) pangeometrical views with his investi- gations respecting the geometrical interpretation of imaginary quantities and the theory of " complex numbers " is apparent. \ Cf. Halstead, Bibliography of Hyper-Space and non-Euclidean Ge- ometry. American Journal of Mathematics, vol. i, pp. 261 seq. and 384 seq. ; ib., vol. ii, p. 65 seq. TKANSCENDENTAL GEOMETRY. 209 impulse to this new departure in the walks of mathe- matical theory was given by Riemann in a remarkable dissertation * read before the philosophical faculty of Goettingen, June 10, 1854 (published by Dedekind in 1866, after Riemann's death), and by Helmholtz in an equally remarkable essay f published two years later. These publications have since been followed by numer- ous articles, pamphlets and books expository of the doctrines thus advanced, and, as was to be expected, there has been no lack of writings in which these doc- trines have met with criticism and denial. The articles of the new geometrical faith are cer- tainly startling. Among them are propositions such as these : that our ordinary u Euclidean " tridimensional and " homaloidal " (flat) space is but one of several pos- sible forms of space ; that the preeminence of this Eu- clidean space among other forms of space can be main- tained upon empirical grounds alone, and, in the sense of the logical and psychological tenets of the sensation- alist school, depends solely upon the accidents of no- tional association, which may be (and, in the opinion of some enthusiastic advocates of the new doctrines, have been) overthrown by the discovery that the existence of additional dimensions is a necessary inference from cer- tain facts of experience which can not otherwise be ex- plained just as the third dimension of space is said to be, not directly perceived, but simply inferred from familiar facts of visual or tactual experience for whose explanation the third dimension is an indispensable * Ueber die Hypothesen welche der Geometrie zu Gnmde liegen (Ab- handlungen der Kgl. Gesellschaft der Wissenschaften zu Goettingen, vol. xiii, p. 133 seq.). f Ueber die Thatsachen die der Geometrie zu Grunde liegen (Nach- richten der Kgl. Gesellschaft der Wissenschaften zu Goettingen, 1865, June 3). 210 CONCEPTS OF MODERN PHYSICS. postulate ; that true and real space, therefore, has, or at least, for aught we know, may have, not three but four or even a greater number of dimensions ; that the space in which we move is, or may be, not homaloidal or flat, but essentially non - homaloidal, curved, spherical or pseudo-spherical, so that every line, which we have hitherto regarded as straight, may upon sufficient pro- longation prove to be a closed curve ; that, by reason of the inherent and essential curvature of space, the uni- verse, though unlimited, may be, and probably is, not infinite, but finite ; that on the supposition of the pseu- do-spherical character of space, a whole pencil of " shortest lines " may be drawn through the same point, all which are parallel to a given other " shortest line " in the sense that they will never intersect w T ith it, how- ever far produced ; that not only the measure of curva- ture of space, as well as the number of its dimensions, may be, and probably is, different in different spatial regions, so that no valid inference can be drawn, from our experiences in the regions in which we happen to dwell, as to the curvature or the dimensions of space immeasurably distant or immeasurably small, but that in any given region both the curvature of space and the degree or number of its dimensions may be, and probably is, undergoing a gradual transformation, and so on.* * The more cautious pangeometers have of late evinced a disposition to stigmatize some of the doctrines above enumerated, particularly those relating to the increase in the number of spatial dimensions and to the local differences and changes in the constitution of space, as inventions of their enemies or as extravagances of persons who are carried away by their enthusiasm. I may be pardoned, therefore, for citing a passage from a lecture of Professor P. G. Tait (who is certainly ready enough, as the book I quote from shows, to insist on sobriety in physics and math- ematics at least, whatever may, in his opinion, be the appropriate frame TRANSCENDENTAL GEOMETRY. - 211 However dissonant from the teachings of our fa- miliar experience these propositions seem, it is claimed that they are by no moans without empirical warrant. of mind for surveying the "Unseen Universe"): "The properties of space," says Tait, " involving (we know not why) the essential element of three dimensions, have recently been subjected to a careful scrutiny by mathematicians of the highest order, such as Riemann and Helmholtz ; and the result of their inquiries leaves it as yet undecided whether space may or may not have precisely the same properties throughout the universe. To obtain an idea of what is meant by such a statement, consider that in crumpling a leaf of paper, which may be taken as representing space of two dimensions, we may have some portions of it plane, and other por- tions more or less cylindrically or conically curved. But an inhabitant of such a sheet, though living in space of two dimensions only, and there- fore, we might say beforehand, incapable of appreciating the third di- mension, would certainly feel some difference of sensations in passing from portions of his space which were less to other portions which were more curved. So it is possible that, in the rapid march of the solar system through space, ice may be gradually passing to regions in which space has not precisely the same properties as we find here where it may have some- thing in three dimensions analogous to curvature in two dimensions some- thing, in fact, which will necessarily imply a fourth-dimension change of form in portions of matter in order that they may adapt themselves to their new locality." P. G. Tait, On Some Recent Advances in Physical Science, p. 5. In keeping with this passage is a note of the distinguished mathe- matician, Professor J. J. Sylvester, to his opening address to the Mathe- matical and Physical Section of the British Association at Exeter, in 1869, as follows: "It is well known, to those who have gone into these views, that the laws of motion accepted as a fact suffice to prove in a general way that the space we live in is a flat or level space (a ' homa- loid '), our existence therein being assimilable to the life of a bookworm in the flat space ; but what if the page should be undergoing a process of gradual bending into a curved form? Mr. W. K. Clifford has indulged in some remarkable speculations as to the possibility of our being able to infer, from certain unexplained phenomena of light and magnetism, the fact of our level space of three dimensions being in the act of under- going in space of four dimensions (space as inconceivable to us as our space to our supposititious bookworm) a distortion analogous to the rum- pling of the page. I know there are many who, like my honored and deeply lamented friend, the late eminent Professor Donkin, regard the alleged notion of generalized space as only a disguised form of algebrai- 212 CONCEPTS OF MODERN PHYSICS. It is insisted that there are numerous optic, magnetic and other physical phenomena of which they yield the only sufficient explanation. Moreover, it is said that they alone afford a clew to the mysteries of modern spiritism, enabling us to bring within the chain of nat- ural causation certain magical performances which we should otherwise be constrained to relegate to the regions of the Supernatural. In the first article of the first num- ber of the American Journal of Mathematics, Professor Simon Newcomb demonstrates analytically that, " if a fourth dimension were added to space, a closed material surface (or shell) could be turned inside out by simple flexure without either stretching or tearing," Felix Klein having shown, some time before, that knots can not exist in a four-dimensional space. Accordingly, Professor Zoellner accounts for the well-known feats of the American " medium " Slade on the principle of the fourth dimension one of these feats, however, strange- ly enough, consisting in the production of real trefoil knots in a rope the ends of which were sealed together and held in Zoellner's hands. And, finally, it is as- serted that the theorems of Lobatschewsky, Riemann, cal formulization ; but the same might be said with equal truth of our notion of infinity in algebra, or of impossible lines, or lines making a zero angle in geometry, the utility of dealing with which as positive sub- stantiated notions no one will be found to dispute. Dr. Salmon, in his extensions of Chasles's theory of characteristics to surfaces, Mr. Clifford in a question of probability, and myself in my theory of partitions, and also in my paper on Barycentric Projection, in the Philosophical Magazine, have all felt and given evidence of the practical utility of handling space of four dimensions as if it were conceivable space. Moreover, it should be borne in mind, that every perspective representation of figured space of four dimensions is a figure in real space, and that the properties of figures admit of being studied, to a great extent, if not completely, in their perspective representations." Nature, vol. i, p. 237 scq. The ital- ics in the above passages are mine. TRANSCENDENTAL GEOMETRY. 213 Helmholtz and Beltrami,* are the only true basis of a proper and exhaustive theory of parallelism. In the fullness of their faith in the impregnability of these positions, the votaries at the shrine of geometrical tran- scendentalism make bold to announce that, with the appearance of Lobatschewsky's " Geometrical Investiga- tions," f a new era has dawned upon the mathematical world, and that in the daylight of this era the whole body of geometrical truths will be reduced to simplicity and order in a way analogous to that in which the the- ory of celestial motions was simplified and cleared up by the great thought of Copernicus. " What Yesalius was to Galen," exclaims Professor Clifford, J "what Copernicus was to Ptolemy, that was Lobatschewsky to Euclid." The debate between the disciples of the new tran- scendental or pangeometrical school and the adherents of the old geometrical faith presents one feature which can not fail to strike the ordinary observer with some amazement. The disciples of the new school take their stand firmly upon empirical ground ; their very first proposition is that all geometrical truths are of empiri- cal origin, and that all we know of space and its prop- erties is what we are taught by sensible experience. This proposition and the consequent denial of the * An Italian mathematician who has investigated the properties of pseudo-spherical surfaces, which are distinguished from other surfaces of constant curvature by the fact that they admit of a sort of parallelism, in the transcendental sense, between their " straightest lines." A refer- ence to Beltrami's writings and a brief exposition of their contents may be found in Helmholtz's essay on " The Origin and Meaning of Geometri- cal Axioms," Mind, vol i, p. 306. f Geometrische Untcrsuchungen zur Theorie der Parallellinien, von Nicolaus Lobatschewsky. Berlin, Fincke'sche Buchhandlung, 1840. J Philosophy of the Pure Sciences, W. K. Clifford's Lectures and Es- says, vol. i, p. 29*7. 214: CONCEPTS OF MODERN PHYSICS. transcendental origin of geometrical axioms are empha- zised by Riemann and Helmholtz alike. And yet, upon this foundation they construct a theory which lands us in the remotest regions of transcendentalism in the realms of a metageoinetrical space in which all our wonted powers of imagination and conception are at fault and in which the facts of every-day experience as well as their natural relations are wholly out of sight. On the other hand, the most conspicuous champions of the old geometrical creed, in their defense of the famil- iar data of sensible experience and in their antagonism to the " vagaries " of transcendental geometry, invoke the doctrine of the non-empirical or transcendental origin of our ideas of space and its essential relations. The pangeometers erect a transcendental structure on empirical foundations, while the ordinary geometers build a system conforming to the data of experience upon transcendental grounds. This circumstance, how- ever, strange as it appears at first sight, will hardly surprise the thoughtful student of the history of theo- ries of cognition, or the intelligent reader of the pre- ceding pages. It is by no means unusual to find that ontological speculations, whether they appear in the guise of physical or in that of metaphysical theories, prove subversive in the end, not merely of the facts for whose explanation they were devised, but of the very supports by which they are supposed to be upheld. Having indicated, generally, the purport and scope of the transcendental theory of space, I now proceed to the examination of the premisses upon which it rests and of the arguments by which it is sought to be sus- tained. Here, at the outset, we find an assumption which obviously lies at the base of the whole theory : the assumption that space is a physically real thing TRANSCENDENTAL GEOMETRY. 215 not merely an object of experience, but an independent object of direct sensation whose properties may be as- certained by the aid of the ordinary instruments of physical and astronomical research whose degree of curvature, for instance, is to be determined by means of the telescope. This assumption is explicitly stated by each of the three great expounders of the theory in question. " The only means at our command," says Lo- batschewsky,* " to determine the accuracy of the prop- ositions (calculations) of ordinary geometry consist in an appeal to astronomical observations." To the same effect Riemann : f "If we assume that bodies exist in- dependently of their location in space, the measure of curvature (of space) is everywhere constant ; and then it follows from astronomical measurements that it is not different from zero." And in the same sense Helmholtz : f " All systems of practical mensuration that have been used for the angles of large rectilinear triangles, and especially all systems of astronomical measurement which make the parallax of immeasura- bly distant fixed stars equal to zero (in pseudo-spherical space the parallax even of infinitely distant points would be positive), confirm empirically the axiom of parallels and show the measure of curvature of our space thus far to be indistinguishable from zero. It remains, however, a question, as Riemann observed, whether the result might not he different if we could use other than our limited base-lines, the greatest of which is the major axis of the earths orbit" The view thus taken of the nature of space and of * Geometrische Untersuchungen, etc., p. 60. \ Ueber die Ilypothesen, etc., Abhandl. der Kgl. Gesellschaft dcr Wissenschaften zu Goettingen, vol. xiii, p. 148. \ "On the Origin and Meaning of Geometrical Axioms," Mind, vol. i, p. 314. 216 CONCEPTS OF MODERN PHYSICS. the origin of our notions concerning it is obviously in- dicative of a decided advance beyond the farthest out- posts of the old sensationalist camp. Nevertheless, it is supported in the main by a reference to the writings of a British thinker, J. S. Mill, who has been repeated- ly referred to in these pages, and who is regarded, es- pecially on the Continent, as the ablest modern ex- pounder and defender of the doctrines of sensational- ism, so far, at least, as they bear upon the subject now under consideration.* Stated in brief words, these doctrines are that the idea or notion of space is directly derived from sensible experience ; that the properties of space are to be determined by observation or ex- periment ; that the fundamental truths of geometry, like all other truths of physical science, are of induc- tive origin and warrant ; and that the certainty to be attributed to geometrical theorems, though possibly different in degree, is not different in kind from that belonging to any general assertion respecting physical facts. The peculiar tenets of pangeometry being thus founded, in great part at least, upon the general sensa- tionalist theory, it will be useful to enter upon a brief examination of this theory before I proceed to discuss the pangeometrical tenets themselves. For this pur- * I do not mean to say that Riemann and Helmholtz themselves di- rectly refer to Mill. But there are few German physicists and mathema- ticians who have not been diligent students of Mill's Logic, particularly since the appearance of Schiel's translation and the extravagant praises of Liebig; and this is quite apparent in most of the writings of the pangeometers. The interest with which each new edition of Mill's Logic has been received by scientific men everywhere is mainly due, doubtless, to its frequent references to scientific methods and results. The fact is that Mill has, for a series of years, been the official logician and meta- physician of the Continental naturalists and mathematicians, and the re- gard in which he is held by contemporary men of science is not unlike that which Aristotle enjoyed among the early mediaeval scholastics. TRANSCENDENTAL GEOMETRY. 217 pose I select an exposition of the theory in the book above referred to, the System of Logic of J. S. Mill, in which the fifth chapter of the second book "On Demonstration and Necessary Truth " contains an elab- orate statement of the author's views on the basis and methods of geometrical science. " The foundation of all sciences, even deductive or demonstrative sciences," says Mill * " is Induction ; every step in the ratiocination of geometry is an act of induction. . . . The character of necessity ascribed to the truths of mathematics, and even (with some reser- vations to be hereafter made) the peculiar certainty at- tributed to them is an illusion ; in order to sustain which it is necessary to suppose that those truths relate to, and express, the properties of purely imaginary ob- jects. It -is acknowledged that the conclusions of ge- ometry are deduced, partly at least, from the so-called Definitions, and that those definitions are assumed to be correct representations, as far as they go, of the ob- jects with which geometry is conversant. Now, we have pointed out that, from a definition as such, no proposi- tion, unless it be one concerning the meaning of a word, can ever follow ; and that what apparently follows from a definition follows in reality from an implied assump- tion that there exists a real thing conformable thereto. This assumption, in the case of the definitions of geom- etry, is not strictly true ; there exist no real things ex- actly conformable to the definitions. There exist no points without magnitude; no lines without breadth, nor perfectly straight ; no circles with all their radii exactly equal, nor squares with all their angles perfectly right. It will perhaps be said that the assumption does not extend to the actual, but only to the possible, exist- * A System of Logic (eighth ed.), p. 168 seq. 12 218 CONCEPTS OF MODERN PHYSICS. ence of such things. I answer that, according to any test we have of possibility, they are not even possible. Their existence, so far as we can form any judgment, would seem to be inconsistent with the physical consti- tution of our planet at least, if not of the universe. To get rid of this difficulty, and at the same time to save the credit of the supposed system of necessary truth, it is customary to say that the points, lines, circles, and squares which are the subject of geometry exist in our conceptions merely, and are part of our minds ; which minds, by working on their own materials, construct an a prim science, the evidence of which is purely men- tal, and has nothing whatever to do with outward ex- perience. By however high authorities this doctrine may have been sanctioned, it appears to me psychologi- cally incorrect. The points, lines, circles, and squares which any one has in his mind are (I apprehend) simply copies of the points, lines, circles, and squares which he has known in his experience. Our idea of a point I apprehend to be simply our idea of the minimum visi- fiile, the smallest portion of surface which we can see. A line, as defined by geometers, is wholly inconceivable. We can reason about a line as if it had no breadth ; be- cause we have a power, when a perception is present to our senses, or a conception to our intellects, of attending to a part only of that perception or conception, instead of the whole. But we can not conceive a line without breadth ; we can form no mental picture of such a line ; all the lines which we have in our minds are lines possessing breadth. If any one doubts this, we may re- fer him to his own experience. I much question if any one, who fancies that he can conceive what is called a mathematical line, thinks so from the evidence of his consciousness : I suspect it is rather because he supposes TRANSCENDENTAL GEOMETRY. 219 that, unless such a conception were possible, mathematics could not exist as a science : a supposition which there will be no difficulty in showing to be entirely ground- less. " Since, then, neither in nature, nor in the human mind, do there exist any objects exactly corresponding to the definitions of geometry, while yet that science cannot be supposed to be conversant about nonentities, nothing remains but to consider geometry as conversant with such lines, angles, and figures, as really exist ; and the definitions, as they are called, must be regarded as some of our first and most obvious generalizations con- cerning those natural objects. The correctness of those generalizations, as generalizations, is without a flaw ; the equality of all the radii of a circle is true of all cir- cles, so far as it is true of any one : but it is not exactly true of any circle ; it is only nearly true ; so nearly that no error of any importance in practice will be incurred by feigning it to be exactly true. When we have occa- sion to extend these inductions, or their consequences, to cases in which the error would be appreciable to lines of perceptible breadth or thickness, parallels which deviate sensibly from equidistance, and the like we correct our conclusions by combining them with a fresh set of propositions relating to the aberration ; just as we also take in propositions relating to the physical or chemical properties of the material if those properties happen to introduce any modification into the result ; which they easily may, even with respect to figure and magnitude, as in the case, for instance, of expansion by heat. So long, however, as there exists no practical necessity for attending to any of the properties of the object except its geometrical properties, or any of the natural irregularities in those, it is convenient to neglect 220 CONCEPTS OF MODERN PHYSICS. the consideration of the other properties and of the ir- regularities, and to reason as if these did not exist : ac- cordingly, we formally announce in the definitions that we intend to proceed on this plan. But it is an error to suppose, because we resolve to confine our attention to a certain number of the properties of an object, that we therefore conceive, or have an idea of, the object, denuded of its other properties. We are thinking, all the time, of precisely such objects as we have seen and touched, and with all the properties which naturally be- long to them ; but, for scientific convenience, we feign them to be divested of all properties, except those which are material to our purpose, and in regard to which we design to consider them. u The peculiar accuracy, supposed to be character- istic of the first principles of geometry, thus appears to be fictitious. The assertions on which the reasonings of the science are founded do not, any more than in other sciences, exactly correspond with the fact; but we suppose that they do so, for the sake of tracing the consequences which follow from the supposition. The opinion of Dugald Stewart respecting the foundations of geometry is, I conceive, substantially correct ; that it is built on hypotheses ; that it owes to this alone the peculiar certainty supposed to distinguish it ; and that in any science whatever, by reasoning from a set of hypotheses, we may obtain a body of conclusions as cer- tain as those of geometry, that is, as strictly in accord- ance with the hypotheses, and as irresistibly compelling assent, on condition that those hypotheses are true." I have quoted this passage, from Mill's Logic, at length, not only because it is the most elaborate and connected statement of the sensationalist theories con- cerning the character of necessary truths and especially TRANSCENDENTAL GEOMETRY. 221 the truths of geometry, but also because this statement exhibits certain peculiarities that are worthy of atten- tion. One of these peculiarities is the concession that the mind has the power of abstraction, and of forming and reasoning about generalizations which, "as generali- zations, are without a flaw." The inconsistency of this admission with the claim that " the points, lines, circles, and squares which any one has in his mind are simply copies of the points, lines, circles, and squares which he has known in his experience," is evident. And this inconsistency has not escaped the notice of other pro- moters of the experiential or sensationalist doctrine, as is shown, for instance, in the writings of Mr. Buckle, who does not hesitate to draw the true conclusions (from which Mill himself appears to recoil) from Mill's premisses. Buckle not only boldly asserts that there are no lines without breadth (he strangely forgets the thickness), but also that the neglect of this breadth by the geometrician vitiates all the results of geometrical inference, the only comfort vouchsafed to us being that the error, after all, is not very considerable. " Since, however," he says,* " the breadth of the faintest line is so slight as to be incapable of measurement, except by an instrument used under the microscope, it follows that the assumption that there can be lines without breadth is so nearly true that our senses, when unas- sisted by art, can not detect the error. Formerly, and until the invention of the micrometer, in the seven- teenth century, it was impossible to detect it at all. Hence, the conclusions of the geometrician approximate so closely to truth that we are justified in accepting them as true. The flaw is too minute to be perceived. * History of Civilization in England, vol. ii, p. 342 (Appletons* Ameri- can edition). 222 CONCEPTS OF MODERN PHYSICS. But that there is a flaw appears to me certain. It ap- pears certain that, whenever something is kept back in the premisses, something must be wanting in the con- clusion. In all such cases, the field of inquiry has not been entirely covered ; and, part of the preliminary facts being suppressed, it must, I think, be admitted that complete truth be unattainable, and that no problem in geometry has been exhaustively solved." "Whether Buckle was able to think of a line as the limit between two surfaces, and whether, in his opinion, such a limit has breadth (i. e., is itself a surface, so that we are driven from limit to limit, ad infinitwrn\ he does not tell us. Nor does he say whether or not, in view of the fact that the breadth of a line depends upon the material out of which it is constructed, or upon which it is drawn, there ought to be a pasteboard geom- etry, a wooden geometry, a stone geometry, and so on, as distinct sciences. To do justice, however, to Mill and the subject under discussion, we must keep before us Mill's own statement. Returning, then, to his exposition, the question arises at once : What does he mean when he says that none of the elements of space exist in fact as they are considered in the science of geometry that, for example, there exist no lines perfectly straight? The only possible meaning is that none of the straight lines, so called, of which we have experiential knowl- edge, are congruent with the straight lines of which we have other knowledge that they do not conform to the standard straight line in our minds. But Mill asserts that " the lines, etc., which any one has in his mind, are simply copies of the lines which he has known in his experience." There is no standard, therefore, with which the lines presented in experience can be compared of THI TRANSCENDENTAL GEOMETRY. and from which, they can be Thus Mill's theory breaks down with the very first fact which he brings to its support.* This is no mere cap- tious criticism. It is a simple exhibition of the utter senselessness of the premisses from which Mill's con- clusions are drawn. The whole foundation of his theory crumbles the moment it is touched. And, upon further examination, it is found that he entirely mistakes the significance of the facts which he adduces. The real import of Mill's assertion just referred to is very differ- ent from that which he ascribes to it. The truth which lies at the bottom of that assertion is that we have no experiential knowledge, in Mill's sense, of lines, circles, squares, at all. We have experiential knowledge of so-called straight rods, cords, edges, or grooves, of spherical and cubical bodies with circular or square sections or sides; but our knowledge of points, lines, surfaces, and geometrical solids is obtained solely by the process of abstraction. Nothing is clearer and more readily demonstrable than that the elements of geomet- rical science the foundations upon which the science of geometry rests can not have been obtained by in- duction, and that, 'a fortiori, it can not be true, as Mill contends, that " every step in the ratiocinations of geometry is an act of induction. 1 ' Induction is a cu- * That so acute a thinker as J. S. Mill was blind to the many incon- sistencies and absurdities with which his Logic and parts of his other writings abound, is explicable solely by the fact that he took his theory of cognition upon trust as a sacred inheritance from his father, who, in turn, had derived it from the French and English nominalists and sensational- ists of the seventeenth and eighteenth centuries. The doctrines of these sensationalists were necessarily crude, because they originated at a time when rational psychology was in its infancy and comparative psychology was not even thought of ; and they were extravagant because they were generated by the antagonisms of an equally extravagant realism. 224 CONCEPTS OF MODERN PHYSICS. rmilation of instances in all of wliicli the same element or feature is found along with other elements or feat- ures. But no one has ever seen two bodies whose edges, though called straight, did not prove to be broken in different degrees when examined with sufficient magnifying power. Experience furnishes no two in- stances presenting the feature of straightness in the same degree. Much less has any one seen a great num- ber of bodies whose edges were exactly coincident. The same thing is, of course, true, mutatis mutandis, of points, curves, surfaces, and solids. The divergences between their figures as well as their magnitudes become more apparent in proportion to the magnifying power with which they are examined. And their true figures are wholly undiscoverable by any magnifying process at our command. The truth is, that we never get sight or come into the actual presence of a true and complete geometrical fact. It is simply nonsense, therefore, to say, with "Mill, that the points, lines, surfaces, solids, etc., with which the science of geometry deals and respecting which it is able to draw valid deductions, are real (i. e., physical) and not imaginary points, lines, surfaces, and solids, and that the points, lines, surfaces, and solids in our minds are copies of them. It is true enough that the geometrical elements are not imaginary, because they have reference to real facts ; nor are they in any proper sense hypothetical, as is contended by Dugald Stewart ; they are conceptual, the results of abstraction. If it were otherwise, deductive geomet- rical ratiocination and, indeed, any other kind of rea- soning properly so called would be utterly impossible. All deductive reasoning depends upon the power of abstraction. And this truth is applicable, not only to geometry and to mathematics generally, but to all TRANSCENDENTAL GEOMETRY. 225 sciences whatever. This is so for two reasons : In the first place, no physical thing (or historical event) ever becomes experientially known to us with all its prop- erties, relations, or incidents ; sensation and perception never furnish the intellect with a complete fact. And, in the second place, as I have heretofore shown, in dealing with the facts, so called, obtained at the hands of sensible experience, the intellect is restricted to cer- tain definite relations which it segregates or abstracts from other relations. In the processes of discursive thought, the intellect never has before it either sensible objects or the whole complement of relations which make up their mental images or representations, but only some single relation or class of relations. It oper- ates along lines of abstraction, the final synthesis of whose results never yields anything more than outlines of the objects represented. During all its operations the intellect is fully aware that neither any one link in the chain of abstraction nor the group of abstract results which we call a concept (in the narrower sense of a collection of attributes representing an object of intui- tion or sensation) is a copy or an exact counterpart of the object represented. It is always conscious that, to bring about true conformity between concepts or any of their constituent attributes with forms of objective reality, the group of relations embodied in these con- cepts would have to be supplemented with an inde- terminate number of other relations which have not been apprehended and possibly are insusceptible of apprehension. But this nowise affects the validity of the intellectual operation. The mathematician, when he determines the properties of a conic section, knows full well that he will never meet with a body whose geometrical outline is an exact exemplification of the 226 CONCEPTS OF MODERN PHYSICS. law of the constancy of the ratio between the distances of any one of its points from a fixed point and a fixed straight line respectively, and that there is in nature no trajectory which strictly coincides with such a curve. But this knowledge does not in the least degree disturb his faith in the perfect validity of his reasoning. When he comes to apply the results of this reasoning to a nat- ural fact, he supplements it, as well as he can, with the results of other processes of reasoning based upon other known relations of the same fact, and thus approaches the fact as nearly as possible, nothing daunted by the ever-present reflection that he will never succeed in coming into the actual presence of the whole fact with all its relations. It is obvious that the conformity of the results of abstract or conceptual reasoning to the data of experi- ence is in direct proportion to the degree of indepen- dence of the relations dealt with from the other relations which constitute the conditions of real existence in the object represented in the operation of thought. Herein lies the preeminence of geometry among the physical sciences. In the physical sciences usually so called the relations about which these sciences are conversant are closely interdependent ; the thermal, electric, magnetic, optical, and chemical properties of a body in various ways determine each other. If the nature and degree of this interdependence were accurately known and could be brought within the reach of exhaustive con- ceptual analysis, these sciences would become deductive to the same extent to which the science of geometry is deductive. All the physical sciences are constantly striving to progress in this direction, but the progress is so slow as to afford little hope that the goal here in- dicated will ever be reached. One reason for this is TRANSCENDENTAL GEOMETRY. 227 that the number of newly discovered relations multiplies at the same (if not at a greater) rate at which the nature and degree of the interdependence between the relations already known are brought to light. And the difficulty of determining the interdependence in question increases geometrically as the number of new relations is aug- mented arithmetically. The foregoing reflections are sufficient, in my judg- ment, to show that the sensationalist views of space and of the nature and warrant of geometrical truth are un- tenable, at least in the form in which they are pro- pounded by Mr. Mill. But these reflections do not in the least degree impugn the general proposition that all our knowledge of the objective world is derived from experience. This proposition appears to me to be undeniable, and is, doubtless, assented to, explicitly or in some mode of implication, by every sane person at the present day, the only controversies respecting it being disputes about the meaning of terms. But the sensationalists, and especially, as I have already shown, the founders and supporters of transcendental geometry, advance a thesis which is to be carefully distinguished from the proposition just stated. They maintain that space is not only objectively real, but a direct and inde- pendent object of sensation whose properties are to be empirically ascertained like those of any other physical thing. This assertion has been met with the counter- assertion, made by the antagonists of geometrical tran- scendentalism, that space, like time, is not an indepen- dent object of sensation, but, as Kant has taught, or is supposed to have taught, a mere form of intuition, a state or condition of the intellect existing independently and in advance of all sensible experience. The contest between the champions of the new doctrine and their 228 CONCEPTS OF MODERN PHYSICS. opponents has been mainly carried on under the belief, common to both the contending parties, that these views are strict alternatives, and that no other view is admis- sible or possible. Let us test these two conflicting as- sertions by facts of cognition about which there is no contest, or which clearly can not be contested on rational grounds. First, as to the assertion of Riemann and Helm- holtz : if space is a physically real object, it certainly is not a thing outside of, coordinate with, and different from, other physical objects. When we say that all things are in space, we do not mean that they are con- tained in it as water is contained in a vessel, but we mean that there is no objectively real thing which is not spatially extended, or, according to the usual form of expression, that spatial extension is a primary prop- erty of all varieties of objective existence. This fact is so obvious that Descartes was led by it to maintain that spatial extension was the only true essence of ob- jective reality. In what way, then, and by what means, do we distinguish between space and physical things ordinarily so called ? Certainly not, or at least not di- rectly, by sensation. Different acts of sensation may present different properties of the same object, and these properties may thus be dissociated. But no act of sensation dissociates the extension of a body from all its other properties and presents the property of exten- sion alone. The sensationalists, however, contend (and here they trench upon the ground of their opponents, the Kantian idealists) that, although there are no phys- ical objects without spatial extension, and although such extension is in a sense a common property of all physical objects, nevertheless these objects do not fill all space, there being pure space between them. The TRANSCENDENTAL GEOMETRY. 229 reply to this is that this assertion, if true, does not help the sensationalists. For acts of sensation are possible only when and where there is objective difference and change ; we have direct sensation of different and vari- able physical qualities ordinarily so called, and not of that which is absolutely homogeneous and invariable. Here comes in Hobbes's law : Sentire semper idem et non sentire ad idem recidunt. It is precisely the fact of its homogeneity and unchangeableness, in addition to that of its invariable presence in all physical objects, which distinguishes the property of spatial extension from all other properties characteristic of a real thing, and enables the sensationalist to speak of the existence of space at all. If this distinction could be obliterated if the cognitive or conceptual barrier which separates the sensations produced by physical action from the states of consciousness representative of space were once broken down there would no longer be any ground whatever for the distinction between the " properties " of space and the properties of matter in any of its va- rieties. We should be constrained to say that the only form or variety of objective existence is either space or matter (it being a mere question of nomenclature which), and that all the properties we now attribute to matter are in truth and in fact properties of space. That all this should have escaped the attention of Riemann and Helmholtz is marvelous, considering the o assumption made by both in order to account for the alleged necessity of attributing to space a constant measure of curvature, and thus limiting the number of the species of space, which, according to their state- ment, are admissible, to three, viz. : spherical space with a positive curvature, pseudo-spherical space with a negative curvature, and flat or homaloidal space with a 230 CONCEPTS OF MODERN PHYSICS. curvature equal to zero.* I allude to the assumption that bodies, in the language of Riemann already quoted, " exist independently of their location in space/' which means, of course, that they are at least different from, if they have not a physical constitution wholly inde- pendent of, space. But for this assumption there can be no valid reason founded upon or consistent with the premisses of the transcendental theory, why space may not be essentially paraboloidal, or hyperboloidal, or polyhedral, or of any other inherent form evolvable from the creative fancy of the next non-homaloidal in- tellect. This brings me to the allegation of the transcenden- talists, that the properties of space, such as the degree and form of its curvature, are to be determined by ex- periment. How would such a determination be ef- fected? Suppose an astronomer, at proper intervals, directed his telescope to some fixed star whose distance from the earth he knew in some way (say from spec- troscopic data) to be far greater than that of Arcturus, for the purpose of ascertaining its parallax. And sup- pose he found this parallax sensibly to exceed that of the less distant star in other words, suppose he found, that the angle of intersection between the lines of his vision was different from that required by the known facts and laws of astronomy and optics : what would be his conclusion ? It is not difficult to anticipate the an- swer to this question, for the case supposed is not with- out precedent in the history of astronomy. Displace- ments in the lines of vision have repeatedly been observed by astronomers, who were unable to account * Felix Klein " (Uebcr die Nicht-Euklidische Geometrie," Mathema- tische Annalen, vol. iv, p. 577) designates these kinds of space as elliptic, parabolic, and hyperbolic. TRANSCENDENTAL GEOMETRY. 231 for them by the facts and natural laws of which they had knowledge. In the early part of the last century, Bradley (with the aid of Molyneux) made a series of telescopic observations of the star 7 Draconis, to the end of determining the amount of its apparent displace- ment due to the orbital motion of the earth, so as to detect the annual parallax of the fixed stars an achieve- ment very desirable in Bradley's time by reason of a standing objection to the Copernican system urged on the ground of the alleged absence of such a parallax. To his surprise he found a displacement different in direction and far greater in degree than that looked for. This anomaly had to be explained ; and Bradley knew of no physical cause to which it could be assigned. He thought for a time of nutation ; then of refraction ; but he soon became satisfied that no explanation was af- forded by either fact. He was finally led, by a careful study of the variations in the direction and rate of the displacement, to look for a solution of the mystery in the composition between the velocity of light and that of the earth's orbital motion, and thus became the dis- coverer of what is now know r n as the aberration of light. Amid all his perplexities, however, it does not appear at any time to have occurred to him that the anomalous phenomenon could be the result of a constitutional crook in space. And it may be asserted with confi- dence that there is no astronomer living to-day who would attribute the anomalous parallax, whose dis- covery I have supposed, to a spatial pseu do sphericity. For, irrespective of all other considerations, the astrono- mer would at once meet every suggestion- of this sort with the objection that an inherent curvature of space presupposes differences between its several parts hete- rogeneities in its internal constitution and that the 232 CONCEPTS OF MODERN PHYSICS. hypothesis suggested, therefore, involved nothing less than the attribution to space of the very properties by the absence of which alone it is distinguishable from matter. The theory of the geometrical transcendentalists is thus invalidated by the absurdity of its fundamental as- sumption. Space is not, can not be, an object of sen- sation. The attribution to space of relations and sen- sible interactions of the kind reflected in sensation is impossible without the assumption of diversities among its constituent parts, the denial of which is the basis of every notion or concept of space, whatever may be the logical or psychological doctrine to which that notion is referred. Are we driven, then, to the counter-assump- tion of the Kantian idealists, that space is a purely sub- jective form of intuition existing in the mind inde- pendently of and antecedently to all acts of sensa- tion to the doctrine of the metaphysical and mathe- matical adversaries of geometrical transcendentalism? Let us see by what arguments this doctrine is en- forced. The Kantian idealists affirm that the idea of space is not only an invariable element of every act of sensation, but a condition precedent to sensation ; that, before we are able to refer any subjective impression to an objec- tive cause, and thus to speak of the existence of objec- tively real things or phenomena at all, the basis of this reference of the relation, not merely between the With- in and the Without, but also between two elements at least of the Without whose interaction produces the sen- sation must already be present in the intellect. Sensa- tion, it is said, is of objects ; it is essentially a step from a subjective affection or feeling to an objective reality. Where is the ground for this step ? Not, contends the TRANSCENDENTAL GEOMETRY. 233 Kantian, in tlie world of objects ; for the objects are reached and become existent in intuition and sensation only by means of the step. It must be in the subject, therefore, in the intellect ; and it must be present there in advance of the act of sensation. That this is so, ap- pears, moreover (it is claimed), from the fact that the idea of space is absolutely indestructible. We can men- tally evacuate space of its sensible contents ; the intel- lect can "think away" everything that is an object of sensation ; but it can not " think away " space itself. Space is an integral part of all states of consciousness whatever. The foregoing exposition is a fair and sufficiently exhaustive statement of the Kantian view. This view has one feature in common with that of the sensation- alists to which I have already made incidental allusion - the assumption that space exists, either as an object of sensation or as a form of intuition, as an independent fact, and is therefore susceptible of objective or subjec- tive apprehension "by itself. I have already shown that this assumption, in the sensationalist sense, is unfounded. And, upon careful examination, it proves equally un- founded in -the sense of the idealists. It is not^ true that we can mentally evacuate space of all its contents, and have in the mind, or before the mind, the form or image of pure space. On the contrary, the idea of space is invariably associated in consciousness with some definite quality of sense. When we attempt to bring space before the mind (or, as it is usually called, to " realize " it) in its visual aspect, it always appears in synthesis with a mental reproduction of some sensation of color, however faint. Similarly, when we make the effort of mentally " realizing " or representing it in its tactual aspect, it proves equally indissociable from a 234 CONCEPTS OF MODERN PHYSICS. reproduction of some form of pressure or feeling.* In this respect the arguments of Hume and Berkeley (which are of necessity simple appeals to consciousness) have never been successfully met. The dissociation be- tween the " idea " of spatial extension and the feeling or feelings constituting sensation which we are able and, for the purposes of discursive reasoning, constrained to effect, is not an intuitional, but a conceptual dissocia- tion. Whenever we contemplate and reason about an objectively real thing, we can, in virtue of the power of abstraction, attend to the property of spatial extension, in total disregard of its sensible qualities ; but whenever we strive to bring its extension before the mind as real to frame a mental image of extension, or to represent it as a distinct form of intuition we are instantly forced to invest or associate it with some one datum of sensa- tion which we interpret as the incident or reaction of a physical process. Intuition (using the word in the Kantian sense) is an integral part of sensation, and ap- pears as such alike in the presentations of sense and their representations or reproductions in the phantasy. This disposes of the Kantian argument that space must be a subjective form of intuition because the mind can not banish it from consciousness. And another simple reflection is equally fatal to the claim that space must be a subjective form existing in advance of all acts of sensation, inasmuch as it is the indispensable ground for the step by which the intellect reaches an object external to itself. The obvious answer to this is that, if space is purely subjective, being wholly in the mind, it certainly can not afford ground for a step out * Cf. Sir William Hamilton's Lectures on Metaphysics, Lect. xxii ; Stumpf, Ueber den psychologischen Urspmng der Raumvorstellungen (Leipzig, Hirzel, 1873), p. 19. TRANSCENDENTAL GEOMETRY. 235 of it. This reflection is the true basis of the post- Kantian species of idealism, such as that of Fichte, and, in a sense, of Schopenhauer. But the whole argument, as well as the idealistic perplexities that have been oc- casioned by it, is founded on the old ontological assump- tion that things or entities exist independently of each other and otherwise than as terms of relations. That this is not true of objectively real things has been suffi- ciently enforced in the preceding pages of this book ; and it is equally untrue of the relation between the cognizing subject and its object. In every act of pri- mary cognition, the objective phenomenon, so called, and its subjective counterpart are born into conscious- ness at the same moment, because the reality of either depends upon that of the other. This is the great pri- mary and irreducible fact of cognition, which is not the less a fact because it has been misinterpreted by the metaphysicians in a variety of ways, and has given rise to a host of absurd cognitive theories. "What, then, is the real nature of space and what is the true source of our knowledge respecting it ? If the preceding considerations are valid and conclusive, this question admits of but one answer. Space is a concept, a product of abstraction. All objects of our sensible ex- perience present the feature of extension in conjunction with a number of different and variable qualities at- tested by sensation ; and, when we have successively abstracted these various sensations, we finally arrive at the abstract or concept of a form of spatial extension. I purposely say form of extension, and not simply ex- tension or space, for the former, and not the latter, is the summum genus of the line of abstraction here indicated. If the word " concept " be used in the sense in which it is representative of a possible object of intuition, a spar 236 CONCEPTS OF MODERN PHYSICS. tially extended form is the last result of the process by which an object or phenomenon can be conceived. The abstract or concept (using the word now in its wider sense) extension generally, or space, is reached by an- other series of abstractions of which I may have some- thing to say hereafter. The failure to discriminate between those concepts, so called, which involve no reference to limits or forms and the true summa genera of the classification of sensible objects is one of the sources of the confusion which everywhere besets the theory of transcendental space, as we shall presently see. The doctrines of the idealists (more properly called intellectualists) respecting the nature of space are, there- fore, as untenable as those of the sensationalists. And the opinion of the disciples of Kant and Schopenhauer, that the teachings of transcendental geometry can be refuted by an appeal to the " Transcendental ^Esthet- ics" of the "Critique of Pure Reason" is a mistake. The proposition that space is a pure subjective form of intuition, if true, could not in the slightest degree shake the position of the geometrical transcendentalist. His simple retort upon the Kantian is, that, if space is an innate form or condition of the intellect determining the apprehension of external objects in a certain order, or according to certain laws, it is again a question of fact, what is that order and what are those laws. Whether space be within the mind or without it, the question of its flatness, sphericity or pseudo-sphericity remains. Whether the form of the lines and surfaces possible in space is the result of its physical constitution outside of the mind, or of the internal constitution of the mind itself in either case the fact is the same, whatever it may prove to be. This is in entire ac- TRANSCENDENTAL GEOMETRY. 237 cord with Kant's own distinct declaration in his " Notes to Transcendental ^Esthetics," * when he says that our mode of intuition is not necessarily confined to the peculiar constitution of our minds, but may be shared by other thinking beings, "although this is a matter which we are unable to decide." From this declaration the inference is unavoidable that the question, what the precise form of intuition is in a given intellect, is purely a question of fact. In this respect, then, Helmholtz f is unquestionably in the right as against Land, Krause, Becker, and the other Kantians. Having reached the conclusion that space is neither a physical object of sensation, nor an innate form of the mind independent of and preexisting to all sensation, but a concept, we are now able to enter upon a series of considerations akin to those already presented against the alleged experimental determinability of the curva- ture of space, by which the true character of the tran- scendental theory of space may be so thoroughly exhib- ited that there can be no rational controversy respecting its merits. The first of these considerations is this : If the doctrines of the transcendentalists are founded in fact, it follows that there is in space a coercive power resulting from its constitution which makes lines and surfaces other than those conforming to its inherent figure impossible. If space is not "flat," but spherical, for instance I assume for the moment, and for the sake of argument, that there is sense in the assertion of the "flatness" of ordinary "Euclidean" space then every line in it necessarily follows a definite course to which it is astricted by the internal law governing the * Kritik der reinen Vernunf t (ed. Rosenkranz), p. 49. f Cf. " The Origin and Meaning of Geometrical Axioms," Mind, vol. iii, p. 212 scq., also Die Thatsachen in der Wahrnchmung, Berlin, 1879. 238 CONCEPTS OF MODERN PHYSICS. arrangement of its parts. From this it is a legitimate and inevitable consequence that, in a space of definite and inherent curvature, lines even of different degrees of curvature are impossible. The measure of curvature of such a space being once determined, all its lines must conform to it. To this it is no answer to say that Lo- batschewsky and Beltrami have shown the practicability of constructing consistent and logically coherent sys- tems of geometry on the principle of the non-parallel- ism of " shortest lines," and that Professor Lipschitz has demonstrated that the laws of motion as dependent on motive forces could also be consistently transferred to spherical or pseudo-spherical space, so that the com- prehensive expression for all the laws of dynamics, Hamilton's principle, may be directly transferred to spaces of which the measure of curvature is other than zero. For the constructions of Lobatschewsky and Bel- trami (which serve also as the basis of Lipschitz's inves- tigations) are all constructions of lines and surfaces ; and these constructions are founded upon postulates utterly inconsistent with the properties of non-Euclidean space. One of these postulates is, that in spherical as well as in pseudo -spherical space it is possible to trace lines of any degree of curvature, and therefore also lines whereof the curvature is zero, that is to say, straight lines in the old sense. How, indeed, could the "meas- ure of curvature " be other wis-e determined ? That measure depends upon the radius of curvature ; accord- ing to Gauss, the measure of curvature belonging to every surface that admits of the motion of the figures lying upon it, without change of any of their lines and angles, measured along it, is the constant reciprocal of the greatest and least radii of curvature. These radii are straight radii, in the old sense ; for, if they are not TRANSCENDENTAL GEOMETRY. 239 straight, they are of some definite degree of curvature, which again can be determined only by reference to another particular radius, and so on, either ad infinitum, or until we come at last to the old Euclidean straight line. The legitimate premisses of the theory of non-Eu- clidean space lead to the inevitable conclusion that the lines of such a space, though curves, have neither tan- gents nor normals, neither radii nor cords, and that on the grounds of non-Euclidean postulates alone they are wholly indeterminable. This is again a curious exem- plification of the ontological error according to which things and forms are determinable in themselves, with- out reference to, or contrast with, correlative things and forms. What is especially remarkable, in this aspect of the doctrine of the transcendentalists, is the ascription to real space of an inherent disjunction be- tween the forms of its alleged curvature the assertion that its measure of curvature must be either positive, or negative, or zero. This assertion is all the more re- markable by reason of the transcendentalist claim that the new doctrine has emancipated the old system of geometry from its arbitrary limitations, and is a widen- ing, a logical expansion, of the idea of space. The source of all the perplexities in which we find ourselves involved by the assumptions and theories of the transcendentalists is so obvious, that it is a wonder how it has come to be completely ignored by the adver- saries of the new doctrine no less than by its adherents. The parent error of this doctrine is the assertion that the space, with which ordinary " Euclidean " geometry deals, is a " flat," and not a spherical or pseudo-spherical space. The truth is that the space whose idea or notion underlies all geometrical constructions whatever, in- 240 CONCEPTS OF MODERN PHYSICS. eluding those of the pangeometers, is neither flat, nor spherical, nor pseudo-spherical, nor of any other inher- ent figure, but is simply the intuitional and conceptual possibility of tracing any or all of the lines characteris- tic of plane, spherical, ellipsoidal, pardboloidal, hyper- l>oloidal,.etc., and, to some extent, pseudo-spherical sur- faces within it a possibility due to the circumstance that it is nothing more nor less than a concept formed by dismissing from our mental representation of physi- cal objects, not only all the attributes constituting their physical properties other than extension, lut also all the determinations of figure ~by which they are distin- guished. This is the only sense in which we have any right to speak of space as even or homaloidal. Space has no internal structure or inherent figure, because it is not a physical object, and therefore has no " proper- ties " which can be ascertained by experiment or obser- vation. Nor has it any properties, rightly so called, that are determinable a priori, by an act of intuition. Space is one of those ultimates of abstraction in which the connotation coincides with the denotation, and in which, therefore, true connotation is at an end. I re- peat : space has no properties, for, considered as an entity, it has no relations, its very essence being a denial of, or abstraction from, all relations. For this reason it is an abuse of terms to define geometry (as is so fre- quently done, and has lately been done by Professor Henrici *) as " the science whose object it is to investi- gate the properties of space." The object of geometry is the investigation of the possible determinations or limi- tations of space, i. e., of the relations between the vari- ous forms of extension or of the properties of figures. f * Encycl. Britan., s. v. Geometry. f In this sense D'Alembert (Elemens do Philosophic, 15 CEuvres, TRANSCENDENTAL GEOMETRY. 241 The whole science of geometry is conversant about that which the concept space of necessity excludes, viz., about determinations or limits. Geometry, indeed, has refer- ence to space, inasmuch as the determinations with which it deals are spatial determinations. Upon this fact arises the difference between the scope of geometry and that of the other branches of pure mathematics, and the inapplicability of many of the methods and results of mathematical analysis to the relations between the forms of space a difference the disregard of which is prolific of so many errors in the reasoning of those who seek to draw conclusions respecting the tf properties " of space (such as the possible number of its dimensions) from the abstract concept quantity. Geometry is undoubt- edly an empirical science, though not in the sense in which the term " empirical " is generally understood, and especially not in the sense in which it is interpreted by Mill and the geometrical transcendentalists. It is an empirical science, inasmuch as it deals with a property of physical things, extension, which is an ultimate, or, rather, primary and irreducible datum of the act of sen- sation just as much such a datum as the sensation of color with which, as I have shown, the visual intuition of space is invariably associated. All attempts, such as those of Herbart, to produce the " idea " of extension by an elaboration of such data of sensation as are com- monly designated as qualitative, are as abortive as the corresponding attempts to deduce the qualitative ele- ments of sensation from forms of extension. The pri- mary datum of extension is the empirical element in the science of geometry. This primary datum is not space, but limited extension, for sensation and intuition are of tome 5, p. 268) defines geometry as " la science des proprietes de 1'eten- due en tant qtfon la considere comme ^implement etcndue etfiguree." 13 242 CONCEPTS OF MODERN PHYSICS. particular bodies, and therefore of limited extension, not of extension generally, or space. Forms of limited extension, however, give rise to the concept space, by the application of the processes of abstraction I have indicated. On the other hand, the conclusions of ge- ometry are not derived from empirical data alone, and are not reached by processes of induction, as Mill con- tends, and in that sense geometry is not an empirical science. Nor is there any geometrical axiom which is purely a datum of sensation, as is asserted ~by the sensationalists, or of intuition, according to the teach- ings of the idealists or intellectualists. All the geomet- rical axioms, which serve as starting-points of deduction, contain two elements : an element of intuition (as a part of sensation) and an element of arbitrary intellect- ual determination which is called definition. The facts of extension and its limits surfaces, lines, and points are given in intuition ; without sensible experience we should not know anything about geometrical solids, surfaces, lines, and points; but nothing is deducible from the existence of these elements, or our intuition of them, until they are defined. This is evident upon a simple inspection of the geometrical axioms. The axiom that between two points but one straight line can be drawn (or, what is the same thing, that two straight lines can not inclose a space) involves the definition of the straight line a definition, by the way, far more difficult on purely geometrical grounds than that of par- allels.* Again : the axiom respecting parallels, in the * The real source of this difficulty lies in a fundamental defect of the current theories of cognition in the failure to see that all processes of deductive reasoning involve an ultimate reference to primary constants which are not given in experience, but established by the intellect. This primary constant in geometry is the straight line, or simple direction. That the difficulties presented by the 10th Euclidean axiom ("two straight TRANSCENDENTAL GEOMETRY. 243 form now generally given to it, viz., that through a given point but one straight line can be drawn parallel to another straight line, presupposes the definition, not only of the straight line, but of parallelism which, in elementary geometry, presents the difficulty of involv- ing the concept of infinite extension, and has given rise to innumerable quandaries (such as that of infinitely distant, and yet real, points of intersection), among which those of the pangeometrical sort are not the least. The Euclidean list of definitions, postulates and axioms is vitiated, not, or, at least, not only, by the fact that his lines of distinction between these several pre- requisites of geometrical reasoning are not correctly drawn that he confounds definitions with axioms and postulates with both,* and, besides, fails to discriminate between axioms of quantity in general and axioms of spatial quantity but by his ignorance or disregard of the fact, to which I have referred, that every axiom, which is geometrically fertile, involves a definition. linos can not inclose a space ") are of the same nature with those of the 12th (usually called the llth the axiom of parallelism) has long been known. " La definition et les proprietes de la ligne droite," says D'Alem- bcrt (Elemens de Philosophic, 12 (Euvres, tome i, p. 280) " ainsi que des lignes paralleles sont done 1'ecueil et, pour ainsi dire, le scandale des elemens de geometric." * Hankel (Vorlesungen ueber die complexen Zahlen und ihre Func- tionen, p. 52) draws attention to the fact that the confusion above referred to is chargeable, not to Euclid, but to his editors and commen- tators. "In all the manuscripts," says Hankel, "which F. Peyrard has collected in preparing his excellent edition of Euclid ((Euvres d'Euclide trad, en Latin et en Fran9ais, tome i, p. 454) the famous llth principle of the theory of parallels appears, not among the KOIVCU swoiui relating to equal and unequal quantities, but as the 5th postulate (ctfrrj^a). Simi- larly the 10th axiom in all these manuscripts appears as the 4th postulate, while the MSS. vary in respect to the 12th axiom, it being thus evident that the three axioms owe the place which unaccountably they still oc- cupy in the list of axioms to a misunderstanding." 244 CONCEPTS OF MODERN PHYSICS. And this ignorance very excusable in Euclid's time unfortunately appears to be snared by the writers of geometrical text-books at the present day. One of the points upon which the debate between Helmholtz and his opponents has largely turned is the question whether or not Beltrami's pseudo-spherical space is conceivable or imaginable (vorstellbar) ; and, to maintain the affirmative, Helmholtz propounds a re- markable definition of imaginability. He defines the power of imagining spatial forms as "the power of fully representing the sense-impressions which the ob- ject would excite in us according to the known laws of our sense-organs under all conceivable conditions of ob- servation, and by which it would be distinguished from other similar objects." * Whatever may be the general merits of this definition, it is certainly obnoxious to the charge of irrelevancy. As the old logicians would say, it is founded upon an ignoratio elenchi, a misapprehen- sion of the question. Granting, for the sake of argu- ment, that the act of imagining a form of space is truly described as an anticipation of sense-impressions, the question as to the existence of the power sought to be defined is, not what would be the nature of these im- pressions, but whether or not they could coexist in the imagination in the required spatial order and form ac- cording to the known laws of the representative faculty. Helmholtz refers to the attempt of Beltrami to make pseudo-spherical space representable by projecting its points, lines and surfaces upon the interior of an ordi- nary spherical surface " whose points correspond to the infinitely distant points of pseudo- spherical space," and claims that this attempt is successful. In the same sense Professor Sylvester, in the note to his Exeter * " Origin and Meaning of Geometrical Axioms," Mind, vol. iii, p. 216. TRANSCENDENTAL GEOMETRY. 245 address already quoted, observes that "every perspec- tive representation of figured space of four dimensions is a figure in real space, and that the properties of fig- ures admit of being studied to a great extent, if not completely, in their perspective representations." And it has become a standing assertion of the pangeometers that the forms of a space of any given dimension may be projected into the space of the next lower dimension. But this assertion, at best, holds good only for the limits of projection, where the resulting point or figure ceases to be an explicit reproduction of the figure projected. When a straight line is projected orthogonally upon another straight line at right angles to it, it appears as a point; a form of the first dimension is, in a sense, reduced to the dimension zero. But the representative point, by itself, does not enable us to reproduce and reason about the line whereof it is the projection. It may be said that we can at least know that the line pro- jected is straight ; but that is a conclusion which fol- lows only from the properties of lines as they are other- wise known ; from the mere inspection of the point it is not even inferable that it is a projection of a line at all. Similarly a plane may be projected upon another plane, so as to appear as a line, a form of two dimen- sions being reduced to a form of one dimension ; but it does not follow that we may study the properties of the plane by merely contemplating or analyzing the line. The so-called projections of solids upon surfaces are in fact projections, upon a normal surface, of several sur- faces making different angles with it, and the inferences from such a projection respecting the properties of geo- metrical solids depend upon our associations of visual with tactual impressions in which our apprehension of geometrical solidity has its origin. There being con- 246 CONCEPTS OF MODERN PHYSICS. fessedly no tactual or other impressions evidencing the existence of a fourth dimension, the analogy upon which the alleged imaginability of transcendental space- forms is founded is without support. But it is of little consequence what ground there is for the claim (which has recently been urged in another form by Felix Klein *) that the resources of projective geometry are sufficient to enable us to represent the properties of a space of more than three dimensions in tridimensional space : for the question of representabil- ity is wholly foreign to the matter in dispute. If it were shown, for instance, that a pseudo-spherical sur- face may be, mentally or really, traced in space, this certainly would not prove, or tend to prove, that space is inherently pseudo-spherical. There is no doubt about the imaginability of a spherical surface, but from this it does not follow that space itself is spherical. To support the conclusion of the immanent pseudo-sphe- ricity of space it would be necessary to maintain that none but pseudo-spherical surfaces can exist, and, there- fore (conformably to the teachings of sensationalism), be represented, or imagined as existing, in it. And, in view of this, the whole argument of Helmholtz not only ceases to be available as a support of geometrical tran- scendentalism, but recoils upon himself. If pseudo- spherical surfaces can be imagined to exist, and there- fore, upon his own principles, are possible in " flat " space, why can not ordinary straight lines and flat sur- faces exist in pseudo-spherical space ? And what, then, * " Ueber die Nicht-Euklidische Geometric." Math. Ann., vol. iv, p. 673. In this article, as in nearly all the writings of the pangeometers, who deal with imaginary arid infinitely distant points ad libitum, analyti- cal representability (by means of symbols among which infinite and imaginary elements are treated as coordinate with real elements) is con- founded with imaginability. TRANSCENDENTAL GEOMETRY. 247 becomes of Ms telescopic test of the curvature of space?. Or am I under a misapprehension as to Helmholtz's true meaning? does he simply contend that pseudo- spherical surfaces would be imaginable by pseudo-spher- ical beings with pseudo-spherical organs of sense, and consequent pseudo-spherical intellects in a pseudo- spherical space, if it existed ? That is a proposition which even Land and Krause would hardly dispute. The history of cognition affords no illustration, per- haps, of the irrepressibility of intellectual traditions which is more instructive than the doctrines of tran- scendental geometry. Glancing back at the contents of the present chapter, we see that even the science of mathematics the exactest of all the sciences, whose methods are said to be as infallible as its foundations are supposed to be permanent, and which, ever since the dawn of human intelligence, has pursued the even tenor of its way amid all the vicissitudes of speculation is not exempt from the prepossessions of ontological realism. The same hypostasis or reification of concepts, which has given rise to the atomo-mechanical theory in physics, has led to the doctrine of pangeometry in math- ematics. The hypostasis of space, by the mathemati- cians, is a strict analogue of the hypostasis of mass and motion by the physicists. The full extent, however, to which the minds of contemporary mathematicians are bewildered by the false light of ontology can be brought into still clearer view by a further examination of the speculative back- ground of transcendental geometry, as it appears in the famous essay of Kiemann already referred to. U1UVBRSIT! CHAPTER XIY. METAGEOMETRICAL SPACE IN THE LIGHT OF MODERN THE essay of Bernhard Riemann, " On the Hypothe- ses which lie at the Base of Geometry," owes its great celebrity to the fact that he was a mathematical analyst of the first order, one of the favorite pupils of Gauss, under the inspiration of whose teachings, if not at his suggestion, the essay was written by whom, in fact, it was presented, in 1854, shortly before his (Gauss's) death to the philosophical faculty of Goettingen, and by whom its cardinal propositions were expressly in- dorsed as an exposition of his own speculative opinions. Every intelligent reader of this essay will agree with me, I think, that its intrinsic merit is not at all com- mensurate with the attention with which it was re- ceived and the interest with which it is still generally considered. Not only are its statements, both of the problem and of the proposed methods of solution, crude and confused, but they bear the impress throughout of Riemann's very imperfect acquaintance with the nature of logical processes and even with the import of logical terms. It is apparent, from the whole tenor of the essay, that its author was an utter stranger to the dis- cussions respecting the nature of space which have been so vigorously carried on by the best thinkers of our time ever since the days of Kant, and that he was so RIEMANFS DISSERTATION. 249 little familiar with the history of logic as to be with- out the faintest suspicion of the manifold ambiguity of such terms as "concept" and "quantity," and of the necessity of their exact definition preliminary to an inquiry respecting the very foundations of human knowledge.* The general argument of the essay is, that the nat- ure of space is to be deduced from its concept ; that the formation of such a concept of necessity involves its subsumption under a higher concept ; that this higher concept is that of a "multiply extended quantity;" that, in order to determine how many kinds of space are possible, it is requisite to ascertain in how many ways quantity may be " multiply extended " (mehrfach ausgedehnf) ; and that, after the number of conceptually * Riemann himself modestly apologizes for the philosophical short- comings of bis essay on the ground of his inexperience in philosophical matters. But the crudeness of his speculations affords a very striking illustration, in my judgment, of the well-known fact that exclusive devo- tion to the labors of the mathematical analyst has a tendency to develop certain special powers of the intellect at the expense of its general grasp and strength. Although Sir William Hamilton, no doubt, overstated the case against the mathematicians, I believe that his suggestions are not wholly unworthy of attention, and that there is force in the words of D'Alembert (referred to by Sir William Hamilton), which it is perhaps safest to quote in the original, without translation : "II semble que les grands geometres devraient etre excellens metaphysiciens, au moins sur les objets dont ils s'occupent ; cependant il s'en f aut bien qu'ils le soient toujours. La logique de quelques uns d'entre eux est renfermee dans Icurs formules et ne s'etend pas au dela. On peut les comparer a un homme qui aurait le sens de la vue contraire a celui du toucher, ou dans lequel le second de ces sens ne se perf ectionnerait qu'aux de*pens de 1'autre. Ccs mauvais metaphysiciens dans une science ou il est si facile de ne le pas etre, le seront a plus forte raison infailliblement, comme 1'experience le prouve, sur les matieres ou ils n'auront pas le calcul pour guide. Ainsi la geometric qui mesure les corps, peut servir en certains cas a mesurer les esprits mSine." D'Alembert, Elemens de Philosophic, 11 ; (Euvres, tome i, p. 276. 250 CONCEPTS OF MODERN PHYSICS. possible varieties or species of multiple extension has thus been fixed, it is a matter of experimental determi- nation which of these varieties or species is represented by our space, i. e., by the space in which the world, as we know it, has its being. After thus asserting that the concept " space " is to be subsumed under the con- cept " quantity," Riemann proceeds to declare that all quantities are in their nature multiples or aggregates (Mannigfaltiglceiten) which are continuous whenever there is continuous transition from one of their several " specializations " to the other, and discrete, when there is no such transition ; that the " specializations" of dis- crete quantities are called points, and those of continu- ous quantities elements / and that continuous quantities are determined by measurement, while discrete quan- tities are determined by numeration. Space, according to Riemann, though a continuous quantity, is a quan- tity of ^-fold (geometrical) extension, and is thus a Multiple or Aggregate,, and therefore a quantity, not- withstanding its continuity. The degree of the multi- plicity of this extension i. e., the fact of its being sim- ple, twofold, threefold, or generally n-fold determines the (logical) extension of the concept space. We have here five distinct propositions, which, for convenience of reference and discussion, may be stated in distinct form as follows : 1. That the nature of space is to be deduced from its concept. 2. That the concept" of space can be formed and determined only by its subsumption under a higher concept. 3. That our space is a " triply extended Multiple or Aggregate," the higher concept under which its con- cept is to be subsumed being that of an "/i-fold ex- RIEMANN'S DISSERTATION. 251 tended Multiple " or a " multiply extended Aggregate " (eine n-fach ausgedehnte MannigfaltigJceit), and that translating Kiemann's phraseology into its plain logical import the (logical) extension of this higher concept determines the number of the possible kinds of space. 4. That the conceptual possibility of space is coex- tensive with its empirical possibility, though not with its empirical reality. 5. That continuous quantities are coordinate with discrete quantities, i. e., are species of the same genus, both being in their nature multiples or aggregates.* * The order and numeration of these propositions is, of course, my own ; in Riemann's essay they appear in very promiscuous order. In proof of the general correctness of my statement of Riemann's doctrines, it is perhaps well to quote the introductory part of his essay in the origi- nal, italicizing the more important passages : " Uebcr die Hypothesen welche der Geometric zu Grunde liegen. " Plan der UntersucJmng. " Bekanntlich setzt die Geometrie sowohl den Begriff des Raumes, nls die ersten Grundbegriffe fuer die Constructionen im Raume als etwas Gegebenes voraus. Sie giebt von ihnen nur Nominaldefinitionen, waeh- rend die wesentlichen Bestimmungen in Form von Axiomen auftreten. Das Yerhaeltniss dieser Voraussetzungen bleibt dabei im Dunkeln ; man sieht weder ob und in wie weit ihre Yerbindung nothwendig, noch a pri- ori^ ob sie moeglich ist. " Dicse Dunkelheit wurde auch von Euklid bis Legendre, urn den be- ruehmtesten neueren Bearbeiter der Geometrie zu nennen, weder von den Mathematikern, noch von den Philosophen, welche sich damit beschaeftig- ten, gehoben. Es hatte dies seinen Grund wohl darin, dass der allge- meine Begriff mchrfach ausgedehnter Groessen, unter welchcn die Raum- groesscn enthalten sind, ganz unbearbeitet blieb. Ich habe mir daher zunacchst die Aufgabe gestellt, den Begriff einer mehrfack ausgedehnten Groesse aus allegcmeincn Groessenbegriffcn zu construiren. Es wird daraus hervorgehen dass eine mehrfach ausgedehnte Groesse verschiedener Maass- vcrhacltnisse faehig ist, und der Raum also nur cinen besondern Fall einer dreifach ausgedehnten Groesse bildet. Hiervon aber ist eine nothwendige Folge, dass die Saetze der Geometrie sich nicht aus allgemeinen Groessen- begriffen ableiten lasscn, sondern dass diejenigen Eigenschaften, durch 252 CONCEPTS OF MODERN PHYSICS. I proceed to consider these propositions in their order. 1. The first proposition is in plain words a statement of the general ontological fallacy (discussed at length in welche sich der Raum von andern denkbaren dreifach ausgedehnten Groessen unterscheidet, nur aus der Erfahrung entnommen werden koen- nen. Hieraus enstcht die Aufgabe, die einfachsten Thatsachen aufzu- suchen, aus denen sich die Maassverhaeltnisse des Raumes bestimmen lassen eine Aufgabe, diederNatur der Sache nach nicht voellig bestimmt 1st ; denn es lassen sich rnehrere Systeme einfacher Thatsachen angeben, welche zur Bestimmung der Maassverhaeltnisse des Raumes hinreichen ; am wichtigsten 1st fuer den gegenwaertigen Zweck das von Euklid zu Grunde gelegte. Dicse Thatsachen sind, wie alle Thatsachen, nicht noth- wendig, sondcrn nur von empirischer, Gewissheit, sie sind Hypothesen, man kann also ihre Wahrscheinlichkeit, welche innerhalb der Grenzen der Beo- bachtung allerdings schr gross ist, untersuchen, und hienach ueber die Zulaessigkeit ihrer Ausdehnung jenseits der Grenzen der Beobachtung sowohl nach der Scite des TJnmessbargrossen, als nach der Seitc des Un- mcssbarklcinen urtheilen. " I. Begriff eincr n-fach ausgedehnten Groesse. " Indem ich nun von diesen Aufgaben zunaechst die erste, die Entwicke- lung des Begriffes mehrfach ausgedehnter Groessen, zu loesen versuche, glaube ichum somehr auf eine nachsichtige BeurtheilungAnspruch mach- en zu duerf en, da ich in dcrgleichen Arbeiten philosophischer Natur, wo die Schwierigkeiten mehr in den Begriffen, als in den Constructionen liegen, wenig geuebt bin und ich ausser einigen ganz kurzen Andeutungen welche Herr Ilof rath Gauss in der zweiten Abhandlung ueber die biquadratischen Reste, in den Goettingischen gelehrten Anzeigen, und in seiner Jubi- laeumschrift darueber veroeffentlicht hat, und einigen philosophischen Untersuchungen Herbart's durchaus keine Yorarbeiten benutzen konnte. " Groessenbegrijfe sind nur da moeglich, wo sich cin allgemeincr Begriff vorftndct, der verschicdene Bestimmungsweisen zulaesst. Je nachdcm untcr diesen JBestimmungsweisen von eincr zu eincr andern cin stetiger Ucbcrgang stattjindet oder nicht, bilden sie eine stetige oder discrete Mannigfaltigkeit ; die einzelnen Bestimmungsweisen heisscn im ersten Fall Punctc, in Ictztcrem JKlemente dieser Mannigfaltigkeit. Begriffe, deren Bestimmungsweisen eine discrete Mannigfaltigkeit bilden, sind so haeufig, dass sich fuer bc- liebig gegebene Dinge wenigstens in den gebildcteren Sprachen immer eiu Begriff auffindcn laesst, unter welchem sic enthalten sind (und die Mathematiker konnten daher in der Lchre von den discreten Groessen RIEMANN'S DISSERTATION. 253 the ninth chapter) that things and their properties are to be deduced from our concepts of them. As I have already said, Eiemann does not define the term " con- cept ; " nor does he inquire how concepts are formed or how they come to be possessions of the intellect. He says, indeed, that concepts of quantity are possible only when they can be subsumed under higher concepts, unbedeiiklich von der Forderung ausgehen, gegebene Dinge als gleich- artig zu betrachten), dagegen sind die Veranlassungen zur Bildung von Begriffen, dercn Bestimrnungsweisen eine stetige Mannigfaltigkeit bilden, im gemeinen Leben so selten, dass die Orte der Sinnengegenstaende und die Farben wohl die einzigen einfachen Begriffe sind, deren Bestimmungs- weisen eine mehrfach ausgedehnte Mannigfaltigkeit bilden. Haeufigere Veranlassung zur Erzeugung und Ausbildung dieser Begriffe findet sich erst in der hoehern Mathematik. "Bestimmte, durch ein Mcrkmal oder cine Grenze unterschiedene Theile einer Mannigfaltigkeit heissen Quanta. Ihre Vergleichung der Quantitaet nach geschieJit bei den discreten Groessen durch Zaehlung, bei den stctigen durch Messung. . . . Fuer den gegenwaertigen Zweck genuegt es, aus dicsem allgemeinen Theile der Lehre von den ausgedehnten Grocssen, wo wciter nichts vorausgesetzt wird, als was in dem Begriffe derselben ent- halten ist, zwei Puncte hervorzuheben, wovon der erste die Erzeugung des Begriffs einer mehrfach ausgedehnlen Mannigfaltigkeit, die ziveite die Zu- rucckfueJirung der Ortsbestimmungen in einer gegebcnen Mannigfaltigkeit auf Quantitaetsbestimmungen betriffi, und das wesentliche Kennzeichen einer n-fachen Ausdehnung deutlich machen wird." I ought to say that my interpretations of several passages of this text are more or less conjectural. There is room for serious doubt, for in- stance, whether the expression " Bestimmungsweisen " is meant to denote the species comprehended by a genus, or the parts constituting a whole. A wretched translation of Riemann's essay, which, by its clumsy literalism, materially adds to the obscurity and confusion of the original, was pub- lished in 1873, by W. K. Clifford (Nature, vol. viii, pp. 14 and 86 seg.). This translation was no doubt made, not by, but for, Professor Clifford, by some one who had a very insufficient knowledge of German. The merits of the translation are not unfairly instanced in the rendering of Rie- mann's term " Mannigfaltigkciten " (varieties, multiplicities, used in the sense of multiples Hclmholtz translates "aggregates") by "manifold- nesses," of " Groessenbegriffe " by " magnitude-notions," etc. Of one passage the whole sense is changed by reading kocnntcn for konnten. 254 CONCEPTS OF MODERN PHYSICS. or, as lie expresses it, " when there is a general concept which admits of different specializations." But the question, where this process of subsumption begins or ends, and what are the nature and origin of the highest concept or swnmum genus of which all inferior genera or species must be specializations, does not occur to him. It is, however, an inevitable conclusion from Kiemann's first proposition itself that he holds this most general concept to be an a priori form or possession of the mind, and that he believes the process of deduction by which its specializations are derived from it to be (in the language of Kant) a series of synthetic judg- ments a priori. In view of this a further consideration of the proposition is unnecessary ; it is refuted by the whole tenor of the preceding chapters of this book. I may be permitted to observe, however, that it is with- out parallel in the entire history of intellectualism (usually called idealism) ; Kant, for example, expressly disclaims all belief in the doctrine that the intellect is aboriginally furnished with ready-made concepts. 2. The second proposition, that concepts of quantity can be formed and determined only by subsumption under more general concepts, is probably a vague remi- niscence of the old logical rule that all definition \%per genus et differentiam. In spite of Eiemann's com- plaint, in the second sentence of his essay, that hitherto the science of geometry has given nominal definitions only of space and constructions in space a complaint, by the way, which, so far as it applies to constructions in space, is unfounded he does not seem to have a very clear insight into the nature of the distinction be- tween definitions and concepts. For, if he had prop- erly realized this distinction, he could not have failed to ask himself the question, what, under his definition, RIEMANN'S DISSERTATION. 255 became of the summum genus " quantity " which is the logical terminus of the processes of subsumption of which he speaks. Is this summum genus also a concept ? Then it must be subsumable, in conformity with his rule, under a still higher concept, which, ex vi termini, it is not, being itself the highest. Or is it something else a datum of experience ? If it is, how then is the second proposition to be reconciled with the first, according -to which everything is to be deduced from, as well as subsumed under, a concept ? Or is this the old case of the hen in Kewmarket which lays an egg, from which the same hen presently comes forth as a chicken ? The proposition here discussed almost at the outset involves our author in the most intolerable perplexity. " Concepts," he says, " whose specializations form a discrete aggregate (or multiple) are so common that, in the more cultivated languages at least, a concept may always be found under which things of whatever kind are subsumable." The meaning of this is, I take it, that of discrete aggregates there are always several similar or connatural kinds or species which may read- ily be subsumed under a higher concept. " But," he continues, " the occasions for the formation of concepts, whose specializations constitute a continuous aggregate, are so rare in ordinary life, that the places of things and colors are probably the only simple concepts whose specializations constitute a multiply extended aggre- gate " that is to say, I suppose, there is but one spe- cies of a continuous aggregate or multiple other than space that admits of coordination and subsumption with it under the concept " multiply extended aggre- gate," viz., color. This singular statement (which, it may be noted parenthetically, is the exact reverse of 256 CONCEPTS OF MODERN PHYSICS. the truth, there being, as we shall hereafter see, but one kind of discrete, quantities, viz., numbers, and in- numerable kinds of continuous quantities) has been elaborated with an extravagant expenditure of analyti- cal power by Benno Erdmann,* who finds that there are two triply extended multiples which are coordinate and subsumable with space of three dimensions under the concept of a " continuous multiply extended ag- gregate : " sound and color. Sound, according to Erd- mann, is a function of three independent variables, acuteness, intensity, and timbre (Klangfarbe). Similarly color depends on the variables tone, degree of saturation (Saettigungsgrad), and intensity. f All this is simply puerile. To imagine that conclu- sions respecting the nature of space and the origin of its concept can be drawn from the mere fact that space is a function of three variables, and may thus in a manner be classified with similar functions, is a mock- ery of all reasoning from which an old scholastic would have turned with the scornful reminder that coordina- tion and subsumption, for the purpose of effectually aiding in the formation of a particular concept, must not only be under a genus, but under a genus proxi- mum. J Weissenborn's remark,* that on the same log- * Die Axiome der Geometrie (Leipzig, 1877) p. 40 seq. f It is significant, in this connection, that according to Helmholtz (who also falls in with Riemann's theory of conception) the three vari- ables of the function " color " are the three primary colors of which each several color is said to be a mixture. " The Origin and Meaning," etc. Mind, vol. i, p. 309. % Of this Erdmann seems to have some inkling, for he notes that space differs from color and sound in the circumstance of the absolute interchangeability of its three dimensions, the " dimensions " of color and sound not being interchangeable. * " Ueber die neucren Ansichten vom Raum," Vierteljahrsschrift fuer wissenschaftliche Philosophic, vol. ii, p. 321. RIEMANN'S DISSERTATION. 257 ical principles space miglit be coordinated with the amount of interest produced by a certain capital, which is a function of the three variables capital, rate of in- terest, and time, is perfectly just. And the number of species coordinate with space in the same sense might be indefinitely increased. For instance, space might be coordinated with the velocity of a railway-train on a straight road, inasmuch as this velocity is a function of the motive power of the engine, the weight of the train, and the grade of the track ; or with the volatility of a liquid, which is a function of the nature of the liquid, its temperature, and the pressure of the atmos- phere ; or with the capacity of a man for labor, which depends on his general health and strength, the quantity of nourishment he has taken, and the amount of sleep he has had ; and so on indefinitely. All this is very absurd, but not more so than the coordination of space with color and sound on the mere basis of the depend- ence of each on three variables which are arbitrarily called " dimensions." 3. I come now to Riemann's third proposition, that space is an " n-fold extended multiple " or a " multiply extended aggregate " (eine mehrfach oder n-fach ausge- dehnte Mannigfaltigkeit "). The term " Mannigfaltig- keit," as here employed, is a standing puzzle to the readers of Riemann's essay. "Weissenborn, who justly objects to the use of an adjective or predicative word in an appellative sense, for the denotation of a substan- tive entity, conjectures * that it was expressly devised by Kiemann for the purpose of bringing the concept " space " within the scope of his second proposition. But this is a mistake. Kiemann adopted the term from Gauss, who was probably the originator of its employ- * L. c., p. 320. 258 CONCEPTS OF MODERN PHYSICS. ment for the designation of " space in general " (as distinguished from " flat space," in the metageometrical sense).* Gauss, in turn, took the expression, no doubt, from Herbart,f to whose attempt at an elaboration of the idea of space from the manifold qualitative data of sense I have already referred, and whose philosophy is, to a great extent, a sort of reproduction of the old Eleatic quandaries about " The One and the Many." Herbart, in fine, had obtained it from Kant, whose dis- ciple he was, or believed himself to be, and whose phrase " Mannigfaltigkeiten der Empfindung " is vari- ously found, not only in his own writings, but also in those of his followers. The only comment which I deem it necessary to make on this proposition is that space is not a " multiple " or " aggregate " at all, but that its very essence is con- tinuity. This, as has been abundantly shown, follows from its conceptual nature as well as from its relativity. The determination of points in space, or " elements " of space, results from the establishment of quantitative relations between its parts, i. e., its purely arbitrary divisions, by means of numbers, in the manner to be considered presently. I have already shown, in the * In his Anzeige of the Theoria residuorum biquadraticorum, Com- mentatio secunda, Gauss says : " Der Verfasser hat sich vorbehalten, den Gegenstand welcher in der vorliegcnden Abhandlung eigentlieh nur gelegentlich beruehrt ist, kuenftig vollstaendig zu bearbeiten, wo dann auch die Frage, warum die Relationcn zwischen Dingen, die eine Mannig- faltigkeit von mehr als zwei Dimensionen darbieten, nicht noch andcre, in der allgemeinen Arithmetic zulaessige Arten von Groessen liefern kocnnen, ihre Beantivortung finden wird." Gauss, Werke, vol. ii, p. 178. This notice appeared originally in the Goettingische Gelehrte Anzcigen of April 25, 1831. f In his Synechologie, e. g., Herbart speaks of " die Mannigfaltighcit der irrationalen FortscJircitungcn in Bezug auf den Raum" Herbart's Werke, vol. iv, p. 153. RIEMANN'S DISSERTATION. 259 last chapter, that ppace itself is not, in any intelligible sense, a quantity. 4. Riemann's fourth proposition is founded on a confusion between conceptual possibility and real or empirical possibility. Conceptual possibility is deter- mined solely by the consistency or inconsistency of the elements of the concept to be formed it is tested sim- ply by the logical law of non-contradiction ; while em- pirical possibility depends upon the consistency of the thing conceived with the various conditions of sensible reality, or, what is the same thing, the laws of nature. This subject, also, has already been discussed to some extent in the last chapter, where it was pointed out that conceivability (in the strict sense of the term) of a thing or phenomenon is no proof of its imaginability or representability under the conditions of our physical and intellectual organization. Upon this distinction depend the utility and scope of the artifice not unfre- quently resorted to, in certain analytical investigations, of supposing the existence of a fourth spatial dimen- sion for the purpose of reducing certain functions to a symmetrical form ; and this distinction, too, is the basis of an observation made by Boole* twenty-six years ago: " Space is presented to us, in perception, as possess- ing the three dimensions of length, breadth, and depth. But in a large class of problems relating to the prop- erties of curved surfaces, the rotation of solid bodies around axes, the vibration of elastic media, etc., this limitation appears in the analytical investigation to be of an arbitrary character, and, if attention were paid to the processes of solution alone^ no reason could be discovered why space should not exist in four, or in any * Laws of Thought, p. 175, note. 260 CONCEPTS OF MODERN PHYSICS. greater number of, dimensions. The intellectual pro- cedure in the imaginary world thus suggested can be apprehended by the clearest light of analogy." Upon the same ground, and in the same sense, Hermann Grassrnann, who is sometimes referred to as one of the founders of transcendental geometry, has developed the theory of extension in its general application to an indefinite number of dimensions, although he certainly did not cherish the delusion (as seems to be supposed by Victor Schlegel *) that this could be the source of inferences respecting the number of actual or empiri- cally possible dimensions of space. , On this subject we have Grassmann's own explicit declaration : f "It is clear," he says, "that the concept of space can in no wise be generated by thought. . . . Whoever maintains the contrary must undertake to derive the dimensions of space from the pure laws of thought a problem which is at once seen to be impossible of solution." 5. Closely akin to his third and fourth propositions is Riemann's fifth proposition, that continuous quanti- ties are coordinate with discrete quantities, both being in their nature multiples or aggregates, and therefore species of the same genus. This pernicious fallacy is one of the traditional errors current among mathema- ticians, and has been prolific of innumerable delusions. It is this error which has stood in the way of the forma- tion of a rational, intelligible, and consistent theory of irrational and imaginary quantities, so called, and has shrouded the true principles of the doctrine of " com- plex numbers " and of the calculus of quaternions in an impenetrable haze. The proposition that discrete and continuous quan- * System der Raumlehre, preface, p. vi. f Die lineare Ausdehnungslehrc (1844) Einleitung, p. 20 scq. EIEMANN'S DISSERTATION, 261 titles are coordinate species of the same genus amounts to nothing less than the thesis that signs are logically coordinate with their significates. There are no " dis- crete quantities" except those which are dealt with in special (common) and general arithmetic, that is to say, numbers. Now, a number is an aggregate or collection of units each of which simply represents an act of ap- prehension, whatever may be the extent or nature of the object apprehended. If this object is designated as a quantity, a number is not a quantity at all, nor a measure of quantity, but simply an intellectual vehicle of quantities a purely subjective instrumentality for their comparison and admeasurement. All the uncer- tainty and confusion which are characteristic of the numerous attempts to define and classify quantities are due to the ignorance or neglect of this elementary truth. Quantity has been defined as "that which is susceptible of augmentation, diminution and division," and as ".the genus of which magnitude and multitude are the species ; " or quantities have been first divided into extensive quantities (space) and intensive quantities (forces, colors, sounds, and all subjective affections), and the extensive quantities have then been subdivided into continuous and discrete. Now, the fact is that all ob- jects of apprehension, including all data of sense, are in themselves, i. e., within the act of apprehension, es- sentially continuous. They become discrete only by being subjected, arbitrarily or necessarily, to several acts of apprehension, and by thus being severed into parts, or coordinated with other objects similarly appre- hended into wholes. To say that a datum of sensation or of subjective feeling is in itself discrete is to assert that it is absolute, and to deny that quantity is essen- tially relative. And to maintain (with those who speak 262 CONCEPTS OF MODERN PHYSICS. of positive, negative, fractional, irrational, imaginary, complex, linear, or directional numbers) that number may be continuous is to ignore the plainest and most unmistakable fact in all our intellectual operations, and to misinterpret all the teachings of the history of mathe- matics. Numbers, in themselves, being mere groups or series of acts of intellectual apprehension without reference to their contents, are not and can not be posi- tive or negative, much less fractional, irrational, or im- aginary. They can, indeed, be applied, not only to data of sensation and of subjective feeling, but also, by anal- ogy, to relations between them, including relations es- tablished by the intellect. They can, therefore, stand, not only for things, but also for their actions and reac- tions and for the operations to which they are subjected. A number may represent motion in a given direction and in the direction opposite to it, thus becoming af- fected by the signs plus and minus / but these signs do not indicate any change in the nature of numbers, but merely a particularity in their application. Similarly numbers may represent ratios and assume the form of fractions ; but the numbers do not thereby cease to be what they are, viz., units or collections of units, and therefore essentially integers. Fractions can be prop- erly called numbers only in the sense that they point to the division, not of the primary units expressive of the original acts of apprehension, hut of the objects appre- hended^ into subordinate units. Again : numbers may be signs of operations upon quantities that can not be successfully performed, such as the reduction of the diagonal and the side of a square to a common measure in other words, the establishment of a definite numer- ical ratio between two quantities which do not admit of such a ratio. In such case the futility of the attempt RIEMANFS DISSERTATION. 263 finds expression in a sign prefixed to a number which, together with its significate, is ordinarily termed an irrational quantity.; but the irrationality lies, not in the number, but in the attempt at its application to incom- mensurable magnitudes. The same thing is true, muta- tis mutandis, of " imaginary quantities " and " complex numbers." The object of the act of apprehension, which is represented by the numerical unit, may be, not only rectilinear motion or transference in a given direction, but also angular motion ; as the calculus of quaternions expresses it, the unit of operation may be a tensor, or a versor, or both ; whence it follows that whenever the attempt is made to represent such an op- eration in terms of linear units with their positive or negative prefixes indicative of a fixed direction in which the motions, whereof the lines are the measures, occur, the attempt again fails, and this fact emerges in the form of the symbol which (being part of a system of symboli- zation that is not comprehensive enough to embrace the new operation) assumes a so-called imaginary form. But here once more, it is not the number which is imagi- nary, ~but the operation as interpreted in conformity with the conventional rules of symbolization, the con- sequence foeing that these rules have to be extended, and that the meaning of the symbols has to be widened. And this again imports a change, not in the nature of the signs, i. e., of the numbers, but in the nature and extent of their significates. In this manner the scope of arithmetical (and, of course, algebraic) symbolization is continually extended, not only by enlarging, but also by wholly changing the things, relations or operations which are successively the objects of intellectual appre- hension. All this is perfectly safe and legitimate, pro- vided that the change in the signification of the symbols 264: CONCEPTS OF MODERN PHYSICS. be made in conformity with the logical canon of con- sistency, and with due regard, moreover, to the effect of such change upon the validity of the rules govern- ing the syntheses and analyses to which the symbols are subjected. In the operation of ordinary arithmetical or algebraic multiplication, for instance, the law of com- mutation is of universal validity. Multiplication being nothing more than an abbreviated addition, the multi- plicand and the multiplier may exchange places or func- tions without any effect upon the result. In the calcu- lus of quaternions the mathematician generalizes the principle of multiplication, defining it as a process of finding a quantity which is produced from, or related to, the multiplicand in the same way in which the mul- tiplier is produced from, or related to, the unit. Under this new definition he multiplies lines and other quan- tities into each other ; but now it appears that the law of commutation is no longer generally applicable. The reason is that the apparent expansion of the principle of multiplication was in fact also a limitation, or rather a shifting of the meaning of the arithmetical or alge- braic symbol a removal of the condition upon which the validity of the law of commutation depended. I may observe here, incidentally, that it is a mistake to say, with Kelland and others, that the calculus of quater- nions grows out of the common arithmetical or alge- braic calculus by the removal of limitations. The example just adduced shows that it may involve an im- position of limitations as well. For this reason Pea- cock's law, which he calls the " principle of the perma- nence of equivalent forms,* viz., that " whatever algebrai- cal forms are equivalent, when the symbols are general in form but specific in value, will be equivalent likewise * Peacock, Symbolical Algebra, p. 59. RIEMANN'S DISSERTATION. 265 when the symbols are general in value as well as in forms," in order to be available as the fundamental principle of the theory of " complex numbers," requires a modification far more serious than is implied in Han- kers new statement of it as " the principle of the per- manence of formal laws." For the expression " formal laws " is ambiguous and leaves us in doubt as to what laws are formal in the sense of being applicable to all the operations which are in any way representable by arithmetical or algebraic symbols. The error respecting the true nature and function of arithmetical and algebraic quantities has become next to ineradicable by reason of the inveterate use of the word " quantity" for the purpose of designating indis- criminately both extended objects or forms of exten- sion and the abstract numerical units or aggregates by means of which their metrical relations are determined. The effect of this indiscriminate use is another illustra- tion of the well-known fact in the history of cognition that words react powerfully upon the thoughts of men, and by this reaction become productive of incalculable error and confusion. It is not to be expected, of course, that mathematicians will cease, at this late day, to speak of arithmetical or algebraic symbols as "quantities;" but there may be a little hope for the suggestion that they might return to the old phrase " geometrical (and other) magnitudes." The mischief lies, not so much in the use of a particular word, as in the employment of the same word for the denotation of objects differing from each other toto genere* * The perplexities occasioned by the use of improper and misleading terms in mathematics are animadverted on by Gauss himself in the notice already cited (Werke, vol. ii, p. 178), where he speaks of the obscurity incident to the interpretation of " negative and imaginary numbers," and H 266 CONCEPTS OF MODERN PHYSICS. The ignorance or oblivion of the distinction here referred to also illustrates a phase in the history of error exemplifications of which have repeatedly been met with in the preceding pages : the confusion between purely conventional forms of thought and speech and forms or laws of objective existence. This confusion, which is at the bottom of the old assumption that our arbitrary or conventional classifications of natural phe- nomena are coincident with essential distinctions be- tween them and can be used as a source of inferences respecting their nature and origin that, as some one has said, the score of the Lord's creation, like that of Haydn's Creation, is crossed with bars has been pro- lific of an endless train of fanciful presumptions by which the progress of science is incessantly obstructed. For the reasons here set forth, the terms " abstract and concrete numbers " are also fallacious and mislead- ing. Numbers, in themselves, are essentially abstract. In another sense they are necessarily concrete : they al- ways stand for some particular object, relation, or opera- tion. They are nothing in themselves. This remark is doubly true of algebraic symbols which require in- terpretation, in the first place, by assigning to them particular numerical values, these, in turn, remaining without significance until the units, of which they con- sist, are referred to their proper objects, relations, or operations. This is, no doubt, Duehring's meaning when he observes, somewhere in his History of the Principles of Mechanics, that algebraic symbolization is radically defective inasmuch as it makes no display of the numerical units which are the essential coefficients observes: "If -f- 1, 1, V~ had not been called positive, negative, imaginary (or even impossible) "units, but, for example, direct, inverse, lateral units, this obscurity would have vanished." RIEMANN'S DISSERTATION. 267 of every literal symbol. He might have extended this observation by adding that the use of letters as alge- braic symbols, i. e., as representatives of numbers, is in itself a serious (though, perhaps, an unavoidable) in- firmity of mathematical notation. In the simple for- mula, for instance, expressive of the velocity of a moving body in terms of space and time ft>= -V the letters have a tendency to suggest to the mathematician that he has before him direct representatives of the things or ele- ments with which he deals, and not merely of their ratios expressible in numbers. In every algebraic opera- tion the use of letters obscures the real nature, both of the processes and of the results, and tends to strengthen ontological prepossessions. The true theory of the relations between arithmet- ical or algebraic quantities and magnitudes of extension was stated long ago, in Germany by Martin Ohm and in England by George Peacock (the Dean of Ely), Au- gustus de Morgan, D. F. Gregory, and others ; but the writings of these thinkers have produced little impres- sion upon contemporary and succeeding generations of mathematicians. This is peculiarly apparent in the books and articles expository of the theories of "im- aginary quantities " and " complex numbers," and of the doctrines of the calculus of quaternions. The im- mense extension of the sphere of analysis since Des- cartes's new application of algebra to the determination of geometrical magnitudes is almost universally attrib- uted to a growing insight into the true character of " arithmetical quantities," and to a progressive explica- tion of the essential implications of number. It is sup- posed that Euclid's denial of the existence of numeri- cal ratios between incommensurable quantities, as well 268 CONCEPTS OF MODERN PHYSICS. as the protests of the early occidental arithmeticians and algebraists against negative or irrational numbers as "numeri dbsurdi infra nil" or " numeri ficti" or the designation by Girolamo Cardano of the negative roots of an equation as " c&stimationes fictce " representing solutions " vere sophisticce" are one and all simply evi- dences of the ignorance of these several writers of the real nature of numbers. It is not at all unusual to meet with the dogma, in treatises on the theory of " complex numbers," that algebra and arithmetic are essentially linear, numeration being impossible except by progres- sion, in equal steps, in the direction of a straight line.* And, I may add, the belief is by no means uncommon that metageometry is an advance beyond the old doc- trines concerning the relations between geometrical forms in ordinary space, in the same sense and by the same logic in and by which the calculus of quaternions is an advance beyond ordinary analytical geometry. The foregoing discussion has brought us to the point where the reader is in a condition, I hope, to realize the great fundamental absurdity of Riemann's endeavor to draw inferences respecting the nature of space and the extension of its concept from algebraic representations of " multiplicities." An algebraic multiple and a spa- tial magnitude are totally disparate. That no conclu- sions about forms of extension or spatial magnitudes are derivable from the forms of algebraic functions is evi- dent upon the most elementary considerations. The same algebraic formula may stand for the most various things. Equations of the second degree, for example, may represent either geometrical areas, or geometrical curves. The equation y a? 2 may represent, either the area of a square whose side is a?, or a parabola (referred * Cf. Riecke, die Rechnung mit Richtungszahlen (Stuttgart, 1856). KIEMANN'S DISSERTATION. 269 to an axis of ordinates) whose parameter is 1. If Rie- mann's argument were fundamentally valid, it could be presented in very succinct and simple form. It would be nothing more than a suggestion that, because alge- braic quantities of the first, second, and third degrees denote geometrical magnitudes of one, two, and three dimensions respectively, there must be geometrical mag- nitudes of four, five, six, etc., dimensions correspond- ing to algebraic quantities of the fourth, fifth, sixth, etc., degree.* It is hardly necessary to say, after all this, that the analytical argument in favor of the existence, or possi- bility, of transcendental space is another flagrant in- stance of the reification of concepts. * It is not unworthy of remark, here, that the practice of reading x 2 and a; 3 as x square and x cube, instead of x of the second or third power, is founded upon the silent or express assumption that an algebraic quantity has an inherent geometric import. The practice is, therefore, mislead- ing, and ought to be disused. Principiis obsta I CHAPTER XY. COSMOLOGICAL AND COSMOGENETIC SPECULATIONS. THE NEBULAE HYPOTHESIS. LIKE all metaphysical theories, the atomo-mechanical theory has its cosmogonies. All metaphysical cosmog- onies are attempts to deduce the universe and its phe- nomena from one or more primordial elements by the application of a few general principles, The cosmogo- nies of the atomo-mechanical theory are attempts to deduce the universe and its phenomena from the ele- ments of mass and motion by the application of mechan- ical principles expressive of the simple laws of motion. As has been shown, the ultimate problem of the atomo- mechanical theory, to whose effectual and complete so- lution the physicists of the day look forward with a greater or less degree of confidence though many of them are clear-sighted enough to regard it as an aspira- tion never to be realized is the exhibition of all vital and organic phenomena as results of ordinary chemical and physical action, and of chemical and physical action, in turn, as exchanges and transferences of mechanical motion between constant and uniform elements of mass. A question necessarily preliminary to cosmological speculations of whatever kind has been extensively mooted, of late, by mathematicians and physicists alike the question respecting the finitude or infinitude of or ran COSMOLOGICAL SPECULATIONS. 2 the universe in time, space, and mass.* A cosmogony properly so called, inevitably involves the presumption that the universe is finite in past time at least, for it is a theory respecting the origin or beginning of the uni- verse. The vision of the cosmogenetic theorist extends backward, either to the absolute nothing, or to a state of physical uniformity wholly destitute of those phe- nomenal differences and changes which are the essential prerequisites of the notion of time. This universal cosmogenetic presumption of the finite duration of the universe in the past has recently been supplemented by the assertion of its limited duration in the future an assertion founded on a variety of physical considerations, the most noteworthy among which is the doctrine of the progressive dissipation of energy. This doctrine is stated in the most intelligible form, perhaps, by Sir William Thomson, f and is embodied in the following propositions : " 1. There is at present in the material world a universal tendency to the dissipation of mechanical en- ergy. " 2. Any restoration of mechanical energy, without more than an equivalent of dissipation, is impossible in inanimate material processes, and is probably never effected by material masses either endowed with vege- table life, or subjected to the will of an animated creat- ure. " 3. Within a finite time past the earth must have been, and within a finite period of time to come the earth must again be, unfit for the habitation of man as * Cf. Wundt, " Ueber das Kosmologische Problem," Vierteljahrs- schrift fuer wissenschaftliche Philosophie, vol. i, p. 80 seq. t " On a Universal Tendency in Nature to the Dissipation of Mechani- cal Energy," Phil. Mag., series iv, vol. iv, p 304 seq. 272 CONCEPTS OF MODERN PHYSICS. at present constituted, unless operations have been, or are to be, performed which are impossible under the laws to which the known operations going on at present in the material world are subject/' The reasoning by which these conclusions (which, it may be noted in passing, are carefully and in terms confined to our planet, or, at least, our planetary sys- tem) are arrived at is that, inasmuch as all the opera- tions of nature, which constitute its life and action, depend upon transformations of energy, and as every such transformation, in conformity with the second law of thermo-dynamics, is in effect (to use the expression of P. Gr. Tait) a degradation from a plane of higher to one of lower transf ormability or availability, the ulti- mate effect must be a conversion of all the energy of the world into heat and a reduction of its temperature to absolute uniformity. From this state of uniformity in the diffusion of heat no restoration of available en- ergy is possible ; for heat admits of transformation into other forms of energy only by passing from a body of higher to one of lower temperature.* * The doctrine of the dissipation of energy has been extensively de- veloped by Clausius, who designates the sum of the possible transforma- tions of the world's energy as its entropy, and announces that " the entropy of the world tends to a maximum." (Pogg. Ann., vol. cxxi, p. 1 ; Abhandlungen ueber die mechanische Waermethcorie, vol. ii, p. 44.) It is to be regretted that Tait, while adopting the word " entropy," under- takes to use it, as he himself says (Thermo-dynamics, 48 ; ib., 178), " in the opposite sense to that in which Clausius employed it,", and that Maxwell (Theory of Heat, pp. 186, 188) follows him. Nothing is more to be reprobated than an arbitrary change in scientific terminology, and especially a deliberate tampering with the received meaning of a term. It ought to be added that Tait does not even succeed in his attempt to reverse Clausius's meaning, and that Maxwell, too, is in error when he says that "Clausius uses the word (entropy) to denote the part of energy which is not available." COSMOLOGICAL SPECULATIONS. 273 It is clear that, if the law of the dissipation of energy applies to the universe at large that is to say, if the dynamics of a finite material system can be legitimately extended to the Cosmos as an infinite whole -there must, sooner or later, be an end of the universe identi- cal with its beginning as assumed by the atomo-mechan- ical theory. The processes of nature must eventuate in a thorough homogeneity of its elements in a com- plete absence of the differences and changes which constitute the attestation of its real or actual existence. This conclusion has been sought to be avoided by the assumption of the finitude of the universe in mass, or in space, or in both. The first impulse in this direction probably came from an article of W. M. Kankine* (published shortly after the appearance of that of Sir William Thomson), in which it was argued that "if there is between the atmospheres of the heavenly bodies an interstellar medium perfectly transparent and dia- thermanous i. e., incapable of converting light and heat from the radiant into the fixed or conductible form, and thus incapable of acquiring any temperature whatever and if this interstellar medium has bounds beyond which there is empty space, the radiant heat of the world will be totally reflected and will ultimately be reconcentrated into foci in which a star (i. e., an extinct mass of inert compounds) would be vaporized and resolved into its elements, a store of chemical force being thus reproduced at the expense of a corresponding amount of radiant heat." The supposition of the finitude of the mass of the universe was not new ; it had often been made before. But here it presented itself in a new form. Hitherto * " On the Reconcentration of the Mechanical Energy of the Uni- verse," Phil. Mag. (iv), vol. iv, p. 358 scq. 274 CONCEPTS OF MODERN PHYSICS. the supposition had been that the mass, though limited, was diffused throughout unlimited space; and in this form it has recently been revived by Wundt, who im- agines that the finitude of a mass may be reconciled with the infinitude of its volume by the assumption of an endlessly progressive increase of its tenuity, the mass being taken as the finite sum of an infinite con- verging series. Rankine, on the contrary, required the physicist to grant that the mass of the universe is finite also in extent and is everywhere surrounded by void space. The conception of a material universe thus bounded in boundless space obviously presents insur- mountable difficulties ; and in view of these difficulties many astronomers and physicists hailed with delight the thesis of the metageometers that space itself, though unlimited by reason of its inherent curvature, is not infinite, and that, therefore, the mass of the universe must be finite, however diffused. This thesis was doubly welcome because it appeared, at first sight, also to afford the means of escape from another diffi- culty raised by the astronomers. In 1826 Olbers * observed that, if the number of bodies in the universe radiating heat and light is infinite, each point in space must receive an infinite number of caloric and lumiiiar rays, and must, therefore, be infinitely hot and bright adding, however, that this consequence could be avoid- ed by supposing an absorption of the greater part of these rays by the dark and cold bodies in space. But this salvo at once appeared questionable, on the reflec- tion that the dark and cold bodies disseminated among the luminous stars must speedily reach the point of incandescence, and that their absorbing power must soon be exhausted. * Bodc's astron. Jahrbuch, 1826, p. 110 scg. Quoted by Zoellner. COSMOLOGICAL SPECULATIONS. 275 There is supposed to be a still further and similar perplexity, growing out of the fact of gravitation, espe- cially in view of its instantaneous action. It is said that a universe consisting of an infinite number of bodies attracting each other would not only be without a definite center of gravity to which all cosmical mo- tions could be referred its center of attraction being everywhere, and therefore nowhere but would result in an infinite pressure (I follow the expression of Wundt, though it would, perhaps, be more correct to say an infinite strain) at every point in space. This difficulty, in particular, is urged by Wundt as insuper- able so long as the mass of the universe is held to be infinite ; it can, in his opinion, be overcome only by the assumption that this mass is limited. It is unnecessary to enter upon a minute examina- tion of the validity of these considerations adduced in support of the theory of the finitude of the material universe. As to the last of them, relating to the ef- fects of radiation and gravitation, it is readily seen, and has been pointed out by Lasswitz,* that they lose their force the moment we recollect that the intensity, both of radiation and gravity, decreases as the square of the distance increases, and that the infinite series expressive of the several effects of heat, light, and gravitation are converging, their summation yielding finite results. And of the application of the doctrine of the dissipa- tion of energy to an infinite universe it is to be said that it is wholly inadmissible. That doctrine is, no doubt, irrecusable in its application to any finite mate- rial system. Every such system must come to an end, as it has had a beginning. And this is true of every such system, whatever its extent. But it is not true of * Viertcljahrs:elmft f. w. P., vol. i, p. 329 seq. 270 CONCEPTS OF MODERN PHYSICS. a universe absolutely unlimited. Neither the law of the conservation of energy, nor that of its dissipation, can be legitimately applied to it. The universe, taken as absolutely infinite, is not a conservative system and is not in any proper sense subject to physical laws. We can not deal with the Infinite as with a physically real thing, because definite physical reality is coexten- sive with action and reaction ; and physical laws can not be applied to it, because they are determinations of the modes of interaction between distinct, finite bodies. The universe, so called, is not a distinct body, and there are no bodies without it with which it could in- teract. Operations with the term Infinite in analogy to operations with finite terms are as illegitimate in physics as they are in mathematics. The Infinite is simply the expression of the essential relativity of all material things and their properties, and is thus, in a sense, inherent in every finite form. It is the basis of all the relations which constitute sensible actuality, but it is not itself a group of such relations. It is the back- ground of all material actions and forms ; no system of elements or forces can exist without it, or is cognizable without reference to it ; and in this sense, and in this sense only, the universe is necessarily infinite in mass as well as in space and in time. It follows that all cosmogonies which purport to be theories of the origin of the universe as an absolute whole, in the light of physical or dynamical laws, are fundamentally absurd. The only question to which a series or group of phenomena gives legitimate rise relates to their filiation and interdependence ; and the attempts to transcend the bases of this filiation and interdepend- ence to determine the conditions of the emergence of physical phenomena beyond the bounds of space and COSMOLOGICAL SPECULATIONS. 277 the limits of time are as futile as (to use the happy simile of Sir William Hamilton) the attempt of the eagle to outsoar the atmosphere in which he floats and by which alone he may be supported. This leads me to a discussion of a cosmogenetic the- ory which has attained to great celebrity and very gen- eral acceptance, under the name of the Nebular Hy- pothesis. As now generally held, this theory may be briefly stated as follows : Primordially the materials, which are at present found, partly at least, conglomerated in the bodies com- posing the stellar, solar, planetary, satellitic, and me- teoric systems, were uniformly dispersed throughout space. In some way, by the action of cosmic (attrac- tive and other) forces, this uniformly diffused and very attenuated matter came to be divided into large nebu- lous spheres which began slowly to rotate, the rotation resulting, perhaps, from the act of division, or from in- ternal differences in their densities and irregularities in their forms, which deflected the lines of gravitation from a strictly radial direction, the centers of attraction no longer coinciding with the centers of figure. In proportion as these spheres parted with their heat they contracted ; and this contraction led to an increase of their velocities of rotation in conformity to a mechani- cal law known as the law of the conservation of areas or of angular momentum This law, in its most gen- eral expression, is simply a corollary from the law of inertia, from which it follows that the resultant angular momentum of any material system can not be changed, either in magnitude or the direction of its axis, by the mutual action of its constituents.* For the purpose of * All mechanical or dynamical laws of conservation the conserva- tion of momentum, of angular momentum and of energy are (as I have 278 CONCEPTS OF MODERN PHYSICS. its application to a rotating nebulous mass, however, the law may be more intelligibly stated in another form, viz., that, whatever change of volume or form may be produced in a material system by the mutual attraction of its constituent elements, the sum of all the areas de- scribed by the radii vectores of the several elements or particles round the center of rotation, in a unit of time, is constant. Now, the areas being proportional to the squares of the diameters, it follows that the angular velocity increased with great rapidity as the contraction of a nebulous mass proceeded. An immediate conse- quence of this increase of velocity was a proportionate increase of the centrifugal force in the equatorial regions of the rotating sphere, so that in course of time this force came to balance, and afterward to exceed, the centripetal gravitation. This led at first to a dispro- portionate contraction of the sphere at the poles and to the assumption, by the sphere, of an oblately spheroidal or lenticular form, and eventually to successive detach- ments of equatorial rings or zones which at first circu- lated round the residual mass in the direction of its original rotation, but which by reason of the instabili- ty of such rings in case of the least departure from ab- solute regularity of form or constitution broke up into parts, forming one or more minor spheres or spheroids. These continued to revolve round the sun with a veloci- ty nearly equal to the rotatory velocity of their mate- already indicated in the sixth chapter) at bottom nothing more than ap- plications of the principle of inertia to complex material systems. It is the great merit of Poinsot to have brought to light the formal analogies (prefigured, to a certain extent, in the writings of Euler) between the laws governing movements of rotation and those determining the forms of ordinary translatory motion. It is hardly necessary to add that the law of the conservation of areas is in form a generalization of Kepler's second law. COSMOLOGICAL SPECULATIONS. 279 rials at the moment of their detachment and congloba- tion. In most cases, probably, the whole mass of such a ring coalesced into a single body, i. e., into a planet, while in some cases several bodies were formed, such as they appear in our planetary system in the zone of asteroids. Each of the planets, while revolving round the residual mass whose condensation is supposed to have produced the sun, also began to rotate on an axis of its own, the direction of this rotation coinciding with that of its revolution. It thus became subject to the same dynamical conditions which determined the evolu- tion of the parent system ; it also threw off rings which either retained their form (as in the case of the Satur- nian rings) or formed into minor satellitic bodies. The arguments which have been advanced in sup- port of this hypothesis are so well known that it is hardly necessary to recapitulate them. Among them are the existence, in the stellar regions, of nebulous masses in various stages of condensation ; the evidences of the increase of temperature from the surface of our planet toward the interior ; the proximate coincidence of the orbital motions of the several planets, both in direction and plane, and the further proximate coinci- dence of this orbital motion with the direction and plane of the sun's rotation ; the similar coincidence of the directions of the orbital motions of the satellites with the axial motions of their planets; the oblately spheroidal form of the earth, and, as far as we know, of the other planets, which peculiar form has not only been theoretically demonstrated, but has also been ex- perimentally shown, by M. Plateau, to be the form necessarily assumed by a rotating body in a liquid or semi-liquid state. These considerations were adduced, al- most in the same order and form, by Kant and Laplace, 280 CONCEPTS OF MODERN PHYSICS. and they have since been supplemented by a variety of other considerations more or less plausible, among which may be mentioned the agreement of the theoretical con- sequences of the fact, that the projection of planetary masses from the parent globe must have taken place with ever-increasing rapidity as the contraction of the globe progressed, with certain well-known features of our own planetary system. Attempts not wholly un- successful have even been made to effect a deduction, from the elements of this theory, of the empirical law respecting the distances of the several planets from the sun which is known as the law of Bode or Titius. The nebular hypothesis, as a theory of the origin, not only of our planetary system, but of stellar and planetary systems throughout the universe, is common- ly ascribed to Laplace, who is supposed to have been unaware of the fact that the hypothesis which he ad- vanced had been published by Kant, in his Naturge- schichte des Himmels, in 1755, nearly half a century before the first appearance of the Exposition du 8ys- teme du Monde, in 1796. But the truth is that the Nebular Hypothesis, in the form in which it is now generally held, is due to Kant, and differs in several essential particulars from the hypothesis of Laplace. This latter hypothesis is limited in terms to our plane- tary system, and there is no indication in any of the writings of the French astronomer certainly none in his Exposition du Systeme du Monde that he ventured to extend it to the entire universe, as was expressly done by Kant. But there is a difference still more im- portant between the hypotheses of the two thinkers. Kant's assumption was that u all the materials compos- ing the spheres that belong to our solar world were, in the beginning of all things, resolved into their element- COSMOLOGICAL SPECULATIONS. 281 ary substance and filled the whole space of the system in which these spheres now move." * This assumption is common to all recent forms of the nebular hypoth- esis that have fallen under my notice they all postu- late a diffusion of the entire mass of the sun, planets, comets, and satellites constituting our planetary system throughout the planetary space. The assumption of Laplace, on the contrary, is simply that the atmosphere of the sun at one time extended beyond the orbits of the farthest planets, and that the formation of the planets and their satellites as well as that of the comets was due to a gradual cooling and contraction of this atmosphere.f It is hardly necessary to say that the Laplacean form of the nebular hypothesis is far too narrow to serve the purposes of a general cosmological theory. Such a theory demands the derivation of the several concre- tions of cosmical matter from some primitive homo- geneous mass. This demand is complied with by the hypothesis of Kant ; but it is very partially, if at all, satisfied by that of Laplace. And this brings us into the presence of a formidable difficulty. It is to be feared that, in proportion to its amplification to cosmo- genetic dimensions, the nebular hypothesis parts with its validity as a physical theory. This subject was ex- * " Ich nehme an, class alle Materie, daraus die Kugeln die zu unserer Sonnenwelt gehoercn, alle Planeten und Kometen bestehen, im Anfang allcr Dingc in ihrcn elcmentarischen Grundstoff aufgeloes't, den ganzen Raum des Weltgebaendes erfuellt haben, darin jetzt diese gebildeten Koerper hcrumlaufen." " Naturgeschichte des Himmels," Kant's Werke, vol. vi, p. 95. f " La consideration des mouvemens plan6taires nous conduit done a pcnser qu'en vertu d'une chaleur excessive Vatmospliere du soleil s'est primitivement etendue au dela des orbes de toutes les planetes, et qu'elle s'est resserree successivement jusqu'k ses limites actuelles." Sys- teme du Monde (2me ed.), p. 345. 282 CONCEPTS OF MODERN PHYSICS. amined, nearly twenty years ago, by M. Babinet, in an article on the Cosmogony of Laplace,* in winch he shows that the actual rotatory velocities of the several planets are in fact vastly greater than the velocities to be deduced, by the aid of the law of the conservation of areas, from the nebular hypothesis, if that hypothesis includes the assumption of a diffusion of the solar mass itself throughout a space coextensive with the limits of our planetary system. " Several persons," says M. Babinet, "have thought that the sun .himself had origi- nally been expanded so as to fill the entire space now occupied by the planets, although Laplace expressly mentions that at the moment of the formation of these bodies it was only the atmospJiere of the sun which had this vast extent. We are able to test this question mathematically, by calculating from the sun's actual pe- riod of rotation, which is twenty-five and three tenths days, what would be the velocity of rotation if, con- serving the sum of the areas described by all its mate- rial points, it were expanded so that its radius, which is now equal to one hundred and twelve times the equa- torial radius of the earth, became equal to the distance from the earth to the sun, or from Neptune to the sun. . . . The calculation on the first of these bases gives 'a rotation of 1,162,000 days, amounting to more than three thousand (3,181) years. The period of revo- lution calculated on the second basis would evidently be nine hundred times greater, that is to say, more than twenty-seven thousand centuries. * " Note sur un Point de la Cosmogonie de Laplace," Comptes Rcndus, vol. lii, p. 481 seq. My attention was drawn to this article by a passage in an interesting little pamphlet of Dr. E. Budde, of Bonn, Zur Kos- mologie der Gegenwart (Bonn, ed. Weber, 1872), to which I shall have occasion to recur hereafter. COSMOLOGICAL SPECULATIONS. 283 " These numbers being infinitely greater than those expressive of the earth's and Neptune's actual periods of revolution, it is plainly impossible to admit that these planets have been formed out of the solar mass itself extended beyond the planetary orbits. This, however, does not preclude the idea that the stars themselves have been formed at the expense of a universal cosmic matter endowed with excessively feeble movements of rotation round the center of gravity of each mass which was in process of formation as an independent sun. " The conclusion is that, if the entire mass of the sun had been expanded to the limits of the planetary system, it must have had a movement of rotation far too feeble to enable the centrifugal force to balance the force of gravity so as to lead to the separation of an equatorial ring from the total mass." The discrepancies here brought to light between the actual orbital periods of the planets and the correspond- ing periods found by calculation in accordance with the postulates of the nebular hypothesis, are so enormous that there appears to be no possibility of accounting for them by the assumption of a progressive contraction of the orbits of the several planets since their projection, and the consequent quickening of their orbital motions. The calculations of M. Babinet do not constitute the only difficulty which besets the nebular hypothesis, either in its general cosmogenetic or in its special La- placean form. In the progress of astronomical dis- covery it has appeared that several of the supposed coincidences between the facts and the hypothesis fail. Thus, there appears to be an exception to the direc- tional uniformity of the axial and orbital motions of the planets and their satellites in the case of Uranus, the orbital planes of whose satellites are nearly perpen- OY THGR r w v v W vi n v H 284 CONCEPTS OF MODERN PHYSICS. dicular to the ecliptic, the circumplanetary motions of the satellite as well as the axial motion of the planet, moreover, being retrograde a fact long since discovered by Sir William Herschel, and confirmed by various sub- sequent observations. Another difficulty has been sug- gested by the recent discovery (1877), by Professor Asaph Hall, of two satellites of the planet Mars, and the proximate determination of their respective distances from the primary, as well as their orbital (circumplane- tary) periods. It was found that the distances of the inner and outer satellites from the center of the planet are about three and six times, respectively, the radius of the planet, and that the periods of revolution of these satellites are 7*65 and 30'25 hours, respectively, while the period of rotation of the planet (Mars) itself is 24-623 hours. It appears, then, that one of the satel- lites revolves about the planet in less than one third of the time required for the planet's axial rotation. At first sight this fact would seem to be wholly in- compatible with the nebular hypothesis. In the light of this hypothesis, the orbital motions of a satellite are con- tinuations of the axial motions of the materials out of which the satellites are formed ; its orbital period ought, therefore, to be equal, proximately at least, to the period in which the planet rotated at the time of the satellite's formation. And that period is of necessity greater than the period of the planet's present rotation, by reason of the acceleration produced by its subsequent contraction. Two attempts have been made, thus far, to recon- cile the anomaly just referred to with the postulates of the nebular hypothesis. One of these is based on the supposition that the orbits of the satellites have been contracted by the resistance of the ethereal medium which was formerly supposed to have shortened the COSMOLOGICAL SPECULATIONS. 285 period of Encke's comet. But this resistance is wholly insufficient to account for the anomaly, even if the very doubtful existence of an interstellar and interplanetary medium capable of offering material resistance to plane- tary motion be granted. The other attempt seeks to refer the anomaly to a retardation of the planet's period of rotation, and a corresponding quickening of the re- volving motion of the satellites by tidal action. "While it is admitted that the retardation of the rotatory period of the planet by the action of the tides raised upon it by a satellite could at most produce a coincidence of that period with the orbital period of the satellite, it is claimed by G. H. Darwin that the period of planetary rotation may be lengthened beyond that of satellitic revolution by the friction of the tides due to solar ac- tion. The effect here alluded to is the result of a con- version of the energy of planetary rotation into heat, and of a transference of the angular momentum of ro- tation to that of orbital revolution of the bodies by whose mutual attraction the tides are raised about their common center of inertia ; and, inasmuch as a larger planet has both more energy of rotation and more an- gular momentum than a small one, the most rapid changes in the periods of rotation and revolution will be produced in the case of the smallest planet. Mars being the smallest planet attended by satellites, it is said that the slowness of his rotation as compared with the revolution of his inner satellite may have been pro- duced by the agency above indicated within the period assigned to the history of our planetary system by the advocates of the nebular hypothesis. Whatever may be the weight of the several objections urged against certain features, more or less essential, of the nebular hypothesis, there is one objection which is 286 CONCEPTS OF MODERN PHYSICS. fundamental : the inadmissibility, already pointed out, of all speculations respecting the origin of the universe as an unlimited whole. But, apart from this, it is plain that the derivation of the forms and movements of the stellar and planetary systems from a primordial homo- geneous mass uniformly diffused throughout space is impossible. In the first place, such a mass must be either at rest or in uniform motion ; and this state of rest or uniform motion, according to the most element- ary principles, could be changed only by extraneous impulses or attractions. And, there being no "with- out " to the all-embracing Cosmos or Chaos, the original state of rest or uniform motion would necessarily be perpetual.* In the second place, such a nebulous universe would be of perfectly uniform temperature ; all parts would be equally hot (or cold), and there could be no radiation or loss of heat resulting in a contraction of any part of the nebulous mass. Its thermo-dynami- cal condition would be constant for the same reason which establishes the permanence of its general dy- namical condition. The cumulation of difficulties presented by the nebular hypothesis has become so great, and is begin- ning to be so extensively realized, as to develop a ten- dency to modify or supplant it by another hypothesis which may be called the hypothesis of meteoric agglom- eration. This hypothesis commends itself to the modern physicist by reason of its apparent exemplification of the general doctrine that, for the purpose of ascer- taining the nature of the agencies which have produced * As Duehring expresses it (Kritische Geschichte der allgemeinen Principien dcr Mechanik, 2d ed., 151), "If ever there had been perfect equilibrium between the parts (of the nebulous mass) it would continue to exist now." COSMOLOGICAL SPECULATIONS. 287 a particular physical system or form, we must in the first instance look to the agencies concerned in its maintenance or destruction a doctrine which might be condensed into a canon : quod sustinet vel delet^for- mavit. This doctrine is in effect nothing more than a new statement of the old law of parsimony which for- bids the unnecessary multiplication of explanatory ele- ments and agencies. It has been extensively and suc- cessfully applied in geology, which now endeavors to account for all the past phases in the history of the earth by the regular and ordinary action of the forces known to be at work in maintaining or modifying its present condition. The theory of meteoric agglomera- tion was first suggested by Julius Eobert Mayer,* and was founded on the reflection that the great annual fall of meteoric masses upon the earth indicates the circula- tion or movement within our planetary space of a vast number of small bodies which must strike large bodies, like the sun, in numbers enormously exceeding those reaching the earth, the number being greater in pro- portion both to the masses and the surfaces of the larger bodies. These meteors, according to Mayer, are in a sense the fuel of the sun, and all bodies within the planetary system are subject to accretions, both of mass and temperature, in consequence of their col- lisions with them. Now, it is supposed that in astro- nomically primeval times the proportion of these me- teoric masses to the masses of the large solar and plan- etary bodies may have been far greater than it is now that, in fact, there may have been a time when the space now occupied by our planetary system presented the appearance of a swarm of such meteors of all sizes * In his Beitraege zur Mcchanik dcs Ilimmels (first published in 1848), Mechanik dcr Waermc, p. 157 seq. 288 CONCEPTS OF MODERN PHYSICS. and of all degrees and forms of consistency and aggre- gation, moving about at all rates of velocity, in all di- rections, and in orbits of every degree of eccentricity. These masses would be consolidated, and movements, both of rotation and revolution, would be generated in the bodies so formed by their collisions. At this point the question obtrudes itself : how can a theory, which seeks to derive the orderly, symmetrical, and harmonious world as we know it from the wildest congestion of aboriginal differences and anomalies > from a spring-head of utter incongruity and confusion be made to account for the regularities and coincL dences whose simple and natural explanation was the conspicuous merit of the hypothesis of Laplace ? An answer to this question is sought, by the advo- cates of the new theory, in an appeal to a principle long since established by Laplace himself. This principle relates to the fact that, amid all the disturbances caused by the mutual attractions of the planetary bodies, there exists an invariable plane passing through the center of gravity of the whole system, about which these bodies perpetually oscillate with but slight deviations on either side. If on this invariable plane we project the areas described by the radii vectores of the several elements of mass in a given time, and multiply each mass into its respective area thus projected, the sum of the products is a maximum, and the rate of its increase is constant.* Such a plane exists, not only for the solar system, but for any system of bodies controlled solely by their mut- ual attractions. Now, it is evident that both the sum and the rate of its increase, of the products of the masses * Cf. Laplace, Mecanique Celeste, lere partie, liv. ii, chap. vii. (" DCS incyalites seculaires des mouvemcns celestes.") The theory was first pub- lished in the Journal de 1'Ecole Polytechnique, 1798. COSMOLOGICAL SPECULATIONS. 289 into the projections of the areas described by tlieir radii vectores, are always less than the sum and the rate of its increase of the products of the masses into the radii vectores themselves, inasmuch as these radii (unless they are parallel to the plane) are shortened by their projec- tion ; and the difference between these two sums is in direct proportion to the deviations of the movements from the direction of the total increase, which direction, for purposes of reference, is taken as positive, the oppo- site direction being, of course, taken as negative. And whenever the several movements meet with resistance, some of the components of the velocities of the moving masses are necessarily destroyed, so that the difference in question is diminished and eventually annulled. When this has happened, the absolute values of the areas described by the radii vectores of the masses in a given time become equal to their maximum projections ; in other words, their planes coincide with or become par- allel to Laplace's invariable plane. From this follows the general principle that the movements of the bodies constituting any finite system, whatever be their origi- nal divergence of direction, tend (except in a very few special cases), by reason of any resistance to these move- ments, to become parallel to or coincident with an inva- riable plane.* Before leaving this subject I may observe that the principle just stated, which admits of a further generali- zation, so as to assume this form that all movements of the elements of a finite material system depending upon * The possible exceptions to this law are, of course, those cases in which the components destroyed are exactly equal and opposite. The improbability of the occurrence of such cases is so great that Budde, who states the law substantially as I have stated it in the text (1. c., p. 30), does not even allude to the possibility of an exception. 15 \ 290 CONCEPTS OF MODERN PHYSICS. the mutual action of such elements tend, in consequence of any permanent interference with or determination of these movements from without, from irregularity and disorder to regularity and order is, in my judgment, one of the most important in the whole range of theo- retical physics. For the condition here assigned that the internal movements of the system be subject to constant interference from without is in fact insepa- rable from every material system, there being no such system which is at any time under the exclusive con- trol of its own internal forces. There is, consequently, in every finite part of the world an ingenerate bias from irregularity to regularity, a natural bent from disorder to order, an inherent tendency from Chaos to Cosmos ; and this tendency is the simple and direct consequence of the relativity of all material forms of the fact that each finite whole is always a part of a still greater whole in short, that the finite exists only on an ever- receding background of infinitude. It is possible even that this principle is more than coextensive with the sphere of physics, and that, to a certain extent, it may have its applications within the domains of those sci- ences which are ordinarily designated as historical. Although attempts at a transference of laws governing the interdependence of phenomena whose lines of con- nection are simple and easily traced (such as the move- ments of inorganic masses) to a class of phenomena whose relations are complicated and imperfectly under- stood (such as the phenomena of organic and vital ac- tion) are perilous in the extreme, and never to be made without a careful reference to the nature and ground of the analogies by which they are induced, it is neverthe- less true that a great part of the progress which is now being made in the several departments of science is COSMOLOGICAL SPECULATIONS. 291 due to liberal exchanges, not only of results, but also of principles and methods.* The theory of meteoric aggregation undertakes to grapple with still further elements of the general prob- lem of explaining the actual features of our planetary system, as, for instance, the comparative inferiority of the sizes of the planets nearest the sun. The reasoning is something like this : Somewhere within the space comprising the various movements of the bodies, whose materials are in process of agglomeration, a mass will probably be formed which is preeminent above all the others. This the nucleus of the future sun of the system must gradually draw to its neighborhood the perihelia of all the moving meteoric masses or groups. In this region, therefore, the movements of all the bodies must have the greatest velocity ; ' here the me- teors must fly past each other with the greatest swift- ness, and their approach and agglomeration must be most difficult a circumstance which also prevents the rapid growth of bodies in this region after their incho- ate formation. Near the confines of the system, on the contrary, where the movements of the meteors are slug- gish, the conditions for the congestion of large masses are comparatively favorable. Similarly, a rough account is given of the fact that the densities of the planets are * Instances of the application of dynamical and, generally, of physi- cal laws, not only to vital, but also to psychological, action are afforded by the recent discussion, by Avenarius, of the evolution of thought in conformity to the principle of least action (Die Philosophic als Denken der Welt gemaess dem Princip des Kleinsten Kraftmaasses, Leipzig, 1876), and the previous discussion, by Schleicher, of the evolution of language in the light of the doctrine of natural selection which, it may be said parenthetically, is not without analogy to the principle discussed in the text (Die Darwin'schc Theorie und die Sprachwissenschaft, Weimar, 1863). 292 CONCEPTS OF MODERN PHYSICS. generally in the inverse ratio to their sizes. A larger body attracts a meteor with greater intensity than a smaller one ; its growth is, therefore, marked by more violent collisions productive of a higher temperature and a corresponding expansion. It is not my purpose to discuss the merits of this theory in detail, or to express an opinion as to its soundness and sufficiency ; but it is proper to say that it appears to me to stand in favorable contrast to the nebular hypothesis precisely by reason of the absence of some of the characteristics to which the general plausibility of this latter hypothesis is due. The nebu- lar hypothesis found ready and almost enthusiastic ac- ceptance, not so much on physical as on metaphysical grounds. The proneness to derive the Multiple from the absolutely Simple, the Various from the absolutely Uniform, has its root in the second of the great structu- ral fallacies which I have discussed in the ninth chap- ter in the assumption that the abstract result of a gen- eralization, i. e., a general concept, may be made available as a starting-point for the evolution of the particular things subsumed under it. The enthusiasm for the nebular hypothesis was, in this respect, an ontological survival. And in another respect it was even more than that it was a recrement of ancient traditions about the origin of the universe from Nothing. The original mist of the nebular hypothesis is assumed to be of extreme tenuity of a density less than the one hundred thousandth part of hydrogen, the lightest gase- ous body known to the chemist. By reason of this sethereal subtilty it was readily substituted, in the con- ceptions of the popular mind, for the old void from which the world was said to have emerged, and, in the imaginations of those who look upon matter as a sort COSMOLOGICAL SPECULATIONS. 293 of inspissation of Mind for the universal antemundane impersonal Spirit. It thus conformed to the assump- tion that, on any hypothesis respecting the mode of the world's formation, it must " in the beginning " have been " without form and void," and at the same time satisfied the mystic yearnings after. the Ethereal and " Spiritualistic," which is the special distinction of that large class of philosophers whose philosophy begins where clear thinking ends. CHAPTER XVI. CONCLUSION. THE considerations presented in the preceding pages lead to the conclusion that the atomo-mechanical theory- is not, and can not be, the true basis of modern physics. On proper examination, this theory appears to be not only, as is generally conceded, incompetent to account for the phenomena of organic life, but it proves to be equally incompetent to serve as an explanation of the most ordinary cases of inorganic physical action. And the claim that, in contradistinction to metaphysical the- ories, it resorts to no assumptions, and operates with no elements save the data of sensible experience, is found to be wholly inadmissible. In announcing this conclusion it is necessary, however, to guard against two fundamen- tal misconceptions. In the first place, the denial of the theory of the atomic constitution of matter, as it is gener- ally held by physicists and chemists, involves no assertion respecting the real constitution of bodies of chemical elements or compounds and certainly does not imply the metaphysical thesis of the absolute continuity of matter. "What is the actual constitution of particular bodies is a question to be determined in each case by experiment and observation. There is, no doubt, a large class of bodies whose constitution is molecular; but from this it does not follow that the molecules CONCLUSION. 295 composing them arc primordial, unchangeable units, existing independently and in advance of all physical action, and therefore absolutely exempt from change. On empirical grounds the inference, from the molecular structure of a body, of the permanent existence of ab- solutely immutable and indestructible atoms or mole- cules is as irrational as would be the assertion that primordially, and in advance of the formation of or- ganic bodies, there existed an indefinite number of elementary cells, because all organic bodies are of cellu- lar structure. In the second place, dissent from the proposition that all physical action is mechanical in the sense of being a transference of motion between distinct masses by collision or impact is not to be construed as a doubt respecting the constancy of physical laws or the univer- sality of their application. What is denied is, not the general dominance of the law of physical causation, but the doctrine that the only form of such causation is the transference of motion by the impact of masses which, in themselves, are absolutely inert. If physical action in conformity with constant and uniform law is des- ignated as mechanical, then all physical action is un- doubtedly mechanical. It may be said that physical action is utterly indeter- minable except on the supposition of the atomic or mo- lecular constitution of matter. This is true only in the sense that we are unable to deal with forms of physical action otherwise than by considering them as modes of interaction between distinct physical terms. Physical action can not be subjected to quantitative determination without a logical insulation of the conceptual elements of matter, and without ultimate reference to conceptual constants of mass and energy. All discursive reasoning 296 CONCEPTS OF MODERN PHYSICS. depends upon the formation of concepts, upon the intel- lectual segregation and grouping of attributes in other words, upon the consideration of phenomena under par- ticular aspects. In this sense the steps to scientific as well as other knowledge consist in a series of logical fic- tions which are as legitimate as they are indispensable in the operations of thought, but whose relations to the phenomena whereof they are the partial and not unfre- quently merely symbolical representations must never be lost sight of. When the old Greek sought to deter- mine the properties of the circle, he began by con- structing a polygon whose sides he subdivided until they were supposed to become infinitely small ; and in his view every line of definite extent and form i. e., every line which could become the subject of mathe- matical investigation was composed of an infinite num- ber of infinitely small straight lines. But, he speedily found that, while this fiction enabled him to deduce a rule for calculating the area of the circle and other- wise to determine a number of its properties, never- theless the circle and its rectilinear diameter were fun- damentally incommensurable, and the quadrature of the circle was impossible. The modern analyst simi- larly determines the locus of a curve by the relation of small increments of coordinates arbitrarily established ; but he is well aware that the curve itself has nothing to do with this arbitrary representation, and he very emphatically asserts the continuity of the curve by dif- ferentiating, or passing to the limit of, his increments at the same time transforming his coordinates by changing their origin or their inclination, or even their system, from bilinears to polars, whenever he finds it convenient, without dreaming that thereby he is in the least affecting the nature of the curve whose properties CONCLUSION. 297 are under discussion. The astronomer, in calculating the attraction of a homogeneous sphere upon a material point, begins by assuming the atomic or molecular con- stitution of the attracting sphere, establishing a series of finite differences as one of the terms of his equation ; but thereupon he takes the series to be infinite and the differences to be infinitely small, and very effectually dis- mantles the molecular scaffolding by integrating instead of effecting a summation of a series of finite differences. Observe : the astronomer begins with two fictions the fiction of a " material point " (which is, in truth, a contradiction in terms), so as to insulate the attractive force and treat it as proceeding from the sphere alone, and the fiction of the finite differences representing the molecular constitution of the sphere ; but the validity of his result depends upon the eventual rescission of these fictions and the rehabilitation of the fact. In like manner the chemist represents the proportions of weight, in which substances combine, as atoms of defi- nite weight, and the resulting compounds as definite groups of such atoms ; and this mythical coinage has been serviceable in many ways. But, apart from the circumstance that the symbols have become wholly in- adequate to the proper representation of the facts, it is important to bear in mind always that the symbol is not the fact. Newton derived many of the leading optical laws from his corpuscular theory of light and from the hypothesis of " fits of easy transmission and reflection." His theory for a time served a good pur- pose ; but it proved, after all, to be but a convenient mode of symbolizing the phenomena with which he was familiar, and had to be discarded when the phe- nomenon of interference was observed. In 1824 Sady Carnot deduced the law of thermic action which still 298 CONCEPTS OF MODERN PHYSICS. bears his name from an hypothesis respecting the nat- ure of heat (supposed by him, as by nearly all the phys- icists of his time, to be imponderable matter), which is now known, or universally believed, to be erroneous. For certain purposes, such as the mathematical deter- mination of gaseous pressure and expansion, thermic phenomena find a convenient representation in the hy- pothesis that a gaseous body is a group of atoms or molecules in a state of incessant motion. Some of the properties of gases have been successfully deduced, by Clausius and others, from formulae founded upon this hypothesis, and Maxwell has even succeeded in predict- ing the phenomenon of the gradual cessation of the oscillatory movement of a disk, suspended between two other disks, in consequence of the friction of a gaseous medium, whatever be the degree of its tenuity, and this prediction has since been experimentally verified ; but neither Clausius',3 formulae nor Maxwell's experi- ments are conclusive as to the real nature of a gas. That no valid inference respecting the real constitution of bodies and the true nature of physical action can be drawn from the forms in which it is found necessary or convenient to represent or to conceive them, is illus- trated by the fact that we habitually resort, not only in ordinary thought and speech, but also for purposes of scientific discussion, to modes of representing natural phenomena which are founded upon views and hy- potheses long since discarded as untenable. Just as we think and speak familiarly of the motions of the sun and stars in terms of the old geocentric doctrine, al- though no one in our day doubts the truth of the he- liocentric theory, so also the modern astronomer would find it difficult to dispense with geocentric fictions in subjecting these motions to mathematical computation. CONC'LUSION. 299 Even the, old epicycles survive in some of the analyti- cal formulae, by means of which that computation is effected. The progress of modern theoretical physics consists in the gradual reduction of the various forms of physi- cal action to the principle of the conservation of energy. For purposes of didactic exposition of this principle we resort to the fiction of systems of molecules or particles, whose motions are simple functions of the distances be- tween them. But, as we have seen, the conflict of this fiction with the facts of experience emerges at once, when we undertake to establish an absolute disjunction between the molecules and their motions. The conser- vation of energy would be impossible, if the ultimate constituents of a material system were in themselves absolutely inert. And the same thing is strikingly ex- hibited in the recent attempts to extend the principle of the conservation of energy to the phenomena of chemi- cal action. These attempts have been prompted by the observation that all chemical action depends upon, or, at least, is attended with, the absorption or liberation of heat, and that the amount of heat absorbed or liber- ated is the measure of such action. The determination of chemical phenomena by means of their thermic inci- dents, which "has until recently been known as thermo- chemistry, and has been treated as a comparatively in- significant part of chemical science, is now coming to be regarded as the true basis of theoretical chemistry. The principles of this new science have already been systematized, to some extent, in several distinct trea- tises, among which may be mentioned Mohr's "Me- chanical Theory of Chemical Affinity," * Naumann's * Friedrich Mohr, Mechanischc Theoric der chcmischen Affinitaet, Braunschweig, 1868. 300 CONCEPTS OF MODERN PHYSICS. " Thermo-Cliemistry," * and Berthelot's " Chemical Mechanics founded on Thermo-Chemistry." f The importance of the part which heat performs in chemical transformations was first distinctly realized upon the announcement, by Dulong and Petit in 1819, of the empirical law, that the specific heats of the ele- ments are inversely proportional to their atomic weights, or, as it is commonly expressed in the language of the atomic theory, that the atoms of all elementary bodies have the same specific heat. Although there are ap- parent exceptions to this law (as in the cases of carbon, boron, and silicon), it holds good in so many cases that there is hope of an explanation of these exceptions on grounds on which they will utimately prove to be con- firmations of the law ; indeed, some progress in this di- rection has already been made. And Neumann, Ke- gnault, and Kopp have shown that the law applies not only to elements, but also to compounds, it appearing that the specific heat of a compound is the sum of the specific heats of its component elements. Dulong and Petit's law, if it were universally valid, would lead to a remarkable law of chemical combina- tion. For, it is obviously identical with the proposition, that chemical elements combine only in so far as they experience the same elevation of temperature in the act of combination. It is not improbable that, if the true relation of the temperature of a body to its total physi- cal and chemical energy were thoroughly understood, this law would become one of the cardinal principles of theoretical chemistry. * Dr. Alexander Naumann, Grundriss der Thcrmochemie, Braun- schweig, 1869. f M. Berthclot, Essai de Mecanique Chimique fondee sur la Thermo- chimie, Paris, 1879. CONCLUSION. 301 The next noteworthy result of thermo-chemical re- search was the discovery that the nature of the chemi- cal reactions between different substances depends upon the relations between the specific energies of the re- agents as determined by the quantities of heat evolved or involved in the progress of these reactions. It was found that there are certain elements oxygen and hydrogen, for example which combine readily, and, under proper conditions, spontaneously, the combina- tion (as Berthelot expresses it) taking place directly, without the aid of extrinsic energy, and being at- tended with the evolution of light, or heat, or both. Such combinations are termed by M. Berthelot, exo- thermic. They result in the formation of compounds, which can not be resolved again into their original elements without a restoration of the amount of en- ergy lost in the combination. On the other hand, there are cases of endothermic combination in which conversely the composition of the elements is attended with an absorption, and the decomposition of the result- ing compound with a liberation, of heat. The com- bination of carbon and sulphur, for instance, is endo- thermic. Carbonic disulphide is formed by passing vaporous sulphur over red-hot charcoal ; the union of carbon and sulphur is possible only on condition of the continuous supply, during the progress of the union, of heat, which is given out again when the disulphide is resolved into its elements. The facts here referred to are explained, by the modern chemist, on the theory that chemical affinity is transformed heat, both heat and affinity being forms of energy ; that in the cases of ex- othermic combination the sum of the specific energies of the component elements exceeds the specific energy of the compound formed, while in endothermic com- 302 CONCEPTS OF MODERN PHYSICS. binations the specific energy of the compound is greater than the aggregated specific energies of the compo- nents. And it has been shown that, whenever we trace a number of elements or compounds through a series of chemical reactions, the total amount of energy (appear- ing, before absorption or after liberation, in the form of heat), which is liberated or absorbed, is exactly equal to the difference between the specific energies of the initial and those of the terminal compounds or elements. It is to be observed that this rule applies, not only to cases of composition and decomposition, so called, but like- wise to cases of allotropy and polymerism, inasmuch as allotropic forms of elements and isomeric forms of' com- pounds are found to be convertible into each other by the addition or withdrawal of definite amounts of heat. A third result of the study of the thermic condition of elements and compounds is the establishment of the remarkable principle that the passage of any body or system of bodies from a condition of a lesser to one of greater stability is always attended with evolution of heat, " whether " (in the language of Odling) " such change be what is commonly called combination, or what is called decomposition " ; and that all chemical action which takes place withoiit the intervention of extrinsic energy tends to the production of a body or bodies whose formation liberates the largest amount of heat.* * A sort of anticipation of this principle is found in one of the well- known laws announced in the early part of this century by M. Bcrthollct, in his " Statique Chimique " in the law that, whenever two soluble salts are mixed in solution, they decompose each other, if the resulting com- pound, or mixture of compounds, is insoluble or less soluble than the salts mixed. The bearing of this law upon the principle, stated in the text, of the maximum evolution of heat, will be understood upon reference to the fact that, generally speaking, the solubility of substances is increased by CONCLUSIO: This brief outline sufficiently indicafe : i := ffi0?acts and generalizations upon which it is proposed to found the new theory of " Chemical Mechanics." Little use has, thus far, been made of the law of Dulong and Petit ; but the other results of experimental induction in the field of thermo-chemistry are summarized by M. Ber- th elot in the introduction to his work,* as follows : " 1. Principle of Molecular Work. The quantity of heat disengaged in any reaction whatever is a meas- ure of the amount of chemical and physical work per- formed in such reaction. " 2. Principle of the Calorific Equivalence of Chem- ical Transformations. If a system of simple or com- pound bodies, taken under determinate conditions, un- dergoes physical or chemical changes capable of bring- ing it to a new state, without producing any mechanical effect outside of the system, the amount of heat liber- ated or absorbed by the effect of these changes depends solely on the initial and final states of the system ; it is the same, whatever be the nature and sequence of the intermediate states. " 3. Principle of Maximum Work. All chemical change effected without the intervention of extraneous energy tends to the production of that body or sys- tem of bodies which liberates the largest amount of heat." This third principle, as Berthelot observes, may also be stated in the form that " all chemical reaction sus- ceptible of being effected without the concurrence of preliminary work and without the intervention of ex- the application of heat. Berthollet's law, however, is subject to excep- tions ; there are cases in which soluble bases are replaced by insoluble bases, the result, nevertheless, being the formation of soluble salts. * " Mecaniquc Chimique," pp. xxviii, xxix. 304: CONCEPTS OF MODERN PHYSICS. trinsic energy will necessarily take place whenever it leads to the evolution of heat." The relation of these propositions to the doctrine of the conservation of energy is apparent. They are obviously applications, to the phenomena of chemical transformation, of the two leading principles which that doctrine embraces, the first and second propositions of Berthelot representing the principle of the correla- tion, equivalence, and mutual convertibility of the sev- eral forms of energy, and the third that of the tendency of all energy to dissipation. The study of chemical changes in the light of the doctrine of the conservation of energy exhibits these changes under an entirely new aspect. It shows that the question as to the possibility of a chemical " com- position," or "decomposition," is as much a question of the definite proportionality of energies as of the definite proportionality of masses; that each element as well as each compound embodies a distinct and invariable amount of energy as well as a distinct and invariable quantity of " matter " (i. e., mass), and that this energy is as constitutive, and as essential a part, of the existence of such element or compound as its weight. And here the question arises: How is all this to be interpreted, by the aid of the ordinary laws of mo- tion and of mechanical principles generally, in con- formity with the assumption that all the phenomena of chemical transformation are reducible to motions of absolutely inert atoms or elements of mass ? For that is the assumption which lies at the base of the new theory of chemical mechanics. Naumann declares in express terms, both in one of the first and in the very last of the sentences of his book that " chemistry in its CONCLUSION. 305 ultimate form must be atomic mechanics." * And Ber- thelot, though he avoids the use of the word atoms, no less explicitly asserts that two data suffice to explain the multiformity of chemical substances : the masses of the elementary particles and the nature of their motion.f The explanation of chemical phenomena by the theory of chemical mechanics is to be effected, then, by reducing them to terms of mass and motion. On what mechanical principles is this reduction possible? The fundamental fact to be accounted for is the con- version of heat into chemical energy. But this con- version implies, not only a change of one kind of motion into another, but also a confinement of a defi- nite amount of this motion to or within a definite mass. According to the mechanical theory, heat, in the form at least in which it is generally supplied to gaseous bodies in process of chemical transformation, consists in recti- linear atomic or molecular motions of all conceivable velocities and directions. The extent of these motions is limited solely by the encounters of the moving masses. By these encounters the range, the velocity, and the direction of the excursion of every atom or molecule are incessantly changed. And, whatever may be the nature of that form of motion which we call chemical energy, we know at least that a definite and invariable amount of it belongs to a definite mass or * " Die Chemie in dcr fucr sic zu crstrebendcn Gestaltung muss sein erne Mcchanik der Atomc," Thermochemie, p. 150. f " La matiere multiforme donl la chimie etudie la diversite obeit aux lois d'une mecanique commune. . . . Au point de vue mecanique, deux donnecs fondamentales caracterisent cette diversite en apparcnce indefinie dcs substances chimiqucs, savoir : la masse des particules elementaires, c*cst- d-dire leur equivalent, et la nature de leurs mouvemcnts. La connaissance de ces deux donnecs doit siiffir pour tout expliquer." Mecanique Chi- mique, tome ii, p. 757. 306 CONCEPTS OF MODERN PHYSICS. number of atoms of any given substance. Whenever heat, therefore, is converted into chemical energy, the motion above described must, of necessity, be so modi- fied that a definite amount of it is brought into some sort of synthesis or union with a definite number of particles. But that is certainly impossible if the par- ticles are mere inert masses, whose motions are deter- mined solely by the impact of other masses, as the mechanical theory assumes. The specialization or in- dividualization of motion, which is required, can be accounted for in no other way than by attributing to the masses themselves some inherent coercive power. Even if an individualization of heat-movements could result mechanically from the collision of inert particles by the conversion of rectilinear into rotatory motion, for instance, as a consequence of oblique impacts there would still remain the impossibility of accounting for the fact that such conversion ' invariably ceased at the precise moment when each atom or moleculo had been supplied with its due amount of energy. In view of all this it is strange to read in the writ- ings of distinguished physicists sentences like these: "The only real things in the physical universe are matter and energy, and of these matter is simply pas- sive," * and, "We see that, whereas (to our present knowledge at least) matter is always the same, though it may be masked in various combinations, energy is constantly changing the form in which it presents it- self. The one is like the eternal, unchangeable Fate or Necessitas of the ancients; the other is Proteus himself in the variety and rapidity of its transforma- tions." f There is little doubt that the principle of the con- * The Unseen Universe, 104. f /., 103. CONCLUSION. 307 servation of energy will prove to be the great theoreti- cal solvent of chemical as well as of physical phenomena ; but thus far, at least, the endeavor to express the laws of chemical action in terms of mass and motion or kinetic energy has been as abortive in chemistry as in physics. To what extent it may be possible, hereafter, to bring the phenomena of chemical action within, the dominion of the mechanical laws controlling the inter- action of solids, it is difficult to determine. There are, however, several well-known facts which appear to in- dicate that, whatever be the nature of chemical energy, it can hardly result from the impact of solid particles. The chemical energies of the elements are proportional neither to their masses as measured by their weights, nor to their volumes ; and their mechanical equivalents are so enormous as to seem out of all analogy to ordi- nary mechanical action. In 1856 "W". Weber and R. Kohlrausch published the results of a series of investi- gations by which they had sought to arrive at a mechan ical measure for the intensity of a galvanic current. They applied these results to the electrolytic decom- position of water, so as to determine the energy repre- sented in the chemical union of hydrogen and oxygen. And they announced their conclusion in the following words * : "If all the particles of hydrogen in one milli- gramme of water contained in a column of the length of one millimetre were attached to a string, the parti- cles of oxygen being attached to another string, each string would have to be under a tension, in a direction opposite to that of the other, of 2,956 cwt. (147,830 kilogrammes), in order to effect a decomposition of the water with a velocity of one milligramme per second." And, looking to the equivalents of chemical energy in * Fogg. Ann., vol. xcix, p. 24. 308 CONCEPTS OF MODERN PHYSICS. terms of units of heat, it lias been found that the com- bination of a gramme of hydrogen with 35*5 grammes of chlorine, so as to form 36'5 grammes of hydrogen chloride, is attended with the liberation of an amount of heat by which the temperature of 24 kilogrammes of water would be raised one degree ; inasmuch, therefore, as the heat required to raise one kilogramme of water one degree is mechanically equivalent to 425 kilogram- metres, the formation of 36*5 grammes of hydrogen chloride gives rise to a power by which a weight of 10,000 kilogrammes can be raised to the height of one metre in a second. INDEX. ABERRATION, 96. Absolute, the, 159. Actio in dislans, 53, 145. Adams, lunar acceleration, 61. ./Ether, ix, x, 113. Akin, history of force, 76. Allotropy, 302. Ampere, atoms, 84. Anderssohn, mechanics of gravita- tion, 58. Antipodes, 144. Arago, gravitation, 60, 63. Aristotle, dynamical theory of mat- ter, 160. hypotheses, 121. maxim of, 158. Astronomy, 96. Atomic theory, 28, 84. Atomic volume, 1 74. Atoms, elasticity of, 41, 42. Avenarius, philosophy as thought in conformity to principle of least action, 291. Avogadro, law of, 34, 35. Axioms, geometrical, 242. Babinct, cosmogony of Laplace, 280. Bacon, matter containing germs of all natural virtue, essence, and action, 154. mathematics, 158. Bain, explanation, 105. "hypotheses, 116. matter, force, and inertia, 161. Barker, G. F., all energy kinetic, xxix. Baumhauer, atomic weights, 173. Beltrami, pseudo-spherical space, 218. Bernoulli, Daniel, motion as result of oscillation, 60. tides, 61. Bernoulli, John, gravitation, 55. conservation of vis viva, 76. Berthelot, chemical mechanics, 300. Berthold, conservation of energy, 69. Berthollet, law of chemical reac- tions, 302. Bohn, conservation of energy, 76. Boltzmann, second law of thermo- dynamics, 27. molecular complexity, 37. kinetic theory of gases, 123. Boole, laws of thought, 259. Boscovich, atoms, 84. Boyle's law, 90, 117, 123, 124. Bradley, aberration of light, 231. Briot, theory of light, 97. Brodie, atomic theory, 103. Buckle, foundations of geometry, 221. Budde, nebular hypothesis, 280, 289. Calorific equivalence of chemical transformations, 303. Cardan, negative roots of an equa- tion, 268. Cauchy, impenetrability, 91, 92, 179. dispersion of light, 93. Causality, law of, xxxvii, 185. Challis, gravitation, 56, 60. Charles, law of, 117, 123, 124. Clarke, F. W., planetary evolution, 33. Clausius, molecular complexity, 37. kinetic theory of gases, 37, 299. 310 INDEX. Clausius, molecular rotation, 123. entropy, 270. Clifford, theory of light, 155. pangeoinetry, 213. Coleridge, bedridden truths, 82. Comte, Auguste, philosophy of, xxxvi. Conceivability, conditions of, 138. Concepts, nature of, 132. Conservative system, 77, 78. Constancy, law of, xxxix. Continuity, law of, xxxix. Cooke, Avogadro's law, 34. atomic weights, 173. Coriolis, dispersion of light, 93. Corpuscular theory of matter, 160. Cosmogenetic speculations, 270. Cotes, matter without inertia, 44. Cournot, atoms,' 102. hypotheses, 113. solidity of matter, 178. mathematical analysis, 108. Croll, force, 168. Curvature, measure of, 238. D'Alembert, gravitation, 56, 60. mathematics, 249. geometry, 243. Darwin, G. H., solar tides, 285. Delboeuf, filiation of ideas and things, 178. Dellingshausen, 159. DC Morgan, algebraic quantities, 267. Descartes, as a philosopher and man of science, iv, vii, 69. conservation of motion, 70. homogeneity of matter, 30. matter nothing but extension, 171, 228. mechanical theory, 16. relativity of motion, 187. Diogenes of Apollonia, 69. Du Bois-Reymond, world formula, v, xlii. causality, xxxviii, xxxix. mechanical theory, 21, 23. gravitation, 58. Duehring, principles of mechanics, 286. Dulong and Petit, law of, 37, 300. Dumas, atomic weights, 33, 173. Dynamical theory of matter, 160. Eddy, second law of thermodynam- ics, 27. Elasticity of atoms, 40. of vortex rings, 43. resulting from rotation generally, 45. Empedocles, 69. Endothermic combination, 801. Energy, conservation of, 66, 304. kinetic, 29, 66, 78, 79. potential, 29, 66, 67. dissipation of, 271. Epicurus, 69. Erdmann, Benno, dimensions of space, 256. Euclid, geometrical axioms, 243. Euler, gravitation, 55. relativity of motion, 192. Evolutionism, 158. Exothermic combination, 302. Explanation, nature of, 105. Faraday, atoms, 84, 160. Fatio de Duillers, gravitation, 63. Fechner, atoms, 84. Fermat, 16. Fichte's idealism, 235. Fictions, logical, 296. Force, definitions of, 166, 167. measure of, 73. Fresnel, polarization of light, 94. Fries, solidity of matter, 182, Fritsch, gravitation, 59. Galilei, laws of motion, 79. Gases, diffusion of, 90. kinetic theory of, 104. Gassendi, 69. Gauss, law of motion under least constraint, xli, 186. transcendental geometry, 208. Geometry, 243. transcendental, 207. Geometrical axioms, 242. inductive origin of, according to Mill, 217. George, 159. Graham, Thomas, equality of atoms, 31. hydrogen, a metal, 136. INDEX. 311 Grassmann, dimensions of space not deducible from laws of thought, 260. Guthric, gravitation, 59. Haeckel, mechanical theory, 20. Ilalley, acceleration of the moon's mean motion, 61. Hall, Asaph, satellites of Mars, 284. Ilalstead, bibliography of hyper- space, 208. Hamilton, Sir William, cognition, 105, hypotheses, 116. definition of concept, 132. relation of judgments to concepts, 135, 136. space, 234. Hamilton, William Rowan, principle of varying action, 27. conical refraction, 115. Hartmann's philosophy, 159. Heat, conversion of, into chemical energy, 305. Hegel's philosophy, 158. Helmholtz, causality, xxxviii, xl. mechanical theory, 18. vortex motion, 39, 43. electro-magnetic theory of light, 97. all-quality relation, 184. transcendental geometry, 208. conceivability of pseudo-spherical space, 243. Henrici, geometry, 240. Hcrakleiitos, 136. Ilerbart, relation of judgments to concepts, 136. Hcrschel, Sir William, satellites of Uranus, 284. Sir John, aether, 114. Hipparchian cycle, 109. Him, G. A., force in space, xxxi. kinetic theory of gases, xxxiii, xxxiv. inconsistencies of physical theo- ries, xii. Hobbes, mechanical theory, 16. sensation, 130. Hudson, wave-theory of light, 114. Hunt, T. Sterry, solution, 136. Iluygens, mechanical theory, 17. gravitation, 55. conservation of vis viva, 76. Inertia of matter, 29, 52, 149, 160. Isenkrehe, causality, xxxviii. gravitation, 49. Jevons, scientific knowledge, 105. nature of hypotheses, 111. - spectroscopy of gases, 127. Kant, measure of force, 72. his anticipation of scientific dis- coveries and theories, 195. pangeometry, 195. absolute center of gravity of the universe, 195. nature of space, 227, 232. nebular hypothesis, 280. Kekule, atomic impacts, 102. Kepler, species immatcriata, 165. Ketteler, dispersion of light, 97. Kirchhoff, mechanical theory, 18. lectures on mathematical physics, 168. rest a special case of motion, 203. Klein, Felix, non-Euclidean geome- try, 230, 246. Kohlrausch, R., intensity of chemi- cal energy, 307. Krause, transcendental geometry, 237. Kroenig, kinetic theory of gases, 42, 104. Kundt and Warburg, ratio of spe- cific heat of mercurial vapor at constant pressure to that at constant volume, 38. Lamb, Horace, basis of statics, xlii. Land, transcendental geometry, 237. Laplace, tidal retardation, 61. nebular hypothesis, 280. invariable plane, 288. lunar acceleration, 61. Lasswitz, cosmic gravitation and heat, 275.' Lea, H. Carey, atomic weights, 173. Leibnitz, law of least action, xli. mechanical theory, 16. conservation of energy, 69, 72, 81. letter to Clarke, 81. 312 INDEX. Leibnitz, simplicity of elements, 1 85. relativity of motion, 188. Le Sage, theory of gravitation, 63, 66. Lewes, G. H., conceivability as a test of truth, 110. Liebig, eonceivability as a test of truth, 140. Light, undulatory theory of, 92, 113. dispersion of, 92, 97. polarization of, 93. Lipschitz, transcendental mechan- ics, 238. Lloyd, conic refraction, 115. Lobatschewsky, pangeometry, 213. Lodge, gravitation, xxii, xxiii, xxv, xxvi. Lorentz, dispersion of light, 97. Lotze, concepts, 133. Lucretius, atoms, 84. conservation of matter, 69. Luebeck, Dr. G , elasticity of atoms, xxvii. MacAlistcr, Donald, criticism of, xix-xxv. Magnus, dilatation of gases, 117. Mansel, theory of conception, 132. Mariotte's law, 90, 177, 123, 124. Mass, relativity of, 87, 205. conservation of, 26, 27. Mathematics, 108, 249. Matter, indestructibility of, 26, 86. solidity of, 171. Maximum work, principle of, 303. Maxwell, James Clerk, mechanical theory, 18. complexity of molecular constitu- tion, 37. frictionless SDther, 39. vortex rings, 44. kinetic theory of gases, 123, 124, 299. molecules, 126. definition of force, 166, 169. entropy, 272. Mayer, Julius Robert, theory of me- teoric agglomeration, 287. Mcndelcjeff, atomic weights, 173. Metaphysics, character of, 137. Meyer, L., atomic weights, 173. Meyer, 0. E., elasticity of atoms, xxvii, xxviii. Mill, J. S., nature of explanation, 110, 116. relativity of knowledge, 130. nominalism and conceptualism, 133. conceivability as a test of truth, 138. inductive origin of geometrical axioms, 217. Mohr, gravitation, 58. mechanical theory of affinity, 299. Molecular work, principle of, 303. Momentum, conservation of, 77. angular, conservation of , 77, 277. Montaigne, hypotheses, 121. Musschenbrock, mechanical theory, 17. Naumann, thermo-chemistry, 300. Nebular hypothesis, 277. Neumann, C., dispersion of light, 97. body Alpha, 198. Newcomb,transcendentalspace,212. Newton, Sir Isaac, inertia, 39, 162. gravitation, 52. letter to Bentley, 53. queries, 54. impenetrability, 89. Tcgula philosophandi) 110. relativity of motion, 189. Norton, W. A., aether, 114. Numbers, complex, 243. Occam's rule, 121. Odling, composition and decompo- sition, 302. Ohm, Martin, relation of numbers to geometrical magnitudes, 267. Olbcrs, cosmical heat, 274. Pangeometry, 207. Parmenides, 69. Peacock, principle of the perma- nence of equivalent forms, 264. Peyrard, edition of Euclid, 243. Poinsot, theory of rotation, 278. resilience of rotating bodies, 46, 47. mathematical analysis, 108. Polymerism, 302. Poisson, inertia, 164. polarization of light, 94. Pott, development of language, 175. INDEX. 313 Quantities, imaginary, 263. Quaternions, theory of, 260, 264, 267. Rankine, second law of thermo- dynamics, 27. definition of force, 170. finitude of material universe, 273. Rayleigh, Lord, electro - magnetic theory of light, 97. Realism, mediseval, 150. Redtenbacher, dispersion of light, 97. absolute weights of atoms, 206. Reductio ad absurdum, 147. Regnault, expansion of gases, 90, 117. Relativity, principle of, 134, 183. Riemann, dissertation on hypothe- ses which lie at the base of ge- ometry, 209, 248. Rohmer, 159. Rudberg, expansion of gases, 117. Sachs, Uranus, 169. Schelling, matter the seed-corn of the universe, 154. dynamical theory of matter, 160. Schlegel, theory of space, 260. Schleicher, Darwin and language, 291. Schopenhauer's philosophy, 159, 235. Schramm, matter and motion, 59. Sigwart, theory of conception, 133. Somoff, definition of force, 167. Space, nature of, 235, 237. properties of, 240. relativity of, 204. a concept, 235. non-homaloidal, 207. Spencer, Herbert, all energy kinetic, xxxiii. equality of elements, 31. symbolic character of thought, 135. conceivability as a test of truth, 138. Spiller, jether, 164. Spinoza, philosophy of, 70. proposition that the order and connection of ideas are identi- cal with those of things, 177. 16 Stas, atomic weights, 173. Stefan, kinetic theory of gases, 123, 124. Stevinus, 16. Stewart, Balfour, gravitation, 58. expansion of gases, 117. Stumpf, origin of concept of space, xvii, 234. Sylvester, transcendental space, 211. Tait, P. G., dead matter, vi. reality indestructibility, vi. all energy kinetic, xxix, 66. gravitation, 58, 66. force and momentum, 166. transcendental space, 210. degradation of energy, 272. entropy, 272. Tauschinsky, theory of conception, 133. Taylor, kinetic theory of gravitation, 58, 66. Thermometer, graduation of, 118. Things per se, 159. Thomson, J. J., electro - magnetic theory of light, 97. Thomson, Sir William, vortex-atom theory, v, xxiii-xxv, 73. elasticity of molecules, 42. vortex rings, 43. atoms, 101. Thought, relation of, to things, 129. Time, relativity of, 204. Tyndall, scientific imagination, vi. Belfast address, 153. Liverpool address, 154. atoms, 154. Universcdia ante rem, 150. in re, 150. Universe, finitude of, 274. Varignon, gravitation, 68. Virtual velocities, principle of, 78. Vis viva, 74. conservation of, 75. Wallace, adaptation by general law, 195. Walter, Arwed, planetary molecule, 122. Weber, W., intensity of chemical energy, 307. 314 INDEX. Werder, 159. Whately, theory t)f conception, 135. Whewell, conceivability as a test of truth, 138. inertia, 162. force, 166. Wittwer, elasticity of atoms, 40. Wrede, aether, 114. Wright, R. A., allotropy of ele- ments, 33. Wundt, mechanical theory, 19. Wundt, equality of atoms, 31. theory of conception, 133. hypotheses, 105. cosmological problem, 271. Young, inertia, 162. force, 166. Zoellner, requisites of valid hypoth- esis, 109. Newton's letter to Bentley, 54. concept of force, 167. U1TI7BRSITT THE END. Scientific Publications. MAN BEFOBE METALS. By N. JOLY, Professor at the Science Faculty of Toulouse ; Correspondent of the Institute. With 148 Illustrations. 12mo. Cloth, $1.75. " The discussion of man's origin and early history, by Professor De Quatrefages, formed one of the most useful volumes in the k International Scientific Series,' and the same collection is now further enriched by a popular treatise on paleontology, by M. N. Joly, Professor in the University of Toulouse. The title of the book, ' Man before Metals,' indicates the limitations of the writer's theme. His object is to bring together the numerous proofs, collected by modern research, of the great age of the human race, and to show us what man was, in respect of customs, industries, and moral or religious ideas, before the use of metals was known to him." New York Sun. " An interesting, not to say fascinating volume." New York Churchman. ANIMAL INTELLIGENCE. By GKOBGB J. ROMANES, F. E. S., Zoological Secretary of the Linnsean Society, etc. 12mo. Cloth, $1.75. " My object in the work as a whole is twofold : First, I have thought it desirable that there should be something resembling a text-book of the facts of Comparative Psychology, to which men of science, and also metaphysicians, may turn whenever they have occasion to acquaint themselves with the particular level of intelligence to which this or that species of animal attains. My second and much more impor- tant object is that of considering the facts of animal intelligence in their relation to the theory of descent." From the Preface. " Unless we are greatly mistaken, Mr. Eomanes's work will take its place as one of the most attractive volumes of the ' International Scientific Series.' Some persons may, indeed, be disposed to say that it is too attractive, that it feeds the popular taste for the curious and marvelous without supplying any commensurate discipline in exact scientific reflection ; but the author has, we think, fully justified himself in his modest preface. The resnlt is the appearance of a collection of facts which will be a real boon to the student of Comparative Psychology, for this is the first attempt to present systematically well-assured observations on the mental life of animals." Sat- urday Review. " The author believes himself, not without ample cause, to have completely bridged the supposed gap between instinct and reason by the authentic proofs here mar- shaled of remarkable intelligence in some of the higher animals. It is the seemingly conclusive evidence of reasoning: powers furnished by the adaptation of means to ends in cases which can not be explained on the theory of inherited aptitude or habit." New York Sun. THE SCIENCE OF POLITICS. By SHELDON AMOS, M. A., author of The Science of Law," etc. 12mo. Cloth, $1.75. " To the political student and the practical statesman it ought to be of great value.* 1 New York Herald. " The author traces the subject from Plato and Aristotle in Greece, and Cicero in Koine, to the modern schools in the English field, not slighting the teachings of the American Be volution or the lessons of the French Kevolution of 1793. Forms of gov- ernment, political terms, the relation of law, written and unwritten, to the subject, a codification from Justinian to Napoleon in France and Field in America, are treated as parts of the subject in hand. Necessarily the subjects of executive and legislative authority, police, liquor, and land laws are considered, and the question ever growing in importance in all countries, the relations of corporations to the state.' 1 New York observer. New York : I). APPLETON & CO., 1, 3, & 5 Bond Street. Scientific Publications. ANTS, BEES, AND WASPS. A Record of Observations on the Habits of the Social Hymenoptera. By Sir JOHN LUBBOCK, Bart., M. P., F. E. S., etc., author of "Origin of Civilization, and the Primitive Condition of Man,'' etc., etc. With Colored Plates. 12mo, cloth, $2.00. " This volume contains the record of various experiments made with ants, bees, and wasps during the last ten years, with a view to test their mental condition and powers of sense. The principal point in which Sir John's mode of experiment differs from those of Huber, Forel, McCook, and others, is that he has carefully watched and marked particular insects, and has had their nests under observation for long periods one of his ants' nests having been under constant inspection ever since 1874. His observations are made principally upon ants because they show more power and flexi- bility of mind; and the value of his studies is that they belong to the department of original research." " We have no hesitation in saying that the author has presented us with the most valuable series of observations on a special subject that has ever been produced, charm- ingly written, full of logical deductions, and, when we consider his multitudinous en- gagements, a remarkable illustration of economy of time. As a contribution to insect psychology, it will be long before this book finds a parallel." London Athenwum. DISEASES OF MEMORY : An Essay in the Positive Psychology. By TH. KIBOT, author of " Heredity," etc. Translated from the French by William Huntington Smith. 12mo, cloth, $1.50. " M. Ribot reduces diseases of memory to law, and his treatise is of extraor- dinary interest." Philadelphia. Press. "Not merely to scientific, but to all thinking men, this volume will prove intensely interesting." New York Observer. "M. Ribot has bestowed the most painstaking attention upon hi? theme, and numerous examples of the conditions considered greatly increase the value and interest of the volume." Philadelphia North American. "To the general reader the work is made entertaining by many illustrations connected with such names as Linnaeus, Newton, Sir Waiter Scott, Horace Ver- net, Gustavo Dore, and many others." Harrisburg Telegraph. "The whole subject is presented with a Frenchman's vivacity cf style." Providence Journal. "It is not too much to say that in no single work have so many curious cases been brought together and interpreted in a scientific manner.'" Boskn Evening Traveller. MYTH AND SCIENCE. By TITO VIGNOLI. 12mo, cloth, price, $1.50. " His book is ingenious ; ... his theory of how science gradually differen- tiated from and conquered myth is extremely well wrought out, and is probably in essentials correct." Saturday Review. "The book is a strong one, and far more interesting to the general reader than its title would indicate. The learning, the acuteness, the strong reasoning power, and the scientific spirit of the author, command admiration." New York Christian Advocate. "An attempt made, with much ability and no small measure of success, to trace tho origin and development of the myth. Ihe author has pursued his inquiry with much patience and ingenuity, and has produced a very readable and luminous treatise." Philadelphia North American. "It is a curious if not startling contribution both to psychology and to the early iistory of man's development." New York World. For sale by att booksellers; or sent by mail, post-paid, on receipt of price. New York: D. APPLETON & CO., 1, 3, & 5 Bond Street. Scientific Publications. THE BRAIN AND ITS FUNCTIONS. By J. LUTB, Physician to the Hospice de la Salpetriere. With Illustrations. 12mo. Cloth, $1.50. "No living physiologist is better entitled to speak with authority upon the structure and functions of the brain than Dr. Luys. His studies on the anatomy of the nervous system are acknowledged to he the fullest and most systematic ev.-r undertaken. Dr. Luys supports his conclusions not only by his own ana- t Miiy.il rc ; earches, but also by many functional observations of various other physiologists', including of course Professor Ferrier's now classical experi- tneatd. 11 St. James's Gazette. ' Dr. Luys, at the head of the great French Insane Asylum, is one of the most e nincut and successful investigators of cerebral science now living; and he has given unquestionably the clearest and most interesting brief account yet made of tiia structure and operations of the brain. We have been fascinated by this vol- u;no more than by any other treatise we have yet seen on the machinery of sen- sibility and thought ; and we have been instructed not only by much that is new, h it by many sagacious practical hints such as it is well for everybody to under- s.au.l.'" The I'opular Science Monthly. THE CONCEPTS AND THEORIES CF MODERN PHYSICS. I y J. B. STALLO. 12mo. Cloth, $1,75. " Judge Stallo's work is an inquiry into the validity of tV.ose mechanical con- ceptions of the universe which are now held as fundamental in physical science. He takes up the leading modern doctrines which are based up(>n this mechanical conception, such as the atomic constitution of matter, the kiuelic theory of gases, the. conservation of energy, the neb'ilar hypothesis, and other views, to find how much stands upon solid empirical irround, and how much rests upon metaphys- ical speculation. Since the appearance of Dr. Draper's ' Religion and Science,' no book has been published in the country calculated to make so deep an impres- si >n on thoughtful and e lucated readers as this volume. . . . The range and mi. in e les* of the author's learaiucr, the s,< uteuess of his reasoning, and the sinxul.ir precision and clearness of his style, are qualities which very seldom havj !>jjiijoinrly exhibited in a scientific treatise." 1 New York San. THIS FORMATION OF VEGETABLE MOULD, THROUGH THE ACTION OF WORMS, WITH OBSERVATIONS ON THEIR HABITS. By CHARLES DARWIN, LL. D., F. R. S., author of " On the Origin of Species," etc., etc. With Illustrations. 12mo, cloth. Price, $1.50. * Mr. Darwin's little volume on the habits and instincts of earth-worms is no \e9A marked than the earlier or more elaborate efforts of his genius by freshness of observation, unfailing power of interpreting and correlating facts, and logical vigor 'n generalizing npon them. The main purpose of the work is to point out tlie square which worms have taken in the formation of the layer of vegetable mould which covers the whole surface of the land in every moderately humid country. All lovers of nature will unite in thanking Mr. Darwin for the new and interesting light he has thrown upon a subject FO long overlooked, yet so full of interest and instruction, as the structure and the labors of the earth-worm." Sat inlay Review. " Respecting worms as among the most useful portions of animate nature, Dr. Darwin relates, in thi? remarkable book, their structure and habits, the part they have played in the burial of ancient buildings and the denudation of the land, in the disintegration of rocks, the preparation of soil for the growth of plants, and in the natural history of the world." .Boston Advertiser. D, APPLETON & CO., Publishers, 1. 3, & 5 Bond Street, New Yo*k. Scientific Publications. THE SUICIDE : An Essay in Comparative Moral Statistics. By HENET MOKSELLI, Pro- fessor of Psychological Medicine in Eoyal University, Turin. 12mo, Cloth, $l.t5. " Suicide " is a scientific inquiry, on the basis of the statistical method, into the laws of suicidal phenomena. Dealing with the subject as a branch of social science, it con- eiders the increase of suicide in different countries, and the comparison of nations, races, and periods in its manifestation. The influences of age, sex, constitution, cli- mate, season, occupation, religion, prevailing ideas, the elements of character, and the .tendencies of civilization, are comprehensively analyzed in their bearing upon the pro- pensity to self-destruction. Professor Morselli is an eminent European authority on .this subject. It is accompanied by colored maps illustrating pictorially the results of statistical inquiries. VOLCANOES : What they Are and what they Teach. By J. W. JUDD, Professor of Geology in the Royal School of Mines (London). With Ninety-six Illustrations. 12mo. Cloth, $2.00. " In no field has modern research been more fruitful than in that of wfiich Professor Judd gives a popular account in the present volume. The great lines of dynamical, geological, and meteorological inquiry converge upon the grand problem of the interior constitution of the earth, and the vast influence of subterranean agencies. . . . Eis book is very far from being a mere dry description of volcanoes and their eruptions ; it is rather a presentation of the terrestrial facts and laws with which volcanic phenomena are associated." Popular Science Monthly. " The volume before us is one of the pleasantest science manuals we have read for eome time." Athenaeum. " Mr. Judd's summary is so full and so concise that it is almost impossible to give a fair idea in a short review." Pali Mall Gazette, 1 SUN. By C. A. YOUNG, Ph. D., LL. D., Professor of Astronomy in the College of New Jersey. With numerous Illustrations. 12mo. Cloth, $2.00. " Professor Young is an authority on ' The Sun,' and writes from intimate knowl- edge. He has studied that great luminary all his life, invented and improved instru- ments for observing it, gone to all quarters of the world in search of the best places and opportunities to watch it, and has contributed important discoveries that have extended our knowledge of it. " It would take a cyclopaedia to represent all that has been done toward clearing up the solar mysteries. Professor Young has summarized the information, and presented it in a form completely available for general readers. There is no rhetoric in his book ; he trusts the grandeur of his theme to kindle interest and impress the feelings. His statements are plain, direct, clear, and condensed, though ample enough for his purpose, and the substance of what is generally wanted will be found accurately given in his pages." Popular /Science Monthly. ILLUSIONS : A Psychological Study. By JAMES SULLY, author of " Sensa- tion and Intuition," etc. 12mo. Cloth, $1.50. This volume takes a wide survey of the field of error, embracing in its view not only the illusions commonly regarded as of the nature of mental aberrations or hallucina- tions, but also other illusions arising from Ifcat capacity for error which belongs essen- tially to rational human nature. The author has endeavored to keep to a strictly scien- tific treatment that is to say, the description and classification of acknowledged errors, and the exposition of them by a reference to their psychical and physical conditions. " This is not a technical work, but one of wide popular interest, in the principles and results of which every one is concerned. The illusions of perception of the senses and of dreams are first considered, and then the author passes to the illusions of introspec- tion, errors of insight, illusions of memory, and illusions of belief. The work is a note- worthy contribution to the original progress of thought, and may b3 relied upon aa representing the present state of knowledge on the important subject to which it is devoted."- Popular Science Monthly. D. APPLETON & CO., Publishers, 1. 3. and 5 Bond Street, New York. Scientific Publications. GENERAL PHYSIOLOGY OF MUSCLES AND NERVES. By Dr. I. ROSENTHAL, Professor of Physiology at the University of Erlangen. With seventy-five WoodcutB. (" International Scientific Series.") 12mo, cloth, $1.50. " The attempt at a connected account of the general physiology of muscles fend nerves is, as far as I know, the first of its kind. The general data for this 'branch of science have been gained only within the past thirty years. 1 ' Extract from Preface. SIGHT : An Exposition of the Principles of Monocular and Binocular Vision By JOSEPH LE CONTE, LL. D., author of "Elements of Geology"; "Re- ligion and Science " ; and Professor of Geology and Natural History in the University of California. With numerous Illustrations. 12mo, cloth, $1.50. a master-hand." *The Nation. ANIMAL, LIFE, as affected by the Natural Conditions of Existence. By KARL SEMPER, Professor of the University of Wiirzburg. With 2 Maps and 106 Woodcuts, and Index. 12mo, cloth, $2.00. " This is in many respects one of the most interesting contributions to zoological literature which has appeared for some time." Nature. THE ATOMIC THEORY. By AD. WUKTZ, Membre de Tlnstitut ; Doyen Honoraire de la Faculte de Medecine ; Professeur a la Faculte des Sciences de Paris. Translated by E. CLEMINSHAW, M. A., F. C.S., F. I. C., Assist ant Master at Sherborne School. 12mo, cloth, $1.50. There was need for a book like this, which discusses the atomic theory both in its historic evolution and in its present form. And perhaps no man of this to perform the task in a masterly . age could have been selected so able to perform the task in a masterly way as the illustrious French chemist, Adolph Wurtz. It is impossible to convey to the reader, in a notice like this, any adequate idea of the scope, lucid instructiveness, and scientific interest of Professor Wurtz' s book. The modern problems of chemistry, which are commonly so obscure from imperfect exposition, are here made wonderfully clear and attractive." The Popular Science Monthly. THE CRAYFISH. An Introduction to the Study of ZoOlogy. By Professor T. H. HUXLEY, F. R. S. With 82 Illustrations. 12mo, cloth, $1.75. " Whoever will follow these pages, crayfish in hand, and will try to verify for himself the statements which they contain, will find himself brought face to face with all the great zoological questions which excite so lively an interest at the present day." " The reader of this valuable monograph will lay it down with a feeling of wonder at the amount and variety of matter which has been got out of so seem- ingly slight and unpretending a subject." Saturday Review. D. APPLETON & CO., Publishers, 1, 3, & 5 BOND STREET, NEW You*. Scientific Publications. THE HUMAN SPECIES. By A. DE QUATREFAGES, Professor of Anthro- pology in the Museum of Natural History, Paris. 12mo, cloth, $2.00. The work treats of the unity, origin, antiquity, and original localization of the human species, peopling of the globe, acclimatization, primitive man, forma- ition of the human races, fossil human races, present human races, and the physi- cal and psychological characters of mankind. STUDENTS' TEXT-BOOK OF COLOR ; or, MODERN CHROMAT- ICS. With Applications to Art and Industry. With 130 Original Illus- trations, and Frontispiece in Colors. By OGDEN N, KOOD, Professor of Physics in Columbia College. 12mo, cloth, $2.00. "In this interesting book Professor Rood, who, as a distinguished Professor of Physics in Columbia College, United States, must be accepted as a competent authority on the branch of science of which he treats, deals briefly and succinctly with what may be termed the scientific rationale of his subject. But the chief value of his work is to be .attributed to the fact that he is himself an accom- plished artist as well as an authoritative expounder of science. 1 ' Edinburgh Review, October, 1879, in an article on " The Philosophy of Color." EDUCATION AS A SCIENCE. By ALEXANDER BAIN, LL. D. 12mo, cloth, $1.75. " This work must be pronounced the most remarkable discussion of educa- tional problems which has been published in our day. We do not hesitate to bespeak for it the widest circulation and the most earnest attention. It should be in the hands of every school-teacher and friend of education throughout the land." New York Sun. A HISTORY OF THE GROWTH OF THE STEAM-ENGINE. By ROBERT H. THURSTON, A.M., C. E., Professor of Mechanical Engineering in the Stevens Institute of Technology, Hoboken, N. J., etc. With 163 Illustrations, including 15 Portraits. 12mo, cloth, $2.50. " Professor Thurston almost exhausts his subject ; details of mechanism are followed by interesting biographies of the more important inventors. If, as is contended, the steam-engine is the most important physical agent in civilizing the world, its history is a desideratum, and the readers of the present work Will agree that it could have a no more amusing and intelligent historian than our author." Boston Gazette. STUDIES IN SPECTRUM ANALYSIS. By J. NORMAN LOCKTER, P. R. S,, Correspondent of the Institute of France, etc. With 60 Illustrations. 12mo, cloth, $2.50. "The study of spectrum analysis is one fraught with a pecnliar fascination, and some of the author's experiments are exceedingly picturesque in their re- sults They are so lucidly described, too, that the reader keeps on, from page to page never flawing in interest in the matter before him, nor putting down toe book until the last page is reached." New York Evening Express. D. APPLETON & CO., Publishers, 1, 8, & 5 BOND STREET, Nw YORK. Scientific Publications. A TREATISE ON CHEMISTRY. By H. E. ROSCOE, F. R. S., ond C. SCHOHLEMMER, P. R. S., Professors of Chemistry in the Victoria University, Owens College, Manchester. Illustrated. Vols. I and II. Inorganic Chemistry. 8vo. Vol. I. Non-Metallic Elements. Price, $5.00. Vol. II. Part I. Metals. Price, $3.00. Vol. II. Part II. Metals. Price, $3.00. Organic Chemistry. 8vo. Vol. III. Part I. The Chemistry of the Hydrocarbons and their Derivatives. Price, $5.00. Vol. III. Part II. Completing the work. (In preparation.) " It is difficult to praise too highly the selection of materials and their arrange- ment, or the wealth of illustrations which explain and adorn the text. Whatever tests of accuracy as to figures and facts we have been able to apply have been satisfactorily met, while in clearness of statement this third volume leaves noth- ing to be desired." London Academy. THE ELEMENTS OF ECONOMICS. By HENRY DUNNING MACLEOD. M. A., of Trinity College, Cambridge, and the Inner Temple, barrister-at- law selected by the Royal Commissioners for the Digest of the Law to pre- pare the digest of the law of bills of exchange, bank notes, etc. Lecturer on Political Economy in the University of Cambridge. In two volumes. Volume I now ready. 12mo, cloth. Price, $1.75. " Mr. Macleod's works on economic science have one great merit, they "belong to the class of books tbat assist inquiry by setting their readers thinking. The views they set forth are not only often valuable in themselves, but they are the generative cause of ideas which may also be valuable in their readers. His books, moreover, are written in the proper way. The subject is divided care- fully in accordance with the opinions held by the author ; all classifications when made are adhered to, and the descriptions and definitions adopted are admirable from his pointpf view, and in some cases from a wider stand-point." The Statist. ADOLPH STRECKER'S SHORT TEXT-BOOK OF ORGANIC CHEMISTRY. By Dr. JOHANNES WISLICENUS. Translated and edited, with Extensive Additions, by W. H. HODGKINSON, Ph. D., and A. J. GREEN- AWAY, F. I. C. 8vo. Cloth, $5.00. " Let ro one suppose that in this ' short text-book * we have to deal with a primer. Everything is comparative, and the term ' short ' here has relation to the enormous development and extent of recent organic chemistry. This solid and comprehensive volume ie intended to represent the present condition of the science in its main facts and leading principles, as demanded by the systematic chemical student. We have here, probably, the best extant text-book of organic chemistry. Not only is it full and comprehensive and remarkably clear and methodical, but it is up to the very latest moment, and it has been, moreover, prepared in a way to secure the greatest excellences in such a treatise." The Popular Science Monthly. THE ORIGIN OF CIVILIZATION AND THE PRIMITIVE CON- DITION OF MAN, Mental and Social Condition of Savages. By Sir JOHN LUBBOCK, Bart., F. R. S., President of the British Association. With Illustrations. Fourth edition, with numerous Additions. 8vo, cloth. Price, $5.00. For sale by all booksellers ; or sent by mail, post-paid, on receipt of price. D. APPLETON & CO., Publishers, 1, 3, & 5 Bond Street, New York. Scientific Publications. THE FUNDAMENTAL, CONCEPTS OF MODERN PHILOSOPHIC THOUGHT, CRITICALLY AND HISTORICALLY CONSID- ERED. By RUDOLPH EUCKEN, Ph. D., Professor in Jena. With an Introduction by NOAH POKTEK, President of Yale College. One vol., 12mo, 804 pages. Cloth. Price, $1.75. President Porter declares of this work that " there are few books within his knowledge which are better fitted to aid the student who wishes to acquaint him- self with the course of modern speculation and scientific thinking, and to form an intelligent estimate of most of the current theories." MIND IN THE LOWER ANIMALS IN HEALTH AND DISEASE. By W. LAUDEB LINDSAY, M. D., F. R. S. E., etc. 2 vols., 8vo. Cloth, $4.00. " The author of this work, which, regarded merely as an accumulation of verified and classified facts, is a unique and precious contribution to the data of comparative psychology, claims that he entered on his inquiry without any theory to defend, support, or illustrate. We are bound to say that, while his general conclusions are boldly and continually avowed, his claim of fairness and caution is justified by his method of examining particular phenomena ; that he seems willing at all times to renounce any impression or belief which is shown to be scientifically untenable." New York Sun. "In this work two volumes of over 500 pages Dr. Lindsay marshals a pro- portionately large number of facts against those philosophers who maintain that the intelligence of man differs in kind aud not simply in degree from that of the lower animals. It is one purpose of bis book to show that the main differences between man and the lower animals exist rather in their physical than in their mental structure. In this way of thinking, all animals possess not the semblance of, but the true substance of mind and will." New York World. " So far as we are aware there has been no treatise upon the subject of animal intelligence so broad in its foundations, so well considered, or so scientific in its methods of inquiry, as that which has been prepared by Dr. W. Lauder Lindsay in two large volumes, the first being devoted to a study of animal mind in health, and the second to animal mind in disease. We may safely say that his work is, in some respects, the most important essay of the kind that has yet been under- taken. His observations have been supplemented by a thorough mastery of the history and literature of the subject, and hence his conclusions rest upon the broadest possible foundation of safe induction. There is a good analytical index to the book, as there ought to be to every work of the kind." New York Evening THE ELEMENTARY PRINCIPLES OF SCIENTIFIC AGRICULT- URE. By N. T. LUPTON, LL. D., Professor of Chemistry in Vanderbilt University, Nashville, Tenu. 18mo. Cloth. Price, 45 cents. A GLOSSARY OF BIOLOGICAL, ANATOMICAL, AND PHYSIO- LOGICAL TERMS. By THOMAS DTJNMAN. Small 8vo. Cloth. 161 pages. Price, $1.00. " It has been the author's task to furnish here a small and convenient but very complete glossary of those terms ; and he has done this so well, both in his choice of terms for definition and in his clear exposition of their etymological and tech- nical meaning, as to leave nothing to be desired in this direction." New York Evening Post. For sale by all booksellers, or any work sent by mail, post-paid, on receipt of price. D. APPLETON & CO., Publishers, 1, 3, and 5 Bond Street, New York. Scientific Publications. ANTHROPOLOGY : An Introduction to the Study of Man and Civilization. By EDWABD B. TTLOB, F. E. 8. With 78 Illustrations. 12rao. Cloth, $2.00. " To take in hand a chaos and reduce it to an orderly plan, to examine the vast com- plex of human life in all ages and all stages, and to show how it may be treated under half a dozen heads this is indeed a difficult task ; yet it is one which Mr. Tylor has performed within very modest limits and with distinguished success. The students who read Mr. Tyler's book may congratulate themselves upon having obtained so easy, pleasant, and workmanlike an introduction to a fascinating and delightful science." London Athenaeum. THE HUMAN SPECIES. By A. DE QITATKEFAGES, Professor of Anthropology in the Museum of Natural History, Paris. 12mo. Cloth, $2.00. NATURAL HISTORY OF MAN : A COUESE OF ELEMENT AEY LEG- TUBES. With an Appendix. By A. DE QTTATBEFAGES, Professor of Anthro- pology in the Museum of Natural History, Paris. 12mo. Cloth, $1.00. " In introducing this work to the public notice in a cheap and convenient form there is much sound judgment. M. de Quatrefages is one of the ablest, as he is one of the most enthusiastic, anthropologists of the day." New York Times. MAN BEFORE METALS. By N. JOLT, Professor at the Science Faculty of Toulouse; Correspondent of the Institute. With 148 Illustrations. 12mo. Cloth, $1.75. " The discussion of man's origin and early history, by Professor De Quatrefages, formed one of the most useful volumes in the ' International Scientific Series,' and the same collection is now further enriched by a popular treatise on paleontology, by M. N. Joly, Professor in the University of Toulouse. The title of the book, ' Man before Metals,' indicates the limitations of the writer's theme. His object is to bring together the numerous proofs, collected by modern research, of the great age of the human race, and to show us what man was, in respect of customs, industries, and moral or religious ideas, before the use of metals was known to him." New York Sun. "An interesting, not to say fascinating volume." New York Churchman. THE RACES OF MAX, AND THEIR GEOGRAPHICAL DISTRI- BUTION. From the German of OSCAB PESCHEL. 12ino. Cloth, $2.25. THE ORIGIN OF CIVILIZATION AND THE PRIMITIVE CON- DITION OF MAN, MENTAL AND SOCIAL CONDITION OF SAVAGES. By Sir JOHN LUBBOCK, Bart., F. E. 8. Fourth edition, with numerous Additions. With Illustrations. Svo. Cloth, $5.00. " This interesting work for it is intensely so in its aim, scope, and the ability of its author treats of what the scientists denominate anthropology, or the natural history of the human species; the complete science of man, body and soul including sex temperament, race, civilization, etc." Providence Press. ' PREHISTORIC TIMES, AS ILLUSTEATED BY ANCIENT EEMAIN9 AND THE MANNEES AND CUSTOMS OF MODEEN SAVAGES. By Sir JOHN LTJBBOCK, Bart., F. E. S. Illustrated. Entirely new revised edition, Svo. Cloth, $5.00. _^ New York : D. APPLETON & CO., 1, 3, & 5 Bond Street. THE ORGANS OF SPEECH, And their Application in the Formation of Articulate Sounds, By GEORG HERMANN VON MEYER, Professor of Anatomy at the University of Zurich, With numerous Illustrations. - - - 12mo, cloth, $1,75. " At once philosophical and practical, suitable as a text-book in a medical col- lege or for reading at home. Persons engaged in philological studies, and all professional musicians, will find it full of extremely useful facts and suggestions." New York Journal qf Commerce. " This volume comprises the author's researches into the anatomy of the vo- cal organs, with special reference to the point of view and needs of the philolo- gist and the trainer of the voice. It seeks to explain the origin of articulate sounds, and to outline a system in which all elements of all languages may be co-ordinated in their proper place. The work has obviously a special value for students in the science of the transmutations of language, for etymologists, elo- cutionists, and musicians." New York Home Journal. " The author perceives in the sounds made by animals meanings analogous to words, and in support of this is the fact that in the legends of all nations an im- portant part is played by wise men who understand the language of the brute creation. With patient thoroughness Professor von Meyer describes minutely the vocal apparatus, and the sounds produced by the complex combinations of its simple parts." Cincinnati Commercial Gazette. " It is surprising to note what different sounds, and shades of tone and mean- ing, can be produced by volition in the use of the human organs of speech." Hartford Daily Times. " The book presents a happy combination of the Teutonic thoroughness of treatment with the method and lucidity of statement which especially distinguish French scientists. The author's expositions are remarkable for their clearness and avoidance of technicalities ; while their meaning is rendered more apparent by the use of numerous diagrams." Edinburgh Scotsman. " The work is a thorough and exhaustive one." Boston Commonwealth. " The author's plan has been to give a sketch of all possible articulate sounds, and to trace upon that basis their relations and capacity for combination." Philadelphia North American. " A treatise of remarkable interest." Boston Transcript, " Ought to be welcomed for the varied, new, and original interpretations con- tained in the book." Harrisburg Telegraph. & 5 Boiid Street. TJFI7BESIT7J >* . 14 DAY USE RETURN TO DESK FROM WHICH BORROWED LOAN DEPT. This book is due on the last date stamped below, or on the date to which renewed. Renewed books are subject to immediate recall. STACKS MAY20'6B LD 21A-60rti-10,'65 (F7763slO)476B General Library University of California Berkeley GENERAL LIBBABY-U.C.BEBKELE iiim^^